sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Each of these risks has an empirical counterpart where the expectation with respect to P is replaced by an average over an i.
Sentence 2: the following empirical version of R φ,σ will be used The main focus of this paper is the analysis of learning algorithms that minimize the empirical φ-risk regularized by the RKHS norm R φ,σ , and their limit as the number of points tends to infinity and the kernel width σ decreases to 0 at a suitable rate when n tends to ∞, λ being kept fixed.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , A N ⊂ V, the random measure H ∼ DP(G, α) has the property This implies for the present construction that renormalized restrictions of the random measure are also Dirichlet process distributed.
Sentence 2: if A ⊂ V and H A is the marginal renormalized random measure on A induced by the DP on V with concentration parameter α and base measure G, then H A ∼ DP(G A /G(A), αG(A)), where G A is the base measure restricted to A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our proposed algorithm integrates the proximal newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees.
Sentence 2: by leveraging a sophisticated characterization of sparse modeling structures (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) quadratic convergence, and eventually obtains a sparse approximate local optimum with optimal statistical properties after only a few convex relaxations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Second, the bound is polynomial in all constants of interest, such as the number of items L and the number of maximization steps K in each episode.
Sentence 2: we would like to stress that it is not linear in the number of contexts Y K at step K, which is exponential in K. Finally, note that our bound captures the shape of the optimized function f . because the function f is adaptive submodular, the upper bound on the gain of the policy π g from step k forward, G k , decreases as k increases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it can be calculated in a time that is no more than a forward-backward step on a single data point.
Sentence 2: in a mini-batch setting with batch size B, if the backpropagation on the mini-batch can be done in time BT , the running time of the Path-SGD on the mini-batch will be roughly (B + 1)T -a very moderate runtime increase with typical mini-batch sizes of hundreds or thousands of points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: They discovered that regularized estimation based on the Schatten 1-norm, which is a popular technique for recovering low-rank matrices via convex optimization, can also be applied to tensor decomposition.
Sentence 2: the  (7,8,9) tensor of dimensions 50 × 50 × 20 from partial measurements; see [23] for the details.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We introduce a new local flatness condition for the objective function (Assumption 1), under which Runge-Kutta discretization obtains convergence rates even faster than O(N −2 ), without requiring high-order integrators.
Sentence 2: we show that if the objective is locally flat around a minimum, by using only gradient information we can obtain a convergence rate of O(N −p ), where p quantifies the degree of local flatness.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, we make the following contributions: • In Section 4, we prove that consistency can be guaranteed even when the integrand f does not belong to the assumed RKHS.
Sentence 2: we derive a convergence rate of the form n −θb , where 0 < θ ≤ 1 is a constant characterizing the (relative) smoothness of the integrand.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We draw y ab repeatedly by choosing a community assignment (z a→b , z a←b ) for a pair of nodes (a, b), and drawing a binary value from a community distribution.
Sentence 2: the conditional probability of a link in MMSB is p(y ab = 1|z a→b,i , z a←b,j , β) = K i=1 K j=1 z a→b,i z a←b,j β ij , where β is the blockmodel matrix of community strength parameters to be estimated.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume now that we have some partial spectral information available.
Sentence 2: fix k ∈ {0, 1, . . . , n − 1} and assume we know eigenvectors u i and eigenvalues λ i for i = 1, . . . , k. We now define a parametric distribution D(α, β 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the sparsity ratio k/n and SNR are both fixed, m can satisfy (10) while growing only linearly with k. In contrast, Theorem 1 shows that with sparsity ratio and SNR • MAR fixed, m = Ω(k log(n−k)) is necessary for reliable sparsity recovery.
Sentence 2: the number of measurements must grow superlinearly in k in the linear sparsity regime with bounded SNR.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For an image volume V = V (x, y, z, c), we propose to calculate the topographical map T (disaffinity map) as where x, y, z denote the spatial coordinates, c denotes the color coordinate, and G x , G y , G z denote the spatial gradients of V (nearest neighbor differencing).
Sentence 2: any edge with significant deviation in any color channel will correspond to a "mountain" in the topographic map.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we show that well-known concepts from statistical learning theory can be directly applied to reason about learning from data an approximately revenue-maximizing auction.
Sentence 2: for a set C of auctions and an arbitrary unknown distribution F over valuations in [1, H] samples from F are enough to learn (up to a 1 ✏ factor) the best auction in C, where d C denotes the pseudo-dimension of the set C (defined in Section 2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, it turns out that if we apply the tapering schedule induced by the construction of our estimator to the population parameter Ω ∈ F α , we may express the tapered population Ω as a sum of block matrices in exactly the same way that our estimator is expressed as a sum of block matrices.
