sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The message for entry d from node ρ i on the tree to its parent is a vector The local likelihood term is then: The logarithm of the proposal density is then: This is not of standard form, and we use an approximation log qlr (t lr ) instead.
Sentence 2: we use a piecewise linear log qlr (t lr ), which can be easily sampled from, and for which the normalization term is easy to compute.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Like its discrete counterpart, Theorem 3.1 provides necessary and sufficient conditions for the continuous relaxation to be tight.
Sentence 2: for the relaxation to be tight, the optimal solution on any Mcover, appropriately scaled, cannot exceed the value of the optimal solution of the MAP problem over G. Theorem 3.1 provides necessary and sufficient conditions for the tightness of the continuous relaxation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, a reasonable ranking rule based on the binary answers is r(x) = y = 1 + min {k : Although the definition can be flexibly applied even when f b is not consistent, a consistent f b is usually desired in order to introduce a good ranking rule r. Furthermore, the ordinal information can help to model the relative confidence in the binary outputs.
Sentence 2: when k is farther from the rank of x, the answer f b (x, k) should be more confident.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The larger cluster is clearly a bent-double cluster: 89 of the 150 objects are more likely to belong to this cluster under the model and 88 out of the 89 objects in this cluster have the bent-double label.
Sentence 2: the unsupervised algorithm has discovered a cluster that corresponds to "strong examples" of bent-doubles relative to the particular feature-space and model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At each round, we either add a new predictor tree or a new state-space trajectory to the model, depending on which results in a greater decrease in the objective.
Sentence 2: with the following definitions: (y t f k 1 (x t , s t )) j (x t ), k,j = 1 T T X t=1 (y t f k 1 (x t , s t ))s t,j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: LWMB splitting is similar to the splitting operation presented in [9] for factor-graph models, but operates by partitioning a group of MLN domain elements rather than a group of variational parameters (as in [9]).
Sentence 2: we partition a single domain ∆ ∈ ∆ into two disjoint domains (∆ (1) ∪ ∆ (2) = ∆ and ∆ (1) ∩ ∆ (2) = ∅).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The basic assumption is that neuronal development 1 is a Markov process governed by local rules [4].
Sentence 2: we assume that (i) each step in dendritic outgrowth only depends on the preceding step and on current local conditions; and (ii) dendrites do not undergo geometrical or topological modification after their formation (see, however, Discussion).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our goal in this paper is to address this stark contrast by providing a theoretical analysis of the performance of affine policies on synthetic instances of the problem generated from a probabilistic model.
Sentence 2: we consider random instances of the two-stage adjustable problem (1) where the entries of the constraint matrix B are random from a given distribution and analyze the performance of affine policies for a large class of distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proof: s = 0 implies [c, c + s 1] = ;, and therefore xp,c exactly consists of all updates until clock c 1.
Sentence 2: ⇤ Our key tool for convergence analysis is to define a reference sequence of states x t , informally referred to as the "true" sequence (this is different and unrelated to the SSPtable server's view): , where u t := u t mod P ,bt/P c . we sum updates by first looping over workers (t mod P ), then over clocks bt/P c. We can now bound the difference between the "true" sequence x t and the noisy views xp,c : for certain constants F, L. Then, under suitable conditions (f t are L-Lipschitz and the distance between two points )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we assume that p j ≤ c/m for a universal (numerical) constant c, then Theorem 5 guarantees that which is the convergence rate of ADAGRAD except for a small factor of min{ √ log T /T, p j } in addition to the usual p j /T rate.
Sentence 2: optimizing by choosing η = R ∞ , and assuming p j 1 T log T , we have convergence guarantee which is minimax optimal by Proposition 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It follows that Q satisfies the differential privacy constraints and the resulting f -divergence is also preserved.
