sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We also discuss comparisons to [9] in Appendix C. We can observe that all methods that use matrix factorization clearly outperform their counterparts that do not, even though the noise mechanism does not obey common noise assumptions in matrix factorization literature.
Sentence 2: the Mahalanobis matching (Match) benefits the most from matrix factorization that simultaneously alleviates the measurement noise and reduces the dimension.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we consider the high dimensional case, and show that if the loss function is a convex generalized linear function, then an -LDP algorithm is achievable with its risk bound depending only on n and the Gaussian Width of C, which is much smaller than the one in [19].
Sentence 2: if C is an 1 norm ball or a distribution simplex, the risk bound depends only on n and log p, instead of p. techniques to other problems.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We develop a new online algorithm, the regularized dual averaging (RDA) method, that can explicitly exploit the regularization structure in an online setting.
Sentence 2: at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient.
Relationship: entailment
Domain: NeurIPS
Sentence 1: geometry and generate new data points from the manifold.
Sentence 2: we focus on the case where the points in X are unevenly sampled from M, and aim to generate a set of M new data points Y = {y 1 , ..., y M } ⊆ R D such that 1. the new points Y approximately lie on the manifold M, and 2. the distribution of points in the combined dataset Z X ∪ Y is uniform.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we encode x as a binary indicator vector z(x) over the bins, z(x) z(y) = 1 if x and y fall in the same bin and zero otherwise, so Now consider shift-invariant kernels that can be written as convex combinations of hat kernels on a compact subset of R × R: k(x, y) = ∞ 0 k hat (x, y; δ)p(δ) dδ.
Sentence 2: if the pitch δ of the grid is sampled from p, z again gives a random map for if the pitch δ of the grid is sampled from p, and the shift u is drawn uniformly from [0, δ] the probability that x and y are binned together is k(x, y).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, our goal is to provide a quantitative description of the content of a subject's visual memory for the display after the delay interval.
Sentence 2: we want to characterize a subject's belief state about s. We suggest that a subject's belief state can be expressed as a random variable ŝ = [ŝ 1 , ŝ2 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, using the conditional autoregression (CAR) representation of Gaussian Markov random fields [16], we can write where −i ≡ {1 . . . p}\i and C = [c i j ] has the same non-zero pattern as the adjacency matrix of the graph A, and amounts to a proper probability distribution for X.
Sentence 2: by Brook's Lemma [16]  Taking limit as ζ → 0, it follows that L and L + correspond to Ω and Σ, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: frequentist coverage) of adaptive credible sets in nonparametric regression may extend to our setting [16].
Sentence 2: [ Theorem 1,16] shows that -under some extrapolation conditions-adapting a GP prior via risk-based empirical Bayes guarantees honest credible sets: investigating the validity of these results in our setting is an interesting topic for future research.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, a disadvantage of the method is that the randomness introduces variance -this is caused by the fact that g t (w (t−1) , ξ t ) equals the gradient ∇P (w (t−1) ) in expectation, but each g t (w (t−1) , ξ t ) is different.
Sentence 2: if g t (w (t−1) , ξ t ) is large, then we have a relatively large variance which slows down the convergence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It does not provide for a 1/T 2 accelerated optimization rate, but even worse, compared to non-accelerated SGD it suffers a slowdown quadratic in the delay, compared to the linear slowdown we would expect.
Sentence 2: the guarantee ( 14) only allows maximum delay of ⌧ = O(T 1/4 ) in order to attain the optimal statistical rate ⇥(L/ p T ), whereas the lower bound allows a delay up to ⌧ = O(T 3/4 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: V and W, which essentially involves solving for their maximum-a-posteriori (MAP) estimates, which are available in closed form.
Sentence 2: as shown in [20], estimating w k requires solving a linear system which, in our case, is of the form where S k = XΩ k X + Γ −1 , d k = Xκ k , Ω k and κ k are defined as in Section 3.1, except that the Pólya-Gamma random variables are replaced by their expectations given by Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose we run Algorithm 1 in the non-realizable case with inputs example oracle U , labelling oracle O, hypothesis class H, confidence-rated predictor P , target excess error � and target confidence δ.
