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2401.09350 | 250 | 93
ââ
# ââ
94
6 Graph Algorithms
# 6.3.3 Approximation
The results of Beaumont et al. [2007a] are encouraging. In theory, so long as we can construct the Delaunay graph, we not only have the optimality guarantee, but we are also guaranteed to have a poly-logarithmic number of hops to reach the optimal answer. Alas, as we have discussed previously, the Delaunay graph is expensive to build in high dimensions.
Moreover, the number of neighbors per node is no longer O(1). So even if we inserted long-range edges into the Delaunay graph, it is not imme- diate if the time saved by skipping Voronoi regions due to long-range edges offsets the additional time the algorithm spends computing dis- tances between each node along the path and its neighbors. | 2401.09350#250 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 250,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "93\nââ\n# ââ\n94\n6 Graph Algorithms\n# 6.3.3 Approximation\nThe results of Beaumont et al. [2007a] are encouraging. In theory, so long as we can construct the Delaunay graph, we not only have the optimality guarantee, but we are also guaranteed to have a poly-logarithmic number of hops to reach the optimal answer. Alas, as we have discussed previously, the Delaunay graph is expensive to build in high dimensions.\nMoreover, the number of neighbors per node is no longer O(1). So even if we inserted long-range edges into the Delaunay graph, it is not imme- diate if the time saved by skipping Voronoi regions due to long-range edges offsets the additional time the algorithm spends computing dis- tances between each node along the path and its neighbors.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 251 | We are back, then, to approximation with the help of heuristics. Beau- mont et al. [2007b] describe one such method in a follow-up study. Their method approximates the Voronoi regions of every node by resorting to a gossip protocol. In this procedure, every node has a list of 3d + 1 of its cur- rent neighbors, where d denotes the dimension of the space. In every iteration of the algorithm, every node passes its current list to its neighbors. When a node receives this information, it takes the union of all lists, and finds the subset of 3d + 1 points with the minimal volume. This subset becomes the nodeâs current list of neighbors. While a na¨ıve implementation of the protocol is prohibitively expensive, Beaumont et al. [2007b] discuss an alternative to estimating the volume induced by a set of 3d + 1 points, and the search for the minimal volume. | 2401.09350#251 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 251,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We are back, then, to approximation with the help of heuristics. Beau- mont et al. [2007b] describe one such method in a follow-up study. Their method approximates the Voronoi regions of every node by resorting to a gossip protocol. In this procedure, every node has a list of 3d + 1 of its cur- rent neighbors, where d denotes the dimension of the space. In every iteration of the algorithm, every node passes its current list to its neighbors. When a node receives this information, it takes the union of all lists, and finds the subset of 3d + 1 points with the minimal volume. This subset becomes the nodeâs current list of neighbors. While a na¨ıve implementation of the protocol is prohibitively expensive, Beaumont et al. [2007b] discuss an alternative to estimating the volume induced by a set of 3d + 1 points, and the search for the minimal volume.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 252 | Malkov et al. [2014] take a different approach. They simply permute the vectors in the collection X , and sequentially add each vector to the graph. Every time a vector is inserted into the graph, it is linked to its k nearest neighbors from the current snapshot of the graph. The intuition is that, as the graph grows, the edges added earlier in the evolution of the graph serve as long-range edges in the final graph, and the more recent edges form an approximation of the k-NN graph, which itself is an approximation of the Delaunay graph. Later Malkov and Yashunin [2020] modify the algorithm by introducing a hierarchy of graphs. The resulting graph has proved successful in practice and, despite its lack of theoretical guarantees, is both effective and highly efficient.
# 6.4 Neighborhood Graphs
In the preceding section, our starting point was the Delaunay graph. We aug- mented it with random long-range connections to improve the transmission
6.4 Neighborhood Graphs | 2401.09350#252 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 252,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Malkov et al. [2014] take a different approach. They simply permute the vectors in the collection X , and sequentially add each vector to the graph. Every time a vector is inserted into the graph, it is linked to its k nearest neighbors from the current snapshot of the graph. The intuition is that, as the graph grows, the edges added earlier in the evolution of the graph serve as long-range edges in the final graph, and the more recent edges form an approximation of the k-NN graph, which itself is an approximation of the Delaunay graph. Later Malkov and Yashunin [2020] modify the algorithm by introducing a hierarchy of graphs. The resulting graph has proved successful in practice and, despite its lack of theoretical guarantees, is both effective and highly efficient.\n# 6.4 Neighborhood Graphs\nIn the preceding section, our starting point was the Delaunay graph. We aug- mented it with random long-range connections to improve the transmission\n6.4 Neighborhood Graphs",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 253 | In the preceding section, our starting point was the Delaunay graph. We aug- mented it with random long-range connections to improve the transmission
6.4 Neighborhood Graphs
rate through the network. Because the resulting structure contains the Delau- nay graph, we get the optimality guarantee of Theorem 6.4 for free. But, as a result of the complexity of the Delaunay construction in high dimensions, we had to approximate the structure instead, losing all guarantees in the process. Frustratingly, the approximate structure obtained by the heuristics we discussed, is certainly not a super-graph of the Delaunay graph, nor is it necessarily its sub-graph. In fact, even the fundamental property of connect- edness is not immediately guaranteed. There is therefore nothing meaningful to say about the theoretical behavior of such graphs. | 2401.09350#253 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 253,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In the preceding section, our starting point was the Delaunay graph. We aug- mented it with random long-range connections to improve the transmission\n6.4 Neighborhood Graphs\nrate through the network. Because the resulting structure contains the Delau- nay graph, we get the optimality guarantee of Theorem 6.4 for free. But, as a result of the complexity of the Delaunay construction in high dimensions, we had to approximate the structure instead, losing all guarantees in the process. Frustratingly, the approximate structure obtained by the heuristics we discussed, is certainly not a super-graph of the Delaunay graph, nor is it necessarily its sub-graph. In fact, even the fundamental property of connect- edness is not immediately guaranteed. There is therefore nothing meaningful to say about the theoretical behavior of such graphs.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 254 | In this section, we do the opposite. Instead of adding edges to the Delaunay graph and then resorting to heuristics to create a completely different graph, we prune the edges of the Delaunay graph to find a structure that is its sub- graph. Indeed, we cannot say anything meaningful about the optimality of exact top-k retrieval, but as we will later see, we can state formal results for the approximate top-k retrieval variantâalbeit in a very specific case. The structure we have in mind is known as the Relative Neighborhood Graph (RNG) [Toussaint, 1980, Jaromczyk and Toussaint, 1992].
In an RNG, G = (V,â¬), for a distance function 6(-,-), there is an undirected edge between two nodes u,v ⬠V if and only if d(u,v) < max (5(u, w),d(w,v)) for all w ⬠V \ {u,v}. That is, the graph guar- antees that, if (u,v) ⬠â¬, then there is no other point in the collection that is simultaneously closer to u and v, than u and v are to each other. Conceptually, then, we can view constructing an RNG as pruning away edges in the Delaunay graph that violate the RNG property. | 2401.09350#254 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 254,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "In this section, we do the opposite. Instead of adding edges to the Delaunay graph and then resorting to heuristics to create a completely different graph, we prune the edges of the Delaunay graph to find a structure that is its sub- graph. Indeed, we cannot say anything meaningful about the optimality of exact top-k retrieval, but as we will later see, we can state formal results for the approximate top-k retrieval variantâalbeit in a very specific case. The structure we have in mind is known as the Relative Neighborhood Graph (RNG) [Toussaint, 1980, Jaromczyk and Toussaint, 1992].\nIn an RNG, G = (V,â¬), for a distance function 6(-,-), there is an undirected edge between two nodes u,v ⬠V if and only if d(u,v) < max (5(u, w),d(w,v)) for all w ⬠V \\ {u,v}. That is, the graph guar- antees that, if (u,v) ⬠â¬, then there is no other point in the collection that is simultaneously closer to u and v, than u and v are to each other. Conceptually, then, we can view constructing an RNG as pruning away edges in the Delaunay graph that violate the RNG property.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 255 | The RNG was shown to contain the Minimum Spanning Tree [Toussaint, 1980], so that it is guaranteed to be connected. It is also provably contained in the Delaunay graph [OâRourke, 1982] in any metric space and in any number of dimensions. As a final property, it is not hard to see that such a graph G comes with a weak optimality guarantee for the best-first-search algorithm: If q = uâ â X , then the greedy traversal algorithm returns the node associated with q, no matter where it enters the graph. That is due simply to the following fact: If the current node u is a local optimum but not the global optimum, then there must be an edge connecting u to a node that is closer to uâ. Otherwise, u itself must be connected to uâ. | 2401.09350#255 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 255,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The RNG was shown to contain the Minimum Spanning Tree [Toussaint, 1980], so that it is guaranteed to be connected. It is also provably contained in the Delaunay graph [OâRourke, 1982] in any metric space and in any number of dimensions. As a final property, it is not hard to see that such a graph G comes with a weak optimality guarantee for the best-first-search algorithm: If q = uâ â X , then the greedy traversal algorithm returns the node associated with q, no matter where it enters the graph. That is due simply to the following fact: If the current node u is a local optimum but not the global optimum, then there must be an edge connecting u to a node that is closer to uâ. Otherwise, u itself must be connected to uâ.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 256 | Later, Arya and Mount [1993] proposed a directed variant of the RNG, which they call the Sparse Neighborhood Graph (SNG) that is arguably more suitable for top-k retrieval. For every node u â V, we apply the following procedure: Let U = V \ {u}. Sort the nodes in U in increasing distance to u. Then, remove the closest node (say, v) from U and add an edge between u to v. Finally, remove from U all nodes w that satisfy δ(u, w) > δ(w, v). The process is repeated until U is empty. It can be immediately seen that the weak optimality guarantee from before still holds in the SNG.
95
96
6 Graph Algorithms
1.0 : 0.8 0.6 0.4 0.2 0.0 U0 0.35 0.50 0.75 1.00
1.0 0.8 0.6 0.4 0.2 0.0 U0 0.35 0.50 0.75 1.00
(a) α = 1 (b) α = 1.1 (c) α = 1.2 (d) α = 1.3 | 2401.09350#256 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 256,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Later, Arya and Mount [1993] proposed a directed variant of the RNG, which they call the Sparse Neighborhood Graph (SNG) that is arguably more suitable for top-k retrieval. For every node u â V, we apply the following procedure: Let U = V \\ {u}. Sort the nodes in U in increasing distance to u. Then, remove the closest node (say, v) from U and add an edge between u to v. Finally, remove from U all nodes w that satisfy δ(u, w) > δ(w, v). The process is repeated until U is empty. It can be immediately seen that the weak optimality guarantee from before still holds in the SNG.\n95\n96\n6 Graph Algorithms\n1.0 : 0.8 0.6 0.4 0.2 0.0 U0 0.35 0.50 0.75 1.00\n1.0 0.8 0.6 0.4 0.2 0.0 U0 0.35 0.50 0.75 1.00\n(a) α = 1 (b) α = 1.1 (c) α = 1.2 (d) α = 1.3",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 257 | Fig. 6.8: Examples of α-SNGs on a dataset of 20 points drawn uniformly from [0, 1]2 (blue circles). When α = 1, we recover the standard SNG. As α becomes larger, the resulting graph becomes more dense.
Neighborhood graphs are the backbone of many graph algorithms for top- k retrieval [Malkov et al., 2014, Malkov and Yashunin, 2020, Harwood and Drummond, 2016, Fu et al., 2019, 2022, Jayaram Subramanya et al., 2019]. While many of these algorithms make for efficient methods in practice, the Vamana construction [Jayaram Subramanya et al., 2019] stands out as it introduces a novel super-graph of the SNG that turns out to have provable theoretical properties. That super-graph is what Indyk and Xu [2023] call an α-shortcut reachable SNG, which we will review next. For brevity, though, we call this graph simply α-SNG.
6.4 Neighborhood Graphs | 2401.09350#257 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 257,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 6.8: Examples of α-SNGs on a dataset of 20 points drawn uniformly from [0, 1]2 (blue circles). When α = 1, we recover the standard SNG. As α becomes larger, the resulting graph becomes more dense.\nNeighborhood graphs are the backbone of many graph algorithms for top- k retrieval [Malkov et al., 2014, Malkov and Yashunin, 2020, Harwood and Drummond, 2016, Fu et al., 2019, 2022, Jayaram Subramanya et al., 2019]. While many of these algorithms make for efficient methods in practice, the Vamana construction [Jayaram Subramanya et al., 2019] stands out as it introduces a novel super-graph of the SNG that turns out to have provable theoretical properties. That super-graph is what Indyk and Xu [2023] call an α-shortcut reachable SNG, which we will review next. For brevity, though, we call this graph simply α-SNG.\n6.4 Neighborhood Graphs",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 258 | Fig. 6.9: The sets Bi and rings Ri in the proof of Theorem 6.8.
# 6.4.1 From SNG to α-SNG
Jayaram Subramanya et al. [2019] introduce a subtle adjustment to the SNG construction. In particular, suppose we are processing a node u, have already extracted the node v whose distance to u is minimal among the nodes in U (i.e., v = arg minwâU δ(u, w)), and are now deciding which nodes to discard from U. In the standard SNG construction, we remove a node w for which δ(u, w) > δ(w, v). But in the modified construction, we instead discard a node w if δ(u, w) > αδ(w, v) for some α > 1. Note that, the case of α = 1 simply gives the standard SNG. Figure 6.8 shows a few examples of α-SNGs on a toy dataset. | 2401.09350#258 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 258,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 6.9: The sets Bi and rings Ri in the proof of Theorem 6.8.\n# 6.4.1 From SNG to α-SNG\nJayaram Subramanya et al. [2019] introduce a subtle adjustment to the SNG construction. In particular, suppose we are processing a node u, have already extracted the node v whose distance to u is minimal among the nodes in U (i.e., v = arg minwâU δ(u, w)), and are now deciding which nodes to discard from U. In the standard SNG construction, we remove a node w for which δ(u, w) > δ(w, v). But in the modified construction, we instead discard a node w if δ(u, w) > αδ(w, v) for some α > 1. Note that, the case of α = 1 simply gives the standard SNG. Figure 6.8 shows a few examples of α-SNGs on a toy dataset.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 259 | That is what Indyk and Xu [2023] later refer to as an α-shortcut reachable graph. They define α-shortcut reachability as the property where, for any node u, we have that every other node w is either the target of an edge from u (so that (u, w) â E), or that there is a node v such that (u, v) â E and δ(u, w) ⥠αδ(w, v). Clearly, the graph constructed by the procedure above is α-shortcut reachable by definition.
# 6.4.1.1 Analysis
Indyk and Xu [2023] present an analysis of the α-SNG for a collection of vectors X with doubling dimension d⦠as defined in Definition 3.2. | 2401.09350#259 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 259,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "That is what Indyk and Xu [2023] later refer to as an α-shortcut reachable graph. They define α-shortcut reachability as the property where, for any node u, we have that every other node w is either the target of an edge from u (so that (u, w) â E), or that there is a node v such that (u, v) â E and δ(u, w) ⥠αδ(w, v). Clearly, the graph constructed by the procedure above is α-shortcut reachable by definition.\n# 6.4.1.1 Analysis\nIndyk and Xu [2023] present an analysis of the α-SNG for a collection of vectors X with doubling dimension d⦠as defined in Definition 3.2.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 260 | Indyk and Xu [2023] present an analysis of the α-SNG for a collection of vectors X with doubling dimension d⦠as defined in Definition 3.2.
For collections with a fixed doubling constant, Indyk and Xu [2023] state two bounds. One gives a bound on the degree of every node in an a-SNG. The other tells us the expected number of hops from any arbitrary entry node to an â¬-approximate solution to top-1 queries. The two bounds together give us an idea of the time complexity of Algorithm 3 over an a-SNG as well as its accuracy. Theorem 6.8 The degree of any node in an a-SNG is O((4a)% log A) if the collection YX has doubling dimension do and aspect ratio A = 6,./6*.
97
98
6 Graph Algorithms
Proof. Consider a node u â V. For each i â [log2 â], define a ball centered at u with radius δâ/2i: Bi = B(u, δâ/2i). From this, construct rings Ri = Bi\Bi+1. See Figure 6.9 for an illustration. | 2401.09350#260 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 260,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Indyk and Xu [2023] present an analysis of the α-SNG for a collection of vectors X with doubling dimension d⦠as defined in Definition 3.2.\nFor collections with a fixed doubling constant, Indyk and Xu [2023] state two bounds. One gives a bound on the degree of every node in an a-SNG. The other tells us the expected number of hops from any arbitrary entry node to an â¬-approximate solution to top-1 queries. The two bounds together give us an idea of the time complexity of Algorithm 3 over an a-SNG as well as its accuracy. Theorem 6.8 The degree of any node in an a-SNG is O((4a)% log A) if the collection YX has doubling dimension do and aspect ratio A = 6,./6*.\n97\n98\n6 Graph Algorithms\nProof. Consider a node u â V. For each i â [log2 â], define a ball centered at u with radius δâ/2i: Bi = B(u, δâ/2i). From this, construct rings Ri = Bi\\Bi+1. See Figure 6.9 for an illustration.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 261 | Because Â¥ has a constant doubling dimension, we can cover each R; with O((4a)*?) balls of radius 6,/a2'*?. By construction, two points in each of these cover balls are at most 6,/a2't! apart. At the same time, the distance from u to any point in a cover ball is at least 6,/2â++. By construction, all points in a cover ball except one are discarded as we form uâs edges in the a-SNG. As such, the total number of edges from u is bounded by the total number of cover balls, which is O((4a)*° log A). ia
Theorem 6.9 If G = (V,â¬) is an a-SNG for collection X, then Algorithm 3 with k = 1 returns an (34 + â¬)-approzimate top-1 solution by visiting O( log, wm) nodes. | 2401.09350#261 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 261,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Because Â¥ has a constant doubling dimension, we can cover each R; with O((4a)*?) balls of radius 6,/a2'*?. By construction, two points in each of these cover balls are at most 6,/a2't! apart. At the same time, the distance from u to any point in a cover ball is at least 6,/2â++. By construction, all points in a cover ball except one are discarded as we form uâs edges in the a-SNG. As such, the total number of edges from u is bounded by the total number of cover balls, which is O((4a)*° log A). ia\nTheorem 6.9 If G = (V,â¬) is an a-SNG for collection X, then Algorithm 3 with k = 1 returns an (34 + â¬)-approzimate top-1 solution by visiting O( log, wm) nodes.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 262 | Proof. Suppose q is a query point and uâ = arg minuâX δ(q, u). Further as- sume that the best-first-search algorithm is currently in node vi with distance δ(q, vi) to the query. We make the following observations: ⢠By triangle inequality, we know that δ(vi, uâ) ⤠δ(vi, q) + δ(q, uâ); and, ⢠By construction of the α-SNG, vi is either connected to uâ or to another
node whose distance to uâ is shorter than δ(vi, uâ)/α.
We can conclude that, the distance from q to the next node the algorithm visits, vi+1, is at most:
δ(vi+1, q) ⤠δ(vi+1, uâ) + δ(uâ, q) δ(vi, uâ) α δ(vi, q) α
By induction, we see that, if the entry node is s â V:
4(s,q) : 5(vi,g) < + (a+ 1)6(q,u") Da j=l < ea) . a+ 5a uâ). (6.1) | 2401.09350#262 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 262,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. Suppose q is a query point and uâ = arg minuâX δ(q, u). Further as- sume that the best-first-search algorithm is currently in node vi with distance δ(q, vi) to the query. We make the following observations: ⢠By triangle inequality, we know that δ(vi, uâ) ⤠δ(vi, q) + δ(q, uâ); and, ⢠By construction of the α-SNG, vi is either connected to uâ or to another\nnode whose distance to uâ is shorter than δ(vi, uâ)/α.\nWe can conclude that, the distance from q to the next node the algorithm visits, vi+1, is at most:\nδ(vi+1, q) ⤠δ(vi+1, uâ) + δ(uâ, q) δ(vi, uâ) α δ(vi, q) α\nBy induction, we see that, if the entry node is s â V:\n4(s,q) : 5(vi,g) < + (a+ 1)6(q,u\") Da j=l < ea) . a+ 5a uâ). (6.1)",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 263 | There are three cases to consider. Case 1: When δ(s, q) > 2δâ, then by triangle inequality, δ(q, uâ) > δ(s, q) â δ(s, uâ) > δ(s, q) â δâ > δ(s, q)/2. Plugging this into Equation (6.1) yields:
6.4 Neighborhood Graphs
Paw) OAT ig, 5(vi, g) < 2 d(q,u*) ~ a 6(v;,q) < ai As such, for any ⬠> 0, the algorithm returns a (<4 in log, 2/e steps.
# atl
# a-l
(<4 +e)-approximate solution
As such, for any ⬠> 0, the algorithm returns a (<4 +e)-approximate solution in log, 2/e steps.
Case 2: 6(s,q) < 26, and 6(q,u*) > janeâ By Equation (6.1), the algorithm returns a (4 + â¬)-approximate solution as soon as 6(s,q)/aâ < â¬0(q,u*). So in this case: | 2401.09350#263 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 263,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "There are three cases to consider. Case 1: When δ(s, q) > 2δâ, then by triangle inequality, δ(q, uâ) > δ(s, q) â δ(s, uâ) > δ(s, q) â δâ > δ(s, q)/2. Plugging this into Equation (6.1) yields:\n6.4 Neighborhood Graphs\nPaw) OAT ig, 5(vi, g) < 2 d(q,u*) ~ a 6(v;,q) < ai As such, for any ⬠> 0, the algorithm returns a (<4 in log, 2/e steps.\n# atl\n# a-l\n(<4 +e)-approximate solution\nAs such, for any ⬠> 0, the algorithm returns a (<4 +e)-approximate solution in log, 2/e steps.\nCase 2: 6(s,q) < 26, and 6(q,u*) > janeâ By Equation (6.1), the algorithm returns a (4 + â¬)-approximate solution as soon as 6(s,q)/aâ < â¬0(q,u*). So in this case:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 264 | δ(vi, q) δ(q, uâ) ⤠⤠2δâ αiδ(q, uâ) 8(α + 1)δâ αi(α â 1)δâ + + α + 1 α â 1 α + 1 α â 1 .
.
As such, the number of steps to reach the approximation level is log, Sete which is O(log, A/(a â 1)e).
4(α+1) δâ. Suppose vi ̸= uâ. Observe that: (a) δ(vi, uâ) ⥠δâ; (b) δ(vi, q) > δ(q, uâ); and (c) δ(q, uâ) < δâ/2 by as- sumption. As such, triangle inequality gives us: δ(vi, q) > δ(vi, uâ)âδ(uâ, q) > δâ â δâ/2 = δâ/2. Together with Equation (6.1), we obtain:
2δâ αi + =â αi ⤠8â =â i ⤠logα 8â.
The three cases together give the desired result. | 2401.09350#264 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 264,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "δ(vi, q) δ(q, uâ) ⤠⤠2δâ αiδ(q, uâ) 8(α + 1)δâ αi(α â 1)δâ + + α + 1 α â 1 α + 1 α â 1 .\n.\nAs such, the number of steps to reach the approximation level is log, Sete which is O(log, A/(a â 1)e).\n4(α+1) δâ. Suppose vi ̸= uâ. Observe that: (a) δ(vi, uâ) ⥠δâ; (b) δ(vi, q) > δ(q, uâ); and (c) δ(q, uâ) < δâ/2 by as- sumption. As such, triangle inequality gives us: δ(vi, q) > δ(vi, uâ)âδ(uâ, q) > δâ â δâ/2 = δâ/2. Together with Equation (6.1), we obtain:\n2δâ αi + =â αi ⤠8â =â i ⤠logα 8â.\nThe three cases together give the desired result.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 265 | 2δâ αi + =â αi ⤠8â =â i ⤠logα 8â.
