sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Uniformity testing is arguably the most extensively studied problem in distribution property testing [GR00, Pan08, VV14, DKN15b, Gol16, DGPP16, DGPP17] and its sample complexity is well understood.
Sentence 2: it is known [Pan08, CDVV14, VV14, DKN15b] that Θ(n 1/2 / 2 ) samples are necessary and sufficient for this task.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Wilson and Ghahramani [20] join machine learning and econometrics efforts by proposing a model for both mean and covariance regression in multivariate time series, improving previous work of Bru [21] on Wishart Processes in terms of computational tractability and scalability, allowing more complex structure of dependence between Σ(t) and Σ(t + ∆).
Sentence 2: they propose a continuous time Generalised Wishart Process (GWP), which defines a collection of positive semi-definite random matrices Σ(t) with Wishart marginals.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although promising directions within the sampling community such as Elliptical Slice Sampling (ESS, [5]) have been proposed, they have been shown to be particularly slow compared to variational methods.
Sentence 2: [6] showed that their automated variational inference (AVI) method can provide posterior distributions that are practically indistinguishable from those obtained by ESS, while running orders of magnitude faster.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each class is associated with a stem distribution and a suffix distribution.
Sentence 2: we assume that stems and suffixes are independent given the class, so we have where c k , t k , and f k are the class, stem, and suffix associated with ℓ k , and t.f indicates the concatenation of t and f . we generate a label by first drawing a class, then drawing a stem and a suffix conditioned on the class.
Relationship: entailment
Domain: NeurIPS
Sentence 1: L "observed features" j ∈ {1, . . . , R} "noise features" j ∈ {R + 1, . . . , 40} However, we can do much better by reorganizing the computations in such a way that the Jacobian matrix J d is never explicitly required.
Sentence 2: substituting ( 6) into (1), ) we observe that it suffices to solve the single linear system, By organizing the computations this way, the number of least squares problems that must be solved is substantially reduced from k to only one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We apply this relabeling operation iteratively to obtain context-dependent atom vectors where h V are shared across layers.
Sentence 2: the final atom representations arise from mimicking the set comparison function in the WL isomorphism test, yielding The set comparison here is realized by matching each rank-1 edge tensor h to a set of reference edges also cast as rank-1 tensors W (0) [k] ⌦ W (1) [k] ⌦ W (2) [k], where W[k] is the k-th row of matrix W. Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, if evolving x t with a numerical integrator using discrete time increment h induces an error bounded by O(h K ), the integrator is called a Kth order integrator, e.g., the popular Euler method used in SGLD [3] is a 1st-order integrator.
Sentence 2: [17] proved the bounds stated in Lemma 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, we can reduce this time to O(log n) by using the JL lemma, which informs us that pairwise distances can be approximately captured by compressive projection onto d = O(log n) dimensions.
Sentence 2: for d > (4 + 2β) log n ǫ 2 /2 − ǫ 3 /3 , let Ω denote an n × d matrix with i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The appealing additive property of the objective function J makes it trackable to analyze each split when the search tree grows.
Sentence 2: we split the k-th leaf node into two child nodes k(1) and k(2) if and only if it increases the overall objective function J k(1) + J k(2) > J k . Moreover, we optimize each split by choosing the one that maximizes J k(1) + J k (2) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the proposed approach, the random variables (RVs) linked by a tree substructure across different levels compete for their label assignments, while in the existing approaches the goal is to cooperate in the label assigns, which is usually achieved by averaging.
Sentence 2: usually the label assignment of a parent node is enforced to be compatible with the label assignment of its children by averaging.
Relationship: entailment
Domain: NeurIPS
Sentence 1: DNNs can be trained greedily, in a layer-by-layer manner, as in Restricted Boltzmann Machines [12] and (sparse) auto-encoders [3,17], or by learning all layers simultaneously, which is relatively efficient if the layers are convolutional [18].
Sentence 2: the advent of massively-parallel GPUs has recently made it possible to train deep convolutional networks on a large scale with excellent performance [7,14].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This means that any regularizer that depends on scale (such as 1 and 2 norms) is effectively meaningless since the minimization in (1) will drive Ω(θ) to 0 (as this doesn't affect the loss).
