sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: To compress M, we consider each D ∈ D independently and learn a new domain D where |D | ≤ D and g : D → D is a surjective mapping, i.e., ∀ µ ∈ D , ∃ C ∈ D such that g(C) = µ.
Sentence 2: each cluster of objects is replaced by its cluster center in the reduced domain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For both neurons i = 1, 2 we maximize information transmission under the constraint that their outputs stay as statistically independent from each other as possible.
Sentence 2: we maximize Since the same terms (up to the sign) are optimized in (7) and ( 15) we can derive a gradient ascent rule for the weights of neuron i, w ij , analogously to section 3: Figure 4 shows the results of an experiment where two neurons receive the same Poisson input with a rate of 20Hz at their 100 synapses.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We modify conjugate gradients to (1) perform linear solves with multiple right hand sides simultaneously, and (2) return tridiagonal matrices corresponding to partial Lanczos tridiagonalizations of K XX with respect to each right hand side.
Sentence 2: 2 mBCG takes as input a matrix [ y z 1 • • • z t ], and outputs: where T1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Tournament solutions have long been used in social choice and voting theory to define winners in general tournaments when no Condorcet winner exists [12,13].
Sentence 2: a tournament solution is any mapping that maps each tournament on K nodes to a subset of 'winning' nodes in [K]; for our purposes, we will define a tournament solution to be any mapping S : P K →2 [K] that maps each preference matrix P (via the induced tournament T P ) to a subset of winning arms S(P) ⊆ [K].
Relationship: entailment
Domain: NeurIPS
Sentence 1: For instance, the non-recalling √ n-run policy consists in exploiting the first arm giving a run larger than √ n. For a uniform mean-reward distribution over [0, 1], the average number of explored arms is √ n and the selected arm is exploited for the equivalent of n time steps with an expected failure rate of 1/ √ n, yielding the regret of 2 √ n. We introduce a second target to improve the expected failure rate of the selected arm, at the expense of a slightly more expensive exploration phase.
Sentence 2: we show that it is optimal to explore n/2 arms on average, resulting in the expected failure rate 1/ √ 2n of the exploited arm, for the equivalent of n time steps, hence the regret of √ 2n.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is conventional to plot the graphon as one typically plots an adjacency matrix: with the origin in the upper-left corner.
Sentence 2: darker shades correspond to higher values of W. Figure 1b depicts a graphon W φ which is weakly isomorphic to W. W φ is the relabeling of W by the measure preserving transformation φ(x) = 2x mod 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Obviously, by the max-flow min-cut theorem, the resulting cut set C mf has the smallest cut.
Sentence 2: c mf,n is the cut set restricted to the graph G n . Lemma 1 Given labeling L mf,n ∈ L mf , S n forms a single connected area within graph G n . ∀i ∈ S n , there is a path with nodes {i, j, k, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recently, Valiant and Valiant [1,2] showed that a class of distributional properties, which includes such practically relevant properties as entropy, the number of distinct elements, and distance metrics between pairs of distributions, can be estimated given a sublinear sized sample.
Sentence 2: given a sample consisting of independent draws from any distribution over at most n distinct elements, these properties can be estimated accurately using a sample of size O(n/ log n).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, to achieve the same error, HYBRID needs 100⇥ less samples than MC.
Sentence 2: to achieve the error of 0.01, (a) shows HYBRID needs 10 3 and MC needs 10 5 samples; (b) shows HYBRID needs 10 4 and MC needs 10 6 samples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: All three paradigms seldom exploit a common trait of real-world optimization problems: instances of the same type of problem are solved again and again on a regular basis, maintaining the same combinatorial structure, but differing mainly in their data.
Sentence 2: in many applications, values of the coefficients in the objective function or constraints can be thought of as being sampled from the same underlying distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The UCI Higgs dataset [1,11] is a dataset with 11 million observations, where the problem is to distinguish between the signal process where Higgs bosons are found, versus the background process that do not produce Higgs bosons.
Sentence 2: we will consider a two-sample test with the ME and SME test on the high level features derived by physicists, as well as a two-sample test on four extremely low level features (azimuthal angular momentum φ measured by four particle jets in the detector).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This makes |S a (w, l, x)| = 2 i , which further implies that or equivalently, Case (ii): If w(a, x) ≤ M 2 l+1 , we have |S a (w, l, x)| = 1.
Sentence 2: hence, the following bound holds in both cases (i) and (ii): The lemma holds by summing up over X and maximizing over A on all sides of Inequality 9.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We often assume that we do not explicitly form matrices A and B i , but we can only compute Ax, B i y i and their adjoints A > and B > i for any given x, y i , and for i = 1, • • • , m. Problem (1) is sufficiently general to cope with many applications in different fields including machine learning, statistics, image/signal processing, and model predictive control.