The three cases together give the desired result.
In addition to the bounds above, Indyk and Xu [2023] present negative re- sults for other major SNG-based graph algorithms by proving (via contrived examples) linear-time lower-bounds on their performance. These results to- gether show the significance of the pruning parameter α in the α-SNG con- struction.
# 6.4.1.2 Practical Construction of α-SNGs
The algorithm described earlier to construct an α-SNG for m points has O(m3) time complexity. That is too expensive for even moderately large val- ues of m. That prompted Jayaram Subramanya et al. [2019] to approximate the α-SNG by way of heuristics.
The starting point in the approximate construction is a random R-regular graph: Every node is connected to R other nodes selected at random. The algorithm then processes each node in random order as follows. Given node u,
99
ââ
100
6 Graph Algorithms | 2401.09350#265 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 265,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "2δâ αi + =â αi ⤠8â =â i ⤠logα 8â.\nThe three cases together give the desired result.\nIn addition to the bounds above, Indyk and Xu [2023] present negative re- sults for other major SNG-based graph algorithms by proving (via contrived examples) linear-time lower-bounds on their performance. These results to- gether show the significance of the pruning parameter α in the α-SNG con- struction.\n# 6.4.1.2 Practical Construction of α-SNGs\nThe algorithm described earlier to construct an α-SNG for m points has O(m3) time complexity. That is too expensive for even moderately large val- ues of m. That prompted Jayaram Subramanya et al. [2019] to approximate the α-SNG by way of heuristics.\nThe starting point in the approximate construction is a random R-regular graph: Every node is connected to R other nodes selected at random. The algorithm then processes each node in random order as follows. Given node u,\n99\nââ\n100\n6 Graph Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 266 | 99
ââ
100
6 Graph Algorithms
it begins by searching the current snapshot of the graph for the top L nodes for the query point u, using Algorithm 3. Denote the returned set of nodes by S. It then performs the pruning algorithm by setting U = S \ {u}, rather than U = V \ {u}. That is the gist of the modified construction procedure.2 Naturally, we lose all guarantees for approximate top-k retrieval as a re- sult [Indyk and Xu, 2023]. We do, however, obtain a more practical algorithm instead that, as the authors show, is both efficient and effective.
# 6.5 Closing Remarks
This chapter deviated from the pattern we got accustomed to so far in the monograph. The gap between theory and practice in Chapters 4 and 5 was narrow or none. That gap is rather wide, on the other hand, in graph-based retrieval algorithms. Making theory work in practice required a great deal of heuristics and approximations.
Another major departure is the activity in the respective bodies of litera- ture. Whereas trees and hash families have reached a certain level of maturity, the literature on graph algorithms is still evolving, actively so. A quick search through scholarly articles shows growing interest in this class of algorithms. This monograph itself presented results that were obtained very recently. | 2401.09350#266 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 266,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "99\nââ\n100\n6 Graph Algorithms\nit begins by searching the current snapshot of the graph for the top L nodes for the query point u, using Algorithm 3. Denote the returned set of nodes by S. It then performs the pruning algorithm by setting U = S \\ {u}, rather than U = V \\ {u}. That is the gist of the modified construction procedure.2 Naturally, we lose all guarantees for approximate top-k retrieval as a re- sult [Indyk and Xu, 2023]. We do, however, obtain a more practical algorithm instead that, as the authors show, is both efficient and effective.\n# 6.5 Closing Remarks\nThis chapter deviated from the pattern we got accustomed to so far in the monograph. The gap between theory and practice in Chapters 4 and 5 was narrow or none. That gap is rather wide, on the other hand, in graph-based retrieval algorithms. Making theory work in practice required a great deal of heuristics and approximations.\nAnother major departure is the activity in the respective bodies of litera- ture. Whereas trees and hash families have reached a certain level of maturity, the literature on graph algorithms is still evolving, actively so. A quick search through scholarly articles shows growing interest in this class of algorithms. This monograph itself presented results that were obtained very recently.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 267 | There is good reason for the uptick in research activity. Graph algorithms are among the most successful algorithms there are for top-k vector retrieval. They are often remarkably fast during retrieval and produce accurate solution sets.
That success makes it all the more enticing to improve their other charac- teristics. For example, graph indices are often large, requiring far too much memory. Incorporating compression into graphs, therefore, is a low-hanging fruit that has been explored [Singh et al., 2021] but needs further investiga- tion. More importantly, finding an even sparser graph without losing accuracy is key in reducing the size of the graph to begin with, and that boils down to designing better heuristics.
Heuristics play a key role in the construction time of graph indices too. Building a graph index for a collection of billions of points, for example, is not feasible for the variant of the Vamana algorithm that offers theoretical guarantees. Heuristics introduced in that work lost all such guarantees, but made the graph more practical.
Enhancing the capabilities of graph indices too is an important practical consideration. For example, when the graph is too large and, so, must rest on disk, optimizing disk access is essential in maintaining the speed of query processing [Jayaram Subramanya et al., 2019]. When the collection of vectors | 2401.09350#267 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 267,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "There is good reason for the uptick in research activity. Graph algorithms are among the most successful algorithms there are for top-k vector retrieval. They are often remarkably fast during retrieval and produce accurate solution sets.\nThat success makes it all the more enticing to improve their other charac- teristics. For example, graph indices are often large, requiring far too much memory. Incorporating compression into graphs, therefore, is a low-hanging fruit that has been explored [Singh et al., 2021] but needs further investiga- tion. More importantly, finding an even sparser graph without losing accuracy is key in reducing the size of the graph to begin with, and that boils down to designing better heuristics.\nHeuristics play a key role in the construction time of graph indices too. Building a graph index for a collection of billions of points, for example, is not feasible for the variant of the Vamana algorithm that offers theoretical guarantees. Heuristics introduced in that work lost all such guarantees, but made the graph more practical.\nEnhancing the capabilities of graph indices too is an important practical consideration. For example, when the graph is too large and, so, must rest on disk, optimizing disk access is essential in maintaining the speed of query processing [Jayaram Subramanya et al., 2019]. When the collection of vectors",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 268 | 2 We have omitted minor but important details of the procedure in our prose. We refer the interested reader to [Jayaram Subramanya et al., 2019] for a description of the full algorithm.
# References
References
is live and dynamic, the graph index must naturally handle deletions and insertions in real-time [Singh et al., 2021]. When vectors come with metadata and top-k retrieval must be constrained to the vectors that pass a certain set of metadata filters, then a greedy traversal of the graph may prove sub- optimal [Gollapudi et al., 2023]. All such questions warrant extensive (often applied) research and go some way to make graph algorithms more attractive to production systems.
There is thus no shortage of practical research questions. However, the aforementioned gap between theory and practice should not dissuade us from developing better theoretical algorithms. The models that explained the small world phenomenon may not be directly applicable to top-k retrieval in high dimensions, but they inspired heuristics that led to the state of the art. Find- ing theoretically-sound edge sets that improve over the guarantees offered by Vamana could form the basis for other, more successful heuristics too.
# References
S. Arya and D. M. Mount. Approximate nearest neighbor queries in fixed dimensions. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 271â280, 1993. | 2401.09350#268 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 268,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "2 We have omitted minor but important details of the procedure in our prose. We refer the interested reader to [Jayaram Subramanya et al., 2019] for a description of the full algorithm.\n# References\nReferences\nis live and dynamic, the graph index must naturally handle deletions and insertions in real-time [Singh et al., 2021]. When vectors come with metadata and top-k retrieval must be constrained to the vectors that pass a certain set of metadata filters, then a greedy traversal of the graph may prove sub- optimal [Gollapudi et al., 2023]. All such questions warrant extensive (often applied) research and go some way to make graph algorithms more attractive to production systems.\nThere is thus no shortage of practical research questions. However, the aforementioned gap between theory and practice should not dissuade us from developing better theoretical algorithms. The models that explained the small world phenomenon may not be directly applicable to top-k retrieval in high dimensions, but they inspired heuristics that led to the state of the art. Find- ing theoretically-sound edge sets that improve over the guarantees offered by Vamana could form the basis for other, more successful heuristics too.\n# References\nS. Arya and D. M. Mount. Approximate nearest neighbor queries in fixed dimensions. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 271â280, 1993.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 269 | Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.
O. Beaumont, A.-M. Kermarrec, L. Marchal, and E. Riviere. Voronet: A scalable object network based on voronoi tessellations. In 2007 IEEE In- ternational Parallel and Distributed Processing Symposium, pages 1â10, 2007a.
O. Beaumont, A.-M. Kermarrec, and ´E. Rivi`ere. Peer to peer multidimen- sional overlays: Approximating complex structures. In Principles of Dis- tributed Systems, pages 315â328, 2007b.
M. Brito, E. Ch´avez, A. Quiroz, and J. Yukich. Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection. Statistics & Probability Letters, 35(1):33â42, 1997. | 2401.09350#269 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 269,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Y. Bachrach, Y. Finkelstein, R. Gilad-Bachrach, L. Katzir, N. Koenigstein, N. Nice, and U. Paquet. Speeding up the xbox recommender system using a euclidean transformation for inner-product spaces. In Proceedings of the 8th ACM Conference on Recommender Systems, page 257â264, 2014.\nO. Beaumont, A.-M. Kermarrec, L. Marchal, and E. Riviere. Voronet: A scalable object network based on voronoi tessellations. In 2007 IEEE In- ternational Parallel and Distributed Processing Symposium, pages 1â10, 2007a.\nO. Beaumont, A.-M. Kermarrec, and ´E. Rivi`ere. Peer to peer multidimen- sional overlays: Approximating complex structures. In Principles of Dis- tributed Systems, pages 315â328, 2007b.\nM. Brito, E. Ch´avez, A. Quiroz, and J. Yukich. Connectivity of the mutual k-nearest-neighbor graph in clustering and outlier detection. Statistics & Probability Letters, 35(1):33â42, 1997.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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6 Graph Algorithms
M. Connor and P. Kumar. Fast construction of k-nearest neighbor graphs for point clouds. IEEE Transactions on Visualization and Computer Graphics, 16(4):599â608, 2010.
B. Delaunay. Sur la sph`ere vide. Bulletin de lâAcad´emie des Sciences de lâURSS. Classe des sciences math´ematiques et na, 1934(6):793â800, 1934. W. Dong, C. Moses, and K. Li. Efficient k-nearest neighbor graph construc- tion for generic similarity measures. In Proceedings of the 20th Interna- tional Conference on World Wide Web, pages 577â586, 2011. | 2401.09350#270 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 270,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "E. Ch´avez and E. S. Tellez. Navigating k-nearest neighbor graphs to solve nearest neighbor searches. In Proceedings of the 2nd Mexican Conference on Pattern Recognition: Advances in Pattern Recognition, pages 270â280, 2010.\nJ. Chen, H.-r. Fang, and Y. Saad. Fast approximate knn graph construc- tion for high dimensional data via recursive lanczos bisection. Journal of Machine Learning Research, 10:1989â2012, 12 2009.\n101\n102\n6 Graph Algorithms\nM. Connor and P. Kumar. Fast construction of k-nearest neighbor graphs for point clouds. IEEE Transactions on Visualization and Computer Graphics, 16(4):599â608, 2010.\nB. Delaunay. Sur la sph`ere vide. Bulletin de lâAcad´emie des Sciences de lâURSS. Classe des sciences math´ematiques et na, 1934(6):793â800, 1934. W. Dong, C. Moses, and K. Li. Efficient k-nearest neighbor graph construc- tion for generic similarity measures. In Proceedings of the 20th Interna- tional Conference on World Wide Web, pages 577â586, 2011.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 271 | H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for In Proceedings of the 8th Annual Symposium on regular triangulations. Computational Geometry, pages 43â52, 1992.
S. Fortune. Voronoi Diagrams and Delaunay Triangulations, pages 377â388. CRC Press, Inc., 1997.
C. Fu and D. Cai. Efanna : An extremely fast approximate nearest neighbor search algorithm based on knn graph, 2016.
C. Fu, C. Xiang, C. Wang, and D. Cai. Fast approximate nearest neighbor search with the navigating spreading-out graph. Proceedings of the VLDB Endowment, 12(5):461â474, 1 2019.
C. Fu, C. Wang, and D. Cai. High dimensional similarity search with satellite system graph: Efficiency, scalability, and unindexed query compatibility. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(8): 4139â4150, 2022. | 2401.09350#271 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 271,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "H. Edelsbrunner and N. R. Shah. Incremental topological flipping works for In Proceedings of the 8th Annual Symposium on regular triangulations. Computational Geometry, pages 43â52, 1992.\nS. Fortune. Voronoi Diagrams and Delaunay Triangulations, pages 377â388. CRC Press, Inc., 1997.\nC. Fu and D. Cai. Efanna : An extremely fast approximate nearest neighbor search algorithm based on knn graph, 2016.\nC. Fu, C. Xiang, C. Wang, and D. Cai. Fast approximate nearest neighbor search with the navigating spreading-out graph. Proceedings of the VLDB Endowment, 12(5):461â474, 1 2019.\nC. Fu, C. Wang, and D. Cai. High dimensional similarity search with satellite system graph: Efficiency, scalability, and unindexed query compatibility. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(8): 4139â4150, 2022.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 272 | S. Gollapudi, N. Karia, V. Sivashankar, R. Krishnaswamy, N. Begwani, S. Raz, Y. Lin, Y. Zhang, N. Mahapatro, P. Srinivasan, A. Singh, and H. V. Simhadri. Filtered-diskann: Graph algorithms for approximate near- est neighbor search with filters. In Proceedings of the ACM Web Conference 2023, pages 3406â3416, 2023.
L. Guibas and J. Stolfi. Primitives for the manipulation of general subdivi- sions and the computation of voronoi. ACM Transactions on Graphics, 4 (2):74â123, 04 1985.
L. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental con- struction of delaunay and voronoi diagrams. Algorithmica, 7(1â6):381â413, 3 1992.
K. Hajebi, Y. Abbasi-Yadkori, H. Shahbazi, and H. Zhang. Fast approximate nearest-neighbor search with k-nearest neighbor graph. In Twenty-Second International Joint Conference on Artificial Intelligence, 2011. | 2401.09350#272 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 272,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Gollapudi, N. Karia, V. Sivashankar, R. Krishnaswamy, N. Begwani, S. Raz, Y. Lin, Y. Zhang, N. Mahapatro, P. Srinivasan, A. Singh, and H. V. Simhadri. Filtered-diskann: Graph algorithms for approximate near- est neighbor search with filters. In Proceedings of the ACM Web Conference 2023, pages 3406â3416, 2023.\nL. Guibas and J. Stolfi. Primitives for the manipulation of general subdivi- sions and the computation of voronoi. ACM Transactions on Graphics, 4 (2):74â123, 04 1985.\nL. J. Guibas, D. E. Knuth, and M. Sharir. Randomized incremental con- struction of delaunay and voronoi diagrams. Algorithmica, 7(1â6):381â413, 3 1992.\nK. Hajebi, Y. Abbasi-Yadkori, H. Shahbazi, and H. Zhang. Fast approximate nearest-neighbor search with k-nearest neighbor graph. In Twenty-Second International Joint Conference on Artificial Intelligence, 2011.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 273 | B. Harwood and T. Drummond. Fanng: Fast approximate nearest neigh- bour graphs. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 5713â5722, 2016.
P. Indyk and H. Xu. Worst-case performance of popular approximate nearest neighbor search implementations: Guarantees and limitations. In Proceed- ings of the 36th Conference on Neural Information Processing Systems, 2023.
J. Jaromczyk and G. Toussaint. Relative neighborhood graphs and their relatives. Proceedings of the IEEE, 80(9):1502â1517, 1992.
References
S. Jayaram Subramanya, F. Devvrit, H. V. Simhadri, R. Krishnawamy, and R. Kadekodi. Diskann: Fast accurate billion-point nearest neighbor search on a single node. In Advances in Neural Information Processing Systems, volume 32, 2019.
S. M. Jeffrey Travers. An experimental study of the small world problem. Sociometry, 32(4):425â443, 1969.
J. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 163â170, 2000. | 2401.09350#273 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 273,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "B. Harwood and T. Drummond. Fanng: Fast approximate nearest neigh- bour graphs. In 2016 IEEE Conference on Computer Vision and Pattern Recognition, pages 5713â5722, 2016.\nP. Indyk and H. Xu. Worst-case performance of popular approximate nearest neighbor search implementations: Guarantees and limitations. In Proceed- ings of the 36th Conference on Neural Information Processing Systems, 2023.\nJ. Jaromczyk and G. Toussaint. Relative neighborhood graphs and their relatives. Proceedings of the IEEE, 80(9):1502â1517, 1992.\nReferences\nS. Jayaram Subramanya, F. Devvrit, H. V. Simhadri, R. Krishnawamy, and R. Kadekodi. Diskann: Fast accurate billion-point nearest neighbor search on a single node. In Advances in Neural Information Processing Systems, volume 32, 2019.\nS. M. Jeffrey Travers. An experimental study of the small world problem. Sociometry, 32(4):425â443, 1969.\nJ. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 163â170, 2000.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 274 | J. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 163â170, 2000.
W. Li, Y. Zhang, Y. Sun, W. Wang, M. Li, W. Zhang, and X. Lin. Approx- imate nearest neighbor search on high dimensional data â experiments, analyses, and improvement. IEEE Transactions on Knowledge and Data Engineering, 32(8):1475â1488, 2020.
J. Liu, X. Yan, X. DAI, Z. Li, J. Cheng, and M. Yang. Understanding and improving proximity graph based maximum inner product search. In AAAI Conference on Artificial Intelligence, 2019.
Y. Malkov, A. Ponomarenko, A. Logvinov, and V. Krylov. Approximate near- est neighbor algorithm based on navigable small world graphs. Information Systems, 45:61â68, 2014.
Y. A. Malkov and D. A. Yashunin. Efficient and robust approximate nearest neighbor search using hierarchical navigable small world graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(4):824â836, 4 2020. | 2401.09350#274 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 274,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "J. Kleinberg. The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 163â170, 2000.\nW. Li, Y. Zhang, Y. Sun, W. Wang, M. Li, W. Zhang, and X. Lin. Approx- imate nearest neighbor search on high dimensional data â experiments, analyses, and improvement. IEEE Transactions on Knowledge and Data Engineering, 32(8):1475â1488, 2020.\nJ. Liu, X. Yan, X. DAI, Z. Li, J. Cheng, and M. Yang. Understanding and improving proximity graph based maximum inner product search. In AAAI Conference on Artificial Intelligence, 2019.\nY. Malkov, A. Ponomarenko, A. Logvinov, and V. Krylov. Approximate near- est neighbor algorithm based on navigable small world graphs. Information Systems, 45:61â68, 2014.\nY. A. Malkov and D. A. Yashunin. Efficient and robust approximate nearest neighbor search using hierarchical navigable small world graphs. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(4):824â836, 4 2020.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 275 | S. Milgram. The Small-World Problem. Psychology Today, 1(1):61â67, 1967. S. Morozov and A. Babenko. Non-metric similarity graphs for maximum inner product search. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 4726â4735, 2018.
G. Navarro. Searching in metric spaces by spatial approximation. The VLDB Journal, 11(1):28â46, 08 2002.
J. OâRourke. Computing the relative neighborhood graph in the l1 and lâ metrics. Pattern Recognition, 15(3):189â192, 1982.
A. Singh, S. J. Subramanya, R. Krishnaswamy, and H. V. Simhadri. Freshdiskann: A fast and accurate graph-based ann index for streaming similarity search, 2021.
G. T. Toussaint. The relative neighbourhood graph of a finite planar set. Pattern Recognition, 12(4):261â268, 1980.
P. M. Vaidya. Ano(n logn) algorithm for the all-nearest-neighbors problem. Discrete and Computational Geometry, 4(2):101â115, 12 1989. | 2401.09350#275 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 275,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S. Milgram. The Small-World Problem. Psychology Today, 1(1):61â67, 1967. S. Morozov and A. Babenko. Non-metric similarity graphs for maximum inner product search. In Proceedings of the 32nd International Conference on Neural Information Processing Systems, pages 4726â4735, 2018.\nG. Navarro. Searching in metric spaces by spatial approximation. The VLDB Journal, 11(1):28â46, 08 2002.\nJ. OâRourke. Computing the relative neighborhood graph in the l1 and lâ metrics. Pattern Recognition, 15(3):189â192, 1982.\nA. Singh, S. J. Subramanya, R. Krishnaswamy, and H. V. Simhadri. Freshdiskann: A fast and accurate graph-based ann index for streaming similarity search, 2021.\nG. T. Toussaint. The relative neighbourhood graph of a finite planar set. Pattern Recognition, 12(4):261â268, 1980.\nP. M. Vaidya. Ano(n logn) algorithm for the all-nearest-neighbors problem. Discrete and Computational Geometry, 4(2):101â115, 12 1989.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 276 | M. Wang, X. Xu, Q. Yue, and Y. Wang. A comprehensive survey and exper- imental comparison of graph-based approximate nearest neighbor search. Proceedings of the VLDB Endowment, 14(11):1964â1978, jul 2021.
Z. Zhou, S. Tan, Z. Xu, and P. Li. M¨obius transformation for fast inner product search on graph. In Advances in Neural Information Processing Systems, volume 32, 2019.
103
Chapter 7 Clustering
Abstract We have seen index structures that manifest as trees, hash tables, and graphs. In this chapter, we will introduce a fourth way of organizing data points: clusters. It is perhaps the most natural and the simplest of the four methods, but also the least theoretically-justified. We will see why that is as we describe the details of clustering-based algorithms to top-k retrieval.
# 7.1 Algorithm | 2401.09350#276 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 276,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "M. Wang, X. Xu, Q. Yue, and Y. Wang. A comprehensive survey and exper- imental comparison of graph-based approximate nearest neighbor search. Proceedings of the VLDB Endowment, 14(11):1964â1978, jul 2021.\nZ. Zhou, S. Tan, Z. Xu, and P. Li. M¨obius transformation for fast inner product search on graph. In Advances in Neural Information Processing Systems, volume 32, 2019.\n103\nChapter 7 Clustering\nAbstract We have seen index structures that manifest as trees, hash tables, and graphs. In this chapter, we will introduce a fourth way of organizing data points: clusters. It is perhaps the most natural and the simplest of the four methods, but also the least theoretically-justified. We will see why that is as we describe the details of clustering-based algorithms to top-k retrieval.\n# 7.1 Algorithm",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 277 | # 7.1 Algorithm
As usual, we begin by indexing a collection of m data points X â Rd. Except in this paradigm, that involves invoking a clustering function, ζ : Rd â [C], that is appropriate for the distance function δ(·, ·), to map every data point to one of C clusters, where C is an arbitrary parameter. A typical choice for ζ is the KMeans algorithm with C = O( m). We then organize X into a table whose row i records the subset of points that are mapped to the i-th cluster: ζ â1(i) â {u | u â X , ζ(u) = i}.