Sentence 2: then with probability 1 − δ, for all k = 1, 2, . . . , k 0 + 1, and for all h ∈ V k , err D (h) ≤ err D (h * (D)) + � k . the ĥ returned at the end of the algorithm satisfies err D ( ĥ) ≤ err D (h * (D)) + �.
Relationship: entailment
Domain: NeurIPS
Sentence 1: • Conductance Testing in Graphs: MSTP-estimation is an essential primitive for conductance testing in large Markov chains [1].
Sentence 2: in regular undirected graphs, Kale et al [6] develop a sublinear bidirectional estimator based on counting collisions between walks in order to identify 'weak' nodes -those which belong to sets with small conductance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each sample k in the training set is a triplet containing (1) the RGB input patch x k , (2) the binary mask corresponding to the input patch m k (with m ij k ∈ {±1}, where (i, j) corresponds to a pixel location on the input patch) and (3) a label y k ∈ {±1} which specifies whether the patch contains an object.
Sentence 2: a patch x k is given label y k = 1 if it satisfies the following constraints: (i) the patch contains an object roughly centered in the input patch (ii) the object is fully contained in the patch and in a given scale range Otherwise, y k = −1, even if an object is partially present.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We construct an arm selection policy whose regret improves upon previous result for a large class of problems.
Sentence 2: our results imply that if X is the unit hypercube in a Euclidean space and the mean-payoff function has a finite number of global maxima around which the behavior of the function is locally Hölder with a known exponent, then the expected regret is bounded up to a logarithmic factor by √ n, i.e., the rate of the growth of the regret is independent of the dimension of the space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Gauss map takes a point θ on the curve and maps it to the unit velocity vector v(θ) = v(θ)/ v(θ) • v(θ).
Sentence 2: the natural metric on S N −1 induces a Gauss metric on the curve, given by g G (θ) = (∂ θ v(θ)) • (∂ θ v(θ)), which measures how quickly the unit tangent vector v(θ) changes as θ changes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These differences between the cases dim(H) = ∞ and dim(H) < ∞ are sharp and manifest themselves in the concrete formulas for the Log-Hilbert-Schmidt metric which we obtain in Sections 4.2 and 5.
Sentence 2: the formulas for the case dim(H) = ∞ are not obtainable from their corresponding finite-dimensional versions when dim(H) → ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that we can use algorithm A to violate property 5 above.
Sentence 2: letting p = µ/ µ 1 for µ a random measure taken from either D or D , we note that with high probability p 2 = Θ(n −1/2 ) and p 3 = Θ(n −2/3 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the sampling rate δ satisfies δ ≥ log 4 m m , the authors show that with probability 1 − exp(− log 4 m), This performance guarantee is derived from Lemma 1.1 and Theorem 1.4 in [AM07].
Sentence 2: to compare (2) and (3), note that our assumptions on the bounded entries of M ensures that: we see that the worst case bound for (3) is √ nm which is always lower than the worst case bound for (2): , our bound is only larger by a logarithmic term in m compared to [AM07].
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , d n can be chosen as −e l (l ∈ S) and e l (l ∈ V \ S).
Sentence 2: now we focus on the case where |S| = k. Define a neighbor S (i,j) of S as the neighbor S (i,j) is given by replacing one of the elements of S with that of V \ S. Note that I S (i,j) − I S = e j − e i for any neighbor S (i,j) of S. Let S (i * ,j * ) be a neighbor that maximizes f (S (i,j) ) among all neighbors of S. Since a subset S of size ) and S * ← S (i * ,j * ) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Random CSPs exhibit sharp phase transitions between satisfiable and unsatisfiable instances and are an important model to analyze the average hardness of CSPs, both in theory and practice [25,26].