Sentence 2: in such discrete setting we need a scale-invariant regularizer, such as the 0 pseudo-norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From both perspectives, redundancy reducing algorithms such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Independent Subspace Analysis (ISA) and Radial Factorization [11; 21] have received considerable interest since they yield image representations that are favorable for compression and image processing and at the same time resemble properties of the early visual system.
Sentence 2: iCA and ISA yield localized, oriented bandpass filters which are reminiscent of receptive fields of simple and complex cells in primary visual cortex [4; 16; 10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2, D-TS and D-TS + have much smaller regret STD than that of ECW-RMED in the non-Condorcet dataset, where certain preference probabilities (for different arms) are close to 1/2.
Sentence 2: the STD of regret for ECW-RMED is almost 200% of its mean value, while it is only 13.16% for D-TS + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is relatively straightforward to include additional model sophistication into (10), one example of which we will consider in the context of the image-processing example.
Sentence 2: in many applications it is inappropriate to assume a Gaussian model for the noise or residual n . In Section 4.3 we consider the following augmented noise model: with gamma priors placed on α λ and α 2 , and with p = 1, . . . , M . The term λ n • m n accounts for "spiky" noise, with potentially large amplitude, and πn represents the probability of spiky noise in data sample n. This type of noise model was considered in [18], with which we compare.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One line of related research studies the communication complexity of convex optimization.
Sentence 2: [40] studied two-processor convex minimization in the same model, provided a lower bound of Ω(n(log n + log(1/ ))) bits on the communication cost of n-dimensional convex problems, and proposed a non-stochastic algorithm for strongly convex problems, whose communication cost is within a log factor of the lower bound.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the Poincaré Ball has a Riemannian manifold structure, we can optimize Equation (2) via stochastic Riemannian optimization methods such as RSGD [6] or RSVRG [39].
Sentence 2: let T θ B denote the tangent space of a point θ ∈ B d . Furthermore, let ∇ R ∈ T θ B denote the Riemannian gradient of L(θ) and let ∇ E denote the Euclidean gradient of L(θ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Hessians r 2 f are M -Lipschitz continuous over the set C, i.e., for any x,  In this section, we introduce a generic framework to reach an (✏, )-SOSP of the non-convex function f over the convex set C, when C has a specific structure as we describe below.
Sentence 2: we focus on the case when we can solve a quadratic program (QP) of the form up to a constant factor ⇢  1 in a finite number of arithmetic operations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also give numerical results which demonstrate good performance on a benchmark problem.
Sentence 2: we show that our algorithm achieves much better performance on this problem than OP-MDP [2] when both algorithms generate the same number of successor states, despite the fact that OP-MDP assumes knowledge of all successor state probabilities in the MDP, whereas our algorithm only samples states from a simulator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: , PCP yields a unique solution which agrees with the correct (L, S) with high probability if there exists a dual certificate Y obeying where = 1 32 p n log n . The proof of the above proposition adapts the idea in [1,12] for uniform errors to non-uniform errors.
Sentence 2: the proof exploits the properties of R ⌦ associated with non-uniform errors, which are presented as Lemma 1 (established in [9]) and Lemma 2 in Appendix A.1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: After each full Gibbs sweep, we obtain a sample z (t) (and corresponding counts N (t) j of the number of points assigned to cluster j), which can be used to infer the Bernoulli parameters µ (t) j .
Sentence 2: we use T of these samples to estimate the probability that a file x i needs to be changed given files in the starter set S. averaging predictions over the T Gibbs samples, we estimate: x n . (6) A restricted Boltzmann machine (RBM) is a Markov random field (MRF) whose nodes are (typically) binary random variables [17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We first consider the scenario where the observations are exact.
Sentence 2: suppose we are given m observations {T ijk } (ijk)∈Ω , where Ω is sampled from uniformly at random from We propose to recover matrices A, B, C and therefore tensor T = Pair(A, B, C) using the following convex program, minimize where M * denotes the nuclear norm of matrix M, which is the sum of singular values of M, and S A , S B , S C is defined in Proposition 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The result also suggests that instead of selecting top few eigenvectors, as is normally done in KPCA, selecting them by our method and then applying L 1 regularization yields better result.