Accompanying the index is a routing function Ï : Rd â [C]â. It takes an arbitrary point q as input and returns â clusters that are more likely to contain the nearest neighbor of q with respect to δ. In a typical instance of this framework Ï (·) is defined as follows:
1 IC-*()| (4) T(q) = argmino| q, iâ¬(C] Ss â). (7.1) uecâ*(i) Bi | 2401.09350#277 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 277,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 7.1 Algorithm\nAs usual, we begin by indexing a collection of m data points X â Rd. Except in this paradigm, that involves invoking a clustering function, ζ : Rd â [C], that is appropriate for the distance function δ(·, ·), to map every data point to one of C clusters, where C is an arbitrary parameter. A typical choice for ζ is the KMeans algorithm with C = O( m). We then organize X into a table whose row i records the subset of points that are mapped to the i-th cluster: ζ â1(i) â {u | u â X , ζ(u) = i}.\nAccompanying the index is a routing function Ï : Rd â [C]â. It takes an arbitrary point q as input and returns â clusters that are more likely to contain the nearest neighbor of q with respect to δ. In a typical instance of this framework Ï (·) is defined as follows:\n1 IC-*()| (4) T(q) = argmino| q, iâ¬(C] Ss â). (7.1) uecâ*(i) Bi",
"title": "Foundations of Vector Retrieval",
"year": 2024
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2401.09350 | 279 | Fig. 7.1: Illustration of the clustering-based retrieval method. The collection of points (left) is first partitioned into clusters (regions enclosed by dashed boundary on the right). When processing a query q using Equation (7.1), we compute δ(q, ·) for the centroid (solid circles) of every cluster and conduct our search over the â âclosestâ clusters.
When processing a query q, we take a two-step approach. We first obtain the list of clusters returned by Ï (q), then solve the top-k retrieval problem over the union of the identified clusters. Figure 7.1 visualizes this procedure. Notice that, the search for top-â clusters by using Equation (7.1) and the secondary search over the clusters identified by Ï are themselves instances of the approximate top-k retrieval problem. The parameter C determines the amount of effort that must be spent in each of the two phases of search: When C = 1, the cluster retrieval problem is solved trivially, whereas as C â â, cluster retrieval becomes equivalent to top-k retrieval over the entire collection. Interestingly, these operations can be delegated to a subroutine that itself uses a tree-, hash-, graph-, or even a clustering-based solution. That is, a clustering-based approach can be easily paired with any of the previously discussed methods! | 2401.09350#279 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 279,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Fig. 7.1: Illustration of the clustering-based retrieval method. The collection of points (left) is first partitioned into clusters (regions enclosed by dashed boundary on the right). When processing a query q using Equation (7.1), we compute δ(q, ·) for the centroid (solid circles) of every cluster and conduct our search over the â âclosestâ clusters.\nWhen processing a query q, we take a two-step approach. We first obtain the list of clusters returned by Ï (q), then solve the top-k retrieval problem over the union of the identified clusters. Figure 7.1 visualizes this procedure. Notice that, the search for top-â clusters by using Equation (7.1) and the secondary search over the clusters identified by Ï are themselves instances of the approximate top-k retrieval problem. The parameter C determines the amount of effort that must be spent in each of the two phases of search: When C = 1, the cluster retrieval problem is solved trivially, whereas as C â â, cluster retrieval becomes equivalent to top-k retrieval over the entire collection. Interestingly, these operations can be delegated to a subroutine that itself uses a tree-, hash-, graph-, or even a clustering-based solution. That is, a clustering-based approach can be easily paired with any of the previously discussed methods!",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 280 | This simple protocolâwith some variant of KMeans as ζ and Ï as in Equa- tion (7.1)âworks well in practice [Auvolat et al., 2015, J´egou et al., 2011, Bruch et al., 2023b, Babenko and Lempitsky, 2012, Chierichetti et al., 2007]. We present the results of our own experiments on various real-world datasets in Figure 7.2. This method owes its success to the empirical phenomenon that real-world data points tend to follow a multi-modal distribution, natu- rally forming clusters around each mode. By identifying these clusters and grouping data points together, we reduce the search space at the expense of retrieval quality.
However, to date, no formal analysis has been presented to quantify the retrieval error. The choice of ζ and Ï , too, have been left largely unexplored, with KMeans and Equation (7.1) as default answers. It is, for example, not known if KMeans is the right choice for a given δ. Or, whether clustering with spillage, where each data point may belong to multiple clusters, might reduce the overall error, as it did in Spill Trees. It is also an open question if, for a particular choice of ζ and δ, there exists a more effective routing
7.2 Closing Remarks
107 | 2401.09350#280 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 280,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "This simple protocolâwith some variant of KMeans as ζ and Ï as in Equa- tion (7.1)âworks well in practice [Auvolat et al., 2015, J´egou et al., 2011, Bruch et al., 2023b, Babenko and Lempitsky, 2012, Chierichetti et al., 2007]. We present the results of our own experiments on various real-world datasets in Figure 7.2. This method owes its success to the empirical phenomenon that real-world data points tend to follow a multi-modal distribution, natu- rally forming clusters around each mode. By identifying these clusters and grouping data points together, we reduce the search space at the expense of retrieval quality.\nHowever, to date, no formal analysis has been presented to quantify the retrieval error. The choice of ζ and Ï , too, have been left largely unexplored, with KMeans and Equation (7.1) as default answers. It is, for example, not known if KMeans is the right choice for a given δ. Or, whether clustering with spillage, where each data point may belong to multiple clusters, might reduce the overall error, as it did in Spill Trees. It is also an open question if, for a particular choice of ζ and δ, there exists a more effective routing\n7.2 Closing Remarks\n107",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 281 | 7.2 Closing Remarks
107
âCURACY 0.6) a ) âpeerin a Feven-MiNLM B04 -@-QUORA-MINILM > Feven-TasB -@-NQ-MiniLM- MS Manco Passace-MiniLM + NO-TasB 1% 3% 1% 2% ah ah me AS PERCENT OF C
1.0 a ) aan Deerisâ4-Gisr =O-GL0VE-200 AMS Tense srr 1% 5% 1% 3% cu; 3% AS PERCENT OF C
(a) MIPS (b) NN
Fig. 7.2: Performance of the clustering-based retrieval method on various real- world collections, described in Appendix A. The figure shows top-1 accuracy versus the number of clusters, â, considered by the routing function Ï (·) as a m, percentage of the number of clusters C. In these experiments, we set C = where m = |X | is the size of the collection, and use spherical KMeans (MIPS) and standard KMeans (NN) to form clusters.
functionâincluding learnt functions tailored to a query distributionâthat uses higher-order statistics from the cluster distributions. | 2401.09350#281 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 281,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "7.2 Closing Remarks\n107\nâCURACY 0.6) a ) âpeerin a Feven-MiNLM B04 -@-QUORA-MINILM > Feven-TasB -@-NQ-MiniLM- MS Manco Passace-MiniLM + NO-TasB 1% 3% 1% 2% ah ah me AS PERCENT OF C\n1.0 a ) aan Deerisâ4-Gisr =O-GL0VE-200 AMS Tense srr 1% 5% 1% 3% cu; 3% AS PERCENT OF C\n(a) MIPS (b) NN \nFig. 7.2: Performance of the clustering-based retrieval method on various real- world collections, described in Appendix A. The figure shows top-1 accuracy versus the number of clusters, â, considered by the routing function Ï (·) as a m, percentage of the number of clusters C. In these experiments, we set C = where m = |X | is the size of the collection, and use spherical KMeans (MIPS) and standard KMeans (NN) to form clusters.\nfunctionâincluding learnt functions tailored to a query distributionâthat uses higher-order statistics from the cluster distributions.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 282 | functionâincluding learnt functions tailored to a query distributionâthat uses higher-order statistics from the cluster distributions.
In spite of these shortcomings, the algorithmic framework above contains a fascinating insight that is actually useful for a rather different end-goal: vector compression, or more precisely, quantization. We will unpack this connection in Chapter 9.
# 7.2 Closing Remarks
This chapter departed entirely from the theme of this monograph. Whereas we are generally able to say something intelligent about trees, hash functions, and graphs, top-k retrieval by clustering has emerged entirely based on our intuition that data points naturally form clusters. We cannot formally deter- mine, for example, the behavior of the retrieval system as a function of the clustering algorithm itself, the number of clusters, or the routing function. All that must be determined empirically.
What we do observe often in practice, however, is that clustering-based top-k retrieval is efficient [Paulev´e et al., 2010, Auvolat et al., 2015, Bruch et al., 2023b, J´egou et al., 2011], at least in the case of Nearest Neighbor search with Euclidean distance, where KMeans is a theoretically appropriate choice. It is efficient in the sense that retrieval accuracy often reaches an acceptable level after probing a few top-ranking clusters as identified by Equation (7.1).
108
7 Clustering | 2401.09350#282 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 282,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "functionâincluding learnt functions tailored to a query distributionâthat uses higher-order statistics from the cluster distributions.\nIn spite of these shortcomings, the algorithmic framework above contains a fascinating insight that is actually useful for a rather different end-goal: vector compression, or more precisely, quantization. We will unpack this connection in Chapter 9.\n# 7.2 Closing Remarks\nThis chapter departed entirely from the theme of this monograph. Whereas we are generally able to say something intelligent about trees, hash functions, and graphs, top-k retrieval by clustering has emerged entirely based on our intuition that data points naturally form clusters. We cannot formally deter- mine, for example, the behavior of the retrieval system as a function of the clustering algorithm itself, the number of clusters, or the routing function. All that must be determined empirically.\nWhat we do observe often in practice, however, is that clustering-based top-k retrieval is efficient [Paulev´e et al., 2010, Auvolat et al., 2015, Bruch et al., 2023b, J´egou et al., 2011], at least in the case of Nearest Neighbor search with Euclidean distance, where KMeans is a theoretically appropriate choice. It is efficient in the sense that retrieval accuracy often reaches an acceptable level after probing a few top-ranking clusters as identified by Equation (7.1).\n108\n7 Clustering",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 283 | 108
7 Clustering
That we have a method that is efficient in practice, but its efficiency and the conditions under which it is efficient are unexplained, constitutes a sub- stantial gap and thus presents multiple consequential open questions. These questions involve optimal clustering, routing, and bounds on retrieval accu- racy.
When the distance function is the Euclidean distance and our objective is to learn the Voronoi regions of data points, the KMeans clustering objective makes sense. We can even state formal results regarding the optimality of the resulting clustering [Arthur and Vassilvitskii, 2007]. That argument is no longer valid when the distance function is based on inner product, where we must learn the inner product Voronoi cones, and where some points may have an empty Voronoi region. What objective we must optimize for MIPS, therefore, is an open question that, as we saw in this chapter, has been partially explored in the past [Guo et al., 2020]. | 2401.09350#283 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 283,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "108\n7 Clustering\nThat we have a method that is efficient in practice, but its efficiency and the conditions under which it is efficient are unexplained, constitutes a sub- stantial gap and thus presents multiple consequential open questions. These questions involve optimal clustering, routing, and bounds on retrieval accu- racy.\nWhen the distance function is the Euclidean distance and our objective is to learn the Voronoi regions of data points, the KMeans clustering objective makes sense. We can even state formal results regarding the optimality of the resulting clustering [Arthur and Vassilvitskii, 2007]. That argument is no longer valid when the distance function is based on inner product, where we must learn the inner product Voronoi cones, and where some points may have an empty Voronoi region. What objective we must optimize for MIPS, therefore, is an open question that, as we saw in this chapter, has been partially explored in the past [Guo et al., 2020].",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 284 | Even when we know what the right clustering algorithm is, there is still the issue of âbalanceâ that we must understand how to handle. It would, for example, be far from ideal if the clusters end up having very different sizes. Unfortunately, that happens quite naturally if the data points have highly variable norms and the clustering algorithm is based on KMeans: Data points with large norms become isolated, while vectors with small norms form massive clusters.
What has been left entirely untouched is the routing machinery. Equa- tion (7.1) is the de facto routing function, but one that is possibly sub- optimal. That is because, Equation (7.1) uses the mean of the data points within a cluster as the representative or sketch of that cluster. When clusters are highly concentrated around their mean, such a sketch accurately reflects the potential of each cluster. But when clusters have different shapes, higher- order statistics from the cluster may be required to accurately route queries to clusters.
So the question we are faced with is the following: What is a good sketch of each cluster? Is there a coreset of data points within each cluster that lead to better routing of queries during retrieval? Can we quantify the probability of errorâin the sense that the cluster containing the optimal solution is not returned by the routing functionâgiven a sketch? | 2401.09350#284 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 284,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Even when we know what the right clustering algorithm is, there is still the issue of âbalanceâ that we must understand how to handle. It would, for example, be far from ideal if the clusters end up having very different sizes. Unfortunately, that happens quite naturally if the data points have highly variable norms and the clustering algorithm is based on KMeans: Data points with large norms become isolated, while vectors with small norms form massive clusters.\nWhat has been left entirely untouched is the routing machinery. Equa- tion (7.1) is the de facto routing function, but one that is possibly sub- optimal. That is because, Equation (7.1) uses the mean of the data points within a cluster as the representative or sketch of that cluster. When clusters are highly concentrated around their mean, such a sketch accurately reflects the potential of each cluster. But when clusters have different shapes, higher- order statistics from the cluster may be required to accurately route queries to clusters.\nSo the question we are faced with is the following: What is a good sketch of each cluster? Is there a coreset of data points within each cluster that lead to better routing of queries during retrieval? Can we quantify the probability of errorâin the sense that the cluster containing the optimal solution is not returned by the routing functionâgiven a sketch?",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 285 | We may answer these questions differently if we had some idea of what the query distribution looks like. Assuming access to a set of training queries, it may be possible to learn a more optimal sketch using supervised learning methods. Concepts from learning-to-rank [Bruch et al., 2023a] seem particu- larly relevant to this setup. To see how, note that the outcome of the routing function is to identify the cluster that contains the optimal data point for a query. We could view this as ranking clusters with respect to a query, where we wish for the âcorrectâ cluster to appear at the top of the ranked list. Given this mental model, we can evaluate the quality of a routing function using
# References
References
any of the many ranking quality metrics such as Reciprocal Rank (defined as the reciprocal of the rank of the correct cluster). Learning a ranking function that maximizes Reciprocal Rank can then be done indirectly by optimizing a custom cross entropy-based surrogate, as proved by Bruch et al. [2019] and Bruch [2021].
Perhaps the more important open question is understanding when clus- tering is efficient and why. Answering that question may require exploring the connection between clustering-based top-k retrieval, branch-and-bound algorithms, and LSH. | 2401.09350#285 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 285,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We may answer these questions differently if we had some idea of what the query distribution looks like. Assuming access to a set of training queries, it may be possible to learn a more optimal sketch using supervised learning methods. Concepts from learning-to-rank [Bruch et al., 2023a] seem particu- larly relevant to this setup. To see how, note that the outcome of the routing function is to identify the cluster that contains the optimal data point for a query. We could view this as ranking clusters with respect to a query, where we wish for the âcorrectâ cluster to appear at the top of the ranked list. Given this mental model, we can evaluate the quality of a routing function using\n# References\nReferences\nany of the many ranking quality metrics such as Reciprocal Rank (defined as the reciprocal of the rank of the correct cluster). Learning a ranking function that maximizes Reciprocal Rank can then be done indirectly by optimizing a custom cross entropy-based surrogate, as proved by Bruch et al. [2019] and Bruch [2021].\nPerhaps the more important open question is understanding when clus- tering is efficient and why. Answering that question may require exploring the connection between clustering-based top-k retrieval, branch-and-bound algorithms, and LSH.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 286 | Take any clustering algorithm, ζ. If one could show formally that ζ behaves like an LSH family, then clustering-based top-k retrieval simply collapses to LSH. In that case, not only do the results from that literature apply, but the techniques developed for LSH (such as multi-probe LSH) too port over to clustering.
Similarly, one may adopt the view that finding the top cluster is a series of decisions, each determining which side of a hyperplane a point falls. Whereas in Random Partition Trees or Spill Trees, such decision hyperplanes were random directions, here the hyperplanes are correlated. Nonetheless, that insight could help us produce clusters with spillage, where data points belong to multiple clusters, and in a manner that helps reduce the overall error.
# References
D. Arthur and S. Vassilvitskii. K-means++: The advantages of careful seed- ing. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1027â1035, 2007.
A. Auvolat, S. Chandar, P. Vincent, H. Larochelle, and Y. Bengio. Clustering is efficient for approximate maximum inner product search, 2015. | 2401.09350#286 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 286,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Take any clustering algorithm, ζ. If one could show formally that ζ behaves like an LSH family, then clustering-based top-k retrieval simply collapses to LSH. In that case, not only do the results from that literature apply, but the techniques developed for LSH (such as multi-probe LSH) too port over to clustering.\nSimilarly, one may adopt the view that finding the top cluster is a series of decisions, each determining which side of a hyperplane a point falls. Whereas in Random Partition Trees or Spill Trees, such decision hyperplanes were random directions, here the hyperplanes are correlated. Nonetheless, that insight could help us produce clusters with spillage, where data points belong to multiple clusters, and in a manner that helps reduce the overall error.\n# References\nD. Arthur and S. Vassilvitskii. K-means++: The advantages of careful seed- ing. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1027â1035, 2007.\nA. Auvolat, S. Chandar, P. Vincent, H. Larochelle, and Y. Bengio. Clustering is efficient for approximate maximum inner product search, 2015.",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 287 | A. Babenko and V. Lempitsky. The inverted multi-index. In 2012 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3069â3076, 2012.
S. Bruch. An alternative cross entropy loss for learning-to-rank. In Proceed- ings of the Web Conference 2021, page 118â126, 2021.
S. Bruch, X. Wang, M. Bendersky, and M. Najork. An analysis of the soft- max cross entropy loss for learning-to-rank with binary relevance. In Pro- ceedings of the 2019 ACM SIGIR International Conference on Theory of Information Retrieval, page 75â78, 2019.
S. Bruch, C. Lucchese, and F. M. Nardini. Efficient and effective tree-based and neural learning to rank. Foundations and Trends in Information Re- trieval, 17(1):1â123, 2023a.
S. Bruch, F. M. Nardini, A. Ingber, and E. Liberty. Bridging dense and sparse maximum inner product search, 2023b.
109
110
7 Clustering | 2401.09350#287 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 287,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "A. Babenko and V. Lempitsky. The inverted multi-index. In 2012 IEEE Con- ference on Computer Vision and Pattern Recognition, pages 3069â3076, 2012.\nS. Bruch. An alternative cross entropy loss for learning-to-rank. In Proceed- ings of the Web Conference 2021, page 118â126, 2021.\nS. Bruch, X. Wang, M. Bendersky, and M. Najork. An analysis of the soft- max cross entropy loss for learning-to-rank with binary relevance. In Pro- ceedings of the 2019 ACM SIGIR International Conference on Theory of Information Retrieval, page 75â78, 2019.\nS. Bruch, C. Lucchese, and F. M. Nardini. Efficient and effective tree-based and neural learning to rank. Foundations and Trends in Information Re- trieval, 17(1):1â123, 2023a.\nS. Bruch, F. M. Nardini, A. Ingber, and E. Liberty. Bridging dense and sparse maximum inner product search, 2023b.\n109\n110\n7 Clustering",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 288 | 109
110
7 Clustering
F. Chierichetti, A. Panconesi, P. Raghavan, M. Sozio, A. Tiberi, and E. Upfal. Finding near neighbors through cluster pruning. In Proceedings of the Twenty-Sixth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 103â112, 2007.
R. Guo, P. Sun, E. Lindgren, Q. Geng, D. Simcha, F. Chern, and S. Kumar. Accelerating large-scale inference with anisotropic vector quantization. In Proceedings of the 37th International Conference on Machine Learning, 2020.
H. J´egou, M. Douze, and C. Schmid. Product quantization for nearest neigh- bor search. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 33(1):117â128, 2011.
L. Paulev´e, H. J´egou, and L. Amsaleg. Locality sensitive hashing: A compar- ison of hash function types and querying mechanisms. Pattern Recognition Letters, 31(11):1348â1358, 2010.
# Chapter 8 Sampling Algorithms | 2401.09350#288 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 288,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "109\n110\n7 Clustering\nF. Chierichetti, A. Panconesi, P. Raghavan, M. Sozio, A. Tiberi, and E. Upfal. Finding near neighbors through cluster pruning. In Proceedings of the Twenty-Sixth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages 103â112, 2007.\nR. Guo, P. Sun, E. Lindgren, Q. Geng, D. Simcha, F. Chern, and S. Kumar. Accelerating large-scale inference with anisotropic vector quantization. In Proceedings of the 37th International Conference on Machine Learning, 2020.\nH. J´egou, M. Douze, and C. Schmid. Product quantization for nearest neigh- bor search. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 33(1):117â128, 2011.\nL. Paulev´e, H. J´egou, and L. Amsaleg. Locality sensitive hashing: A compar- ison of hash function types and querying mechanisms. Pattern Recognition Letters, 31(11):1348â1358, 2010.\n# Chapter 8 Sampling Algorithms",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 289 | # Chapter 8 Sampling Algorithms
Abstract Nearly all of the data structures and algorithms we reviewed in the previous chapters are designed specifically for either nearest neighbor search or maximum cosine similarity search. MIPS is typically an afterthought. It is often cast as NN or MCS through a rank-preserving transformation and subsequently solved using one of these algorithms. That is so because inner product is not a proper metric, making MIPS different from the other vector retrieval variants. In this chapter, we review algorithms that are specifically designed for MIPS and that connect MIPS to the machinery underlying multi- arm bandits.
# 8.1 Intuition
That inner product is different can be a curse and a blessing. We have already discussed that curse at length, but in this chapter, we will finally learn some- thing positive. And that is the fact that inner product is a linear function of data points and can be easily decomposed into its parts, thereby opening a unique path to solving MIPS.
The overarching idea in what we refer to as sampling algorithms is to avoid computing inner products. Instead, we either directly approximate the likelihood of a data point being the solution to MIPS (or, equivalently, its rank), or estimate its inner product with a query (i.e., its score). As we will see shortly, in both instances, we rely heavily on the linearity of inner product to estimate probabilities and derive bounds. | 2401.09350#289 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 289,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# Chapter 8 Sampling Algorithms\nAbstract Nearly all of the data structures and algorithms we reviewed in the previous chapters are designed specifically for either nearest neighbor search or maximum cosine similarity search. MIPS is typically an afterthought. It is often cast as NN or MCS through a rank-preserving transformation and subsequently solved using one of these algorithms. That is so because inner product is not a proper metric, making MIPS different from the other vector retrieval variants. In this chapter, we review algorithms that are specifically designed for MIPS and that connect MIPS to the machinery underlying multi- arm bandits.\n# 8.1 Intuition\nThat inner product is different can be a curse and a blessing. We have already discussed that curse at length, but in this chapter, we will finally learn some- thing positive. And that is the fact that inner product is a linear function of data points and can be easily decomposed into its parts, thereby opening a unique path to solving MIPS.\nThe overarching idea in what we refer to as sampling algorithms is to avoid computing inner products. Instead, we either directly approximate the likelihood of a data point being the solution to MIPS (or, equivalently, its rank), or estimate its inner product with a query (i.e., its score). As we will see shortly, in both instances, we rely heavily on the linearity of inner product to estimate probabilities and derive bounds.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 290 | Approximating the ranks or scores of data points uses some form of sam- pling: we either sample data points according to a distribution defined by inner products, or sample a dimension to compute partial inner products with and eliminate sub-optimal data points iteratively. In the former, the more frequently a data point is sampled, the more likely it is to be the so- lution to MIPS. In the latter, the more dimensions we sample, the closer we
111
112
8 Sampling Algorithms
get to computing full inner products. Generally, then, the more samples we draw, the more accurate our solution to MIPS becomes.
An interesting property of using sampling to solve MIPS is that, regard- less of what we are approximating, we can decide when to stop! That is, if we are given a time budget, we draw as many samples as our time budget allows and return our best guess of the solutions based on the information we have collected up to that point. The number of samples, in other words, serves as a knob that trades off accuracy for speed.
The remainder of this chapter describes these algorithms in much greater detail. Importantly, we will see how linearity makes the approximation- through-sampling feasible and efficient.
