sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In fact, our distribution measures framework encompasses optimization objectives for the expected, robust, and percentile MDP problems as well as for VaR and CVaR.
Sentence 2: if µ is the Lebesgue measure (i.e., a uniform density over the unit interval), all percentiles are equally weighted and the µ-robust policy will optimize the expected cumulative reward over the distribution P(M).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Based on the semantic concepts such as right-next-to, holding, etc., it generates the positions for arranging latent distributions of two children nodes.
Sentence 2: we first sample a tuple l = (x, y) from p(l|z s ), indicating the offsets between two children's bounding boxes in latent space, where z s ∼ p(z s ) and p(z s ) is generated by our concept mapping operator τ concept taking the relation word of layout node.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For c ≥ 0, denote 2 by Möbius addition.
Sentence 2: the Möbius addition of x and y in D n c is defined as when c = 0, one recovers the Euclidean addition of two vectors in R n . Note that without loss of generality, the case c > 0 can be reduced to c = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will run no-regret dynamics on this zero-sum game, where at every iteration t = 1, . . . , T , the adversary will pick a distribution over functions and subsequently the learner picks a solution x t . For simpler notation we will denote with w t the probability density function on [m] associated with the distribution of the adversary.
Sentence 2: w t [i] is the probability of picking function L i 2 L. The adversary picks a distribution w t based on some arbitrary no-regret learning algorithm on the m actions in L. For concreteness consider the case where the adversary picks a distribution based on the multiplicative weight updates algorithm, i.e., Subsequently the learner picks a solution x t that is the output of the ↵-approximate stochastic oracle on the distribution selected by the adversary at time-step t. Write ✏(T ) = q 2 log(m) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, Krauthgamer, Nadler and Vilenchik [45] and Berthet and Rigollet [41] have shown that for semi-definite programs (SDP) this bound is tight.
Sentence 2: the natural SDP cannot even solve the detection problem: to distinguish the data from covariance Σ = λvv T + I from the null hypothesis in which no sparse vector is planted, namely the n samples are drawn from the Gaussian distribution with covariance matrix I.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although we cannot enforce nonadaptivity, and it is sometimes unreasonable to expect that analysts are entirely non-adaptive, we intend the mechanism to be used for validation.
Sentence 2: analysts should do their discovery, training, tuning, development, and adaptive data analysis on unrestricted "training" or "discovery" datasets, and only use the protected database when they wish to receive a stamp of approval on their model, predictor, or discovery.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our first goal will be to learn which parameters (α, β) are most consistent with our observations-our unlabeled training set-using maximum likelihood estimation.
Sentence 2: to do this for a particular training set S ⊂ X, we will solve the problem we are maximizing the probability that the observed labels produced on our training examples occur under the generative model in (1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Besides being used as a standalone analysis tool, this approach can inform model-based fMRI analyses to investigate whether the model correctly tracks the brain's internal mechanism.
Sentence 2: if a brain region is found to be important in the current analysis, but not using the model-based fMRI analysis, this could mean that the model used to extract neural information is not representing all of the relevant neural signals involved in decision-making and requires further modification.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In these policies, D is the partial realization that we have observed so far, and X D {x ∈ X | (x, y) ∈ D for some y ∈ Y} is the domain of D (i.e., the set of selected items in D).
Sentence 2: for any item x, we write δ(x | D) to denote the worst-case utility gain if x is selected after we observe D. In this definition, note that we have extended the utility function f to take a partial realization as the second parameter (instead of a full realization).
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is due to the nature of the code R . (iii) As discussed in Appendix E, Theorem 4 can also be used to establish an approximate rule of thumb for pricing.
Sentence 2: considering two query schemes k 1 IC and k 2 IC, the query price π is to be set as π(k1) π(k2) ≈ k1 k2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we hunt for the stable members of the FGSD that are robust against the perturbation or noise in the datasets.
Sentence 2: we will look at the stability of R and F based on FGSD from f (λ) perspective by first analyzing its influence on a single edge perturbation (or in other words analyzing rank one modification of Laplacian matrix).
Relationship: entailment
Domain: NeurIPS
Sentence 1: All the results for uniform γ and weights α t = 1 hold in this case as well.
Sentence 2: by the arguments leading to (5), if we want to minimize m i=1 k l=2 e {ft(i,l)−ft(i,1)} , where f t is as defined in (1), then the following strategy is optimal: in round t output the cost matrix This will ensure that the expression m i=1 k l=2 e {ft(i,l)−ft(i,1)} changes by a factor of at most κ(γ t , α t ) in each round.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These assumptions hold for many target applications, including the ones explored in Section 5.
