sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In an ambient space Ω, we say that a set C separates the sets A and B if any continuous path in Ω with endpoints in A and B intersects C. Theorem 3.3.
Sentence 2: under Assumptions 3.2, let (µ t ) t≥0 be a Wasserstein gradient flow of F such that, for some 0 < r a < r b , the support of µ 0 is contained in B(0, r b ) and separates the spheres if (u m (t)) m∈N,t≥0 is a sequence of classical gradient flows initialized in B(0, r b ) such that µ m,0 converges weakly to µ 0 then (limits can be interchanged) A proof and stronger statements are presented in Appendix C. There, we give a criterion for Wasserstein gradient flows to escape neighborhoods of non-optimal measures-also valid in the finite-particle setting-and then show that it is always satisfied by the flow defined above.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose problem (1) satisfies the HEB condition with θ ∈ (0, 1/2] and F (x 0 )−F * ≤ 0 , where x 0 is an initial solution.
Sentence 2: let η = 1/(36L), and by running Algorithm 1 with R = log 2 0 , we have E[F (x (R) ) − F * ] ≤ , and the computational complexity for achieving an -optimal solution in expectation is O(n log( 0 / ) + Lc 2 max{ 1 1−2θ , log( 0 / )}).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We obtain the following probability model: q c = e f * c . This formulation is closely related to the maximum-likelihood estimate with conditional model q c = e fc / K k=1 e f k (logistic regression).
Sentence 2: if we choose a function class such that the normalization condition K k=1 e f k = 1 holds, then the two formulations are identical.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that most research on VSTM is concerned with characterizing how subjects encode a single item in VSTM (for instance, the precision with which a single item can be encoded [1,2]) and, thus, does not consider the joint encoding of multiple items.
Sentence 2: we are not aware of any previous work attempting to experimentally probe and characterize exactly how the encoding of an item influences the encoding of other items, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the performance of EBC is clearly datadependent, SMA (purple lines in Figure 4) exhibits dramatic performance improvement compared to alternatives in all instances (see the Supplement, Section 3, for extensive additional experiments, including comparisons to other classifiers, and varying adversary's budget constraints).
Sentence 2: figure 5 (left) looks deeper at the nature of SMA solution vectors w. we consider how the adversary's strength, as measured by the query budget, affects the sparsity of solutions as measured by w 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: To obtain a more detailed analysis, we focused on a subset of datasets but extended the configuration budget for optimizing all methods from one hour to one day and to two days for AUTO-SKLEARN.
Sentence 2: we clustered our 140 datasets with g-means [27] based on the dataset meta-features and used one dataset from each of the resulting 13 clusters (see Table 6 in the supplementary material for the list of datasets).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Based on this derivation we believe that the natural way of relating R H and C H would be to replace the p 0 in Herbster's potential formulation by q 0 such that 1/p 0 +1/q 0 = 1.
Sentence 2: one would have to consider C H q 0 and then define b R H p 0 := 1/C H q 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We study the problem of constructing a hypothesis test for an unknown data-generating function h : R p → R, when h is estimated with a black-box algorithm (specifically, Gaussian Process regression) from n observations {y i , x i } n i=1 .
Sentence 2: we are interested in testing for the hypothesis: where H 0 , H a are the function spaces for h under the null and the alternative hypothesis.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main goal of this paper is to provide finite-sample analysis of these estimators, via unified analysis of the estimator after bias correction.
Sentence 2: we show conditions under which, for β-Hölder continuous (β ∈ (0, 2]) densities on D dimensional space, the bias of fixed-k estimators decays as O n −β/D and the variance decays as O n −1 , giving a mean squared error of O n −2β/D + n −1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the sake of interpretability, we now specialize to the case where the region graph has only two layers, with the first layer corresponding to vertices and the second layer corresponding to hyperedges.
Sentence 2: for r, s 2 R, we have r ✓ s only if |r| = 1, and R = V [ F , where F is the set of hyperedges and V denotes the set of singleton vertices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Now, define the estimate of prediction loss Note that y t can be readily computed from `t(•), which is available to the algorithm (equivalently, we may assume that the algorithm observed y t ).