# 8.2 Approximating the Ranks | 2401.09350#290 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 290,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Approximating the ranks or scores of data points uses some form of sam- pling: we either sample data points according to a distribution defined by inner products, or sample a dimension to compute partial inner products with and eliminate sub-optimal data points iteratively. In the former, the more frequently a data point is sampled, the more likely it is to be the so- lution to MIPS. In the latter, the more dimensions we sample, the closer we\n111\n112\n8 Sampling Algorithms\nget to computing full inner products. Generally, then, the more samples we draw, the more accurate our solution to MIPS becomes.\nAn interesting property of using sampling to solve MIPS is that, regard- less of what we are approximating, we can decide when to stop! That is, if we are given a time budget, we draw as many samples as our time budget allows and return our best guess of the solutions based on the information we have collected up to that point. The number of samples, in other words, serves as a knob that trades off accuracy for speed.\nThe remainder of this chapter describes these algorithms in much greater detail. Importantly, we will see how linearity makes the approximation- through-sampling feasible and efficient.\n# 8.2 Approximating the Ranks",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 291 | # 8.2 Approximating the Ranks
We are interested in finding the top-k data points with the largest inner product with a query q â Rd, from a collection X â Rd of m points. Suppose that we had an efficient way of sampling a data point from X where the point u â X has probability proportional to â¨q, uâ© of being selected.
If we drew a sufficiently large number of samples, the data point with the largest inner product with q would be selected most frequently. The data point with the second largest inner product would similarly be selected with the second highest frequency, and so on. So, if we counted the number of times each data point has been sampled, the resulting histogram would be a good approximation to the rank of each data point with respect to inner product with q.
That is the gist of the sampling algorithm we examine in this section. But while the idea is rather straightforward, making it work requires addressing a few critical gaps. The biggest challenge is drawing samples according to the distribution of inner products without actually computing any of the inner products! That is because, if we needed to compute â¨q, uâ© for all u â X , then we could simply sort data points accordingly and return the top-k; no need for sampling and the rest. | 2401.09350#291 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 291,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# 8.2 Approximating the Ranks\nWe are interested in finding the top-k data points with the largest inner product with a query q â Rd, from a collection X â Rd of m points. Suppose that we had an efficient way of sampling a data point from X where the point u â X has probability proportional to â¨q, uâ© of being selected.\nIf we drew a sufficiently large number of samples, the data point with the largest inner product with q would be selected most frequently. The data point with the second largest inner product would similarly be selected with the second highest frequency, and so on. So, if we counted the number of times each data point has been sampled, the resulting histogram would be a good approximation to the rank of each data point with respect to inner product with q.\nThat is the gist of the sampling algorithm we examine in this section. But while the idea is rather straightforward, making it work requires addressing a few critical gaps. The biggest challenge is drawing samples according to the distribution of inner products without actually computing any of the inner products! That is because, if we needed to compute â¨q, uâ© for all u â X , then we could simply sort data points accordingly and return the top-k; no need for sampling and the rest.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 292 | The key to tackling that challenge is the linearity of inner product. Follow- ing a few simple derivations using Bayesâ theorem, we can break up the sam- pling procedure into two steps, each using marginal distributions only [Loren- zen and Pham, 2021, Ballard et al., 2015, Cohen and Lewis, 1997, Ding et al., 2019]. Importantly, one of these marginal distributions can be computed of- fline as part of indexing. That is the result we will review next.
8.2 Approximating the Ranks
# 8.2.1 Non-negative Data and Queries
We wish to draw a data point with probability that is proportional to its inner product with a query: P[u | q] â â¨q, uâ©. For now, we assume that u, q ⪰ 0 for all u â X and queries q.
Let us decompose this probability along each dimension as follows:
d Plu| ql = >> Plt | J Ple| ta), (8.1) t=1
where the first term in the sum is the probability of sampling a dimension t â [d] and the second term is the likelihood of sampling u given a particular dimension. We can model each of these terms as follows:
Plt|qlx Ss Us = Ss Uts (8.2) UuEx Ucrk
and, | 2401.09350#292 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 292,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The key to tackling that challenge is the linearity of inner product. Follow- ing a few simple derivations using Bayesâ theorem, we can break up the sam- pling procedure into two steps, each using marginal distributions only [Loren- zen and Pham, 2021, Ballard et al., 2015, Cohen and Lewis, 1997, Ding et al., 2019]. Importantly, one of these marginal distributions can be computed of- fline as part of indexing. That is the result we will review next.\n8.2 Approximating the Ranks\n# 8.2.1 Non-negative Data and Queries\nWe wish to draw a data point with probability that is proportional to its inner product with a query: P[u | q] â â¨q, uâ©. For now, we assume that u, q ⪰ 0 for all u â X and queries q.\nLet us decompose this probability along each dimension as follows:\nd Plu| ql = >> Plt | J Ple| ta), (8.1) t=1\nwhere the first term in the sum is the probability of sampling a dimension t â [d] and the second term is the likelihood of sampling u given a particular dimension. We can model each of these terms as follows:\nPlt|qlx Ss Us = Ss Uts (8.2) UuEx Ucrk\nand,",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 293 | Plt|qlx Ss Us = Ss Uts (8.2) UuEx Ucrk
and,
Plu At | q] ur ut x . Plt | q] Ut ocx Ut ocx Ut Plu | tq] (8.3)
ut vâX vt vâX vt ̸= 0; if that sum is 0 we can
Plt | q] Ut In the above, we have assumed that },,<y simply discard the t-th dimension.
In the above, we have assumed that },,<y vr 4 0; if that sum is 0 we can simply discard the t-th dimension.
What we have done above allows us to draw a sample according to P[u | q] by, instead, drawing a dimension t according to P[t | q] first, then drawing a data point u according to P[u | t, q].
Sampling from these multinomial distribution requires constructing the distributions themselves. Luckily, P[u | t, q] is independent of q. Its distribu- tion can therefore be computed offline: we create d tables, where the t-th table has m rows recording the probability of each data point being selected given dimension t using Equation (8.3). We can then use the alias method [Walker, 1977] to draw samples from these distributions using O(1) operations. | 2401.09350#293 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 293,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Plt|qlx Ss Us = Ss Uts (8.2) UuEx Ucrk\nand,\nPlu At | q] ur ut x . Plt | q] Ut ocx Ut ocx Ut Plu | tq] (8.3)\nut vâX vt vâX vt ̸= 0; if that sum is 0 we can\nPlt | q] Ut In the above, we have assumed that },,<y simply discard the t-th dimension.\nIn the above, we have assumed that },,<y vr 4 0; if that sum is 0 we can simply discard the t-th dimension.\nWhat we have done above allows us to draw a sample according to P[u | q] by, instead, drawing a dimension t according to P[t | q] first, then drawing a data point u according to P[u | t, q].\nSampling from these multinomial distribution requires constructing the distributions themselves. Luckily, P[u | t, q] is independent of q. Its distribu- tion can therefore be computed offline: we create d tables, where the t-th table has m rows recording the probability of each data point being selected given dimension t using Equation (8.3). We can then use the alias method [Walker, 1977] to draw samples from these distributions using O(1) operations.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 294 | The distribution over dimensions given a query, P{t | g], must be computed online using Equation (8.2), which requires O(d) operations, assuming we compute >> ,,¢y Ur offline for each t and store them in our index. Again, using he alias method, we can subsequently draw samples with O(1) operations. The procedure described above provides us with an efficient mechanism to perform the desired sampling. If we were to draw S samples, that could be done in O(d + $), where O(d) term is needed to construct the multinomial distribution that defines P{t | q].
As we draw samples, we maintain a histogram over the m data points, counting the number of times each point has been sampled. In the end, we can identify the top-kⲠ(for kⲠ⥠k) points based on these counts, compute their inner products with the query, and return the top-k points as the final
113
114
8 Sampling Algorithms
solution set. All these operations together have time complexity O(d + S + m log kâ² + kâ²d), with S typically being the dominant term.
# 8.2.2 The General Case
When the data points or queries may be negative, the algorithm described in the previous section will not work as is. To extend the sampling framework to general, real vectors, we must make a few minor adjustments. | 2401.09350#294 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 294,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The distribution over dimensions given a query, P{t | g], must be computed online using Equation (8.2), which requires O(d) operations, assuming we compute >> ,,¢y Ur offline for each t and store them in our index. Again, using he alias method, we can subsequently draw samples with O(1) operations. The procedure described above provides us with an efficient mechanism to perform the desired sampling. If we were to draw S samples, that could be done in O(d + $), where O(d) term is needed to construct the multinomial distribution that defines P{t | q].\nAs we draw samples, we maintain a histogram over the m data points, counting the number of times each point has been sampled. In the end, we can identify the top-kâ² (for kⲠ⥠k) points based on these counts, compute their inner products with the query, and return the top-k points as the final\n113\n114\n8 Sampling Algorithms\nsolution set. All these operations together have time complexity O(d + S + m log kâ² + kâ²d), with S typically being the dominant term.\n# 8.2.2 The General Case\nWhen the data points or queries may be negative, the algorithm described in the previous section will not work as is. To extend the sampling framework to general, real vectors, we must make a few minor adjustments.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 295 | First, we must ensure that the marginal distributions are valid. That is easy to do: In Equations (8.2) and (8.3), we replace each term with its absolute value. So, P[t | g] becomes proportional to }>,<y|qeus|, and Plu | toa] & lusl/ Cyexltel
We then use the resulting distributions to sample data points as before, but every time a data point u is sampled, instead of incrementing its count in the histogram by one, we add Sign(qtut) to its entry. As the following lemma shows, in expectation, the final count is proportional to â¨q, uâ©.
Lemma 8.1 Define the random variable Z as 0 if data point u ⬠X is not sampled and SIGN(quz) if it is for a query q ⬠R¢ and a sampled dimension t. Then E[Z] = (q,u)/ Diy Dexia:
Proof.
ry us| uo Doves te _ SIGN(qut) lquue| _ deur ocalare| ocx luv E[Z | t] = SIGN(qruz) P{u | t] = Sicn(
Taking expectation over the dimension t yields: | 2401.09350#295 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 295,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "First, we must ensure that the marginal distributions are valid. That is easy to do: In Equations (8.2) and (8.3), we replace each term with its absolute value. So, P[t | g] becomes proportional to }>,<y|qeus|, and Plu | toa] & lusl/ Cyexltel\nWe then use the resulting distributions to sample data points as before, but every time a data point u is sampled, instead of incrementing its count in the histogram by one, we add Sign(qtut) to its entry. As the following lemma shows, in expectation, the final count is proportional to â¨q, uâ©.\nLemma 8.1 Define the random variable Z as 0 if data point u ⬠X is not sampled and SIGN(quz) if it is for a query q ⬠R¢ and a sampled dimension t. Then E[Z] = (q,u)/ Diy Dexia:\nProof.\nry us| uo Doves te _ SIGN(qut) lquue| _ deur ocalare| ocx luv E[Z | t] = SIGN(qruz) P{u | t] = Sicn(\nTaking expectation over the dimension t yields:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 296 | Taking expectation over the dimension t yields:
E[Z] =E(E(Z| 4] = Ya Plt | 4] vex |dUe| d UU Dover lave ,] a t=1 Vvea lure Viet Voex lari (q,u) ~Sd a), Viet Vocal urel
ââ
8.2 Approximating the Ranks
# 8.2.3 Sample Complexity
We have formalized an efficient way to sample data points according to the distribution of inner products, and subsequently collect the most frequently- sampled points. But how many samples must we draw in order to accurately identify the top-k solution set? Ding et al. [2019] give an answer in the form of the following theorem for top-1 MIPS.
Before stating the result, it would be helpful to introduce a few shorthands. Let N = al vex |ave| be a normalizing factor. For a vector u ⬠¥, denote by A, the scaled gap between the maximum inner product and the inner product of u and q: A, = (q,u* â u)/N. | 2401.09350#296 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 296,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Taking expectation over the dimension t yields:\nE[Z] =E(E(Z| 4] = Ya Plt | 4] vex |dUe| d UU Dover lave ,] a t=1 Vvea lure Viet Voex lari (q,u) ~Sd a), Viet Vocal urel\nââ\n8.2 Approximating the Ranks\n# 8.2.3 Sample Complexity\nWe have formalized an efficient way to sample data points according to the distribution of inner products, and subsequently collect the most frequently- sampled points. But how many samples must we draw in order to accurately identify the top-k solution set? Ding et al. [2019] give an answer in the form of the following theorem for top-1 MIPS.\nBefore stating the result, it would be helpful to introduce a few shorthands. Let N = al vex |ave| be a normalizing factor. For a vector u ⬠¥, denote by A, the scaled gap between the maximum inner product and the inner product of u and q: A, = (q,u* â u)/N.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 297 | If S$ is the number of samples to be drawn, for a vector u, denote by Z,,i a random variable that is 0 if u was not sampled in round 7, and otherwise SIGN(qu,) if t is the sampled dimension. Once the sampling has concluded, the final value for point wu is simply Z, = >; Zu,:. Note that, from Lemma 8.1, we have that E[Z,,;] = (q,u)/N. Given the notation above, let us also introduce the following helpful
let us also introduce the following helpful lemma.
Lemma 8.2 Let Cy = ye latte for a data point u. Then for a pair of distinct vectors u,v ⬠X:
Cut Cy E[(Zus- Zea) | = 3,
and,
Cut Cy â (g,u-v)? Var [2 â Z| = $| yo |.
Proof. The proof is similar to the proof of Lemma 8.1.
Theorem 8.1 Suppose u* is the exact solution to MIPS over m points in X for query q. Define 02 = Var [Zu _ Z| and let A = minuex Ay. For 6 ⬠(0,1), if we drew S samples such that: | 2401.09350#297 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 297,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "If S$ is the number of samples to be drawn, for a vector u, denote by Z,,i a random variable that is 0 if u was not sampled in round 7, and otherwise SIGN(qu,) if t is the sampled dimension. Once the sampling has concluded, the final value for point wu is simply Z, = >; Zu,:. Note that, from Lemma 8.1, we have that E[Z,,;] = (q,u)/N. Given the notation above, let us also introduce the following helpful\nlet us also introduce the following helpful lemma.\nLemma 8.2 Let Cy = ye latte for a data point u. Then for a pair of distinct vectors u,v ⬠X:\nCut Cy E[(Zus- Zea) | = 3,\nand,\nCut Cy â (g,u-v)? Var [2 â Z| = $| yo |.\nProof. The proof is similar to the proof of Lemma 8.1.\nTheorem 8.1 Suppose u* is the exact solution to MIPS over m points in X for query q. Define 02 = Var [Zu _ Z| and let A = minuex Ay. For 6 ⬠(0,1), if we drew S samples such that:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 298 | S ⥠max u̸=uâ (1 + âu)2 uh( âu(1+âu) Ï2 Ï2 u ) log m δ ,
where h(x) = (1 + x) log(1 + x) â x, then
P[Zuâ > Zu â u ̸= uâ] ⥠1 â δ.
Before proving the theorem above, let us make a quick observation. Clearly Ï2 u ⤠O(dâu) and (1 + âu) â 1. Because h(·) is monotone increasing in its argument ( âh
115
ââ
116 8 Sampling Algorithms ay Ae to) = (62 4 A+ A,)) log (1+ HOEY) â a4 A.) _ AL + A)? AL A aw AT > F+ a= 015).
Plugging this into Theorem 8.1 gives us S ⤠O( d
# â log m
δ ). | 2401.09350#298 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 298,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "S ⥠max u̸=uâ (1 + âu)2 uh( âu(1+âu) Ï2 Ï2 u ) log m δ ,\nwhere h(x) = (1 + x) log(1 + x) â x, then\nP[Zuâ > Zu â u ̸= uâ] ⥠1 â δ.\nBefore proving the theorem above, let us make a quick observation. Clearly Ï2 u ⤠O(dâu) and (1 + âu) â 1. Because h(·) is monotone increasing in its argument ( âh\n115\nââ\n116 8 Sampling Algorithms ay Ae to) = (62 4 A+ A,)) log (1+ HOEY) â a4 A.) _ AL + A)? AL A aw AT > F+ a= 015).\nPlugging this into Theorem 8.1 gives us S ⤠O( d\n# â log m\nδ ).",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 299 | Plugging this into Theorem 8.1 gives us S ⤠O( d
# â log m
δ ).
Theorem 8.1 tells us that, if we draw O( d δ ) samples, we can iden- tify the top-1 solution to MIPS with high probability. Observe that, â is a measure of the difficulty of the query: When inner products are close to each other, â becomes smaller, implying that a larger number of samples would be needed to correctly identify the exact solution.
Proof of Theorem 8.1. Consider the probability that the registered value of a data point u is greater than or equal to the registered value of the solution uâ once sampling has concluded. That is, P[Zu ⥠Zuâ ]. Let us rewrite that quantity as follows:
P [Zu > Zu-| =P [x Lua - Zuri > 0| Ss Yua Yu
Notice that E[Y,,,;] = 0 and that Y,,,;âs are independent. Furthermore, Y,,,; < 1+ A,. Letting Y, = 30; Yu, we can apply Bennettâs inequality to bound the probability above:
32 P[Y. > yu) < exp ( _ 5% a(S *840)), a+A, So2 u
Setting the right-hand-side to δ m , we arrive at: | 2401.09350#299 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 299,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Plugging this into Theorem 8.1 gives us S ⤠O( d\n# â log m\nδ ).\nTheorem 8.1 tells us that, if we draw O( d δ ) samples, we can iden- tify the top-1 solution to MIPS with high probability. Observe that, â is a measure of the difficulty of the query: When inner products are close to each other, â becomes smaller, implying that a larger number of samples would be needed to correctly identify the exact solution.\nProof of Theorem 8.1. Consider the probability that the registered value of a data point u is greater than or equal to the registered value of the solution uâ once sampling has concluded. That is, P[Zu ⥠Zuâ ]. Let us rewrite that quantity as follows:\nP [Zu > Zu-| =P [x Lua - Zuri > 0| Ss Yua Yu\nNotice that E[Y,,,;] = 0 and that Y,,,;âs are independent. Furthermore, Y,,,; < 1+ A,. Letting Y, = 30; Yu, we can apply Bennettâs inequality to bound the probability above:\n32 P[Y. > yu) < exp ( _ 5% a(S *840)), a+A, So2 u\nSetting the right-hand-side to δ m , we arrive at:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 300 | Setting the right-hand-side to δ m , we arrive at:
So (1+ Ay)Au 6 vol a+ Teh 2 ) < m u Aull + Au) m = S(1+ Au) ?o2h( > log 5: o
It is easy to see that for x > 0, h(x) > 0. Observing that âu(1 + âu)/Ï2 u is positive, that implies that h(âu(1 + âu)/Ï2 u) > 0, and therefore we can re-arrange the expression above as follows:
8.3 Approximating the Scores
1+A,)? S> â + Au) log â¢. (8.4) o3h( Aa4e)) )
We have thus far shown that when S satisfies the inequality in (8.4), then m . Going back to the claim, we derive the following bound P[Yu ⥠yu] ⤠δ using the result above:
P[Zuâ > Zu âu â X ] = 1 â P[â u â X s.t. Zuâ ⤠Zu] ⥠1 â m δ m = 1 â δ,
where we have used the union bound to obtain the inequality.
# 8.3 Approximating the Scores | 2401.09350#300 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 300,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Setting the right-hand-side to δ m , we arrive at:\nSo (1+ Ay)Au 6 vol a+ Teh 2 ) < m u Aull + Au) m = S(1+ Au) ?o2h( > log 5: o\nIt is easy to see that for x > 0, h(x) > 0. Observing that âu(1 + âu)/Ï2 u is positive, that implies that h(âu(1 + âu)/Ï2 u) > 0, and therefore we can re-arrange the expression above as follows:\n8.3 Approximating the Scores\n1+A,)? S> â + Au) log â¢. (8.4) o3h( Aa4e)) )\nWe have thus far shown that when S satisfies the inequality in (8.4), then m . Going back to the claim, we derive the following bound P[Yu ⥠yu] ⤠δ using the result above:\nP[Zuâ > Zu âu â X ] = 1 â P[â u â X s.t. Zuâ ⤠Zu] ⥠1 â m δ m = 1 â δ,\nwhere we have used the union bound to obtain the inequality.\n# 8.3 Approximating the Scores",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 301 | where we have used the union bound to obtain the inequality.
# 8.3 Approximating the Scores
The method we have just presented avoids the computation of inner products altogether but estimates the rank of each data point with respect to a query using a sampling procedure. In this section, we introduce another sampling method that approximates the inner product of every data point instead.
Let us motivate our next algorithm with a rather contrived example. Sup- pose that our data points and queries are in R2, with the first coordinate of vectors drawing values from N (0, Ï2 1) and the second coordinate from N (0, Ï2 2). If we were to compute the inner product of q with every vector u â X , we would need to perform two multiplications and a sum: u1q1 + u2q2. That gives us the exact âscoreâ of every point with respect to q. But if Ï2 1 â« Ï2 2, then by computing q1u1 for all u â X , it is very likely that we have a good approximation to the final inner product. So we may use the partial inner product as a high-confidence estimate of the full inner product. | 2401.09350#301 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 301,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "where we have used the union bound to obtain the inequality.\n# 8.3 Approximating the Scores\nThe method we have just presented avoids the computation of inner products altogether but estimates the rank of each data point with respect to a query using a sampling procedure. In this section, we introduce another sampling method that approximates the inner product of every data point instead.\nLet us motivate our next algorithm with a rather contrived example. Sup- pose that our data points and queries are in R2, with the first coordinate of vectors drawing values from N (0, Ï2 1) and the second coordinate from N (0, Ï2 2). If we were to compute the inner product of q with every vector u â X , we would need to perform two multiplications and a sum: u1q1 + u2q2. That gives us the exact âscoreâ of every point with respect to q. But if Ï2 1 â« Ï2 2, then by computing q1u1 for all u â X , it is very likely that we have a good approximation to the final inner product. So we may use the partial inner product as a high-confidence estimate of the full inner product.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 302 | That is the core idea in this section. For each data point, we sample a few dimensions without replacement, and compute its partial inner product with the query along the chosen dimensions. Based on the scores so far, we can eliminate data points whose full inner product is projected, with high confidence, to be too small to make it to the top-k set. We then repeat the procedure by sampling more dimensions for the remaining data points, until we reach a stopping criterion.