Sentence 2: (1) covers convex empirical risk minimization, support vector machine, LASSO-type, matrix completion, compressive sensing problems as representative examples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The analysis shows more problematic mixing for BCR with respect of LBCR.
Sentence 2: in LBCR the 95% of the chains have a potential reduction factor lower than 1.35, with a median equal to 1.11, while in BCR the 95th quantile is 1.44 and the median equals to 1.18.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since this maximization problem belongs to the zero-one integer problem (i.e., NP-hard problem), an exhaustive search is prohibitive for a large L. Therefore, we solve this problem approximately with the help of the following greedy-search algorithm.
Sentence 2: first, only one y l1 value is set to 1 so that P (y|x * ; Θ) is maximized.
Relationship: entailment
Domain: NeurIPS
Sentence 1: BP was studied in a number of papers which have gradually increased our understanding of the convergence properties and accuracy of the algorithm.
Sentence 2: recent developments show that the stable fixed points are local minima of the Bethe free energy [10,1], which paved the way for more accurate "generalized belief propagation" algorithms and convergent alternatives to BP [11,6].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The essential difference from the statistical dimension for learning is that we must additionally bound the average covariances of indicator functions (or, rather, continuous analogues of indicators) on the outputs of functions in C. The essential claim in our lower bounds is therefore in showing that a typical pair of (indicator functions on outputs of) s-wave functions has small covariance.
Sentence 2: to prove Theorem 1.1, it suffices to upper-bound the quantity for most pairs f m,S , f m,T of s-wave functions, where χ is some smoothed version of an indicator function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The central computational problem in topic modeling is to compute the posterior distribution of the latent structure conditioned on an observed collection of documents.
Sentence 2: our goal is to compute the posterior topics, topic transitions, per-document topic weights, per-word topic assign-ments, and top-level weights conditioned on a set of documents, each of which is a collection of parse trees.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To define this concept, first consider repeatedly playing an extensive game.
Sentence 2: let σ t i be the strategy used by player i on round t. The average overall regret of player i at time T is: Moreover, define σt i to be the average strategy for player i from time 1 to T . for each information set I ∈ I i , for each a ∈ A(I), define: There is a well-known connection between regret, average strategies, and Nash equilibria.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we consider binary classification in GLM with the convex loss function (w, (x, y)) =  In this section we derive a bound for an algorithm that optimizes a surrogate of the loss directly.
Sentence 2: we consider the Bayes loss for linear LGM with conjugate likelihood p(y|f and where a, b, and σ 2 are functions of x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper focuses on paired comparison, in which there are exactly two objects to choose from, and its solution using linear estimation.
Sentence 2: it is concerned with the environmental structures that make it possible to mimic the decisions of the linear estimator using less information and less computation, asking two questions: How much of the linear estimator do we need to know to mimic its decisions, and under what conditions?
Relationship: entailment
Domain: NeurIPS
Sentence 1: The GAMP method reduces the complexity of loopy BP by exploiting the linear nature of the relations between the variables w and z.
Sentence 2: it is shown that when each term z[t] is a linear combination of a large number of terms w j [ ], the belief messages across the factor node for the linear constraints can be approximated as Gaussians and the factor node updates can be computed with a central limit theorem approximation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Set α t = a, and L t = aµt + L + a −1 (µ − L) + , where a ∈ (0, 1) is a constant.
Sentence 2: then the regret of Algorithm 2 can be bounded as with a = 1 2 , the regret bound reduces to 3µ 2 + L D 2 + 2Q 2 µ log(N + 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We write this error loc (x k+s ) and define it as loc (x k+s ) x k+s − x(t k+s ), where x k+s is computed using the real solution x(t k ), ..., x(t k+s−1 ).
Sentence 2: the difference between x k and y k can be described as follows We now write a complete definition of stability, inspired by Definition 6.3.1 from Gautschi [2011].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The training of the R 2 SVM is then carried out in a purely feed-forward way.
Sentence 2: we train a linear SVM for the l-th layer, and then compute the input of the next layer as the addition of the original feature space and the random projection of previous layers' outputs, which is then passed through a simple sigmoid function: ) where θ l are the linear SVM parameters trained with x l , and W l+1 is the concatenation of l random projection matrices [W l+1,1 , W l+1,2 , • • • , W l+1,l ], one for each previous layer, each being a random matrix sampled from N (0, 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The affinity matrix G l is then combined with the current features to compute the next-layer features at each position t, where v is a compositional function such as a GRU cell [8] or a linear layer with residual connections.
Sentence 2: in other words, the feature at each position is computed as a weighted sum of other features, where the weights are determined by the graph G l , followed by transformation function v. Objective Function At the top layer, we obtain the features F L . At each position t, we use the feature f L t to initialize the hidden states of an RNN decoder, and employ the decoder to predict the units following x t . the RNN decoder maximizes the conditional log probability log P (x t+1 , • • • , x t+D |x t , f l t ) using an auto-regressive factorization as in standard language modeling [47] (also see Figure 2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that our motivating example in (9) is an example of a consistently ordered 2-cyclic matrix, for which Theorem 4.7 is applicable.