Sentence 2: the tapering schedule is presented next.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To address this point, we consider the two variants of Algorithm 1 from Section 3.2 and 3.3 with linear approximators.
Sentence 2: we consider the family of linearly parameterized value functions Q w (s, a) = w T φ(s, a) with bounded weights w 2 ≤ w max and uniformly bounded features φ(s, a) 2 ≤ φ max .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In terms of collaborative filtering/matrix completion, there is another line of established research that focuses on robust matrix completion, in which a small portion of elements or rows in the underlying low-rank matrix is assumed to be arbitrarily perturbed [18,19,20,21].
Sentence 2: the stability of alternating minimization solutions was analyzed with respect to malicious data manipulations in [22].
Relationship: entailment
Domain: NeurIPS
Sentence 1: A "Biased" scale-free model that combines the preferred attachment model [7] with a mixed-membership model.
Sentence 2: we generated M = 60, 000 1-edges as follows: (a) pick a vertex i with probability proportional to its degree; (b) randomly pick a destination community k from θ i ; (c) find the set V k of all vertices v such that θ vk is the largest element of θ v (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A key idea of the proposed method is that because the outcome of an intervention depends on both the inherent effects of the intervention and the participants' expectations, the effectiveness should be inferred in a like-for-like fashion.
Sentence 2: the response observed in, say, the sub-group of participants assigned to the control group whose feedback professes belief in the control group assignment should be compared with the response of only the sub-group of the treatment group who equally professed belief in the control group assignment.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It was suggested that a new notion of connectivity called propagation connectivity is a sufficient condition for the solution of random MAX-3LIN problem with a planted solution to be unique [26,Proposition 2].
Sentence 2: it is claimed that if the hypergraph corresponding to an instance of MAX-3LIN is propagation connected, then the optimal solution for MAX-3LIN is unique and there is an efficient algorithm that finds it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , F n (x)) and F n (y) = (F 1 (y), .
Sentence 2: . . , F n (y)) behaves like where h 1 , h 2 : [0, 1] → R + are continuous decreasing functions, and we would like to map D-dimensional real vectors into n-bit binary strings in a locality-sensitive manner, where the notion of locality is induced by the kernel K. We will achieve this goal by drawing F n appropriately at random.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since f (•) = h(A(•)) by ( 2) and h(•) is strictly convex on dom(h) by Assumption 2.1(b), it is easy to verify that the map X → A(X) is invariant over the optimal solution set X .
Sentence 2: there exists a z ∈ dom(h) such that for any X * ∈ X , we have A(X * ) = z.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A permutation π is a bijective function π : {1, . . . , n} → {1, . . . , n} associating with each item i ∈ {1, . . . , n} a rank π(i) ∈ {1, . . . , n}.
Sentence 2: π(i) denotes the rank given to item i and π −1 (i) denotes the item assigned to rank i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Methods of the second family are based on a maximum likelihood (ML) estimator with an 1 penalty (Yuan & Lin, 2007;Banerjee et al., 2008;Friedman et al., 2008).
Sentence 2: if S denotes the empirical covariance matrix, the solution is the matrix Θ which solves the optimization problem An example of an extension to both models (the regression and ML approaches), and the first to explicitly consider additional classical network properties, is the work by Liu & Ihler (2011), which modifies the 1 penalty to derive a non-convex optimization problem that favors scale-free networks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 3) is essentially the maximization of the similarity between the stochastic part of V , S V with the stochastic matrix formed as the product of the K stochastic matrices X 1 , • • • , X K , weighted by D V . Before describing the algorithm solving (3), we first derive the solution to another related problem that we term as the stochastic matrix sandwich (SMS) problem, from which we can construct a solution to (3).
Sentence 2: in an SMS problem one minimizes the following objective function with regards to an m × n stochastic matrix X, as where A and B are two known stochastic matrices of dimensions m × m and n × n, respectively, and C is an m × n nonnegative matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We tested our approach against Clique Averaging algorithm (CAVERAGE), since it was the best performing approach in [2] on the same type of experiments.
Sentence 2: cAVERAGE outperformed Clique Expansion [10] combined with Normalized cuts, Gibson's Algorithm under sum and product model [9], kHMeTiS [11] and Cascading RANSAC [2].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In all cases, the adversary can cause at most γ 0 regret and it is necessary for the adversary to play x T 1 ∈ ABC(Σ, γ 0 ), which implies that (MMS) is optimal.
Sentence 2: for x T 1 ∈ X = ABC(Σ, γ 0 ) and y T 1 ∈ Y = L(B t ), we have sup for all T > 0, which implies the result.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The key idea is to introduce weights related to β 1 (s) to the loss functions so as to make them tighter bounds of (1−NDCG).