The process above saves us time by shrinking the set of data points and computing only partial inner products in each round. But we must decide how we should sample dimensions and how we should determine which data points to discard. The objective is to minimize the number of samples needed to iden- tify the solution set. These are the questions that Liu et al. [2019] answered in their work, which we will review next. We note that, even though Liu et al. [2019] use the Bandit language [Lattimore and Szepesv´ari, 2020] to describe
117
ââ
118
8 Sampling Algorithms
# Algorithm 4: The BoundedME algorithm for MIPS. | 2401.09350#302 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 302,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "That is the core idea in this section. For each data point, we sample a few dimensions without replacement, and compute its partial inner product with the query along the chosen dimensions. Based on the scores so far, we can eliminate data points whose full inner product is projected, with high confidence, to be too small to make it to the top-k set. We then repeat the procedure by sampling more dimensions for the remaining data points, until we reach a stopping criterion.\nThe process above saves us time by shrinking the set of data points and computing only partial inner products in each round. But we must decide how we should sample dimensions and how we should determine which data points to discard. The objective is to minimize the number of samples needed to iden- tify the solution set. These are the questions that Liu et al. [2019] answered in their work, which we will review next. We note that, even though Liu et al. [2019] use the Bandit language [Lattimore and Szepesv´ari, 2020] to describe\n117\nââ\n118\n8 Sampling Algorithms\n# Algorithm 4: The BoundedME algorithm for MIPS.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 303 | 117
ââ
118
8 Sampling Algorithms
# Algorithm 4: The BoundedME algorithm for MIPS.
Input: Query point q ⬠R¢; k > 1 for top-k retrieval; confidence parameters ¢,6 ⬠(0,1); and data points Y Cc R* Result: (1 â 5)-confident e-approximate top-k set to MIPS with respect to q. Li 1 2: Xi << X ; > Initialize the solution set to %. t 8: 9: 10: 11: 12: 13: 14: ei + § and 6; â & 2 Ay +0 Vue X;;> A is a score accumulator. o+ 0 while |%;| > k do tp onl 21 2(1:1-h) ~ (3 oe (iS) z for u ⬠X% do Let J be (t; â t;-1) dimensions sampled without replacement Au Aut Djez Usd §> Compute partial inner product. end for Let a be the peeâ) th score in A Kini â {u ⬠X% s.t. Au > a} ei41 ei, bina &, andi¢+i+l 15: end while 16: return X;
their algorithm, we find it makes for a clearer presentation if we avoided the Bandit terminology. | 2401.09350#303 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 303,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "117\nââ\n118\n8 Sampling Algorithms\n# Algorithm 4: The BoundedME algorithm for MIPS.\nInput: Query point q ⬠R¢; k > 1 for top-k retrieval; confidence parameters ¢,6 ⬠(0,1); and data points Y Cc R* Result: (1 â 5)-confident e-approximate top-k set to MIPS with respect to q. Li 1 2: Xi << X ; > Initialize the solution set to %. t 8: 9: 10: 11: 12: 13: 14: ei + § and 6; â & 2 Ay +0 Vue X;;> A is a score accumulator. o+ 0 while |%;| > k do tp onl 21 2(1:1-h) ~ (3 oe (iS) z for u ⬠X% do Let J be (t; â t;-1) dimensions sampled without replacement Au Aut Djez Usd §> Compute partial inner product. end for Let a be the peeâ) th score in A Kini â {u ⬠X% s.t. Au > a} ei41 ei, bina &, andi¢+i+l 15: end while 16: return X;\ntheir algorithm, we find it makes for a clearer presentation if we avoided the Bandit terminology.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 304 | their algorithm, we find it makes for a clearer presentation if we avoided the Bandit terminology.
# 8.3.1 The BoundedME Algorithm
The top-k retrieval algorithm developed by Liu et al. [2019] is presented in Algorithm 4. It is important to note that, for the algorithm to be correctâas we will explain laterâeach partial inner product must be bounded. In other words, for query q, any data point u â X , and any dimension t, we must have that qtut â [a, b] for some fixed interval. This is not a restrictive assumption, however: q can always be normalized without affecting the solution to MIPS, and data points u can be scaled into the hypercube. In their work, Liu et al. [2019] assume that partial inner products are in the unit interval.
This iterative algorithm begins with the full collection of data points and removes almost half of the data points in each iteration. It terminates as soon as the total number of data points left is at most k.
In each iteration of the algorithm, it accumulates partial inner products for all remaining data point along a set of sampled dimensions. Once a dimension has been sampled, it is removed from consideration in all future iterationsâ hence, sampling without replacement.
The number of dimensions to sample is adaptive and changes from it- eration to iteration. It is determined using the quantity on Line 7 of the | 2401.09350#304 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 304,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "their algorithm, we find it makes for a clearer presentation if we avoided the Bandit terminology.\n# 8.3.1 The BoundedME Algorithm\nThe top-k retrieval algorithm developed by Liu et al. [2019] is presented in Algorithm 4. It is important to note that, for the algorithm to be correctâas we will explain laterâeach partial inner product must be bounded. In other words, for query q, any data point u â X , and any dimension t, we must have that qtut â [a, b] for some fixed interval. This is not a restrictive assumption, however: q can always be normalized without affecting the solution to MIPS, and data points u can be scaled into the hypercube. In their work, Liu et al. [2019] assume that partial inner products are in the unit interval.\nThis iterative algorithm begins with the full collection of data points and removes almost half of the data points in each iteration. It terminates as soon as the total number of data points left is at most k.\nIn each iteration of the algorithm, it accumulates partial inner products for all remaining data point along a set of sampled dimensions. Once a dimension has been sampled, it is removed from consideration in all future iterationsâ hence, sampling without replacement.\nThe number of dimensions to sample is adaptive and changes from it- eration to iteration. It is determined using the quantity on Line 7 of the",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 305 | The number of dimensions to sample is adaptive and changes from it- eration to iteration. It is determined using the quantity on Line 7 of the
8.3 Approximating the Scores
algorithm, where the function h(·) is defined as follows:
(8.5) lta etal, A(z) = min { I+a/dâ1+a/d
At the end of iteration i with the remaining data points in Xi, the algo- rithm finds the â |Xi|âk â-th (i.e., close to the median) partial inner product accumulated so far, and discards data points whose score is less than that threshold. It then updates the confidence parameters ϵ and δ, and proceeds to the next iteration.
It is rather obvious that, the total number of dimensions along which the algorithm computes partial inner products for any given data point can never exceed d. That is simply because once Line 10 is executed, the dimensions in the set J defined on Line 9 are never considered for sampling in future iterations. As a result, in the worst case, the algorithm computes full inner products in O(md) operations. | 2401.09350#305 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 305,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The number of dimensions to sample is adaptive and changes from it- eration to iteration. It is determined using the quantity on Line 7 of the\n8.3 Approximating the Scores\nalgorithm, where the function h(·) is defined as follows:\n(8.5) lta etal, A(z) = min { I+a/dâ1+a/d\nAt the end of iteration i with the remaining data points in Xi, the algo- rithm finds the â |Xi|âk â-th (i.e., close to the median) partial inner product accumulated so far, and discards data points whose score is less than that threshold. It then updates the confidence parameters ϵ and δ, and proceeds to the next iteration.\nIt is rather obvious that, the total number of dimensions along which the algorithm computes partial inner products for any given data point can never exceed d. That is simply because once Line 10 is executed, the dimensions in the set J defined on Line 9 are never considered for sampling in future iterations. As a result, in the worst case, the algorithm computes full inner products in O(md) operations.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 306 | As for the time complexity of Algorithm 4, it can be shown that it re- quires o(@4 flog(1/5)) operations. That is simply due to the fact that in each iteration, the number of data points is cut in half, combined with the inequality h(x) < O(Vdz) for x > 0.
Theorem 8.2 The time complexity of Algorithm 4 is O( m
â
o(2Â¥4 /log(1/5)).
# d
# ϵ
Theorem 8.2 says that the time complexity of Algorithm 4 is linear in the number of data points m, but sub-linear in the number of dimensions d. That is a fundamentally different behavior than all the other algorithms we have presented thus far throughout the preceding chapters.
Proof of Theorem 8.2. Let us first show the following claim: h(x) ⤠O( dx) for x > 0. To prove that, observe that h(x) is the minimum of two positive values a and b. As such, h(x) ⤠ab. Substituting a and b with the right expressions from Equation (8.5):
â | 2401.09350#306 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 306,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "As for the time complexity of Algorithm 4, it can be shown that it re- quires o(@4 flog(1/5)) operations. That is simply due to the fact that in each iteration, the number of data points is cut in half, combined with the inequality h(x) < O(Vdz) for x > 0.\nTheorem 8.2 The time complexity of Algorithm 4 is O( m\nâ\no(2Â¥4 /log(1/5)).\n# d\n# ϵ\nTheorem 8.2 says that the time complexity of Algorithm 4 is linear in the number of data points m, but sub-linear in the number of dimensions d. That is a fundamentally different behavior than all the other algorithms we have presented thus far throughout the preceding chapters.\nProof of Theorem 8.2. Let us first show the following claim: h(x) ⤠O( dx) for x > 0. To prove that, observe that h(x) is the minimum of two positive values a and b. As such, h(x) ⤠ab. Substituting a and b with the right expressions from Equation (8.5):\nâ",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 307 | â
l+a x+a/d _ 1 Jal Fay + t/a) ho < fe ipa x(1 + 2)(1+ 1/d) â Of) | dx ~ 1+2/d = OTF) < O(Vaz).
Note that, in the i-th iteration there are at most m/2i data points to examine. Moreover, for each data point that is eliminated in round i, we will have computed at most ti partial inner products (see Line 7 of Algorithm 4). Using these facts, we can calculate the time complexity as follows:
119
120
8 Sampling Algorithms
â
logm logm -ââ m m mid 1 > Sih(ti) < > Vat, < 0 (M4 hog 5):
# 8.3.2 Proof of Correctness
Our goal in this section is to prove that Algorithm 4 is correct, in the sense that it returns the ϵ-approximate solution to k-MIPS with probability at least 1 â δ:
Theorem 8.3 Algorithm 4 is guaranteed to return the ϵ-approximate solu- tion to k-MIPS with probability at least 1 â δ.
The proof of Theorem 8.3 requires the concentration inequality due to Bar- denet and Maillard [2015], repeated below for completeness. | 2401.09350#307 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 307,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "â\nl+a x+a/d _ 1 Jal Fay + t/a) ho < fe ipa x(1 + 2)(1+ 1/d) â Of) | dx ~ 1+2/d = OTF) < O(Vaz).\nNote that, in the i-th iteration there are at most m/2i data points to examine. Moreover, for each data point that is eliminated in round i, we will have computed at most ti partial inner products (see Line 7 of Algorithm 4). Using these facts, we can calculate the time complexity as follows:\n119\n120\n8 Sampling Algorithms\nâ\nlogm logm -ââ m m mid 1 > Sih(ti) < > Vat, < 0 (M4 hog 5):\n# 8.3.2 Proof of Correctness\nOur goal in this section is to prove that Algorithm 4 is correct, in the sense that it returns the ϵ-approximate solution to k-MIPS with probability at least 1 â δ:\nTheorem 8.3 Algorithm 4 is guaranteed to return the ϵ-approximate solu- tion to k-MIPS with probability at least 1 â δ.\nThe proof of Theorem 8.3 requires the concentration inequality due to Bar- denet and Maillard [2015], repeated below for completeness.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 308 | The proof of Theorem 8.3 requires the concentration inequality due to Bar- denet and Maillard [2015], repeated below for completeness.
Lemma 8.3 Let J â [0, 1] be a finite set of size d with mean µ. Let {J1, J2, . . . , Jn} be n < d samples from J without replacement. Then for any n ⤠d and any δ â [0, 1] it holds:
P(t s Voi 85) >1-6,
where Ïn is defined as follows:
. ne 1 7 f 1 Pn min {1 d (1 pu | )}.
The lemma above guarantees that, with probability at least 1 â δ, the empirical mean of the samples does not exceed the mean of the universe by a specific amount that depends on δ. We now wish to adapt that result to derive a similar guarantee where the difference between means is bounded by an arbitrary parameter ϵ. That is stated in the following lemma.
Lemma 8.4 Let J â [0, 1] be a finite set of size d with mean µ. Let {J1, J2, . . . , Jn} be n < d samples from J without replacement. Then for any ϵ, δ â (0, 1), if we have that: | 2401.09350#308 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 308,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "The proof of Theorem 8.3 requires the concentration inequality due to Bar- denet and Maillard [2015], repeated below for completeness.\nLemma 8.3 Let J â [0, 1] be a finite set of size d with mean µ. Let {J1, J2, . . . , Jn} be n < d samples from J without replacement. Then for any n ⤠d and any δ â [0, 1] it holds:\nP(t s Voi 85) >1-6,\nwhere Ïn is defined as follows:\n. ne 1 7 f 1 Pn min {1 d (1 pu | )}.\nThe lemma above guarantees that, with probability at least 1 â δ, the empirical mean of the samples does not exceed the mean of the universe by a specific amount that depends on δ. We now wish to adapt that result to derive a similar guarantee where the difference between means is bounded by an arbitrary parameter ϵ. That is stated in the following lemma.\nLemma 8.4 Let J â [0, 1] be a finite set of size d with mean µ. Let {J1, J2, . . . , Jn} be n < d samples from J without replacement. Then for any ϵ, δ â (0, 1), if we have that:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 309 | l+2 etal > mi ; n> min{ ap 1l+a/d
where x = log(1/δ)/2ϵ2, then the following holds:
PE a-n<d >1-6. t=1
ââ
8.3 Approximating the Scores
Proof. By Lemma 8.3 we can see that:
so long as:
log 1 δ ⤠ϵ =â n Ïn ⥠1 2ϵ2 log 1 δ .
There are two cases to consider. First, if Ïn = 1 â (n â 1)/d, then:
n 1 1 n > log = 2x Pn 2e? 85 joa" © xt+a/d => > : me 1+a/d
In the second case, Ïn = (1 â n/d)(1 + 1/n), which gives:
ny Ly 1 n S > og = >x Pn ~ 22° 6 (a-8)a+4) x 1 1 = n>fir--"* | n d 25 n? n n nx +x â x x ~ d d = (1+ 5)n?â(w-5)nâ2
20.
)n â x ⥠0. | 2401.09350#309 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 309,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "l+2 etal > mi ; n> min{ ap 1l+a/d\nwhere x = log(1/δ)/2ϵ2, then the following holds:\nPE a-n<d >1-6. t=1\nââ\n8.3 Approximating the Scores\nProof. By Lemma 8.3 we can see that:\nso long as:\nlog 1 δ ⤠ϵ =â n Ïn ⥠1 2ϵ2 log 1 δ .\nThere are two cases to consider. First, if Ïn = 1 â (n â 1)/d, then:\nn 1 1 n > log = 2x Pn 2e? 85 joa\" © xt+a/d => > : me 1+a/d\nIn the second case, Ïn = (1 â n/d)(1 + 1/n), which gives:\nny Ly 1 n S > og = >x Pn ~ 22° 6 (a-8)a+4) x 1 1 = n>fir--\"* | n d 25 n? n n nx +x â x x ~ d d = (1+ 5)n?â(w-5)nâ2\n20.\n)n â x ⥠0.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 310 | 20.
)n â x ⥠0.
To make the closed-form solution more manageable, Liu et al. [2019] relax the problem above and solve n in the following problem instead. Note that, any solution to the problem below is a valid solution to the problem above.
(1+ yn? -(@=â)n-2-130 = [a+ 5)n-2-1][n +1] 20 â 1 + & â ne l+a/d
By combining the two cases, we obtain:
n ⥠min{ 1 + x 1 + x/d , x + x/d 1 + x/d }.
121
ââ
122
8 Sampling Algorithms
Lemma 8.4 gives us the minimum number of dimensions we must sample so that the partial inner product of a vector with a query is at most ϵ away from the full inner product, with probability at least 1 â δ.
Armed with this result, we can now proceed to proving the main theorem.
Proof of Theorem 8.3. Denote by ζi the k-th largest full inner product among the set of data points Xi in iteration i. If we showed that, for two consec- utive iterations, the difference between ζi and ζi+1 does not exceed ϵi with probability at least 1 â δi, that is: | 2401.09350#310 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 310,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "20.\n)n â x ⥠0.\nTo make the closed-form solution more manageable, Liu et al. [2019] relax the problem above and solve n in the following problem instead. Note that, any solution to the problem below is a valid solution to the problem above.\n(1+ yn? -(@=â)n-2-130 = [a+ 5)n-2-1][n +1] 20 â 1 + & â ne l+a/d\nBy combining the two cases, we obtain:\nn ⥠min{ 1 + x 1 + x/d , x + x/d 1 + x/d }.\n121\nââ\n122\n8 Sampling Algorithms\nLemma 8.4 gives us the minimum number of dimensions we must sample so that the partial inner product of a vector with a query is at most ϵ away from the full inner product, with probability at least 1 â δ.\nArmed with this result, we can now proceed to proving the main theorem.\nProof of Theorem 8.3. Denote by ζi the k-th largest full inner product among the set of data points Xi in iteration i. If we showed that, for two consec- utive iterations, the difference between ζi and ζi+1 does not exceed ϵi with probability at least 1 â δi, that is:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 311 | P [<i â Gini < «| 21-46, (8.6)
then the theorem immediately follows:
P [61 â Gogm S¢] 21-6,
because:
logm log m 5 5 Y= La sd%
and,
ala Kee in Me alo a Kio Na L | logm log m
# Equation
So we focus on proving Equation (8.6).
Suppose we are in the i-th iteration. Collect in Zϵi every data point in u â Xi such that ζi â â¨q, u⩠⤠ϵi. That is: Zϵi = {u â Xi | ζi â â¨q, u⩠⤠ϵi}. If at least k elements of Zϵi end up in Xi+1, the event ζi â ζi+1 ⤠ϵi succeeds. So, that event fails if there are more than â |Xi|âk â data points in Xi \ Zϵi with partial inner products that are greater than partial inner products of the data points in Zϵi . Denote the number of such data points by β. | 2401.09350#311 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 311,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "P [<i â Gini < «| 21-46, (8.6)\nthen the theorem immediately follows:\nP [61 â Gogm S¢] 21-6,\nbecause:\nlogm log m 5 5 Y= La sd%\nand,\nala Kee in Me alo a Kio Na L | logm log m\n# Equation\nSo we focus on proving Equation (8.6).\nSuppose we are in the i-th iteration. Collect in Zϵi every data point in u â Xi such that ζi â â¨q, u⩠⤠ϵi. That is: Zϵi = {u â Xi | ζi â â¨q, u⩠⤠ϵi}. If at least k elements of Zϵi end up in Xi+1, the event ζi â ζi+1 ⤠ϵi succeeds. So, that event fails if there are more than â |Xi|âk â data points in Xi \\ Zϵi with partial inner products that are greater than partial inner products of the data points in Zϵi . Denote the number of such data points by β.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 312 | What is the probability that a data point u in ¥; \ Z., has a higher partial inner product than any data point in Z,,? Assuming that u* is the data point that achieves ¢;, we can write: PA, >A, Vue Z.) < P[A. >A, |
PA, >A, Vue Z.) < P[A. >A, | <P[A, > (gu) +S V Au SG 3] <P[A. > (qu) + S$] +P [Aw <6 - $].
We can apply Lemma 8.4 to obtain that, if the number of sampled dimensions is equal to the quantity on Line 7 of Algorithm 4, then the probability above would be bounded by:
8.4 Closing Remarks
â |Xi|âk 2 â + 1 |Xi| â k δi.
Using this result along with Markovâs inequality, we can bound the prob- ability that β is strictly greater than â |Xi|âk â as follows:
2
Xi) âk E[8 piss! 5 ' 1 s oo 2
That completes the proof of Equation (8.6) and, therefore, the theorem. ââ
# 8.4 Closing Remarks | 2401.09350#312 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 312,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "What is the probability that a data point u in Â¥; \\ Z., has a higher partial inner product than any data point in Z,,? Assuming that u* is the data point that achieves ¢;, we can write: PA, >A, Vue Z.) < P[A. >A, |\nPA, >A, Vue Z.) < P[A. >A, | <P[A, > (gu) +S V Au SG 3] <P[A. > (qu) + S$] +P [Aw <6 - $].\nWe can apply Lemma 8.4 to obtain that, if the number of sampled dimensions is equal to the quantity on Line 7 of Algorithm 4, then the probability above would be bounded by:\n8.4 Closing Remarks\nâ |Xi|âk 2 â + 1 |Xi| â k δi.\nUsing this result along with Markovâs inequality, we can bound the prob- ability that β is strictly greater than â |Xi|âk â as follows:\n2\nXi) âk E[8 piss! 5 ' 1 s oo 2\nThat completes the proof of Equation (8.6) and, therefore, the theorem. ââ\n# 8.4 Closing Remarks",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 313 | That completes the proof of Equation (8.6) and, therefore, the theorem. ââ
# 8.4 Closing Remarks
The algorithms in this chapter were unique in two ways. First, they directly took on the challenging problem of MIPS. This is in contrast to earlier chap- ters where MIPS was only an afterthought. Second, there is little to no pre- processing involved in the preparation of the index, which itself is small in size. That is unlike trees, hash buckets, graphs, and clustering that require a generally heavy index that itself is computationally-intensive to build.
The approach itself is rather unique as well. It is particularly interesting because the trade-off between efficiency and accuracy can be adjusted during retrieval. That is not the case with trees, LSH, or graphs, where the con- struction of the index itself heavily influences that balance. With sampling methods, it is at least theoretically possible to adapt the retrieval strategy to the hardness of the query distribution. That question remains unexplored. | 2401.09350#313 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 313,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "That completes the proof of Equation (8.6) and, therefore, the theorem. ââ\n# 8.4 Closing Remarks\nThe algorithms in this chapter were unique in two ways. First, they directly took on the challenging problem of MIPS. This is in contrast to earlier chap- ters where MIPS was only an afterthought. Second, there is little to no pre- processing involved in the preparation of the index, which itself is small in size. That is unlike trees, hash buckets, graphs, and clustering that require a generally heavy index that itself is computationally-intensive to build.\nThe approach itself is rather unique as well. It is particularly interesting because the trade-off between efficiency and accuracy can be adjusted during retrieval. That is not the case with trees, LSH, or graphs, where the con- struction of the index itself heavily influences that balance. With sampling methods, it is at least theoretically possible to adapt the retrieval strategy to the hardness of the query distribution. That question remains unexplored.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 314 | Another area that would benefit from further research is the sampling strategy itself. In particular, in the BoundedME algorithm, the dimensions that are sampled next are drawn randomly. While that simplifies analysisâ which follows the analysis of popular Bandit algorithmsâit is not hard to argue that the strategy is sub-optimal. After all, unlike the Bandit setup, where reward distributions are unknown and samples from the reward dis- tributions are revealed only gradually, here we have direct access to all data points a priori. Whether and how adapting the sampling strategy to the underlying data or query distribution may improve the error bounds or the accuracy or efficiency of the algorithm in practice remains to be studied.
123
124
8 Sampling Algorithms
# References
G. Ballard, T. G. Kolda, A. Pinar, and C. Seshadhri. Diamond sampling for approximate maximum all-pairs dot-product (mad) search. In 2015 IEEE International Conference on Data Mining, pages 11â20, 2015.
R. Bardenet and O.-A. Maillard. Concentration inequalities for sampling without replacement. Bernoulli, 21(3):1361â1385, 2015.
E. Cohen and D. D. Lewis. Approximating matrix multiplication for pat- tern recognition tasks. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 682â691, 1997. | 2401.09350#314 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 314,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Another area that would benefit from further research is the sampling strategy itself. In particular, in the BoundedME algorithm, the dimensions that are sampled next are drawn randomly. While that simplifies analysisâ which follows the analysis of popular Bandit algorithmsâit is not hard to argue that the strategy is sub-optimal. After all, unlike the Bandit setup, where reward distributions are unknown and samples from the reward dis- tributions are revealed only gradually, here we have direct access to all data points a priori. Whether and how adapting the sampling strategy to the underlying data or query distribution may improve the error bounds or the accuracy or efficiency of the algorithm in practice remains to be studied.\n123\n124\n8 Sampling Algorithms\n# References\nG. Ballard, T. G. Kolda, A. Pinar, and C. Seshadhri. Diamond sampling for approximate maximum all-pairs dot-product (mad) search. In 2015 IEEE International Conference on Data Mining, pages 11â20, 2015.\nR. Bardenet and O.-A. Maillard. Concentration inequalities for sampling without replacement. Bernoulli, 21(3):1361â1385, 2015.\nE. Cohen and D. D. Lewis. Approximating matrix multiplication for pat- tern recognition tasks. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 682â691, 1997.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 315 | Q. Ding, H.-F. Yu, and C.-J. Hsieh. A fast sampling algorithm for maxi- mum inner product search. In K. Chaudhuri and M. Sugiyama, editors, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 3004â3012, 4 2019.