Sentence 2: we note that assumption (A0) can be often satisfied with a careful choice of C. It has been studied that (projected) SGD has a sublinear convergence rate for a smooth strongly-convex objective if the variance of gradient estimates is uniformly bounded [23,21].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consequently, the pdf in (3) can be treated as a probability mass function (pmf) over s ∈ κ(σ (i) (1 : t − 1)) and sampling from it becomes equivalent to sampling Hence, we can interpret a particle x (i) σ(1:t−1) as a sequence of indices of selected graph vertices σ(1:t−1) is a vector of ones assigned to RVs with indices in σ (i) (1 : t − 1).
Sentence 2: it holds ind(x , where ind : {0, 1} n → 2 S is a function that assigns to x a set of indices of coordinates of x that are equal to one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To compute subsequent semi-supervised eigenvectors, i.e., at steps t = 2, . . . , k if one ultimately wants a total of k semi-supervised eigenvectors, then one lets Q be the n × (t − 1) matrix with first column equal to 1 and with j th column, for i = 2, . . . , t − 1, equal to x * j−1 (where we emphasize that x * j−1 is a vector not an element of a vector).
Sentence 2: , where x * i are successive semi-supervised eigenvectors, and the projection matrix due to the degree-weighted inner norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The improvement of AdaNCG + over [4]'s algorithm is brought by reducing the number of Hessian-vector products for performing each iteration of AdaNCG.
Sentence 2: the number of Hessian-vector products of each NCD step in [4] is O(1/ √ 2 ), which becomes O(1/ max( 2 , ∇f (x j ) 2/3 )) for each AdaNCD step in AdaNCG + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Notice that this expression does not depend on q T -in fact, this cost precisely corresponds to the scenario where the firm had not set up a CLM and instead used the initial estimate q 0 to encode.
Sentence 2: for α = 1, the firm is entirely neutral to the quality of the estimate q T ; even if the CLM provided an estimate q T which performed worse than q 0 , the cost increase due to the bad choice of q is recouped from payments of the ill-informed participants.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we extend this result and show how the Martingale type of a pair (W , X ) are related to existence of certain uniformly convex functions.
Sentence 2: the following theorem shows that the notion of Martingale type of pair (W , X ) is equivalent to the existence of a non-negative function that is uniformly convex w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can then obtain estimated values of the parameter priors under the null hypothesis by matching the moments of each Gamma distribution to their historical values.
Sentence 2: we set the expectation and variance of the Gamma distribution Ga(α all , β all ) to the sample expectation and variance of C all B all observed in past data: , where C out (S) and B out (S) are respectively the total count ∑ G−S c i and total baseline ∑ G−S b i outside the region.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the data were generated by LIM, AIC or BIC fared much more poorly whereas MDL recovered the correct model (LIM) across both sample sizes.
Sentence 2: under AIC or BIC, FLMP was selected over LIM half of the time for N = 20 (51 % vs. 49%), though such errors were reduced for N = 150 (17 % vs 83 %).
Relationship: entailment
Domain: NeurIPS
Sentence 1: V is a set {V 1 , V 2 , ..., V n } of observable variables (also called endogenous), that are determined by variables in the model (i.e., determined by variables in U ∪ V ), 3.
Sentence 2: f is a set of functions {f 1 , f 2 , ..., f n } such that each f i is a mapping from the respective domains of U i ∪ P A i to V i , where U i ⊆ U and P A i ⊆ V \ V i and the entire set F forms a mapping from U to V . each f i in v i ← f i (pa i , u i ), i = 1, ..., n, assigns a value to V i that depends on the values of the select set of variables (U i ∪ P A i ), and 4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose to use the Inexact Proximal point method for Optimal Transport (IPOT) algorithm to compute the optimal transport matrix T * , which provides a solution to the original optimal transport problem (3) [59].
Sentence 2: iPOT iteratively solves the following optimization problem: where Algorithm 1 IPOT algorithm [59] 1: end for 9: T (t+1) = diag(δ)Qdiag(σ) 10: end for Here the Bregman divergence D h serves as a proximity metric and β is the proximity penalty.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ALPACA offers a simple and natural formulation of the patient staging problem as a probabilistic inference query.
Sentence 2: given the measurements corresponding to a particular patient, we perform patient staging by: (1) computing a posterior distribution over the event ordering σ (j) , then (2) computing a posterior distribution over the most recent element of the snapshot set τ (j) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We construct a "local-geometric-context" vector for each part, and obtain landmark mixture labels by grouping landmark instances with similar local geometry.
Sentence 2: for each landmark i and image n, we construct a K-element vector g in that defines the 2D relative location of a landmark with respect to the other K landmarks in instance n, normalized for the size of that training instance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, the linear conjugate gradient (CG) method works as the inner iterative algorithm for solving the linearized Gauss-Newton normal equation, whereas the outer nonlinear algorithm repeatedly takes so-called "Krylov-dogleg" steps, relying only on matrix-vector multiplication without explicitly forming the Jacobian matrix or the Gauss-Newton model Hessian.