Sentence 2: we consider two such CSPs: k-SAT where constraints are restricted to disjunctions involving exactly k (possibly negated) variables [3] and XORSAT which substitutes disjunctions in k-SAT for XOR constraints of fixed length.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Lemma 3.1 Given the log-transformed observation x t = log |z t | generated by Equations ( 6), if the assumptions A 1 ∼ A 2 hold, by solving simple linear equations involving the autocovariances of x t , the covariance Σ η and AB τ can be uniquely determined, where A = (I − A 0 ) −1 ; furthermore, A and B τ can be identified up to some rotation transformations.
Sentence 2: suppose that two models with parameters (A, Non-Gaussianity of the innovations ϵ t makes the model fully identifiable, as seen in the following theorem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Secondly, we find that Vapnik's key inequalities become equalities once "the misclassification error" is replaced by "the margin error" (defined as the rate of data instances incurring positive costs).
Sentence 2: we find that for margin errors the following three expectations: (i) of the empirical risk, (ii) of the the true risk and (iii) of the leave-one-out risk estimator are equal to each other.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main issue is to find that appropriate direction f ∈ T θ1 (D) such that a geodesic from θ 1 in that direction passes through θ 2 at time t = 1.
Sentence 2: the problem is to solve for an f ∈ T θ1 (D) such that Ψ(θ 1 , 0, f ) = θ 1 and Ψ(θ 1 , 1, f ) = θ 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is a cardinality potential that depends only on the number of variables y i assigned to a particular label.
Sentence 2: ignoring constant terms, the lower bound can be written as a summation of cardinality factors (one for each previous solution y ∈ S): d lb (y|S) = With this approximation, the greedy step means performing MAP inference in a factor graph augmented with cardinality potentials: argmax y r(y) + λd lb (y|S).
Relationship: entailment
Domain: NeurIPS
Sentence 1: when t xt ) t xt equals to the number of leaves of the decision tree ) S ud ~b y , the regularized risks are exactly equal to the "cost complexity" employed to prune decision trees by CART algorithm [4].
Sentence 2: the basis of the pruning algorithm in CART is the minimisation of the regularised risk in the class of subtrees of the maximal tree, with the regularisation constant G selected by a heuristic applied to a validation set.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , v Next, we conducted a series of experiments to confirm that MiME can indeed capture the relationship between Dx codes and treatment codes, thus producing robust performance in small datasets.
Sentence 2: we created three small datasets D 1 , D 2 , D 3 from the original data such that each dataset consisted of patients with varying degree of Dx-treatment interactions (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our known-L max algorithm is FTRL with regularizers ψ t (w) = ψ(w)/η t , where ψ(w) = ( w + 1) log( w + 1) − w and η t is a scale-factor that we adapt over time.
Sentence 2: we set η −1 Lt , beginning a new epoch.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in [20], it is possible to re-write this objective of optimizing an infinite horizon discounted reward MDP (where the reward happens at each step) as one of optimizing an infinite mixture of finite horizon MDPs (where the reward only happens at the last time step).
Sentence 2: we note that by introducing the trans-dimensional probability distribution on {k} × X k × A k × R + given by (2) we can easily rewrite V π µ (θ) as an infinite mixture model of finite horizon MDPs, with the reward happening at the last horizon step; namely at k. Specifically we have: for a randomized policy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For (predictive) log-likelihood, a positive value means that normal model fits better than P-L, whereas for AIC and BIC, a negative number means that normal model fits better than P-L. Predictive likelihood is different from likelihood in the sense that we compute the likelihood of the estimated parameters for a part of the data that is not used for parameter estimation.
Sentence 2: 3 we compute predictive likelihood for a randomly chosen subset of 100 votes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For simplicity, we assume that the deformations are restricted to an arbitrary rotation of the image, where the angle of rotation is not known beforehand.
Sentence 2: this angle (which takes a value from a finite discrete set) is modeled as the hidden variable h. We specify the joint feature vector as Φ(x, y, h) = (0 y(m+1) ; θ h (x) 1; 0 (9−y)(m+1) ), where θ h (x) is the vector representation  of the image x rotated by the angle corresponding to h. the joint feature vector is the rotated image of the digit which is padded in the front and back with the appropriate number of zeroes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that this procedure is inefficient in that it samples each particle φ k merely based on the first observation with label k. Therefore, we use this procedure for bootstrapping, and then run a Gibbs sampling scheme that iterates between parameter update and label update.