T. Lattimore and C. Szepesv´ari. Bandit Algorithms. Cambridge University Press, 2020.
R. Liu, T. Wu, and B. Mozafari. A bandit approach to maximum inner product search. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence, 2019.
S. S. Lorenzen and N. Pham. Revisiting wedge sampling for budgeted maxi- mum inner product search. In Proceedings of the 30th International Joint Conference on Artificial Intelligence, pages 4789â4793, 8 2021.
A. J. Walker. An efficient method for generating discrete random variables with general distributions. ACM Transactions on Mathematical Software, 3(3):253â256, 9 1977.
Part III Compression
Chapter 9 Quantization | 2401.09350#315 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 315,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Q. Ding, H.-F. Yu, and C.-J. Hsieh. A fast sampling algorithm for maxi- mum inner product search. In K. Chaudhuri and M. Sugiyama, editors, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 3004â3012, 4 2019.\nT. Lattimore and C. Szepesv´ari. Bandit Algorithms. Cambridge University Press, 2020.\nR. Liu, T. Wu, and B. Mozafari. A bandit approach to maximum inner product search. In Proceedings of the 33rd AAAI Conference on Artificial Intelligence, 2019.\nS. S. Lorenzen and N. Pham. Revisiting wedge sampling for budgeted maxi- mum inner product search. In Proceedings of the 30th International Joint Conference on Artificial Intelligence, pages 4789â4793, 8 2021.\nA. J. Walker. An efficient method for generating discrete random variables with general distributions. ACM Transactions on Mathematical Software, 3(3):253â256, 9 1977.\nPart III Compression\nChapter 9 Quantization",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 417 | Substituting t into Equation (C.1) gives the desired result.
# C.4 Hoeffdingâs Inequality
We need the following result, known as Hoeffdingâs Lemma, to present Ho- effdingâs inequality.
Lemma C.5 Let X be a zero-mean random variable that takes values in [a, b]. For any t > 0:
E [el*] < exp (eo).
Proof. By convexity of etx and given x â [a, b] we have that:
179
ââ
180
C Concentration of Measure
etx ⤠b â x b â a eta + x â a b â a etb.
Taking the expectation of both sides, we arrive at:
b a E [e'*| < ta tb. e} ba. baâ
# E
To conclude the proof, we first write the right-hand-side as exp(h(t(b â a))) where:
a / b a. h(x) boat t 8 (53 - pat ).
By expanding h(x) using Taylorâs theorem, it can be shown that h(x) ⤠x2/8. ââ That completes the proof.
We are ready to present Hoeffdingâs inequality. | 2401.09350#417 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 417,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Substituting t into Equation (C.1) gives the desired result.\n# C.4 Hoeffdingâs Inequality\nWe need the following result, known as Hoeffdingâs Lemma, to present Ho- effdingâs inequality.\nLemma C.5 Let X be a zero-mean random variable that takes values in [a, b]. For any t > 0:\nE [el*] < exp (eo).\nProof. By convexity of etx and given x â [a, b] we have that:\n179\nââ\n180\nC Concentration of Measure\netx ⤠b â x b â a eta + x â a b â a etb.\nTaking the expectation of both sides, we arrive at:\nb a E [e'*| < ta tb. e} ba. baâ\n# E\nTo conclude the proof, we first write the right-hand-side as exp(h(t(b â a))) where:\na / b a. h(x) boat t 8 (53 - pat ).\nBy expanding h(x) using Taylorâs theorem, it can be shown that h(x) ⤠x2/8. ââ That completes the proof.\nWe are ready to present Hoeffdingâs inequality.",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 418 | We are ready to present Hoeffdingâs inequality.
Lemma C.6 Let {Xi}n mean µ and suppose Xi â [a, b] almost surely. For all ϵ > 0: i=1 be a sequence of iid random variables with finite
2 _â 2ne P [pe â | > | < 2exp (a ap):
Proof. Let X = 1/n >,
i Xi â µ. Observe by Markovâs inequality that:
P[X >¢/=P [e'* > e] <e~ Efeâ).
By independence of Xiâs and the application of Lemma C.5:
t(Xi-4) Te a Te [-*) Too ("5") 2(h â a)2 â *), E{e'*] =E = exp (
We have shown that:
2(bâ a)? P[LX > < exp (-te+ ren). 8n
That statement holds for all values of t and in particular one that minimizes the right-hand-side. Solving for that value of t gives us t = 4nϵ/(b â a2), which implies:
C.5 Bennetâs Inequality | 2401.09350#418 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 418,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "We are ready to present Hoeffdingâs inequality.\nLemma C.6 Let {Xi}n mean µ and suppose Xi â [a, b] almost surely. For all ϵ > 0: i=1 be a sequence of iid random variables with finite\n2 _â 2ne P [pe â | > | < 2exp (a ap):\nProof. Let X = 1/n >,\ni Xi â µ. Observe by Markovâs inequality that:\nP[X >¢/=P [e'* > e] <e~ Efeâ).\nBy independence of Xiâs and the application of Lemma C.5:\nt(Xi-4) Te a Te [-*) Too (\"5\") 2(h â a)2 â *), E{e'*] =E = exp (\nWe have shown that:\n2(bâ a)? P[LX > < exp (-te+ ren). 8n\nThat statement holds for all values of t and in particular one that minimizes the right-hand-side. Solving for that value of t gives us t = 4nϵ/(b â a2), which implies:\nC.5 Bennetâs Inequality",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 419 | C.5 Bennetâs Inequality
P[X ⥠ϵ] ⤠eâ 2nϵ2 By a symmetric argument we can bound P[X ⤠âϵ]. The claim follows by ââ the union bound over the two cases.
# C.5 Bennetâs Inequality
Lemma C.7 Let {Xi}n with zero mean and finite variance Ï2 for all i. Then: i=1 be a sequence of independent random variables i . Assume that |Xi| ⤠a almost surely
2 [Sx zd] <oo(-Sa(S)).
where h(x) = (1+ 2x) log(1 +2) â2 and o? =),
i Ï2 i .
Proof. As usual, we take advantage of Markovâs inequality to write:
P [Lx > â| <eME [a=] =eE Ie] =e Ifs [e*']
Using the Taylor expansion of ex, we obtain:
ple") =8[5 â8 OO \koryayk~3 AY E[X? Xk?) =1+ Y âr 2 a? BS Mak =l+ a k=2 2 o? = 1+ 5(e â1âAa) 2 < exp (25 (e â 1 Aa))
181
182
C Concentration of Measure | 2401.09350#419 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 419,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "C.5 Bennetâs Inequality\nP[X ⥠ϵ] ⤠eâ 2nϵ2 By a symmetric argument we can bound P[X ⤠âϵ]. The claim follows by ââ the union bound over the two cases.\n# C.5 Bennetâs Inequality\nLemma C.7 Let {Xi}n with zero mean and finite variance Ï2 for all i. Then: i=1 be a sequence of independent random variables i . Assume that |Xi| ⤠a almost surely\n2 [Sx zd] <oo(-Sa(S)).\nwhere h(x) = (1+ 2x) log(1 +2) â2 and o? =),\ni Ï2 i .\nProof. As usual, we take advantage of Markovâs inequality to write:\nP [Lx > â| <eME [a=] =eE Ie] =e Ifs [e*']\nUsing the Taylor expansion of ex, we obtain:\nple\") =8[5 â8 OO \\koryayk~3 AY E[X? Xk?) =1+ Y âr 2 a? BS Mak =l+ a k=2 2 o? = 1+ 5(e â1âAa) 2 < exp (25 (e â 1 Aa))\n181\n182\nC Concentration of Measure",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 420 | 181
182
C Concentration of Measure
Putting it all together:
2 âX Od P[> Xi2t|<e âTex (3 (e-1-da)) i i âAt ova =e exp (Sle -â1 da)).
# P
This inequality holds for all values of λ, and in particular one that minimizes the right-hand-side. Setting the derivative of the right-hand-side to 0 and ââ solving for λ leads to the desired result.
Appendix D Linear Algebra Review
Abstract This appendix reviews basic concepts from Linear Algebra that are useful in digesting the material in this monograph.
# D.1 Inner Product
Denote by H a vector space. An inner product â¨Â·, ·⩠: H à H â R is a function with the following properties: | 2401.09350#420 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 420,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "181\n182\nC Concentration of Measure\nPutting it all together:\n2 âX Od P[> Xi2t|<e âTex (3 (e-1-da)) i i âAt ova =e exp (Sle -â1 da)).\n# P\nThis inequality holds for all values of λ, and in particular one that minimizes the right-hand-side. Setting the derivative of the right-hand-side to 0 and ââ solving for λ leads to the desired result.\nAppendix D Linear Algebra Review\nAbstract This appendix reviews basic concepts from Linear Algebra that are useful in digesting the material in this monograph.\n# D.1 Inner Product\nDenote by H a vector space. An inner product â¨Â·, ·⩠: H à H â R is a function with the following properties:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2401.09350 | 421 | # D.1 Inner Product
Denote by H a vector space. An inner product â¨Â·, ·⩠: H à H â R is a function with the following properties:
â u â H, â¨u, u⩠⥠0; ⢠â u â H, â¨u, uâ© = 0 â u = 0; ⢠â u, v â H, â¨u, vâ© = â¨v, uâ©; and, ⢠â u, v, w â H, and α, β â R, â¨Î±u + βv, wâ© = αâ¨u, wâ© + βâ¨v, wâ©.
We call H together with the inner product (-,-) an inner product space. As an example, when H = R%, given two vectors u = al ue; and v = d > : ; a . Yi-1 vie, where e;âs are the standard basis vectors, the following is an inner product:
(u,v) = Ss UiV;- i=1
We say two vectors u and v in an inner product space are orthogonal if their inner product is 0: â¨u, vâ© = 0.
# D.2 Norms | 2401.09350#421 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 421,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# D.1 Inner Product\nDenote by H a vector space. An inner product â¨Â·, ·⩠: H à H â R is a function with the following properties:\nâ u â H, â¨u, u⩠⥠0; ⢠â u â H, â¨u, uâ© = 0 â u = 0; ⢠â u, v â H, â¨u, vâ© = â¨v, uâ©; and, ⢠â u, v, w â H, and α, β â R, â¨Î±u + βv, wâ© = αâ¨u, wâ© + βâ¨v, wâ©.\nWe call H together with the inner product (-,-) an inner product space. As an example, when H = R%, given two vectors u = al ue; and v = d > : ; a . Yi-1 vie, where e;âs are the standard basis vectors, the following is an inner product:\n(u,v) = Ss UiV;- i=1\nWe say two vectors u and v in an inner product space are orthogonal if their inner product is 0: â¨u, vâ© = 0.\n# D.2 Norms",
"title": "Foundations of Vector Retrieval",
"year": 2024
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] |
2401.09350 | 422 | # D.2 Norms
A function Φ : H â R+ is a norm on H if it has the following properties: ⢠Definiteness: For all u â H, Φ(u) = 0 â u = 0;
183
184
D Linear Algebra Review
Homogeneity: For all u â H and α â R, Φ(αu) = |α|Φ(u); and, ⢠Triangle inequality: â u, v â H, Φ(u + v) ⤠Φ(u) + Φ(v).
Examples include the absolute value on R, and the Lp norm (for p ⥠1)
on Rd denoted by â¥Â·â¥p and defined as:
d a lullp = (Solusl) *- i=l
Instances of Lp include the commonly used L1, L2 (Euclidean), and Lâ norms, where â¥uâ¥â = maxi|ui|.
Note that, when H is an inner product space, then the function â¥u⥠=
Note that, when H is an inner product space, then the function ||u|| = /(u,u) is a norm.
# D.3 Distance | 2401.09350#422 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 422,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "# D.2 Norms\nA function Φ : H â R+ is a norm on H if it has the following properties: ⢠Definiteness: For all u â H, Φ(u) = 0 â u = 0;\n183\n184\nD Linear Algebra Review\nHomogeneity: For all u â H and α â R, Φ(αu) = |α|Φ(u); and, ⢠Triangle inequality: â u, v â H, Φ(u + v) ⤠Φ(u) + Φ(v).\nExamples include the absolute value on R, and the Lp norm (for p ⥠1)\non Rd denoted by â¥Â·â¥p and defined as:\nd a lullp = (Solusl) *- i=l\nInstances of Lp include the commonly used L1, L2 (Euclidean), and Lâ norms, where â¥uâ¥â = maxi|ui|.\nNote that, when H is an inner product space, then the function â¥u⥠=\nNote that, when H is an inner product space, then the function ||u|| = /(u,u) is a norm.\n# D.3 Distance",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 423 | Note that, when H is an inner product space, then the function ||u|| = /(u,u) is a norm.
# D.3 Distance
A norm on a vector space induces a notion of distance between two vectors. Concretely, if H is a normed space equipped with â¥Â·â¥, then we define the distance between two vectors u, v â H as follows:
δ(u, v) = â¥u â vâ¥.
# D.4 Orthogonal Projection
Lemma D.1 Let H be an inner product space and suppose u â H and u ̸= 0. Any vector v â H can be uniquely decomposed along u as:
v = v⥠+ vâ¥,
such that â¨vâ¥, vâ¥â© = 0. Additionally:
v⥠= â¨u, vâ© â¨u, uâ© u,
and v⥠= v â vâ¥.
Proof. Let v⥠= αu and v⥠= v â vâ¥. Because v⥠and v⥠are orthogonal, we deduce that: | 2401.09350#423 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 423,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Note that, when H is an inner product space, then the function ||u|| = /(u,u) is a norm.\n# D.3 Distance\nA norm on a vector space induces a notion of distance between two vectors. Concretely, if H is a normed space equipped with â¥Â·â¥, then we define the distance between two vectors u, v â H as follows:\nδ(u, v) = â¥u â vâ¥.\n# D.4 Orthogonal Projection\nLemma D.1 Let H be an inner product space and suppose u â H and u ̸= 0. Any vector v â H can be uniquely decomposed along u as:\nv = v⥠+ vâ¥,\nsuch that â¨vâ¥, vâ¥â© = 0. Additionally:\nv⥠= â¨u, vâ© â¨u, uâ© u,\nand v⥠= v â vâ¥.\nProof. Let v⥠= αu and v⥠= v â vâ¥. Because v⥠and v⥠are orthogonal, we deduce that:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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2401.09350 | 424 | Proof. Let v⥠= αu and v⥠= v â vâ¥. Because v⥠and v⥠are orthogonal, we deduce that:
â¨vâ¥, vâ¥â© = 0 =â â¨Î±u, vâ¥â© = 0 =â â¨u, vâ¥â© = 0.
That implies:
D.4 Orthogonal Projection
â¨v, uâ© = αâ¨u, uâ© =â α = â¨u, vâ© â¨u, uâ© ,
so that:
v⥠= â¨u, vâ© â¨u, uâ© u.
We prove the uniqueness of the decomposition by contradiction. Suppose there exists another decomposition of v to vⲠ⥠+ vâ² â¥. Then: | 2401.09350#424 | Foundations of Vector Retrieval | Vectors are universal mathematical objects that can represent text, images,
speech, or a mix of these data modalities. That happens regardless of whether
data is represented by hand-crafted features or learnt embeddings. Collect a
large enough quantity of such vectors and the question of retrieval becomes
urgently relevant: Finding vectors that are more similar to a query vector.
This monograph is concerned with the question above and covers fundamental
concepts along with advanced data structures and algorithms for vector
retrieval. In doing so, it recaps this fascinating topic and lowers barriers of
entry into this rich area of research. | http://arxiv.org/pdf/2401.09350 | Sebastian Bruch | cs.DS, cs.IR | null | null | cs.DS | 20240117 | 20240117 | [] | {
"authors": "Sebastian Bruch",
"chunk_id": 424,
"doc_id": "2401.09350",
"primary_category": "cs.DS",
"published": 20240117,
"source": "http://arxiv.org/pdf/2401.09350",
"summary": "Vectors are universal mathematical objects that can represent text, images,\nspeech, or a mix of these data modalities. That happens regardless of whether\ndata is represented by hand-crafted features or learnt embeddings. Collect a\nlarge enough quantity of such vectors and the question of retrieval becomes\nurgently relevant: Finding vectors that are more similar to a query vector.\nThis monograph is concerned with the question above and covers fundamental\nconcepts along with advanced data structures and algorithms for vector\nretrieval. In doing so, it recaps this fascinating topic and lowers barriers of\nentry into this rich area of research.",
"text": "Proof. Let v⥠= αu and v⥠= v â vâ¥. Because v⥠and v⥠are orthogonal, we deduce that:\nâ¨vâ¥, vâ¥â© = 0 =â â¨Î±u, vâ¥â© = 0 =â â¨u, vâ¥â© = 0.\nThat implies:\nD.4 Orthogonal Projection\nâ¨v, uâ© = αâ¨u, uâ© =â α = â¨u, vâ© â¨u, uâ© ,\nso that:\nv⥠= â¨u, vâ© â¨u, uâ© u.\nWe prove the uniqueness of the decomposition by contradiction. Suppose there exists another decomposition of v to vⲠ⥠+ vâ² â¥. Then:",
"title": "Foundations of Vector Retrieval",
"year": 2024
} | [
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] |
2311.04254 | 2 | # ABSTRACT
Recent advancements in Large Language Models (LLMs) have revolutionized decision-making by breaking down complex problems into more manageable lan- guage sequences referred to as âthoughtsâ. An effective thought design should consider three key perspectives: performance, efficiency, and flexibility. How- ever, existing thought can at most exhibit two of these attributes. To address these limitations, we introduce a novel thought prompting approach called âEverything â of existing thought of Thoughtsâ (XOT) to defy the law of âPenrose triangle paradigms. XOT leverages pretrained reinforcement learning and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge into thoughts, thereby enhancing LLMsâ capabilities and enabling them to generalize to unseen problems efficiently. Through the utilization of the MCTS-LLM collaborative thought revision framework, this approach autonomously produces high-quality comprehensive cognitive mappings with minimal LLM interactions. Additionally, XOT empowers LLMs to engage in unconstrained thinking, allowing for flexible cognitive mappings for problems with multiple solutions. We evaluate XOT on several challenging problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our results demonstrate that XOT significantly outperforms existing approaches in various dimensions, showcasing its remark- able proficiency in addressing complex problems across diverse domains. | 2311.04254#2 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 2,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "# ABSTRACT\nRecent advancements in Large Language Models (LLMs) have revolutionized decision-making by breaking down complex problems into more manageable lan- guage sequences referred to as âthoughtsâ. An effective thought design should consider three key perspectives: performance, efficiency, and flexibility. How- ever, existing thought can at most exhibit two of these attributes. To address these limitations, we introduce a novel thought prompting approach called âEverything â of existing thought of Thoughtsâ (XOT) to defy the law of âPenrose triangle paradigms. XOT leverages pretrained reinforcement learning and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge into thoughts, thereby enhancing LLMsâ capabilities and enabling them to generalize to unseen problems efficiently. Through the utilization of the MCTS-LLM collaborative thought revision framework, this approach autonomously produces high-quality comprehensive cognitive mappings with minimal LLM interactions. Additionally, XOT empowers LLMs to engage in unconstrained thinking, allowing for flexible cognitive mappings for problems with multiple solutions. We evaluate XOT on several challenging problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our results demonstrate that XOT significantly outperforms existing approaches in various dimensions, showcasing its remark- able proficiency in addressing complex problems across diverse domains.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 3 | 1
# INTRODUCTION
Recent advancements in Large Lan- guage Models (LLMs) have greatly ad- vanced problem solving in diverse do- mains such as mathematical reasoning Frieder et al. (2023), knowledge rea- soning Omar et al. (2023), root cause analysis Chen et al. (2023) and causal inference Kıcıman et al. (2023), etc.. This progress can be largely attributed to the technique of decomposing intri- cate problems into smaller language se- quences referred to as âthoughtsâ. Through a step-by-step inference process involving the use of prompts, each thought functions as an intermediate stage, contributing to the simplification of tack- ling complex problems to fulfill the problemâs ultimate objective.
Table 1: Comparisons of different prompting paradigms. Paradigm Performance Efficiency Flexibility IO CoT CoT-SC ToT GoT XOT
Effective design of thought steps toward complex problem-solving and reasoning, whether for hu- mans or LLMs, should prioritize three crucial aspects, namely:
⢠Performance. Performance is the accuracy of the solution to a problem, including the precision of each thought at intermediate stages. This metric holds paramount importance for problem-solving.
1 | 2311.04254#3 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 3,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "1\n# INTRODUCTION\nRecent advancements in Large Lan- guage Models (LLMs) have greatly ad- vanced problem solving in diverse do- mains such as mathematical reasoning Frieder et al. (2023), knowledge rea- soning Omar et al. (2023), root cause analysis Chen et al. (2023) and causal inference Kıcıman et al. (2023), etc.. This progress can be largely attributed to the technique of decomposing intri- cate problems into smaller language se- quences referred to as âthoughtsâ. Through a step-by-step inference process involving the use of prompts, each thought functions as an intermediate stage, contributing to the simplification of tack- ling complex problems to fulfill the problemâs ultimate objective.\nTable 1: Comparisons of different prompting paradigms. Paradigm Performance Efficiency Flexibility IO CoT CoT-SC ToT GoT XOT\nEffective design of thought steps toward complex problem-solving and reasoning, whether for hu- mans or LLMs, should prioritize three crucial aspects, namely:\n⢠Performance. Performance is the accuracy of the solution to a problem, including the precision of each thought at intermediate stages. This metric holds paramount importance for problem-solving.\n1",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 4 | ⢠Performance. Performance is the accuracy of the solution to a problem, including the precision of each thought at intermediate stages. This metric holds paramount importance for problem-solving.
1
⢠Efficiency. Efficiency relates to the number of LLM inference calls required to solve a single problem. Minimizing this aspect is crucial due to the high computational cost associated with LLM inference, thereby reducing the overall number of cost.