Sentence 2: our iterative dogleg algorithm can reduce both operational counts and memory space by a factor of O(n) (the number of parameters) in comparison with a direct linear-equation solver.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now provide a formal statement of the problem studied in this paper in a very general setting, and then define notation required for our specific results.
Sentence 2: formal Problem Statement: Let P ∈ F G be an unknown probability measure on a sample space X , from which we observe n IID samples X 1:n = X 1 , ..., X n IID ∼ P . In this paper, we are interested in using the samples X 1:n to estimate the measure P , with error measured using the adversarial loss d F D . for various choices of spaces F D and F G , we seek to bound the minimax rate of estimating distributions assumed to lie in a class F G , where the infimum is taken over all estimators P (i.e., all (potentially randomized) functions P : X n → F G ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Intuitively (1) and ( 2) have similar effects since large values of p(x i , y i |θ)p(x i |θ , σ i , x i ) dominate the summation in p(x i , y i |θ)p(x i |θ , σ i , x i )dx i .
Sentence 2: both methods prefer a parameter configuration such that the product p(x i , y i |θ)p(x i |θ , σ i , x i ) is large for some x i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We incorporate them into the edge feature function.
Sentence 2: we define the (and the only) edge feature function as where R i,j = r(x i , x j ) denotes the parent-child relation: r(x i , x j ) = 1 if document x i is the parent of x j , and r(x i , x j ) = 0 for other cases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let π be an arbitrary policy and let δ ∈ [0, 1].
Sentence 2: we define the oracle-infused policy π + δ as follows: when choosing an action, π + δ explores the policy space with probability 1 − δ (according to its current model), but with probability δ, we force it to take an oracle action.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The framework of gyrovector spaces provides an elegant non-associative algebraic formalism for hyperbolic geometry just as vector spaces provide the algebraic setting for Euclidean geometry [2,25,26].
Sentence 2: these operations are used in special relativity, allowing to add speed vectors belonging to the Poincaré ball of radius c (the celerity, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by recently developed first-order methods for optimizing composite functions [6,7,8], the regularized dual averaging (RDA) method we develop exploits the full regularization structure at each online iteration.
Sentence 2: at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the whole regularization term, not just its subgradients.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose a generic architecture, called BAI-MCTS, that builds on a Best Arm Identification (BAI) algorithm and on confidence intervals on the node values in order to solve the best action identification problem in a tree of arbitrary depth.
Sentence 2: we study two specific instances, UGapE-MCTS and LUCB-MCTS, that rely on confidence-based BAI algorithms [8,13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The L(w, S) term is related to the size of the set {y|m ρ (x i , y * i , y, w) ≤ M i }, which can be controlled by margin m ρ 2 (x i , y i , yi ).
Sentence 2: for underestimation, It implies that the larger m ρ 2 (x i , y i , yi ), the lower L(w, S).
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the condition number is roughly equal to the sample size (κ = Θ(n)), then Quartz does better than both ASDCA and SPDC as long as n/τ + ω ≤ n/ √ τ .
Sentence 2: this is the case when the data is sparse: ω ≤ n/ √ τ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we describe the embedding generation, or forward propagation algorithm (Algorithm 1), which assumes that the model has already been trained and that the parameters are fixed.
Sentence 2: we assume that we have learned the parameters of K aggregator functions (denoted AGGREGATE k , ∀k ∈ {1, ..., K}), which aggregate information from node neighbors, as well as a set of weight matrices W k , ∀k ∈ {1, ..., K}, which are used to propagate information between different layers of the model or "search depths".
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the first step, we keep D fixed and minimize over W . This consists in solving the problem This minimization can be carried out independently for each task since the regularizer decouples when D is fixed.
Sentence 2: introducing new variables for (F (D)) 1 2 w t yields a standard L 2 regularization problem for each task with the same kernel K(x, z) = x (F (D)) −1 z, x, z ∈ IR d . In other words, we simply learn the parameters w t -the columns of matrix W -independently by a regularization method, for example by an SVM or ridge regression method, for which there are well developed tool boxes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, in contrast to the Gaussian model, non-linear classifiers can achieve a significantly improved robustness.
Sentence 2: consider the following thresholding operation T : R d → R d which is defined element-wise as It is easy to see that for ε < 1, the thresholding operator undoes the action of any ∞ -bounded adversary, i.e., we have T (B ε ∞ (x)) = {x} for any x ∈ {±1} d . Hence we can combine the thresholding operator with the classifier learned from a single sample to get the following upper bound.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In each time period, the algorithm samples an action from the posterior distribution p t (a) = P t (A * = a) of the optimal action.