Sentence 2: (Parameter update): We resample each particle ψ k from its source distribution conditioned on all samples with label k. for k ∈ [1, m] with r k > 0, we draw ψ k ∼ T (φ k , •)|{x i : l i = k}, and for k ∈ [m + 1, m], we draw ψ k ∼ p|{x i : l i = k}, where p = p qµ or p ν , depending which source ψ k was initially sampled from.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, to maximize L with respect to parameters Θ of the encoding distribution p (r|x, Θ), we repeat the following steps until convergence: • For fixed Θ, find {q new (y|r) , q new (r)} = arg max {q(y|r),q(r)}∈{Q y|r ,Qr} L • For fixed q (y|r) and q (r), find Θ = arg max Θ L. We note that using a simple approximation for the decoding distribution, q(y|r), can carry additional benefits, besides rendering the IB algorithm tractable.
Sentence 2: while an advantage of mutual information is its generality, in certain cases this can also be a drawback.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An instance of partial monitoring is defined by the loss matrix L∈R N×M and the feedback table H ∈Σ N×M , where N and M are the cardinality of the action set and the outcome set, respectively, while Σ is some alphabet of symbols.
Sentence 2: if learner chooses action i while the outcome is j, the loss suffered by the learner is L[i,j], and the feedback received is For an action 1 ≤ i ≤ N, let i denote the column vector given by the i th row of L. Let ∆ M denote the M-dimensional probability simplex.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper proposes a new algorithm for the following task: given a sparse undirected unweighted graph, partition the nodes into disjoint clusters so that the density of edges within clusters is higher than the edges across clusters.
Sentence 2: we are interested in settings where even within clusters the edge density is low, and the density across clusters is an additive (or small multiplicative) constant lower.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The eigenvalues of subgraphs with boundary conditions are studied in [13].
Sentence 2: let S be any (connected) subnetwork of G , and denote by δ S the boundary of S in G . The Neumann boundary condition states that for every x ∈ δ S, ∑ y:{x,y}∈δ S ( f (x) − f (y)) = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let Σ d be the diagonal matrix formed from the top d singular values, and let U d and V d be the matrices produced by selecting the corresponding columns from U and V . The matrix is the matrix of rank d that best approximates the original matrix M , in the sense that it minimizes the approximation errors.
Sentence 2: when using SVD, the dot-products between the rows of W = U d • Σ d are equal to the dot-products between rows of M d . In the context of word-context matrices, the dense, d dimensional rows of W are perfect substitutes for the very high-dimensional rows of M d . Indeed another common approach in the NLP literature is factorizing the PPMI matrix M PPMI with SVD, and then taking the rows of W SVD = U d • Σ d and C SVD = V d as word and context representations, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We initialized the network to a random but self-predicting Here for computational efficiency we used a simplified network dynamics where the compartmental potentials are updated only in two steps before applying synaptic changes.
Sentence 2: for each presented MNIST image, both pyramidal and interneurons are first initialized to their bottomup prediction state (3), u k = v B,k , starting from layer 1 up to the top layer N . Output layer neurons are then nudged towards their desired target u trgt N , yielding updated somatic potentials To obtain the remaining final compartmental potentials, the network is visited in reverse order, proceeding from layer k = N 1 down to k = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 5, one could attempt to estimate the distribution parameters Γ from a set of sampled particles, then apply the LR gradient using the estimated distribution q(x; Γ).
Sentence 2: we will approximate the density as a Gaussian by estimating the mean μ = P i x i /P and variance Σ = P i (x i − μ) 2 /(P − 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can use this probability to estimate the number of highly confident connections.