⢠Flexibility. Flexibility in thought topology refers to the diverse structures that can be employed by LLMs when organizing thoughts for problem-solving. These structures may include chains, trees, or even graphs, mirroring human thought processes. Enabling more flexible thought struc- tures enhances the capacity of LLMs for divergent and creative thinking, which is particularly advantageous in addressing complex problems, especially those with multiple potential solutions. | 2311.04254#4 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 4,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "⢠Performance. Performance is the accuracy of the solution to a problem, including the precision of each thought at intermediate stages. This metric holds paramount importance for problem-solving.\n1\n⢠Efficiency. Efficiency relates to the number of LLM inference calls required to solve a single problem. Minimizing this aspect is crucial due to the high computational cost associated with LLM inference, thereby reducing the overall number of cost.\n⢠Flexibility. Flexibility in thought topology refers to the diverse structures that can be employed by LLMs when organizing thoughts for problem-solving. These structures may include chains, trees, or even graphs, mirroring human thought processes. Enabling more flexible thought struc- tures enhances the capacity of LLMs for divergent and creative thinking, which is particularly advantageous in addressing complex problems, especially those with multiple potential solutions.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 5 | There exist several thought generation paradigms, such as Chain-of-Thought (CoT) Wei et al. (2022), Tree-of-Thought (ToT) Yao et al. (2023), and Graph-of-Thought (GoT), etc.. However, these paradigms each have their limitations and cannot simultaneously achieve all the three desired attributes, as illustrated in Table 1. Specifically, direct Input-Output (IO) prompting is suitable pri- marily for simple problem-solving scenarios with single-step processes, lacking both in performance and flexibility. CoT and self-consistency CoT (CoT-SC) enable step-by-step problem solving, result- ing in modest performance improvements, but they are confined to linear thought structures, limiting their flexibility. In contrast, ToT and GoT permit more versatile thought topologies, accommodating tree-like or graph-like structures. However, these paradigms require the evaluation of intermediate thought steps through LLM itself, incurring significant computational costs and inefficiencies due to multiple LLM calls. These paradigms are constrained by a law analogous to the âPenrose triangle â, wherein they can achieve a maximum of two out of the three attributes, and none of them can | 2311.04254#5 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 5,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "There exist several thought generation paradigms, such as Chain-of-Thought (CoT) Wei et al. (2022), Tree-of-Thought (ToT) Yao et al. (2023), and Graph-of-Thought (GoT), etc.. However, these paradigms each have their limitations and cannot simultaneously achieve all the three desired attributes, as illustrated in Table 1. Specifically, direct Input-Output (IO) prompting is suitable pri- marily for simple problem-solving scenarios with single-step processes, lacking both in performance and flexibility. CoT and self-consistency CoT (CoT-SC) enable step-by-step problem solving, result- ing in modest performance improvements, but they are confined to linear thought structures, limiting their flexibility. In contrast, ToT and GoT permit more versatile thought topologies, accommodating tree-like or graph-like structures. However, these paradigms require the evaluation of intermediate thought steps through LLM itself, incurring significant computational costs and inefficiencies due to multiple LLM calls. These paradigms are constrained by a law analogous to the âPenrose triangle â, wherein they can achieve a maximum of two out of the three attributes, and none of them can",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 6 | We propose a novel solution called âEverything of Thoughtsâ (XOT) to address the limitations of conventional thought frameworks, enhancing essential attributes of thought generation, includ- ing performance, efficiency, and flexibility for LLM inference.1 XOT leverages reinforcement learning (RL) Li (2017) and Monte Carlo Tree Search (MCTS) Silver et al. (2017), in conjunc- tion with lightweight policy and value networks, to pretrain on specific tasks for thought search- ing and subsequently generalize to new problems. This pretraining effectively integrates external domain knowledge into the âthoughtsâ provided to LLMs, expanding their problem-solving capa- bilities, and thereby significantly improving Performance. Once trained, XOT efficiently performs thought searching using MCTS with cost-effective policy and value networks for exploration and au- tonomously generates complete cognitive mappings for LLMs. It then employs a MCTS-LLM col- laborative thought revision process to further improve the thought quality while minimizing LLM interactions. This eliminates the need for LLMs to explore and evaluate thoughts themselves, as required by ToT and GoT, enhancing XOTâs Efficiency. Furthermore, | 2311.04254#6 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 6,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "We propose a novel solution called âEverything of Thoughtsâ (XOT) to address the limitations of conventional thought frameworks, enhancing essential attributes of thought generation, includ- ing performance, efficiency, and flexibility for LLM inference.1 XOT leverages reinforcement learning (RL) Li (2017) and Monte Carlo Tree Search (MCTS) Silver et al. (2017), in conjunc- tion with lightweight policy and value networks, to pretrain on specific tasks for thought search- ing and subsequently generalize to new problems. This pretraining effectively integrates external domain knowledge into the âthoughtsâ provided to LLMs, expanding their problem-solving capa- bilities, and thereby significantly improving Performance. Once trained, XOT efficiently performs thought searching using MCTS with cost-effective policy and value networks for exploration and au- tonomously generates complete cognitive mappings for LLMs. It then employs a MCTS-LLM col- laborative thought revision process to further improve the thought quality while minimizing LLM interactions. This eliminates the need for LLMs to explore and evaluate thoughts themselves, as required by ToT and GoT, enhancing XOTâs Efficiency. Furthermore,",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 7 | This eliminates the need for LLMs to explore and evaluate thoughts themselves, as required by ToT and GoT, enhancing XOTâs Efficiency. Furthermore, MCTS demonstrates remark- able Flexibility as it can explore various thought topologies, including graph structures akin to those employed in human mind mapping processes Faste & Lin (2012); Jamieson (2012). This enables diverse and creative thinking for LLMs, making it particularly valuable when dealing with complex thought structures or tasks featuring multiple potential solutions. By concurrently achieving supe- rior performance, efficiency, and flexibility, XOT challenges the constraints posed by the âPenrose triangle | 2311.04254#7 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 7,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "This eliminates the need for LLMs to explore and evaluate thoughts themselves, as required by ToT and GoT, enhancing XOTâs Efficiency. Furthermore, MCTS demonstrates remark- able Flexibility as it can explore various thought topologies, including graph structures akin to those employed in human mind mapping processes Faste & Lin (2012); Jamieson (2012). This enables diverse and creative thinking for LLMs, making it particularly valuable when dealing with complex thought structures or tasks featuring multiple potential solutions. By concurrently achieving supe- rior performance, efficiency, and flexibility, XOT challenges the constraints posed by the âPenrose triangle",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 8 | We comprehensively evaluate XOT across a diverse range of challenging problem-solving tasks, namely Game of 24, 8-Puzzle, and Pocket Cube. Our experimental results consistently showcase XOTâs superior performance, and its capacity to provide multiple solutions to problems efficiently with just a few LLM calls. These findings establish XOT as an effective thought generation ap- proach, paving the way for new avenues in LLMsâ problem-solving capabilities.
# 2 BACKGROUND
Thought for LLMs. Addressing complex problems often entails breaking down the overarching ob- jective into multiple intermediary steps. The outcomes or cognitive processes associated with each step are thoughts, which can be expressed as linguistic prompt sequences for LLMs to facilitate problem-solving. Structures of these thought may take various forms, including linear chains, hier- archical trees, or interconnected graphs, depending on how the thoughts are organized to advance towards a solution.
1We named it âEverything of Thoughtsâ to signify its three comprehensive thought generation capabilities.
2
(a)lo (b) CoT (9 Corse (d) ToT (f) XoT ic ic) 1 thoughts - Unevalt Hl DD âtought Positive thought Policy/Value ' ' Network on) i ©} mative thought
Figure 1: Comparison of XOT versus other prompting paradigms. | 2311.04254#8 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 8,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "We comprehensively evaluate XOT across a diverse range of challenging problem-solving tasks, namely Game of 24, 8-Puzzle, and Pocket Cube. Our experimental results consistently showcase XOTâs superior performance, and its capacity to provide multiple solutions to problems efficiently with just a few LLM calls. These findings establish XOT as an effective thought generation ap- proach, paving the way for new avenues in LLMsâ problem-solving capabilities.\n# 2 BACKGROUND\nThought for LLMs. Addressing complex problems often entails breaking down the overarching ob- jective into multiple intermediary steps. The outcomes or cognitive processes associated with each step are thoughts, which can be expressed as linguistic prompt sequences for LLMs to facilitate problem-solving. Structures of these thought may take various forms, including linear chains, hier- archical trees, or interconnected graphs, depending on how the thoughts are organized to advance towards a solution.\n1We named it âEverything of Thoughtsâ to signify its three comprehensive thought generation capabilities.\n2\n(a)lo (b) CoT (9 Corse (d) ToT (f) XoT ic ic) 1 thoughts - Unevalt Hl DD âtought Positive thought Policy/Value ' ' Network on) i ©} mative thought\nFigure 1: Comparison of XOT versus other prompting paradigms.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 9 | Figure 1: Comparison of XOT versus other prompting paradigms.
Input-Output (IO) Prompting (Fig. 1 (a)). The IO method is the most straightforward approach to instruct LLMs to address a problem without the provision of any intermediate thought processes.
Chain-of-thought (CoT) Wei et al. (2022) (Fig. 1 (b)). CoT decomposes problem-solving into a sequential chain of thoughts, allowing LLMs to approach complex problems step by step.
Self-consistency CoT (CoT-SC) Wang et al. (2023a) (Fig. 1 (c)). CoT-SC employs multiple in- stances of the CoT to generate multiple outputs from LLMs. It selects the the best results from multiple LLM outputs, offering more robust and consistent inference compared to the vanilla CoT.
Tree-of-thought (ToT) Yao et al. (2023) (Fig. 1 (d)). ToT organizes thoughts in a tree-like structure and utilizes search algorithms (e.g., Breadth-First Search, Depth-First Search) to expand the tree in pursuit of an optimal solution. However, thought evaluation in ToT relies on LLMs themselves, necessitating multiple costly and inefficient LLM inference calls. | 2311.04254#9 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 9,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Figure 1: Comparison of XOT versus other prompting paradigms.\nInput-Output (IO) Prompting (Fig. 1 (a)). The IO method is the most straightforward approach to instruct LLMs to address a problem without the provision of any intermediate thought processes.\nChain-of-thought (CoT) Wei et al. (2022) (Fig. 1 (b)). CoT decomposes problem-solving into a sequential chain of thoughts, allowing LLMs to approach complex problems step by step.\nSelf-consistency CoT (CoT-SC) Wang et al. (2023a) (Fig. 1 (c)). CoT-SC employs multiple in- stances of the CoT to generate multiple outputs from LLMs. It selects the the best results from multiple LLM outputs, offering more robust and consistent inference compared to the vanilla CoT.\nTree-of-thought (ToT) Yao et al. (2023) (Fig. 1 (d)). ToT organizes thoughts in a tree-like structure and utilizes search algorithms (e.g., Breadth-First Search, Depth-First Search) to expand the tree in pursuit of an optimal solution. However, thought evaluation in ToT relies on LLMs themselves, necessitating multiple costly and inefficient LLM inference calls.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 10 | Graph-of-thought (GoT) Besta et al. (2023) (Fig. 1 (e)). GoT extends the ToT approach by en- abling the generation of graph-like thought structures through thought aggregation and refinement during intermediate search phases. Although this method permits more flexible thought structures, it still demands multiple LLM inference calls for evaluation, incurring significant computational costs.
# 3 XOT: EVERYTHING OF THOUGHTS
XOT serves as an LLM-MCTS collaborative framework designed to enhance the thought generation process, thereby assisting LLMs in resolving complex problems. It leverages MCTS for proficient and efficient thought exploration while harnessing the capabilities of LLMs to refine and amend the thoughts derived from MCTS. This synergistic interaction creates a mutually beneficial arrangement, ultimately enabling the successful resolution of intricate problems characterized by high levels of performance, efficiency, and flexibility.
3.1 XOT IN A NUTSHELL
We present an overview of the architecture of XOT in Fig. 1 (f). XOT comprises two key compo- nents: (i) a MCTS module guided by policy/value networks; and (iii) an LLM solver for thought revision and inference. The MCTS and policy/value networks need to be trained and then generalize to the inference process. | 2311.04254#10 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 10,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Graph-of-thought (GoT) Besta et al. (2023) (Fig. 1 (e)). GoT extends the ToT approach by en- abling the generation of graph-like thought structures through thought aggregation and refinement during intermediate search phases. Although this method permits more flexible thought structures, it still demands multiple LLM inference calls for evaluation, incurring significant computational costs.\n# 3 XOT: EVERYTHING OF THOUGHTS\nXOT serves as an LLM-MCTS collaborative framework designed to enhance the thought generation process, thereby assisting LLMs in resolving complex problems. It leverages MCTS for proficient and efficient thought exploration while harnessing the capabilities of LLMs to refine and amend the thoughts derived from MCTS. This synergistic interaction creates a mutually beneficial arrangement, ultimately enabling the successful resolution of intricate problems characterized by high levels of performance, efficiency, and flexibility.\n3.1 XOT IN A NUTSHELL\nWe present an overview of the architecture of XOT in Fig. 1 (f). XOT comprises two key compo- nents: (i) a MCTS module guided by policy/value networks; and (iii) an LLM solver for thought revision and inference. The MCTS and policy/value networks need to be trained and then generalize to the inference process.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 11 | During the training phase, MCTS is harnessed to explore potential thought structures for a spe- cific task through simulated scenarios. This process entails the recording of states, values, and the visitation frequencies of thought nodes in each simulation. These recorded data are subsequently employed to iteratively train the policy and value estimation model, enabling it to assimilate domain knowledge and comprehend the world model.
Once trained, the estimated policy and value are utilized to guide the MCTS to systematically search for a thought trajectory provided to aid LLMs in problem-solving. Note that thoughts extracted only play a supporting role, assisting LLMs in gathering knowledge from external sources. These thoughts do not provide LLMs with definitive or error-free answers, as they may contain inaccu- racies or suboptimal solutions. LLMs are responsible for review and refining these thoughts when they seem erroneous or require adjustments. They continue MCTS the search process if needed
3
(a) Select (b) Expand & Evaluate (c) Backpropagation (d) Thought inference Extracted so a a a ee E Ext aN a ace iS g i hl 6 © 6 OB ARS a Tem ee oe oo s A (P,v) = fa : : a : a } Pt = Q J K~ $3 ratatate BR sews 2 Potente JOR ae | 2311.04254#11 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 11,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "During the training phase, MCTS is harnessed to explore potential thought structures for a spe- cific task through simulated scenarios. This process entails the recording of states, values, and the visitation frequencies of thought nodes in each simulation. These recorded data are subsequently employed to iteratively train the policy and value estimation model, enabling it to assimilate domain knowledge and comprehend the world model.\nOnce trained, the estimated policy and value are utilized to guide the MCTS to systematically search for a thought trajectory provided to aid LLMs in problem-solving. Note that thoughts extracted only play a supporting role, assisting LLMs in gathering knowledge from external sources. These thoughts do not provide LLMs with definitive or error-free answers, as they may contain inaccu- racies or suboptimal solutions. LLMs are responsible for review and refining these thoughts when they seem erroneous or require adjustments. They continue MCTS the search process if needed\n3\n(a) Select (b) Expand & Evaluate (c) Backpropagation (d) Thought inference Extracted so a a a ee E Ext aN a ace iS g i hl 6 © 6 OB ARS a Tem ee oe oo s A (P,v) = fa : : a : a } Pt = Q J K~ $3 ratatate BR sews 2 Potente JOR ae",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 12 | Figure 2: An illustration of iterative phases in MCTS for thought searching ((a)-(c)) and thought inference in problem resolution (d).
and eventually formulate the final answers by integrating these external thoughts with their internal knowledge.
3.2 THOUGHT SEARCHING FORMULATION
The fundamental objective of employing the thought generation paradigm for LLMs is to identify the optimal decomposition of a complex problem into several manageable sub-steps. Each sub-step aims to alter the current status of the problem, eventually culminating in the successful resolution of the overarching problem. This approach, as seen in ToT and GoT, hinges on well-defined state tran- sitions and clear final objectives. Consequently, it is natural to conceptualize the thought-searching process as a Markov Decision Process (MDP) Puterman (1990), in which:
⢠State st: Represents the current status of the problem. The initial state s0 corresponds to the original problem, while intermediate states are characterized by either decomposed sub-problems or the results stemming from their resolution.
⢠Action at: Signifies the one-step solution or action associated with tackling a problem, leading to a transition to a new state, by incorporating their outcomes.
⢠Reward r: Reflects the comprehensive evaluation of the solution to the original problem, assess- ing whether it has been effectively resolved through the process of problem decomposition. | 2311.04254#12 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 12,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Figure 2: An illustration of iterative phases in MCTS for thought searching ((a)-(c)) and thought inference in problem resolution (d).\nand eventually formulate the final answers by integrating these external thoughts with their internal knowledge.\n3.2 THOUGHT SEARCHING FORMULATION\nThe fundamental objective of employing the thought generation paradigm for LLMs is to identify the optimal decomposition of a complex problem into several manageable sub-steps. Each sub-step aims to alter the current status of the problem, eventually culminating in the successful resolution of the overarching problem. This approach, as seen in ToT and GoT, hinges on well-defined state tran- sitions and clear final objectives. Consequently, it is natural to conceptualize the thought-searching process as a Markov Decision Process (MDP) Puterman (1990), in which:\n⢠State st: Represents the current status of the problem. The initial state s0 corresponds to the original problem, while intermediate states are characterized by either decomposed sub-problems or the results stemming from their resolution.\n⢠Action at: Signifies the one-step solution or action associated with tackling a problem, leading to a transition to a new state, by incorporating their outcomes.\n⢠Reward r: Reflects the comprehensive evaluation of the solution to the original problem, assess- ing whether it has been effectively resolved through the process of problem decomposition.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 13 | ⢠Reward r: Reflects the comprehensive evaluation of the solution to the original problem, assess- ing whether it has been effectively resolved through the process of problem decomposition.
⢠Thought Ï : A one-step thought is a combination of one-step state and action, i.e., Ï = {s, a}. This formulation naturally encapsulates the process of decomposing a complex problem into multiple sub-tasks, each accompanied by their respective outcomes.
The detailed definitions of state, action, reward and thought for each task are shown in Table 1. The generation of complete thoughts T = {Ï1, · · · , ÏN }, can be construed as the endeavor to discover a thought trajectory to maximize the accumulated reward to address the overall problem.
3.3 THOUGHTS SEARCHING WITH MCTS | 2311.04254#13 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"id": "2308.09687"
},
{
"id": "2303.16563"
},
{
"id": "2302.00923"
},
{
"id": "2310.08118"
},
{
"id": "2303.11366"
},
{
"id": "2302.06466"
}
] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 13,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "⢠Reward r: Reflects the comprehensive evaluation of the solution to the original problem, assess- ing whether it has been effectively resolved through the process of problem decomposition.\n⢠Thought Ï : A one-step thought is a combination of one-step state and action, i.e., Ï = {s, a}. This formulation naturally encapsulates the process of decomposing a complex problem into multiple sub-tasks, each accompanied by their respective outcomes.\nThe detailed definitions of state, action, reward and thought for each task are shown in Table 1. The generation of complete thoughts T = {Ï1, · · · , ÏN }, can be construed as the endeavor to discover a thought trajectory to maximize the accumulated reward to address the overall problem.\n3.3 THOUGHTS SEARCHING WITH MCTS",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 14 | 3.3 THOUGHTS SEARCHING WITH MCTS
The formulation above naturally aligns the thought within LLM as a state-action pair. This approach facilitates the effective exploration of its optimal trajectory using a combination of MCTS and RL. This adheres to an iterative simulation cycle that encompasses three key phases: selection, expansion & evaluation, and backpropagation. It heavily depends on the utilization of neural networks fθ, which simultaneously estimate the value and action probability for a given state st. The aim is to reduce the number of rollouts and accelerate the search process, similar to the approach employed in AlphaGo Zero Silver et al. (2017). We provide a visual representation of an iteration of the MCTS in Fig. 2 (a)-(c) by taking Pocket Cube as an example and detail each process below.
Selection. In the selection phase, the algorithm initiates at the root node and proceeds to choose an action aâ from the available set A(s) for single-step thought generation in the current state s. This process continues until a leaf node within the current tree is reached. The selection is guided by the PUCT algorithm Rosin (2011), aiming to maximize the Upper Confidence Bound (UCB) Garivier
4
& Moulines (2011), as follows: | 2311.04254#14 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 14,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "3.3 THOUGHTS SEARCHING WITH MCTS\nThe formulation above naturally aligns the thought within LLM as a state-action pair. This approach facilitates the effective exploration of its optimal trajectory using a combination of MCTS and RL. This adheres to an iterative simulation cycle that encompasses three key phases: selection, expansion & evaluation, and backpropagation. It heavily depends on the utilization of neural networks fθ, which simultaneously estimate the value and action probability for a given state st. The aim is to reduce the number of rollouts and accelerate the search process, similar to the approach employed in AlphaGo Zero Silver et al. (2017). We provide a visual representation of an iteration of the MCTS in Fig. 2 (a)-(c) by taking Pocket Cube as an example and detail each process below.\nSelection. In the selection phase, the algorithm initiates at the root node and proceeds to choose an action aâ from the available set A(s) for single-step thought generation in the current state s. This process continues until a leaf node within the current tree is reached. The selection is guided by the PUCT algorithm Rosin (2011), aiming to maximize the Upper Confidence Bound (UCB) Garivier\n4\n& Moulines (2011), as follows:",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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2311.04254 | 15 | 4
& Moulines (2011), as follows:
aâ = arg max aâA(s) Q(s, a) + w · Pθ(s, a) N (s) 1 + N (s, a) . (1)
Here, Q(s, a) denotes the Q-value of a state-action pair (s, a). The term Pθ(s, a) denotes the pre- dicted prior probability of selecting action a given the state s obtained from a neural network fθ, and N (s, a) represents the count of times action a has been chosen in state s. The parameter w con- trols the trade-off between exploration and exploitation. The selection process will continue until an unexplored node is encountered. | 2311.04254#15 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 15,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "4\n& Moulines (2011), as follows:\naâ = arg max aâA(s) Q(s, a) + w · Pθ(s, a) N (s) 1 + N (s, a) . (1)\nHere, Q(s, a) denotes the Q-value of a state-action pair (s, a). The term Pθ(s, a) denotes the pre- dicted prior probability of selecting action a given the state s obtained from a neural network fθ, and N (s, a) represents the count of times action a has been chosen in state s. The parameter w con- trols the trade-off between exploration and exploitation. The selection process will continue until an unexplored node is encountered.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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2311.04254 | 16 | Evaluation and Expansion. Upon reaching a previously unselected leaf node, we expand to the state s for the next step for new thought exploration. This expansion involves the evaluation of its value and action probability on the state, which are modeled by neural networks parameterized by θ, i.e., (Pθ(s), vθ(s)) = fθ(s). Here Pθ(s) is the prior probabilities for all actions on s, and vθ(s) denotes its predicted state value. These two values are retained and stored for backup purposes, and state s is masked as âvisitedâ.
Backpropagation. Following the expansion of a leaf node in the above phases, which could be either an unexplored or terminal state, the algorithm proceeds to update all the Q(s, a) values via backpropagation. For unexplored nodes, this update involves computing the mean of its estimated value vθ, while for terminated nodes, itâs based on the true reward r. These updates occur as infor- mation is backpropagated along the trajectory to subsequent nodes. Additionally, the visit count for each state-action pair is also incremented as follows: N (s, a) = N (s, a) + 1. | 2311.04254#16 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 16,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Evaluation and Expansion. Upon reaching a previously unselected leaf node, we expand to the state s for the next step for new thought exploration. This expansion involves the evaluation of its value and action probability on the state, which are modeled by neural networks parameterized by θ, i.e., (Pθ(s), vθ(s)) = fθ(s). Here Pθ(s) is the prior probabilities for all actions on s, and vθ(s) denotes its predicted state value. These two values are retained and stored for backup purposes, and state s is masked as âvisitedâ.\nBackpropagation. Following the expansion of a leaf node in the above phases, which could be either an unexplored or terminal state, the algorithm proceeds to update all the Q(s, a) values via backpropagation. For unexplored nodes, this update involves computing the mean of its estimated value vθ, while for terminated nodes, itâs based on the true reward r. These updates occur as infor- mation is backpropagated along the trajectory to subsequent nodes. Additionally, the visit count for each state-action pair is also incremented as follows: N (s, a) = N (s, a) + 1.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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] |
2311.04254 | 17 | A simulation is completed after a sequence of selection, evaluation, expansion, and backpropagation steps. After conducting multiple simulations, we proceed to the next step by selecting an action at state s using a probability distribution defined as εa â N (s, a)1/γ, where γ is a temperature constant that regulates the level of exploration.
Policy and Value Networks Training. The simulations described above allow us to compile a dataset for each sample state s containing (s, ε(s), v(s)), where ε(s) = {εa | a â A(s)}, and v(s) represents the ground truth value obtained by accumulating rewards along the trajectory starting from state s. Subsequently, we can train a combined policy and value network fθ to minimize the discrepancy between the predicted value vθ(s) and the actual value v(s), while also maximizing the alignment between the action probabilities produced by the neural network Pθ(s) and the search probabilities ε(s). This can be achieved by minimizing the following loss function:
L = (v(s) â vθ(s))2 + ε(s)T log Pθ(s)). This training iterates alongside the simulation process to continually enhance the performance of fθ, resulting in progressive improvements in thought searching capabilities.