Sentence 2: we introduce weights to the pairwise and listwise losses in the following way, It can be proved (see Proposition 1 in [4]) that the above weighted losses are still upper bounds of (1−NDCG) and they are lower bounds of the original pairwise and listwise losses.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Roughly, B t (a) represents the greatest mean reward value that is statistically plausible.
Sentence 2: b t (a) is typically constructed so that B t (a) → E y∼p * a [R(y)] as data about action a accumulates, but with high probability Like Thompson sampling, many UCB algorithms only select actions that are optimal under some p ∈ P. Consider an algorithm that constructs at each time t a confidence set P t ⊂ P containing the set of distributions that are statistically plausible given observed data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (2) Only if n → ∞ and m i → ∞ simultaneously does the two-layer generalization bound uniformly converge.
Sentence 2: if the number of documents for some query is finite, there will always exist document-layer error no matter how many queries have been used for training; if the number of queries is finite, then there will always exist query-layer error, no matter how many documents per query have been used for training.
Relationship: entailment
Domain: NeurIPS
Sentence 1: , and use the peaks for that pair to initialize our source activity array ↵.
Sentence 2: we compute the likelihood of each phase observation under a set of von Mises distributions centered at each peak (along with a uniform distribution for the garbage source), then smooth these likelihoods over time and frequency using a Gaussian blur, and initialize ↵ from the logarithm of these smoothed likelihoods.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At any given point in our compilation we have some frontier F, which is the set of variables that have some but not all factors included in the current SDD.
Sentence 2: then we select the variable in our current SDD that is, on the graph of our model, closest to the "center" induced by F. we use argmin X 2X max F 2F distance(X , F ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this experiment we consider the regime of clustered eigenvalues described in the introduction and summarized in Table 3.
Sentence 2: we construct a synthetic matrix A ∈ R 30×30 with the smallest 15 eigenvalues clustered in the interval (5, 5 + ∆) and the largest 15 eigenvalues clustered in the interval (θ, θ + ∆).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This linesearch guarantees that the algorithm produces a sufficiently large decrease of the cost function at each iteration, but it may require several function evaluations to find l. This algorithm is guaranteed to converge to a critical point of (8) as long as the step sizes are chosen commensurate with the norm of the Hessian [8].
Sentence 2: nesterov has recently shown that if f has a Lipschitz-continuous gradient with Lipschitz constant L, then the algorithm will converge at a rate of 1/k where k is the iteration counter [15].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The representation in (6) constitutes a proper generative construction for {✓ (c) jn } in the presence of spikes that co-occur within the refractory time, but it complicates inference.
Sentence 2: recall that G (c) In the original construction, (4) and ( 5), in which refractory-time violations are not account for, the Gibbs update equations for {U (c) ji } are analytic, due to model conjugacy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the convex representation, the ξ parameters do not actually represent a probabilistic quantity, but rather arise as parameters in a variational inequality.
Sentence 2: we write, The algorithm is then equivalent to that in §4.1.2 except that the expectation is taken of x 2 as the E step, and the diagonal weighting matrix becomes, where σ i = E (x 2 i |y; ξ i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The central idea behind the TS-ReLU network is to learn a linear classifier directly from the rich, expanded representation Z 1 , rather than collapsing it back to the lower dimensional X 1 .
Sentence 2: in a standard ReLU network, the hidden layer activity X 1 is sent through a linear classifier f X (W X X 1 ) trained to minimize some loss function L X (f X ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our setup, q is given by ( 1) where we let L go to infinity but we tie the parameters of all models after a specific level T (with T ≪ L).
Sentence 2: we impose the condition that for all components with index i > T the variational distributions for the stick-lengths q vi (v i ) and the variational distributions for the components q ηi (η i ) are equal to their corresponding priors, i.e., q vi (v i ; In our model we define the free energy F as the limit F = lim L→∞ F L , where F L is the free energy defined by q in (1) and a truncated DP mixture at level L (justified by the almost sure convergence of an L-truncated Dirichlet process to an infinite Dirichlet process when L → ∞ [6]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we study the problem of learning the graph structure of an underlying GLM graphical model given iid samples.
Sentence 2: we assume that we are given n samples X n 1 = {X (i) } n i=1 , from a GLM graphical model: We have removed node-wise terms for simplicity, noting that our analysis extends to the general case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator.
Sentence 2: we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The dataset consists of the propagation of short textual phrases, referred to as Memes, via the publication of blog posts and main-stream media news articles between March 2011 and February 2012.