Sentence 2: due to the linearity of expectation, we have ˜t(w) provides an unbiased estimate of the loss `t(w) for any w 2 W. Hence, by adding a feature cost term we get ˜t(w) + h s(w), c i as an estimate of the loss that the learner would have suffered at round t had he chosen the weight vector w. Algorithm 1 The LQDEXP3 Algorithm Parameters: Real numbers 0  ⌘, 0 <  1, W 0 ⇢ W finite set, a distribution µ over W 0 , horizon T > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Agnostic-degree-profiling algorithm (see Section 3.2) solves exact recovery in any SBM(n, p, ln(n)Q/n) for which exact recovery is solvable, using no input except the graph in question, and runs in o(n 1+ ) time for all > 0.
Sentence 2: exact recovery is efficiently and universally solvable whenever it is information-theoretically solvable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we propose to use the Fisher score to construct an attribute contribution matrix A ∈ R M ×L , where an entry A d,l represents the discrimination capability of attribute a d for differentiating the class pair (i, j) indexed by u l (i, j).
Sentence 2: given the attribute a d and class pair (i, j), let µ d k and σ d k be the mean and standard deviation of k-th class and µ d be the mean of samples from both classes i and j corresponding to d-th attribute.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As we observe, The best performance among the four algorithms belongs to Ada Newton.
Sentence 2: ada Newton is able to achieve the accuracy of R N (w) R ⇤ N < 1/N by 2.4 passes over the dataset which is very close to theoretical result in Theorem 1 that guarantees accuracy of order O(1/N ) after ↵/(↵ 1) = 2 passes over the dataset.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, we prove the impossibility of bounding performance of k-nearest neighbors only in terms of the diameter and the cut size.
Sentence 2: we give a parameterized family of graphs for which the number of mistakes of the perceptron is upper bounded by a fixed constant independent of the graph size while k-nearest neighbors provably incurs mistakes linearly in the graph size.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithmically, if initialized at a point with small enough objective value, then a descent algorithm always decreases the objective value and hence stays in this region.
Sentence 2: for a generic kernel 3 a 0 ∈ S k−1 , if the sparsity rate 4 θ k −2/3 and the number of measurement m poly(k), initializing at some preconditioned k consecutive entries of y, and applying any descent method that converges to a local minimizer under a strict saddle hypothesis [17,18], produces a near shift-truncation of the ground truth.
Relationship: entailment
Domain: NeurIPS
Sentence 1: With this uniform distribution over trees, we have Ï¡ for each edge, and 9 41 2 @ e f£ n A ¡ y e , so that £ E i p C £ y e . With the set-up of the previous section, the basic form of the upper bounds follows by applying Jensen's inequality [6].
Sentence 2: for any pair £ E i t C £ y e s , we have the upper bound £ y e ¡ 9 41 2 4@ £ ge f£ n BA .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The breakthrough of gradient cost for finding second-order stationary points were achieved in 2016/2017, when the two recent lines of literatures, namely FastCubic [1] and CDHS [8] as well as their stochastic versions [2,35], achieve a gradient cost of Õ(min(n −1.5 +n 3/4 −1.75 , −3.5 )) which serves as the best-known gradient cost for finding an ( , O( 0.5 ))-approximate second-order stationary point before the initial submission of this paper.
Sentence 2: 3 4 Agarwal et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To ensure online learnability, it is sufficient to consider an assumption on the dependence of N ∞ (α, F) on α.
Sentence 2: an obvious example of such a class is a VC-type class with N ∞ (α, F) ≤ (c/α) d for some c which can depend on n. Assuming that F ⊂ [0, 1] X , the value of the game is upper bounded by 2D T (F) ≤ 4 √ dT log c. for binary prediction, using the Sauer-Shelah lemma ensures that the value of the game is at most 4 dT log(eT ), matching the result of [15] up to a constant 2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we present some applications of our theorems and algorithms in machine learning.