Sentence 2: using the fact that and thus the corresponding f -divergence is invariant: We can naturally define equivalence classes for staircase mechanisms that are equivalent up to a permutation of columns and merging/splitting of columns with the same pattern: [Q] = {Q ∈ S ε | exists a sequence of permutations and merge/split of columns from Q to Q} .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let b M be constants such that where m k is the multinomial coefficient and a(k, m, M m . Thus, φ M is an M -degree polynomial approximation to φ logit (yx • θ) with the d+M d monomials of degree at most M serving as sufficient statistics derived from yx.
Sentence 2: we have a exponential family model with where k is taken over all k ∈ N d such that j k j ≤ M . We next discuss the calculation of the b and the choice of M . Choosing the polynomial approximation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we learn to synthesize realistic 2D images given projected 2.5D sketches that encode both the viewpoint and the object shape.
Sentence 2: we learn a texture network G texture that takes a randomly sampled texture code z texture and the projected 2.5D sketches v 2.5D as input, and produces a 2D image x = G texture (v 2.5D , z texture ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The counts n ¬ij jk• , n ¬ij •kxij , n ¬ij •k• act as fields through which z ij interacts with other variables.
Sentence 2: averaging both sides of (8) by p(z ¬ij |x, α, β) gives us the Callen equations, a set of equations that the true posterior must satisfy: Since the latent variables are already weakly dependent on each other, it is possible to replace (9) by a set of mean field equations where latent variables are assumed independent and still expect these equations to be accurate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (1993), we can prove the following basic result: then no RBM network of size m, whose weights are bounded in magnitude by C, can compute a function which represents ndimensional IP with margin δ.
Sentence 2: for C and 1/δ bounded by polynomials in n, for n sufficiently large, this condition is satisfied whenever m < 2 (1/9− )n for some > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then the algorithm move to the node with the third smallest label and so on.
Sentence 2: the algorithm is summarized in Algorithm 1 in Appendix D. Algorithm 1 returns a list r * (s) per node s ∈ V, which contains information about distance to the smallest reachable labels from s. each list contains pairs of distance and random labels, (d, r), and these pairs are ordered as where {•} (l) denotes the l-th element in the list.
Relationship: entailment
Domain: NeurIPS
Sentence 1: L 3 is unique in being universally rooted.
Sentence 2: for any integer r > 1 other than r = 3, we constructively demonstrate a hypergraph G(V, E) with |V | = r + 1 variables for which Theorem 21 examines L 0 r+1 (∇G) and L i r+1 (∇G), which by Theorem 17 are the uprooted equivalents of L r (G) and L r (G i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here, we show that privileged information can naturally be treated as noise in the latent function of a Gaussian process classifier (GPC).
Sentence 2: in contrast to the standard GPC setting, the latent function is not just a nuisance but a feature: it becomes a natural measure of confidence about the training data by modulating the slope of the GPC probit likelihood function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At the same time this is likely to come with a large number of useless features, which makes it suitable for sparsification.
Sentence 2: we include the player hand value, indicator functions for each possible player hand value and dealer hand value, and a large description of the cards not dealt yet (corresponding to the history of the game), under the form of indicator functions for various ranges.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the challenge of ALM is that it produces a series of bound-constrained quadratic problems that, in the traditional sense, are harder to solve than linear system produced by IPM or Simplex methods [17].
Sentence 2: in a Projected-CG approach [18], one needs to solve several linear systems via CG to find solution to the bound-constrained quadratic program, while there is no guarantee on how many iterations it requires.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, sufficient conditions for the convergence and correctness of Gaussian belief propagation (GaBP) are known for these models.
Sentence 2: gaBP converges to the optimal solution if the positive definite matrix A is walk-summable, scaled diagonally dominant, or log-concave decomposable [22; 7; 8; 9].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Short-term coding: r(θ) ∼ I (f (θ), Σ(θ)), where r i ∈ {0, 1} and I(µ, Σ) is the maximum entropy distribution consistent with the constraints provided by µ and Σ, the Ising model [16].