Sentence 2: we were able to find 50% more connections in the perturbed dataset than the spontaneous dataset, with a greater than 0.95 probability (1940 vs 1204); twice times as many highly confident connections with probability 0.99 or higher (1107 vs 535); and five times as many with the probability 0.999 or higher (527 vs 101).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The mean aggregator is nearly equivalent to the convolutional propagation rule used in the transductive GCN framework [17].
Sentence 2: we can derive an inductive variant of the GCN approach by replacing lines 4 and 5 in Algorithm 1 with the following: 4 We call this modified mean-based aggregator convolutional since it is a rough, linear approximation of a localized spectral convolution [17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note a recent independent work of Bailey and Piliouras [2], who show the same result in a more general setting: first, they generalize to discrete Follow-The-Regularized-Leader (FTRL) dynamics, and second, they allow the step-sizes to be mildly decreasing.
Sentence 2: they show that if a FTRL dynamic guarantees every update must stay in the interior of the state space (MW is an example of such FTRL dynamic), then the dynamic converges toward the boundary; otherwise, they show that the FTRL dynamic gets arbitrarily close to the boundary infinitely often (this is weaker than "convergence toward the boundary").
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 3) is attained by decomposing the log-likelihood into two terms, log p(y, , which can be easily verified by substituting the product, p(y, v, h v |h id ) = p(y, v |h id )p(h v |y, v), into the right hand side of the decomposition.
Sentence 2: the first term is the KL-divergence [10] between the true posterior and the distribution q(h v ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To this end, we introduce a new layer that encodes the notion of self-expressiveness.
Sentence 2: let Θ denote the auto-encoder parameters, which can be decomposed into encoder parameters Θ e and decoder parameters Θ d . Furthermore, let Z Θe denote the output of the encoder, i.e., the latent representation of the data matrix X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is a forced interpretation, but it is useful in understanding the implicit prior over counts that is imposed when using LR.
Sentence 2: if p(k) is defined as be a particular function of θ (known as a Poisson-Binomial distribution [2]): , where Z(θ) = k Z k (θ), then the joint probability p(y, k; θ) becomes equivalent to a LR model in the following sense.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we present our DCRF that incorporates plane and edge predictions for depth and surface normal regularization.
Sentence 2: the field of variables we optimize are depths, D = {d i } K i=1 , where K is number of the pixels, and normals, N = {n i } K i=1 , where n i = [n ix , n iy , n iz ] T indicates the 3D normal direction.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To achieve this goal, one part of the model needs to discriminate the generated samples from the training data, and another part should mislead the former part, analogous to the generator and discriminator in GANs.
Sentence 2: we select the approximate inference model (or encoder) of VAEs as the discriminator of GANs and the generator model of VAEs as the generator of GANs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Passing messages through this function requires summing over all edge configurations that correspond to each possible degree, d i , and summing over d i .
Sentence 2: the message, µ Ii→e ik (e ik ), sent from function I(d i , j e ij ) to edge variable e ik is given by where µ eij →Ii (e ij ) is the message sent from e ij to function I(d i , j e ij ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is defined, for every matrix A, as where ψ is a gauge function, namely a function which is symmetric and invariant under permutations.
Sentence 2: if ψ is the ℓ 1 norm then problem (7) corresponds to tensor trace norm regularization, whereas if ψ = ω * * α it implements the proposed regularizer.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Further, we identify a few simple statistics about generated samples to act as proxies for sample quality and diversity.
Sentence 2: we report the number of unique n-grams as a proxy for the latter and the fraction of "valid" n-grams as a proxy for former (see Supplementary   Table 8: Samples from our GAN trained to conditionally transform a given hypothesis and a label as either (E)ntailment, (C)ontradiction or (N)eutral into a premise that satisfies the specified relationship Let P be the data distribution and Q be our model's distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An easy application of L'Hôpital's rule shows that as (N log N )/T , we see that the right-hand side of (5) tends to 2 p T N log N . However the choice ↵ !
Sentence 2: 1 is clearly not the optimal choice, as we show in the following statement, which directly follows from the theorem once we see that the choice of ↵ = 1 2 gives a regret of no more than 4 p NT .
Relationship: entailment