Sentence 2: in case of IONOSPHERE and BREAST-CANCER data sets top |N | (5 and 3 respectively) eigenvectors do not contain the representative ones.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A small modification to the unconstrained formulation (7) extends our tractable Stein discrepancy computation to any domain defined by coordinate boundary constraints, that is, to 1 for all j.
Sentence 2: for each dimension j, we augment the j-th coordinate linear program of (7) with the boundary compatibility constraints max , and k 6 = j.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The "polyhedral lemma", found in [5, Theorem 5.2] and [13, Lemma 2], proves that the distribution of η Y , after conditioning on {Y ∈ A} and on P ⊥ η Y , is given by a truncated normal distribution, with density The interval endpoints a 1 (Y ), a 2 (Y ) depend on Y only through P ⊥ η Y and are defined to include exactly those values of r that are feasible given the event Y ∈ A.
Sentence 2: the interval contains all values r such that r • η + P ⊥ η Y ∈ A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Below, we present such a network that takes both the given conditional variables and an extra set of random variables as inputs, and then passes through a deep neural network with nonlinear transformations to produce the samples of the target variables.
Sentence 2: our network is built on the fact that for any distribution P on sample space K and any continuous distribution Q on L that are regular enough, there is a function G : L → K such that G(x) ∼ P, where x ∼ Q [12].
Relationship: entailment
Domain: NeurIPS
Sentence 1: If it is possible to "encode" members of a class of distributions using a carefully chosen subset of the samples, then this yields an upper bound on the sample complexity of distribution learning for that class.
Sentence 2: by constructing compression schemes for mixtures of axisaligned Gaussians and general Gaussians, we obtain new upper bounds on the sample complexity of learning with respect to these classes, which we prove to be optimal up to logarithmic factors.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To mitigate this problem we develop an importance sampling approach, where sequences are drawn from a proposal distribution q(•) that has an increased likelihood of generating extensions in which X e occurs, and then uses a weighted empirical estimate.
Sentence 2: we will sample extensions The Poisson Superposition Importance Sampler (PSIS) is an importance sampler whose proposal distribution q is based on Poisson superposition.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Following the delayed and perturbed leader More generally, follow-the-perturbedleader optimizes over all information available to the algorithm, plus some additional noise to smoothen the worst-case analysis.
Sentence 2: if the cost vectors are delayed, we naturally interpret followthe-perturbed-leader to optimize over all cost vectors c t delivered in time for round t when picking its point x t . the tth leader becomes the best choice with respect to all cost vectors delivered in the first t rounds: (we use the superscript d to emphasize the delayed setting).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To do so, we build upon tools in amortized variational inference for developing a probabilistic meta-learning approach.
Sentence 2: our method builds on model-agnostic meta-learning (MAML) [9], a few shot metalearning algorithm that uses gradient descent to adapt the model at meta-test time to a new few-shot task, and trains the model parameters at meta-training time to enable rapid adaptation, essentially optimizing for a neural network initialization that is well-suited for few shot learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead of drawing exactly m samples from a distribution p, we first draw m 0 ⇠ Poisson(m), and then draw m 0 samples from p. As a result, the number of times different elements in the support of p occur in the sample become independent, giving much simpler analyses.
Sentence 2: the number of times we will observe domain element i will be distributed as Poisson(mp i ), independently for each i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Again, the standard large deviation bound implies that there exists a universal constant c > 0 such that for all j ≥ 1: .
Sentence 2: with this choice, there exists a constant c > 0 such that There exists a constant c > 0 such that for any L: if we choose s(Z n 1 ) = m −1 e (2k) −1 cn ∆(Z n 1 ) 2 in Figure 1, then ∆R(Epoch-Greedy, H, T ) ≤ 2L + 1 + c k∆ −2 ≤ 2 8k(ln m + ln(T + 1)) c∆ 2 + 1 + c k∆ −2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For bounds directly on function values, see [26].
Sentence 2: we now consider θn = 1 n n−1 k=0 θ k and, following [4,5], we make extra assumptions regarding the smoothness of each f n and the fourth-order moment of the driving noise: (H6) For each n 1, the function f n is almost surely twice differentiable with Lipschitz-continuous Hessian operator f ′′ n , with Lipschitz constant M . for all θ 1 , θ 2 ∈ H and for all n 1, , where • is the operator norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To demonstrate this, we adapt the proposed DirBN structure to the following models: PFA+DirBN Poisson Factor Analysis (PFA) is a popular framework for topic analysis (DPFA , DPFM , GBN (Zhou et al., 2016) can be viewed as a deep extension to PFA).