Sentence 2: for short-term population coding, we assume the population acitivity to be binary with each neuron either emitting one spike or none.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to evaluate the generality of our framework, we simulate data from three different instances of our model: those given by Gaussian, Bernoulli (Ising), and Poisson node-conditional distributions.
Sentence 2: we assume the true conditional distribution, P (Y |X), follows (7) with the parameters: ✓ we permit both the mean, ✓ s (X) and the covariance or edge-weights, ✓ st (X), to depend on the covariates.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For stochastic optimization problems, another important evaluation criterion is the confidence level of the objective value.
Sentence 2: it is of great interest to find (N, δ) as a monotonically decreasing function in both N and δ ∈ (0, 1) such that the solution x N +1 satisfies Pr (φ(x N +1 ) − φ(x * ) ≥ (N, δ)) ≤ δ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 's work [2], this notion of structure is distilled further by the use of "atoms."
Sentence 2: given a set A of very "simple" objects, called atoms, we can say that a vector x is simple (with some low-dimensional structure) if it can be written as a linear combination of few atoms: x = LZ~I C,8;, where k is small relative to the ambient dimensionality.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To show the interpretability of LDR, we visualize representative individuals, each of which suffers from an inferred sub-risk.
Sentence 2: for each inferred sub-risk k under risk j, we find the representative by evaluating a weighted average of all uncensored observations as i w ijk x i / i w ijk , where , λ ijk ∼ Gamma(r jk , e x i βjk ), and rjk and βjk are the estimated values of r jk and β jk , respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , k t < k are the episodes where (x, a) was taken at step h. Proof of Lemma 4.3.
Sentence 2: for each fixed (x, a, h) ∈ S × A × [H], let us denote k 0 = 0, and denote k i is the episode of which (x, a) was taken at step h for the ith time (or k i = K + 1 if it is taken for fewer than i times).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We first review a well-known method for active learning in the realizable case, refered to as CAL after its discoverers Cohn, Atlas, and Ladner [8].
Sentence 2: some n of this size suffices to guarantee that, with probability 1 − δ, the returned classifier ĥ has er( ĥ) ≤ ǫ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let (h, g) be a selective classifier selected after observing the training set S m . We say that (h, g) is a weakly optimal selective classifier if, for any 0 < δ < 1, with probability of at least 1 − δ over random choices of S m , R(h, g) ≤ R(h * , g).
Sentence 2: with high probability our classifier is at least as good as the true risk minimizer over its region of activity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the field of combinatorial optimization, it is well-known that submodular maximization is NP-hard and the approximation factor of (1 − 1/e) (≈ 0.63) achieved by the greedy algorithm [19] is the theoretical limit of a polynomial-time algorithm for positive and nondecreasing submodular functions [3].
Sentence 2: in the worst case, any polynomial-time algorithm cannot give a solution whose function value is more than (1 − 1/e) times larger than the optimal value unless P=NP.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An integrator is said to be a Kth-order local integrator if for any smooth and bounded function f , the corresponding Kolmogorov operator P h satisfies the following relation: The second approximation error is manifested when handling large data.
Sentence 2: the SGLD and SGHMC use stochastic gradients in the 1st and 2nd-order LDs, respectively, by replacing in F and L the full negative log-posterior U with a scaled log-posterior, Ũl , from the l-th minibatch.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, V (α; w, λ, I, J) ≤ U (α; w, λ, I, J) and V (0; w, λ, I, J) = U (0; w, λ, I, J).
Sentence 2: u is an upper bound on V and the bound is exact at α = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There are several advantages by using our new definition: (1) The new definition leads to a unified analysis for both VGD and NAG.