Sentence 2: we consider the regularized problems with a smooth loss: where (a i , b i ), i = 1, . . . , n denote a set of training examples, R(x) could be the 1 norm x 1 , the ∞ norm x ∞ , or a huber norm [28], or the 1,p norm K k=1 x k p , where k is the k-th component vector of x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is worth noting that for k-sparse matrices with k constant, this distance does not scale with p or q.
Sentence 2: if the absolute value of the elements of Θ (1) and Θ (2) are bounded by Θ max then d(Θ (1) , Θ (2) ) ≤ 2 Having the estimator Θ (i) the algorithm constructs the confidence set for episode i as where > 0 is an input parameter to the algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One could straightforwardly instead use, for example, either a mixture of Gaussians, or, normalizing flows (Rezende & Mohamed, 2015;Kingma et al., 2016), to parametrize each inference network factor in order to improve expressivity, at the cost of additional implementational complexity.
Sentence 2: if one were to use a provably universal density estimator to parameterize each inference network factor, such as that introduced in Huang et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By properly tuning the elimination thresholds ε r of Algorithm 3 in accordance with the target accuracy ε, we can establish an explicit trade-off between the number of communication rounds and the number of arm pulls each player needs.
Sentence 2: we can design a multi-player algorithm that terminates after at most R communication rounds, for any given parameter R > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Common techniques include global eigenmodes for linearized dynamics (see, e.g., [3]), discrete Fourier transforms, POD for nonlinear dynamics [30,12], and balancing modes for linear systems [24] as well as multiple variants of these techniques, such as those using shift modes [22] in conjunction with POD modes.
Sentence 2: pOD, which is in principle equivalent to principal component analysis, has been extensively applied to the analysis of physical phenomena [5,22] even though it suffers from numerous known issues, including the possibility of principal directions in a set of data may not necessarily correspond to the dynamically important ones.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the calcium imaging, we use a standard model [7], where the concentration of fluorescent Calcium has a fast initial rise upon an action potential followed by a slow exponential decay.
Sentence 2: we let z k i = [Ca 2+ ] k be the concentration of fluorescent Calcium in neuron i in time bin k and assume it evolves as first-order auto-regressive AR(1) model, where α CA is the Calcium time constant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We state these technical bounds in Appendix E. 3 The wave-filtering algorithm Our online algorithm (Algorithm 1) runs online projected gradient descent [Zin03] on the squared loss Here, each M t is a matrix specifying a linear map from featurized inputs Xt to predictions ŷt .
Sentence 2: after choosing a certain bank of k filters {φ j }, Xt ∈ R nk+2n+m consists of convolutions of the input time series with each φ j (scaled by certain constants), along with x t−1 , x t , and y t−1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Neurophysiological findings support the existence of remembering memories and their involvement in credit assignment and learning in biological systems.
Sentence 2: hippocampal recordings in rats indicate that brief sequences of prior experience are replayed both in the awake resting state and during sleep, both of which conditions are linked to memory consolidation and learning (Foster & Wilson, 2006;Davidson et al., 2009;Gupta et al., 2010;Ambrose et al., 2016).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We cropped out patches of different sizes, in order to simulate different amounts of visual data that may be processed by the toddler's learning system.
Sentence 2: we started with crops of 30 width (roughly corresponding to the area with high acuity) and increased the size in 10 increments up to 70 (the field of view of the head camera); examples are shown in Figures 2a and 2b.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given matrices A, B, C 2 C m⇥k , the tensor product Another view of tensor is that it defines a multi-linear mapping.
Sentence 2: for given dimension m C is defined as: for a 2 C m , we use V (I, I, a) to denote the projection of tensor V along the 3rd dimension.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use the K-method to interpolate between the 1-norm and the ∞-norm, and to interpolate between the 2-norm and the ∞-norm.