Sentence 2: for the example in (9), we can apply Theorem 4.7 with µ = 1 1/n leading to ⇢(C) = 1/n 2 , which exactly coincides with our previous computations of ⇢(C) in Section 3.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We demonstrate the error bounds for perturbed messages with a set of Monte Carlo trials.
Sentence 2: for each trial we construct a binary-valued 5 × 5 grid with uniform potential strengths, which are either (1) all positively correlated, or (2) randomly chosen to be positively or negatively correlated (equally likely); we also assign random single-node potentials to each x s . We then run a quantized version of BP, rounding each log-message to discrete values separated by 2 log δ (ensuring that the newly introduced error satisifies d (e) ≤ δ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Accordingly, we denote the last n − k columns of U by U S c . We let the diagonal elements of Σ S , ΣS , Σ S c , ΣS c be identified by their column indices.
Sentence 2: the diagonal entries σ S,c of Σ S and σS,c of ΣS are indexed by c ∈ {1, . . . , k}; the diagonal entries σ S c ,c of Σ S c and σS c ,c of ΣS c are indexed by c ∈ {1, . . . , n}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in [21,22] (and in many other works) there is an intimate connection between the smallest eigenvalues of H b and mean exit times out of these metastable states.
Sentence 2: in the asymptotic limit of small noise D 1, exit times are exponentially distributed and the first non-trivial eigenvalue (after µ 0 = 0) is given by µ 1 = 1/τ where τ is the mean exit time to overcome the highest potential barrier on the way to the deepest potential well.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (Our results easily generalize to random initial states and random costs.)
Sentence 2: let the space of admissible histories up to time t be H Let Π H,t be the set of all history-dependent policies with the property that at each time t the randomized control action is a function of h t . Π H,t := µ 0 : H 0 → P(A), µ 1 : H 1 → P(A), .
Relationship: entailment
Domain: NeurIPS
Sentence 1: (7-8), consider the example of external-regret matching.
Sentence 2: since Y is the unit simplex in R d , Ȳ is the positive orthant in R d . So, with W 2 (ȳ) = ȳ 2 2 /2, the optimization problem (8) will be equivalent to ȳ = arg min ȳ is the projection of s onto R d + by minimum Euclidean distance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If γ 2 > γ 2 c , the optimal Z is a certain point between 0 and M on the real axis.
Sentence 2: for M = 2 the optimal configuration of the unit-length connected bars is unique (up to flipping about x-axis), meaning that the encoding/decoding vectors are symmetric about the principal axis; for M ≥ 3, there are infinitely many configurations of the bars starting from the origin and ending at the optimal Z, and nothing can be added about their regularity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Intuitively, this assumption allows the loss to be parameterized in terms of the propensity matrix Θ Z and the unnormalized output kernel Θ Θ (hence the superscript of L u ).
Sentence 2: the (i, j)-th component of Θ Z stands for the linear response value of example j with respect to the label of the example i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume that for each object O i ∈ {M, S, P }, the stock price O i .Stock, centered at the mean, is given by where each i is a Gaussian random variable.
Sentence 2: the fact that there are several hidden common causes between M and S can be modeled by the covariance of m and s , σ ms . unlike in standard directed Gaussian models, σ ms is allowed to be non-zero.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The elements u j of u have the effect of directly modifying the transition probabilities of an uncontrolled Markov chain.
Sentence 2: given an uncontrolled transition probability matrix P with elements p ij , we de ne the controlled transition probabilities as Note that P (0) = P . In some sense this is the most general notion of "control" one can imaginewe are allowing the controller to rescale the underlying transition probabilities in any way it wishes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , ψ m } each associated with the elements of W . The factor graph implies that p(Y ) factorizes as p(y 1 , .
Sentence 2: . . , where Z is a normalizing partition function (the dependence on parameters is suppressed here) and Y c is a subset of the random variables that are associated with the neighbors of node c. Y c = {y i |i ∈ Ne(c)} where Ne(c) is the set of vertices that are neighbors of c. Inference in graphical models requires the evaluation and the optimization of Z.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this case the time series consists mainly of transitions within a class, whereas switching between the two classes is relatively rare.
Sentence 2: therefore the covariance of time derivatives is mostly determined by the within-class scatter of the two point sets, and both matrices become approximately proportional: ẋ ẋT t ≈ 2/N • S W . Moreover, if we assume that S W (and therefore ẋ ẋT t ) is positive definite, we can rewrite the SFA objective (2) as the weight vector that optimizes the SFA objective (2) also optimizes the FLD objective (4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We see that the skip-based setting consistently outperforms the conventional setting, and the gains obtained are moderate to high depending on the underlying distribution of the workers' errors.