Sentence 2: if we follow the same notion as [16], we need to redefine t = kη for VGD, which is different from t = k √ η for NAG; (2) The new definition is more flexible, and leads to a unified analysis for both gradient-type (VGD and NAG) and coordinate-gradient-type algorithms (RCGD and ARCG), regardless of their different step sizes, e.g η = 1/L for VGD and NAG, and η = 1/L max for RCGD and ARCG; (3) The new definition is equivalent to [16] only when h = √ η.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The RMSProp optimization algorithm has a significant advantage for backpropagation-based planning because of its ability to perform gradient normalization that avoids exploding gradients and additionally deals with piecewise gradients [Balduzzi et al., 2016] that arise naturally as conditional transitions in many nonlinear domains (e.g., the Navigation domain of Figure 1 has different piecewise transition dynamics depending on the starting region).
Sentence 2: instead of naively updating action a itj through equation 1, RMSProp maintains a decaying root mean squared gradient value G for each variable, which averages over squared gradients of previous epochs and updates each action variable through Here, the gradient is relatively small and consistent over iterations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Second, though standard RL only updates the chosen option's value and DBM updates the posterior probability of all options, assuming maximization decision rule (Blakely, Starin, & Poling, 1988), both models will lead to qualitatively similar choice preference.
Sentence 2: the mostly rewarded choice will have an increasing value in RL model, and increasing reward probability in DBM, while the often-unrewarded choice will lead to a decreasing value in RL model, and decreasing reward probability in DBM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To study the consequences of Theorem 3, observe that when the covariance matrix Σ is generated randomly, the amount of correlation is much larger than in the idealized case that Σ is diagonal.
Sentence 2: for a fixed value of tr(Σ), the quantity tr(D −1 σ )/ |||R||| F , is much smaller in in the presence of correlation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also note that the model we evaluated was chosen based on overall performance on a withheld validation set, using a single seed.
Sentence 2: we did not run multiple replicas with the best hyperparameter settings (as was done in other models, such as the Sparse DNC, which demonstrated performance fluctuations with a standard deviation of more than ±3 tasks passed for the best choice of hyperparameters).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a set of images {I j } K j=1 , we first extract their features {v j } K j=1 with a pretrained CNN, and then average {v j } K j=1 to obtain v. The image encoder converts v into a context vector h v ∈ R D which is used as the visual input for all subsequent modules.
Sentence 2: the image encoder is parameterized as a fully-connected layer, and the visual features are extracted from the last convolution layer of a DenseNet [15] or VGG-19 [29].
Relationship: entailment
Domain: NeurIPS
Sentence 1: With the opponent model τ , the similarity between opponent's different policies τ 1 and τ 2 can be measured using a KL divergence, denoted by D KL ( τ1 , τ2 ) ≈ E (s,a) log τ1(s,a) τ2(s,a) , whereby opponent's different policies with same belief can be further distinguished by utilizing the corresponding opponent model.
Sentence 2: during online interaction, line 5 in Algorithm 1 estimates the opponent's online policy τo , thus the posterior probability (unnormalized) of the opponent using policy τ can be defined as follows: p(τ Here, τi is the previously learned approximate policy of the known opponent's policy τ i stored in T and the KL divergence is measured using observed state-action pairs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This proves that σ (D) and σ (D) are independent when D is drawn from X(n, 1 / 2 ).
Sentence 2: this reasoning easily extends to s XORs in Q s and we have that Evaluating this using the union bound and pairwise independence shows p one,s (σ ) > c(α) 2 −s .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to elucidating possible computational mechanisms in the brain, implementing "usable computation" with the dynamics of a neural network holds a number of advantages over conventional digital computation, including massive parallelism and fault tolerance.
Sentence 2: by following such dynamical systems approach, we can exploit the rich behavior of physical devices such as transistors to directly emulate these dynamics, and obtain more dense and power efficient computation [12].
Relationship: entailment
Domain: NeurIPS
Sentence 1: To address the task of following natural language instructions, we rely on two models: an instructionfollower model of the kind considered in previous work, and a speaker model-a learned instruction generator that models how humans describe routes in navigation tasks.
Sentence 2: we base our follower model on the sequence-to-sequence model [1], computing a distribution P F (r | d) over routes r (state and action sequences) given route descriptions d. The follower encodes the sequence of words in the route description with an LSTM [23], and outputs route actions sequentially, using an attention mechanism [5] over the description.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, we rely on a larger outside dataset for training, as did by all recent highperformance face verification algorithms [4,2,21,20,13].