Sentence 2: thompson sampling uses a policy π t = p t . It is easy to see that this is equivalent to sampling a model index θt from the posterior distribution of models and then selecting an action A t = arg max a∈A R θt (a) that optimizes the sampled model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For each object v 2 V, we have a pool of predefined d features, each of which is a real number in [ 1,1].
Sentence 2: we can represent each v 2 V as a vector of features x 2 [ 1,1] d . We note that the mapping from v to x can be non-linear and that d can be very large.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Approximate Learning: Exact maximum likelihood learning in this model is intractable, but efficient approximate learning of DBMs can be carried out by using a mean-field inference to estimate data-dependent expectations, and an MCMC based stochastic approximation procedure to approximate the model's expected sufficient statistics [7].
Sentence 2: consider approximating the true posterior P (h|v; θ) with a fully factorized approximating distribution over the three sets of hidden units: k |v), where µ = {µ (1) , µ (2) , µ (3) } are the mean-field parameters with q(h i for l = 1, 2, 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose G = (V, E) is an undirected graph over p nodes corresponding to the p response variables.
Sentence 2: given the underlying graph G, and the set of cliques (fully-connected sub-graphs) C of the graph G, the corresponding conditional random field (CRF) is a set of distributions over the response conditioned on the covariates that satisfy Markov independence assumptions with respect to the graph G. letting { c (Y c , X)} c2C denote a set of clique-wise sufficient statistics, any strictly positive distribution of Y conditioned on X within the conditional random field family takes the form: With a pair-wise conditional random field distribution, the set of cliques consists of the set of nodes V and the set of edges E, so that A key model specification question is how to select the class of sufficient statistics, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The regularized GAN also manages to perfectly classify the unregularized GAN's samples as fake (cross-testing FP 0.0) whereas the unregularized GAN classifies the samples of the regularized GAN as real (cross-testing FP 1.0).
Sentence 2: the regularized model is able to fool the unregularized one, whereas the regularized variant cannot be fooled.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These are analogous to the standard radial basis function kernels exp[−(x − x ) 2 /(2σ 2 )], but we find that they have surprising properties on large graphs.
Sentence 2: while loops in large random graphs are long and can be neglected for many purposes, by approximating the graph structure as locally tree-like, here this leads to a non-trivial limiting form of the kernel for σ → ∞ that is not constant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the DRMM, the solution I ⇤ c (L) of the activity maximization for class c (L) can be derived as the sum of individual activity-maximizing patches I ⇤ Pi , each of which is a function of the learned DRMM parameters (see Appendix E).
Sentence 2: i ⇤ c (L) ⌘ This implies that I ⇤ c (L) contains multiple appearances of the same object but in various poses.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For algorithms minimizing a surrogate loss, the question of consistency reduces to the question of calibration of the surrogate loss with respect to the target loss of interest [5][6][7]16]; in general, one is interested in convex surrogates that can be minimized efficiently.
Sentence 2: the existence (and lack thereof) of convex calibrated surrogates for various subset ranking problems, with target losses including for example the discounted cumulative gain (DCG), mean average precision (MAP), mean reciprocal rank (MRR), and pairwise disagreement (PD), has received significant attention recently [9,[11][12][13][15][16][17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( with initialization w 0,j = 1, for all j = 1, . . . , K. Lemma 4.
Sentence 2: [7] For any 0 < ǫ ≤ 1 2 , the MWU algorithm provides the following regret bound for the online learning problem described above: for ǫ = log(d+1) T , we have R(T ) ≤ log(d + 1)T For the rest of the paper, we refer to the MWU algorithm with ǫ = log(d+1) T as the online learning (OL) algorithm, and the update in (5) as the OL update at time t + 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Generally, however, the commitment by P 1 creates new dependencies that render the problem inherently non-Markovian with respect to the state s t and thus harder to solve.
Sentence 2: even though the dynamics of the environment are Markovian with respect to the state s t , the MVDP only becomes Markov in the Stackelberg setting with respect to the hyper-state η t = (s t , π t:T,1 ) where π t:T,1 is the commitment by P 1 for steps t, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This method builds a representation which is a frequency count of all possible contiguous subsequences of length p. This produces a mapping with features φ and 0 otherwise.
Sentence 2: to define a feature space for outputs we count the number of contiguous subsequences of length p on the input that, if starting in position q, have the same element of the alphabet at position q + (p − 1)/2 in the output, for odd values of p. φ We can then learn pre-images using a window also of size p as described in Section 3.2, e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In Section 2.3, we will discuss the sparse case where the column T :,j,k may be entirely zero.
Sentence 2: when that case does not arise, entries of P can be interpreted as the transition probabilities of a second-order Markov chain (Z t ) t∈N : If the last two states were j and k, then the next state is i with probability P i,j,k .
Relationship: entailment