Sentence 2: the gains are quite striking under the hammer-spammer model: this result is not surprising since the mechanism (ideally) screens the spammers out and leaves only the hammers who answer perfectly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, it is easy to see [2] that two characterizers We call two characterizers similar if this holds, and write for some G satisfying GV = 0 and 1G = 0.
Sentence 2: the similarity equivalence class of some characterizer C is the set {C + G|GV = 0, 1G = 0}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the undiscounted setting, then ||h π || p(•|s,a) � span(h π ), and thus C π M � span(h π ).
Sentence 2: we define the class of C-"hard" MDPs by the class of MDPs with optimal policy having a low environmental-value norm, or for short, MDPs with low environmental norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An interesting observation is that algorithm 1 (User Average) and algorithm 2 (Item Average) have similar pattern of sorting order in Figure 3 (right column).
Sentence 2: these two algorithms are similar in nature, and are relatively strong or weaker in similar regions of Ω. Algorithm 8 (NLPMF) on the other hand, has a very different relative strength pattern.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The modeling need motivates us to consider a generalization of SR measures that allows for easy tuning of the relative importance given to quality and diversity considerations.
Sentence 2: we introduce the class of Exponentiated Strongly Rayleigh (ESR) measures, which are distributions of the form ν(S) ∝ µ(S) p , where S is a set, p > 0 is a parameter and µ is an SR measure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A and q W r# sE q $# # Yh © $# # t $# for all $# u $# # Y£ and P£ ¥ $# # with $# Y£ § !© © v# # u .
Sentence 2: the heuristics q D# approximate the best-case plan-execution cost from to ¢# .
Relationship: entailment
Domain: NeurIPS
Sentence 1: for x 2 {0, 1} n one has F (x) = f ({i : x i = 1}), and (3) it can be computed efficiently and deterministically given access to the sets P u , u 2 U .
Sentence 2: we can compute the value of F (x) using at most O(|U | ⇥ |V |) operations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we study the impact of replacing uniform with leverage score sampling.
Sentence 2: we prove that the derived method still achieves optimal learning bounds but the time and memory is now e O(nd eff ), and e O(d eff 2 ) respectively, where d eff is the effective dimension which and is never larger, and possibly much smaller, than p n. To the best of our knowledge this is the best currently known computational guarantees for a kernel ridge regression solver.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We considered a route prediction task where we predict the next road segment, given the route taken so far; see, e.g., [Letchner et al., 2006, Simmons et al., 2006, Krumm, 2008.
Sentence 2: for each route of the testing set, we consider one edge at a time and try to predict the next edge, given the edges observed so far.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This has been a major area of machine learning research in the past decade [19], with several exciting developments on understanding its underlying statistical principles [27,18,4,3,29,21,15,7,31,10,34,6].
Sentence 2: several general characterizations have been developed for describing when active learning can in principle have an advantage over the classic passive supervised learning paradigm, and by how much.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A more general model, proposed in [6,16], employs a hierarchical prior for P (u) with adapted parameters tuned to the global patterns in higher-order dependencies.
Sentence 2: the logarithm of the variances of P (u) is assumed to be a linear function of the higher-order random variables v, Conditional on the higher-order variables, the joint distribution of coefficients is factorisable, as in GSM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For clarity of presentation, we will use a binary 1-of-K representation for the latent state assignments.
Sentence 2: we will write the event of the latent state at time step t being k as z tk = 1 and z tl = 0 ∀l = 1...K, l = k. Then the transition probabilities can be written as T ij = Pr(z tj = 1|z t−1,i = 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The second is the recent work in econometrics and computational economics that aims to estimate critical concepts directly from a limited data set, and reason about the sample complexity of the computational problems.
Sentence 2: in all of the above papers, the algorithm must be able to query or compute C(S) for an arbitrary set S ✓ N . In our work, we are instead given a collection of samples from some distribution; importantly the algorithm does not know C(S) for sets S that were not sampled.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At each time t, given the history h t , the agent can compute the posterior distribution µ t given by µ t (Θ) = P(θ * ∈ Θ|h t ) for any set Θ.
Sentence 2: upon applying the action a t and observing the new state s t+1 , the posterior distribution at t + 1 can be updated according to Bayes' rule as Let N t (s, a) be the number of visits to any state-action pair (s, a) before time t. With these notations, TSDE is described as follows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using our framework we also show how the asymptotic theory of Liu and Ihler applies more generally to distributed composite likelihood estimators.
Sentence 2: the Strong LAP Condition provides a sufficient condition to guarantee the validity of a core assumption made in the theory of Liu and Ihler, namely that each local estimate for the parameter of a clique is a consistent estimator of the corresponding clique parameter in the joint distribution.
Relationship: entailment