Sentence 2: we use the CelebFaces+ dataset [20] for training, which contains 202, 599 face images of 10, 177 identities (celebrities) collected from the Internet.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, posterior inference via the Bayes' theorem is equivalent to solving the following optimisation problem [15]: where P is the space of probability distribution 1 , q(Θ) is the required posterior (here and afterwards we will drop the dependency on X for notation simplicity).
Sentence 2: the Bayesian posterior p(Θ|X) is identical to the optimal solution to (1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Any local maximizer x * must satisfy the Karush-Kuhn-Tucker (KKT) condition [14], i.e., the firstorder necessary conditions for local optimality.
Sentence 2: since x * i , µ i and β i are all nonnegative for all i's, Hence, the KKT conditions can be rewritten as: According to x, the vertices set V can be divided into three disjoint subsets, The Equation ( 9) characterizes the properties of the solution of ( 5), which are further summarized in the following theorem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, choosing π(d|x) to be the same distribution as the log-likelihood for our statistical model, the s samples from the extended state space D may be thought of as pseudo-data, from which we can deterministically calculate an estimate of the Expected Fisher Information to use as the covariance of a proposal distribution.
Sentence 2: each sampled pseudo-data can be used deterministically to give a sample of ∂L dθ given the current θ, all of which may then be used deterministically to obtain an approximation of the covariance of tangent vectors at the current point.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then for i ∈ {0, 1}, Recall the discussion of expectation, variance, pairwise independence, and 3-wise independence in Section 2.
Sentence 2: when a number of random variables are 3-wise independent, the conditional variance of their sum (conditioned on one of these variables) equals the sum of their individual conditional variances.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any r ≥ 1 and series of vertices v 0 , ..., v m , let W r ((v 0 , ..., v m )) be 1 if v 0 , ..., v m is an r-nonbacktracking walk and 0 otherwise.
Sentence 2: also, for any r ≥ 1, series of vertices v 0 , ..., v m and c 0 , ..., c m ∈ R m+1 , let W (c0,...,cm)[r] ((v 0 , ..., v m )) = (i0,...,i m )∈(0,...,m) W (c0,...,cm)[r] ((v 0 , ..., v m )) is the sum over all subsequences of (v 0 , ..., v m ) that form r-nonbacktracking walks of the products of the negatives of the c i /n corresponding to the elements of (v 0 , ..., v m ) that are not in the walks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Previous work has shown that combining multiple heuristics for NP-hard problems into a portfolio can dramatically reduce average-case running time [8,9,19].
Sentence 2: algorithms based on chronological backtracking often exhibit heavy-tailed run length distributions, and periodically restarting them with a fresh random seed can reduce the mean running time by orders of magnitude [8].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then have to solve the prediction problem, i.e.,  We now notice that if the scalar c j = c(j), where c is a non-increasing function of rank position j, then ( 17) can be solved simply by sorting the values of h i , ✓i in decreasing order.
Sentence 2: 1 the matching problem becomes one of ranking the values h i , ✓i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: when the prediction task is regression or multi-class classification) this limits the benefit landscape that can be expressed, but in the important special case of binary classification, the following Proposition establishes that this restriction is without loss of generality 4 .
Sentence 2: we can attach an arbitrary combination of benefit values to the four possible (y, ŷ)-pairs (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More recent work employs the method-of-moments, which permit the means of the mixture components to be arbitrarily close, provided that the sample size is sufficiently large (Kalai et al., 2010;Belkin and Sinha, 2010;Hsu and Kakade, 2013;Hardt and Price, 2015).
Sentence 2: hardt and Price (2015) characterize the information-theoretic limits of parameter estimation for mixtures of two Gaussians, and that they are achieved by a variant of the original method-of-moments of Pearson (1894).
Relationship: entailment