3.4 THOUGHT INFERENCE WITH MCTS | 2311.04254#17 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 17,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "A simulation is completed after a sequence of selection, evaluation, expansion, and backpropagation steps. After conducting multiple simulations, we proceed to the next step by selecting an action at state s using a probability distribution defined as εa â N (s, a)1/γ, where γ is a temperature constant that regulates the level of exploration.\nPolicy and Value Networks Training. The simulations described above allow us to compile a dataset for each sample state s containing (s, ε(s), v(s)), where ε(s) = {εa | a â A(s)}, and v(s) represents the ground truth value obtained by accumulating rewards along the trajectory starting from state s. Subsequently, we can train a combined policy and value network fθ to minimize the discrepancy between the predicted value vθ(s) and the actual value v(s), while also maximizing the alignment between the action probabilities produced by the neural network Pθ(s) and the search probabilities ε(s). This can be achieved by minimizing the following loss function:\nL = (v(s) â vθ(s))2 + ε(s)T log Pθ(s)). This training iterates alongside the simulation process to continually enhance the performance of fθ, resulting in progressive improvements in thought searching capabilities.\n3.4 THOUGHT INFERENCE WITH MCTS",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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] |
2311.04254 | 18 | 3.4 THOUGHT INFERENCE WITH MCTS
Once trained, we utilize the fθ to guide the MCTS in generating a thought for a new problem, which assists the LLM in solving it. Specifically, MCTS is utilized to perform K simulations aimed at thought searching and problem-solving, as illustrated in Fig.2 (d). In each simulation, fθ is em- ployed to guide the MCTS in its search for a thought trajectory. Throughout the training process, fθ incorporates external information related to the state and action quality. This information helps LLMs understand the world model, enhancing their long-term reasoning and planning abilities, which are areas they may not excel in Stechly et al. (2023); Valmeekam et al. (2023), thereby ensur- ing the performance of thought generation. Once the simulation concludes, we record the visiting count N (s, a) and the thought trajectory is obtained based on the number of solutions required:
⢠Single solution. starting from each state s, the action with the highest visiting count N (s, a) is selected.
⢠Multiple solution. we sample M thought trajectories following the probability distribution εa â N (s, a) and remove duplicates. | 2311.04254#18 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 18,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "3.4 THOUGHT INFERENCE WITH MCTS\nOnce trained, we utilize the fθ to guide the MCTS in generating a thought for a new problem, which assists the LLM in solving it. Specifically, MCTS is utilized to perform K simulations aimed at thought searching and problem-solving, as illustrated in Fig.2 (d). In each simulation, fθ is em- ployed to guide the MCTS in its search for a thought trajectory. Throughout the training process, fθ incorporates external information related to the state and action quality. This information helps LLMs understand the world model, enhancing their long-term reasoning and planning abilities, which are areas they may not excel in Stechly et al. (2023); Valmeekam et al. (2023), thereby ensur- ing the performance of thought generation. Once the simulation concludes, we record the visiting count N (s, a) and the thought trajectory is obtained based on the number of solutions required:\n⢠Single solution. starting from each state s, the action with the highest visiting count N (s, a) is selected.\n⢠Multiple solution. we sample M thought trajectories following the probability distribution εa â N (s, a) and remove duplicates.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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] |
2311.04254 | 19 | ⢠Multiple solution. we sample M thought trajectories following the probability distribution εa â N (s, a) and remove duplicates.
This results in one or multiple thought trajectories T â that consist of a sequence of state-action pairs for problem-solving. The trajectories for multi-solution problems may intertwine and converge at
5
MCTS LLM LLM _ââ Identified Extract error state Extract Extracted Simulations hought Additional L Revised thoughts Simulations thoughts (ference)
Figure 3: An illustration of thought revision process in XOT.
the same goal state, resulting in a graph-like thought structure. This demonstrates that XOT is capable of generating thought structures with flexibility. These trajectories are then transformed into text sequences that are concatenated to form a prompt sequence provided to LLMs. Note that the thought trajectory is concatenated into a single prompt, even in the case of problems with multiple solutions. Therefore, we only require a single LLM inference call at this stage. Given that the fθ network is relatively lightweight, this ensures the efficiency of XOT. | 2311.04254#19 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 19,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "⢠Multiple solution. we sample M thought trajectories following the probability distribution εa â N (s, a) and remove duplicates.\nThis results in one or multiple thought trajectories T â that consist of a sequence of state-action pairs for problem-solving. The trajectories for multi-solution problems may intertwine and converge at\n5\nMCTS LLM LLM _ââ Identified Extract error state Extract Extracted Simulations hought Additional L Revised thoughts Simulations thoughts (ference)\nFigure 3: An illustration of thought revision process in XOT.\nthe same goal state, resulting in a graph-like thought structure. This demonstrates that XOT is capable of generating thought structures with flexibility. These trajectories are then transformed into text sequences that are concatenated to form a prompt sequence provided to LLMs. Note that the thought trajectory is concatenated into a single prompt, even in the case of problems with multiple solutions. Therefore, we only require a single LLM inference call at this stage. Given that the fθ network is relatively lightweight, this ensures the efficiency of XOT.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
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2311.04254 | 20 | Thought Revision. It is important to acknowledge that that MCTS may not always provide the globally optimal thought trajectory to directly solve the problem flawlessly. Therefore, the thoughts extracted from MCTS serve as a reference thinking process for the problem, aiding LLMs in a sup- portive capacity. The LLMs will leverage their internal knowledge to review the extracted thought, identify errors in the thought trajectory, and then ground its knowledge in collaboration with the MCTS to revise and refine the thought. | 2311.04254#20 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 20,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Thought Revision. It is important to acknowledge that that MCTS may not always provide the globally optimal thought trajectory to directly solve the problem flawlessly. Therefore, the thoughts extracted from MCTS serve as a reference thinking process for the problem, aiding LLMs in a sup- portive capacity. The LLMs will leverage their internal knowledge to review the extracted thought, identify errors in the thought trajectory, and then ground its knowledge in collaboration with the MCTS to revise and refine the thought.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 21 | The revision process is iterative in nature, as shown in Fig. 3. Initially, upon obtaining the extracted thought, we instruct the LLM to detect any errors in the thought generated by MCTS using its in- ternal knowledge. If the LLM identifies an error, it results in an error state denoted as se within the thought. If no error is found, the thought remains unchanged. Starting from the parent state of se, MCTS conducts an additional set of L simulations, ultimately yielding a revised thought for the LLM. In scenarios involving multiple solutions, each solution undergoes this process individually. Upon the completion of the revision, we supply the LLMs with the revised thoughts for problem- solving. The revision process can be repeated several times to enhance the reliability of the answer. This collaborative MCTS-LLM framework nurtures a mutually beneficial process for both compo- nents, ultimately contributing to the overall performance of problem-solving. Since LLMs are solely utilized for identifying errors during the revision process with only one call, the efficiency of XOT is effectively maintained. | 2311.04254#21 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 21,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "The revision process is iterative in nature, as shown in Fig. 3. Initially, upon obtaining the extracted thought, we instruct the LLM to detect any errors in the thought generated by MCTS using its in- ternal knowledge. If the LLM identifies an error, it results in an error state denoted as se within the thought. If no error is found, the thought remains unchanged. Starting from the parent state of se, MCTS conducts an additional set of L simulations, ultimately yielding a revised thought for the LLM. In scenarios involving multiple solutions, each solution undergoes this process individually. Upon the completion of the revision, we supply the LLMs with the revised thoughts for problem- solving. The revision process can be repeated several times to enhance the reliability of the answer. This collaborative MCTS-LLM framework nurtures a mutually beneficial process for both compo- nents, ultimately contributing to the overall performance of problem-solving. Since LLMs are solely utilized for identifying errors during the revision process with only one call, the efficiency of XOT is effectively maintained.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 22 | The collaborative revision framework harnesses the strengths of both MCTS and LLMs. MCTS efficiently and flexibly generates candidate thoughts for LLMs through simulations, while LLMs use their internal knowledge to revise and ground these thoughts within the MCTS framework, effectively turning MCTS into a world model for LLMs. This process ensures the generation of high-quality thoughts for problem-solving.
# 4 EXPERIMENT
We conduct an extensive evaluation of our XOT approach2 in comparison to several baseline meth- ods across three challenging tasks: the Game of 24, the 8-Puzzle (with a 3 Ã 3 grid), and the 2 Ã 2 Pocket Cube. An overview of these tasks is provided in Table 2. These tasks are characterized by their complexity, requiring multiple steps for completion and potentially having multiple solutions. To assess the effectiveness of our proposed XOT, we compare it against IO, CoT, CoT-SC, ToT, and GoT methodologies. We employ both GPT-3.5 Ouyang et al. (2022) and GPT-4 OpenAI (2023) for these evaluations. Note that temperature and top p are set to 0.0 for all LLM invoked.
2Code and dataset to reproduce this work will be shared in the near future, following compliance with the affiliation policy.
6
Table 2: An overview of tasks employed in this study. | 2311.04254#22 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 22,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "The collaborative revision framework harnesses the strengths of both MCTS and LLMs. MCTS efficiently and flexibly generates candidate thoughts for LLMs through simulations, while LLMs use their internal knowledge to revise and ground these thoughts within the MCTS framework, effectively turning MCTS into a world model for LLMs. This process ensures the generation of high-quality thoughts for problem-solving.\n# 4 EXPERIMENT\nWe conduct an extensive evaluation of our XOT approach2 in comparison to several baseline meth- ods across three challenging tasks: the Game of 24, the 8-Puzzle (with a 3 Ã 3 grid), and the 2 Ã 2 Pocket Cube. An overview of these tasks is provided in Table 2. These tasks are characterized by their complexity, requiring multiple steps for completion and potentially having multiple solutions. To assess the effectiveness of our proposed XOT, we compare it against IO, CoT, CoT-SC, ToT, and GoT methodologies. We employ both GPT-3.5 Ouyang et al. (2022) and GPT-4 OpenAI (2023) for these evaluations. Note that temperature and top p are set to 0.0 for all LLM invoked.\n2Code and dataset to reproduce this work will be shared in the near future, following compliance with the affiliation policy.\n6\nTable 2: An overview of tasks employed in this study.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 23 | Objective Input Output Thought State Action Game of 24 Use four numbers on playing cards to make the number 24 through +, â, Ã, or ÷. 4 numbers ranging from 1 to 13, e.g., (4, 6, 10, 10). An equation to reach 24, e.g., 4 à 6 + 10 â 10 = 24. 3 intermediate equations. The remaining 1-4 numbers. Picking two number and a operation to compose an equation. 8-Puzzle Rearrange the tiles in the 3 à 3 puzzle from an scrambled state to a goal state . A scrambled 3 à 3 digital puzzle, e.g., . The slide sequence of the â-â tile, e.g., (Up, Down, Left, Right · · · ). The step-by-step sliding, and the puzzle state after the move. The current number layout of the puzzle. The one-step moving action of the â-â tile. Pocket Cube Rotating the faces of a 2 à 2 pocket cube until each face of the cube is a uniform color A scrambled 2 à 2 . pocket cube, e.g., . Colors represented as numbers for LLMs. The rotation move sequence of the cube, e.g., (F, R2, Uâ · · · ). The | 2311.04254#23 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 23,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Objective Input Output Thought State Action Game of 24 Use four numbers on playing cards to make the number 24 through +, â, Ã, or ÷. 4 numbers ranging from 1 to 13, e.g., (4, 6, 10, 10). An equation to reach 24, e.g., 4 à 6 + 10 â 10 = 24. 3 intermediate equations. The remaining 1-4 numbers. Picking two number and a operation to compose an equation. 8-Puzzle Rearrange the tiles in the 3 à 3 puzzle from an scrambled state to a goal state . A scrambled 3 à 3 digital puzzle, e.g., . The slide sequence of the â-â tile, e.g., (Up, Down, Left, Right · · · ). The step-by-step sliding, and the puzzle state after the move. The current number layout of the puzzle. The one-step moving action of the â-â tile. Pocket Cube Rotating the faces of a 2 à 2 pocket cube until each face of the cube is a uniform color A scrambled 2 à 2 . pocket cube, e.g., . Colors represented as numbers for LLMs. The rotation move sequence of the cube, e.g., (F, R2, Uâ · · · ). The",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 24 | . Colors represented as numbers for LLMs. The rotation move sequence of the cube, e.g., (F, R2, Uâ · · · ). The step-by-step rotation, and the cube state after the move. Colors of each face of the pocket cube. The one-step rotation action of cube. Reward 1 if the number of the final number is equal to 24 otherwise -1. The negative minimum step on solving the current puzzle state toward the goal state. The negative minimum moving step on solving current cube state toward the goal state. | 2311.04254#24 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 24,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": ". Colors represented as numbers for LLMs. The rotation move sequence of the cube, e.g., (F, R2, Uâ · · · ). The step-by-step rotation, and the cube state after the move. Colors of each face of the pocket cube. The one-step rotation action of cube. Reward 1 if the number of the final number is equal to 24 otherwise -1. The negative minimum step on solving the current puzzle state toward the goal state. The negative minimum moving step on solving current cube state toward the goal state.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 25 | Policy/Value Networks Configurations. The policy and value networks in our model utilize a shared multi-layer perceptron (MLP) architecture with two layers and hidden units arranged as (128, 256). Two heads connected to the MLP are responsible for predicting vθ(s) and Pθ(s) separately. This design results in a considerably smaller model compared to LLM, making it much more ef- ficient. We train this model through three iterations, with each iteration comprising 10 self-play episodes for MCTS.
Evaluation Metric. For each task, we assess the accuracy of each approach on the test set. Addi- tionally, we track the number of LLM invocations required for all approaches to solve a problem, as well as the number of times fθ is invoked in the case of XOT. Itâs important to note that fθ is a considerably smaller model compared to LLMs. In the context of multi-solution scenarios, ac- curacy is computed as the percentage of problems for which any of the answers provided by each approach is correct. Multi-solution Accuracy (MultiAcc) is calculated as the average percentage of correctness across all solutions offered. Furthermore, we capture the total count of distinct solutions provided by each approach, regardless of their correctness, represented as #Sol. Note that we set the maximum solution number to 3 for all problems in multi-solution scenarios. | 2311.04254#25 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 25,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "Policy/Value Networks Configurations. The policy and value networks in our model utilize a shared multi-layer perceptron (MLP) architecture with two layers and hidden units arranged as (128, 256). Two heads connected to the MLP are responsible for predicting vθ(s) and Pθ(s) separately. This design results in a considerably smaller model compared to LLM, making it much more ef- ficient. We train this model through three iterations, with each iteration comprising 10 self-play episodes for MCTS.\nEvaluation Metric. For each task, we assess the accuracy of each approach on the test set. Addi- tionally, we track the number of LLM invocations required for all approaches to solve a problem, as well as the number of times fθ is invoked in the case of XOT. Itâs important to note that fθ is a considerably smaller model compared to LLMs. In the context of multi-solution scenarios, ac- curacy is computed as the percentage of problems for which any of the answers provided by each approach is correct. Multi-solution Accuracy (MultiAcc) is calculated as the average percentage of correctness across all solutions offered. Furthermore, we capture the total count of distinct solutions provided by each approach, regardless of their correctness, represented as #Sol. Note that we set the maximum solution number to 3 for all problems in multi-solution scenarios.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 26 | 4.1 GAME OF 24
The Game of 24 presents a arithmetic challenge wherein the goal is to employ four numbers within the range of 1 to 13, in conjunction with basic arithmetic operations, (i.e., +, â, Ã, ÷), to attain a final result of 24. This game may possess multiple valid solutions.
# 4.1.1 TASK SETUP
We collect a dataset from 4nu, comprising 1,362 games ranked by human solving time, spanning a range of difficulty levels from easy to hard. For our testing phase, we randomly selected 137 games, ensuring coverage of various difficulty intervals. The remaining 1,225 problems were used to train the policy/value networks with MCTS. In the context of this task, as outlined in Table 1, the thoughts refer to the three intermediate equations, while the state encompasses the available numbers (ranging
7
Table 3: Performance comparison on Game of 24. | 2311.04254#26 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 26,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "4.1 GAME OF 24\nThe Game of 24 presents a arithmetic challenge wherein the goal is to employ four numbers within the range of 1 to 13, in conjunction with basic arithmetic operations, (i.e., +, â, Ã, ÷), to attain a final result of 24. This game may possess multiple valid solutions.\n# 4.1.1 TASK SETUP\nWe collect a dataset from 4nu, comprising 1,362 games ranked by human solving time, spanning a range of difficulty levels from easy to hard. For our testing phase, we randomly selected 137 games, ensuring coverage of various difficulty intervals. The remaining 1,225 problems were used to train the policy/value networks with MCTS. In the context of this task, as outlined in Table 1, the thoughts refer to the three intermediate equations, while the state encompasses the available numbers (ranging\n7\nTable 3: Performance comparison on Game of 24.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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] |
2311.04254 | 27 | 7
Table 3: Performance comparison on Game of 24.
Model IO CoT CoT-SC (n=10) ToT (b=1) ToT (b=3) GoT (k=1) XoT (w/o revise) XoT (w/ revise) GPT-3.5 Acc. [%] LLM invoked 6.57 2.19 2.19 5.84 10.22 2.92 61.31 79.56 1.00 1.00 10.00 22.11 43.96 7.00 1.00 1.39 GPT-4 fθ invoked Acc. [%] LLM invoked - - - - - - 68.73 92.15 10.22 4.38 4.38 34.31 60.58 10.95 63.50 74.45 1.00 1.00 10.00 23.50 39.83 7.00 1.00 1.38 fθ invoked - - - - - - 68.69 88.20
from 1 to 4) for creating the equations. Actions involve the selection of two numbers and an operator to form an equation, and the reward is set to 1 if the final equation is both valid and results in the number 24, utilizing each of the input numbers exactly once, otherwise it is set to -1. Performance is measured by calculating the success rate across the 137 test games. | 2311.04254#27 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 27,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "7\nTable 3: Performance comparison on Game of 24.\nModel IO CoT CoT-SC (n=10) ToT (b=1) ToT (b=3) GoT (k=1) XoT (w/o revise) XoT (w/ revise) GPT-3.5 Acc. [%] LLM invoked 6.57 2.19 2.19 5.84 10.22 2.92 61.31 79.56 1.00 1.00 10.00 22.11 43.96 7.00 1.00 1.39 GPT-4 fθ invoked Acc. [%] LLM invoked - - - - - - 68.73 92.15 10.22 4.38 4.38 34.31 60.58 10.95 63.50 74.45 1.00 1.00 10.00 23.50 39.83 7.00 1.00 1.38 fθ invoked - - - - - - 68.69 88.20\nfrom 1 to 4) for creating the equations. Actions involve the selection of two numbers and an operator to form an equation, and the reward is set to 1 if the final equation is both valid and results in the number 24, utilizing each of the input numbers exactly once, otherwise it is set to -1. Performance is measured by calculating the success rate across the 137 test games.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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2311.04254 | 28 | 4.1.2 BASELINES & XOT SETUP
The IO prompt is supported by five in-context examples. In the case of CoT, we augment each input-output pair by including three intermediate equations. As for ToT, we solicit one-step thought candidates from the LLM at each step, subsequently instructing the LLM to categorize each thought candidate for intermediate selection. For experimental comparison, we conduct experiments on both the top-1 candidate (with b=1) and the top-3 candidates (with b=3) being retained, where b indicates the branches retained for exploration at each step. For GoT, we employ LLM to generate one-step thought candidates in the same manner as ToT, then we direct the LLM to select the top-1 thought from all candidates for merging the thoughts. We also examine a CoT-SC baseline, which derives the majority output from 10 CoT samples. For XOT, we perform 200 simulations for each action taken, and this count is increased to 500 during the thought revision process. | 2311.04254#28 | Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation | Recent advancements in Large Language Models (LLMs) have revolutionized
decision-making by breaking down complex problems into more manageable language
sequences referred to as ``thoughts''. An effective thought design should
consider three key perspectives: performance, efficiency, and flexibility.
However, existing thought can at most exhibit two of these attributes. To
address these limitations, we introduce a novel thought prompting approach
called ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle
of existing thought paradigms. XoT leverages pretrained reinforcement learning
and Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge
into thoughts, thereby enhancing LLMs' capabilities and enabling them to
generalize to unseen problems efficiently. Through the utilization of the
MCTS-LLM collaborative thought revision framework, this approach autonomously
produces high-quality comprehensive cognitive mappings with minimal LLM
interactions. Additionally, XoT empowers LLMs to engage in unconstrained
thinking, allowing for flexible cognitive mappings for problems with multiple
solutions. We evaluate XoT on several challenging multi-solution
problem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our
results demonstrate that XoT significantly outperforms existing approaches.
Notably, XoT can yield multiple solutions with just one LLM call, showcasing
its remarkable proficiency in addressing complex problems across diverse
domains. | http://arxiv.org/pdf/2311.04254 | Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang | cs.AI, cs.LG | 17 pages, 5 figures | null | cs.AI | 20231107 | 20231112 | [
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] | {
"authors": "Ruomeng Ding, Chaoyun Zhang, Lu Wang, Yong Xu, Minghua Ma, Wei Zhang, Si Qin, Saravan Rajmohan, Qingwei Lin, Dongmei Zhang",
"chunk_id": 28,
"doc_id": "2311.04254",
"primary_category": "cs.AI",
"published": 20231107,
"source": "http://arxiv.org/pdf/2311.04254",
"summary": "Recent advancements in Large Language Models (LLMs) have revolutionized\ndecision-making by breaking down complex problems into more manageable language\nsequences referred to as ``thoughts''. An effective thought design should\nconsider three key perspectives: performance, efficiency, and flexibility.\nHowever, existing thought can at most exhibit two of these attributes. To\naddress these limitations, we introduce a novel thought prompting approach\ncalled ``Everything of Thoughts'' (XoT) to defy the law of ``Penrose triangle\nof existing thought paradigms. XoT leverages pretrained reinforcement learning\nand Monte Carlo Tree Search (MCTS) to incorporate external domain knowledge\ninto thoughts, thereby enhancing LLMs' capabilities and enabling them to\ngeneralize to unseen problems efficiently. Through the utilization of the\nMCTS-LLM collaborative thought revision framework, this approach autonomously\nproduces high-quality comprehensive cognitive mappings with minimal LLM\ninteractions. Additionally, XoT empowers LLMs to engage in unconstrained\nthinking, allowing for flexible cognitive mappings for problems with multiple\nsolutions. We evaluate XoT on several challenging multi-solution\nproblem-solving tasks, including Game of 24, 8-Puzzle, and Pocket Cube. Our\nresults demonstrate that XoT significantly outperforms existing approaches.\nNotably, XoT can yield multiple solutions with just one LLM call, showcasing\nits remarkable proficiency in addressing complex problems across diverse\ndomains.",
"text": "4.1.2 BASELINES & XOT SETUP\nThe IO prompt is supported by five in-context examples. In the case of CoT, we augment each input-output pair by including three intermediate equations. As for ToT, we solicit one-step thought candidates from the LLM at each step, subsequently instructing the LLM to categorize each thought candidate for intermediate selection. For experimental comparison, we conduct experiments on both the top-1 candidate (with b=1) and the top-3 candidates (with b=3) being retained, where b indicates the branches retained for exploration at each step. For GoT, we employ LLM to generate one-step thought candidates in the same manner as ToT, then we direct the LLM to select the top-1 thought from all candidates for merging the thoughts. We also examine a CoT-SC baseline, which derives the majority output from 10 CoT samples. For XOT, we perform 200 simulations for each action taken, and this count is increased to 500 during the thought revision process.",
"title": "Everything of Thoughts: Defying the Law of Penrose Triangle for Thought Generation",
"year": 2023
} | [
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