Sentence 2: to gain some intuition on the behavior of these interpolationnorms, first note that for any p ≥ 1 and any v ∈ R m it holds that max the interesting range of t for the Next, we prove a theorem which states that interpolating a p-norm with the ∞-norm is approximately equivalent to restricting that p-norm to the absolutely-largest components of the vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To efficiently compute terms of this type, we define Like the forward and backward matrices, α ⋆ (i, j) and β ⋆ (i, j) may be calculated via dynamic programming.
Sentence 2: we have the base cases α ⋆ (i, 1) = 1{i = y ⋆ 1 } • α(i, 1) • Q ′ 1 (w) and β ⋆ (i, L) = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We provide in Theorem 3.2 a bound of the form R T (✓) 6 Õ(D(✓, ⇥ 0 )/ p T ) for a pseudo-metric D and any ✓ 2 B 1 .
Sentence 2: we say that this bound is accelerable as the rate may decrease if D(✓, ⇥ 0 ) decreases with T . it yields an oracle bound of the form R T (✓) 6 O(k✓k 1 / p T ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast, a flower's beauty is greatest from up close, but can also be experienced from a range of distances and perspectives.
Sentence 2: ρ g and δ g determine the scale and shape of the state reward function, with r i (S) = max(ρ g (1 − distance(S, i, G)/δ g ), 0), where distance(S, i, G) is the geodesic distance between agent i and the goal.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We therefore evaluate our algorithm by using the first convolutional layers of the vanilla topologies as "feature extractors" (mostly below 50% of the vanilla network size) and then learning a deep structure, "learned head", from their output.
Sentence 2: the deepest layers of the vanilla network, "vanilla head", is removed and replaced by a structure which is learned, in an unsupervised manner, by our algorithm 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is shown in Figure 5, which compares D-EM to the EM implementations of the GENIE/SMILE and SAMIAM systems, 3 both of which were represented in previous inference evaluations [4].
Sentence 2: we ran these systems on a chain network X 0 → • • • → X 100 , where each variable has 10 states, and using datasets with alternating observed and hidden variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will view Π(s, g) as the set of actions that the agent considers acceptable (or equally good) in state s for goal g. The agent policy π always selects a i after its helper action h i whenever a i is acceptable.
Sentence 2: π((s, h i ), g) = a i whenever a i ∈ Π(s, g).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Streeter and Golovin [26] study an online submodular maximization problem in the oblivious adversary model.
Sentence 2: their result can cover the stochastic K-MAX bandit problem as a special case, and an O(K √ mT log m) upper bound on the (1 − 1/e)-regret can be shown.
Relationship: entailment
Domain: NeurIPS
Sentence 1: X and symmetric linearly independent A i (otherwise, consider optimization over a larger matrix W X X Z with A operating symmetrically on the off-diagonal blocks).
Sentence 2: this setting covers problems including matrix completion (where A i are indicators, [5]), matrix reconstruction from linear measurements [15] and multi-task training (where each column of X is a predictor for a different task and A i have a single non-zero column, [2,1]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: where C M (G) is the set of all M -covers of G. Theorem 2.2 claims that the optimal value of the MAP LP is equal to the supremum over all MAP assignments over all graph covers, appropriately scaled.
Sentence 2: the proof of this result shows that, under mild conditions, there exists an M -cover H of G and an assignment x H such that In this section, we will describe how to extend the previous results from discrete to continuous MRFs (i.e., X = R) using graph covers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our approach is to simultaneously estimate the rigid motion and learn the shape PDF.
Sentence 2: we estimate R, D, θ, and σ 2 to maximize ∝ p(P|R, D, S, σ 2 )p(S|θ)dS The key idea is that we can estimate shape and motion while learning the parameters of the PDF p(S|θ) over shapes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A detailed proof is provided in Appendix C. The main tool of the proof of this rather surprising result is Lindsay's (1983) geometric analysis of the likelihood of nonparametric mixtures, based on Minkovski's theorem.
Sentence 2: lindsay's (1983) Theorem 3.1 ensures that, when the trace of the likelihood curve is compact, the likelihood function is maximised for a finite mixture.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The observation that recent memories are more likely to be lost than old memories in these cases has generally been interpreted as evidence that the role of these medial temporal lobe structures in the storage and/orretrieval of declarative memories is only temporary.