Sentence 2: the dataset contains seven groups of cascades corresponding to the propagation of Memes with certain keywords, namely "apple and jobs", "tsunami earthquake", "william kate marriage"', "occupy wall-street", "airstrikes", "egypt" and "elections."
Relationship: entailment
Domain: NeurIPS
Sentence 1: Both take into account the last sample of the immediate reward, r t . They differ in how they account for the reward from the "next state": either, for Q(1), as q t (the last reward received from the state visited after the last visit to S t ) or, for model-based RL, as p t (the last sample of the reward at the true successor state).
Sentence 2: while TD(1) will incorporate the reward observed following R t , regardless of the state, a model-based system will instead consider the expected successor state [21].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Arguing about the ∞-norm margin is convenient when dealing with sparse hyperplanes, i.e., with hyperplanes having only a small number of relevant features (e.g., [14]).
Sentence 2: we say that a training set S = ((x 1 , y 1 ), ..., (x m , y m )) is linearly separable with margin γ > 0 when there exists a hyperplane w with ||w|| q ≤ 1 such that if the underlying margin is an ∞-norm margin (and α is not close to 1) then w tends to share the sparsity properties of w. See also Section 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a result, their model can be trained by optimizing a convex objective function through a variant of Blahut-Arimoto alternating minimization algorithm, whereas our model is more complex and the objective function becomes non-convex.
Sentence 2: training a simple chain structured CRF model [19] in our framework turns out to be intractable even if using Blahut-Arimoto's type of alternating minimization algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Interestingly, the response time pattern across image groups (Figure Supp.2a)) was inversely correlated to the perceptual bias pattern (Figure 2a)) (Pearson correlation = 1, p < 0.01; two-tailed Pearson correlation test).
Sentence 2: subjects made quicker decisions for the hazard group, then pets group, and then vegetables group.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assumption 1 does not hold for the hinge loss (see Appendix A for details), but it does hold for a modified hinge loss, where we apply the hinge separately to each view: Example 2 (Modified Hinge Loss).
Sentence 2: suppose that L(θ; x, y) L is the sum of 3 hinge losses, one for each view.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By taking = ˜ q✏ 2 p 7/2 /2 in Proposition 4.1 we get a ˜ -approximation to µ(0) with computation bounded by a polynomial in p, 1/˜ .
Sentence 2: the existence of an FPRAS for the mapping µ 7 !
Relationship: entailment
Domain: NeurIPS
Sentence 1: Abbeel and Ng [1] provide several refinements of the analysis in Theorem 1.
Sentence 2: suppose that each sample trajectory has length H ≥ (1/(1 − γ)) ln(1/( H (1 − γ))), and that an P -optimal policy is found in each iteration of the projection algorithm (see Step 1 above).
Relationship: entailment
Domain: NeurIPS
Sentence 1: It arises frequently in applications such as machine learning and signal processing; see a recent survey [7].
Sentence 2: each smooth functions {g i } N i=1 can represent: 1) a mini-batch of loss functions modeling data fidelity, such as the 2 loss, the logistic loss, etc; 2) nonconvex activation functions for neural networks, such as the logit or the tanh functions; 3) nonconvex utility functions used in signal processing and resource allocation, see [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here, we formulate a Coupled Topic Model by extending LDA [2] to model multiple groups of documents.
Sentence 2: it associates the t-th group with a mixture of topics, characterized by a DP sample D t . With this given, the words in a document are generated independently, each from a topic drawn from D t . To exploit the statistical dependency between groups, we further introduce a set of latent DPs to link between these mixtures, as described above.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We translate the high-level Problem 1 into the following question: Given r, c , what is the probability that the entry H(i, j) of the hidden dataset takes on value 1?
Sentence 2: for each entry H(i, j) we are interested in computing the quantity: Here, Pr(H ) encodes the prior probability distribution over all hidden matrices in X .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, our analysis focuses on E[log w (i) ].
Sentence 2: assuming perfect transitions, the expected log weights are given by: each log w (i) can be seen as a biased estimator of log Z b , where the bias δ = log Z b − E[log w (i) ] is given by the sum of KL divergences ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a data matrix X = [x 1 , .
Sentence 2: . . , x n ] with n data points in R p and the corresponding response vector y ∈ R n , the goal of RLSR is to learn a ŵ such that, ( ŵ, Ŝ) = arg min we wish to simultaneously determine the set of corruption free points Ŝ and also estimate the best model parameters over the set of clean points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in the table, we provide matching upper and lower bounds (up to a log factor) for all function and algorithm classes.
Sentence 2: our bounds establish the optimality (up to log factors) of accelerated SDCA, SVRG, and SAG for randomized finite-sum optimization, and also the optimality of our deterministic smoothing algorithms for non-smooth composite optimization.
Relationship: entailment