Sentence 2: several investigators have advocated the general idea that, in the course of a relatively long time period (from several days in rats up to decades in humans), memories are reorganized (or consolidated) so that memories whose successful recall initially depends on the hippocampus gradually become independent of this structure (see Refs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The other important difference between ELBO and RELBO is the residual term which is expressed through E s [log p] E s [log q t ].
Sentence 2: maximizing this term amounts to looking for a density with low cross entropy with the joint p and high cross entropy with the current iterate q t . the next component s t needs to be as close as possible to the target p but also sufficiently different from the current approximation q t . Indeed, s t should capture the aspects of the posterior that the current mixture could not approximate yet.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We consider the stepsize η needed to reach L min (under idealized assumptions) by satisfying We linearize L (around θ) and then, after denoting g = ∇ θ L, we solve for η: First of all, note the clear separation between g, the estimator of the gradient of L and v, the proposed update step.
Sentence 2: moreover, it is easily seen that the final update step ηv is independent of the magnitude of v. the adaptation method only takes into account the "direction" of the proposed update.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We first define an initial grammar that generates the exact set of training samples.
Sentence 2: for each training sample x i ∈ X, there is an Or-rule S → A i in the initial grammar where S is the start symbol and A i is an And-node, and the probability of the rule is 1 X where X is the number of training samples; for each x i there is also an And-rule A i → a i1 a i2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: if (v, P ) is a valid parent set choice for G with some P ⊇ P ∪ {v } then if (v, P ) is valid parent set choice for G and s(v, P ) > s(v, P ) for all P ∈ C then CMSL solution (i.e., under the current branch).
Sentence 2: before the actual branch and bound, we precompute a DP table which stores, for each subset of vertices V ⊂ V of the problem instance, the score of the so-called BN extensions of V , i.e., the optimal BN structures over U = V \ V where we additionally allow the vertices in U to also take parents from V . This guarantees that the BN extensions are compatible with the vertex order in the current branch of the branch-and-bound search tree, and thereby the sum of the score of the current partial CMSL solution over V and the score of the optimal BN extensions of V is a valid upper bound.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similarly, the presented bound for r ∞ also implies sparsity, though it is the sparsity of w k − w. In fact, this is directly related with the sparsity of w k , considering that the classifiers can be postprocessed to subtract their average simultaneously whilst the classification decision won't change for any possible input.
Sentence 2: theorem 3.2 also partially suits linear DNN-based classifications.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose an adaptive CNN (ACNN) that uses side information (e.g., the perspective weight) as an auxiliary input to adapt the CNN to different scene contexts (e.g., appearance changes from high/low angle perspectives, and scale changes due to distance).
Sentence 2: we consider the filter weights in each convolutional layer as points on a low-dimensional manifold, which is modeled using a sub-network where the side information is the input and the filter weights are the outputs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the other hand, notice that by definition of ξ t−1 , w(e) ≤ U t (e), ∀e ∈ E under event ξ t−1 .
Sentence 2: using the monotonicity of f in the probability weight, and the fact that ORACLE is an (α, γ)-approximation algorithm, we have The next observation is that, from the linearity of expectation, the gap f (S t , U t ) − f (S t , w) decomposes over nodes v ∈ V \ S t . for any source node set S ⊆ V, any probability weight function w : E → [0, 1], and any node v ∈ V, we define f (S, w, v) as the probability that node v is influenced if the source node set is S and the probability weight is w. Hence, we have  In the appendix, we show that under any weight function, the diffusion process from the source node set S t to the target node v can be modeled as a Markov chain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In an attempt to equalize that factor, we also ran an experiment where the Local method was allowed to check more candidate k values than our method.
Sentence 2: it could generate as many (proximity-based) candidate neighborhoods at test time as would fit in the run-time required by our approach, where k ranges from 20 up to 6,000 in increments of 10.
Relationship: entailment