Sentence 2: we use the Bayesian nonparametric version of PFA named BGGPFA (Zhou et al., 2012), where θ d is constructed from a negative binomial process and φ k is drawn from a Dirichlet distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For instance, it follows from Proposition 1 that for standard Gaussian measurements, τ 2 (T) = 0 with high probability once n < δ m /2.
Sentence 2: the situation can be much better for random spd measurements ( 14) as exemplified for measurements ∼ N (0, I) in Appendix H. it turns out that τ 2 (T) = Ω(1/r) as long as n = Ω(m • r).
Relationship: entailment
Domain: NeurIPS
Sentence 1: There is some universal constant c 0 ≥ 1, such that for any c ≥ c 0 and any ∈ (0, 1), if we run algorithm 1, using parameters η, T satisfying then after S = log 4 (9/ ) epochs of the algorithm, wS+1 satisfies E[F ( wS+1 )−min w F (w)] ≤ .
Sentence 2: by taking η = Θ(1) and T = Θ(1/λ), the algorithm attains an -accurate solution after O(log(1/ )/λ) stochastic iterations of without-replacement sampling, and O(log(1/ )) sequential passes over the data to compute gradients of F (•).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The standard approach to parameter estimation in statistics is through maximum likelihood, which chooses parameters ✓ by maximising (To keep the notation light, we reserve n to index the data samples.
Sentence 2: x n denotes the n-th |V|-dimensional data vector and x mn refers to the n-th observation of node m.) This estimator has played a central role in statistics as it has many desirable properties including consistency, efficiency and asymptotic normality.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These two classes of matrix factorization problems have attracted significant interest in the research community.
Sentence 2: convex formulations of noisy matrix factorization have been shown to admit strong theoretical recovery guarantees [1,2,3,20], and a variety of algorithms (e.g., [15,16,23]) have been developed for solving both matrix completion and robust matrix factorization via convex relaxation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: But we can also see the iteration as a noisy attempt to reach the parallel coordinate optimum λ VB t . While λ is an unbiased estimate of this quantity, we will show that Eq 1 uses a biased estimate but with reduced variance.
Sentence 2: this new perspective opens the door to other ways of updating λ t based on the noisy estimates of λ VB t . we use a Kalman filter to track the progress of λ t based on the sequence of noisy coordinate updates.
Relationship: entailment
Domain: NeurIPS
Sentence 1: M-step We update the parameters, L i , T i and V i , of π i,l to obtain π � i,l based on the expectation in the E-step.
Sentence 2: take log of the likelihood P r(r T = 1, z 0:T i , T ; π i,l ) with π i,l substituted with π � i,l and focus on terms involving the parameters of π � i,l : In order to update, L i , we partially differentiate the Q-function of Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To resolve the curse of dimensionality, in many practical applications, we approximate the value of each state by some linear map of its feature s 2 < m , where d < S to reduce the dimension.
Sentence 2: we assume that V ⇡ (s) ⇡ T s w ⇤ for some w ⇤ 2 < m . To compute w ⇤ , we formulate the problem as a Bellman residual minimization problem that w); < 1 is a discount factor, and r s,s 0 is the random reward of transition from state s to state s 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Sadly, through upper and lower bounds we show that for the most biased coin problem this estimate-then-explore approach requires quadratically more flips than our proposed algorithm that adapts to these unknown parameters.
Sentence 2: we show that when ✓ 1 ✓ 0 is sufficiently small one cannot use a static estimation step to determine whether ↵ = 0 or ↵ > 0 unless a number of samples quadratic in the optimal sample complexity are taken.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, the matrices X and − Ḡ can be simultaneously singular-value-decomposed by Ū and V . Moreover, the set X c ⊂ X , which is defined as is a non-empty convex compact set.
Sentence 2: by Proposition 4.2, for every k ≥ 0, the point X k has a unique projection Xk ∈ X c onto X c . Let Since we have Q k F = 1 for all k ≥ 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then η is also an admissible natural parameter, because we can write p(x) = h(x) exp( t(x) � η − Ã(η)), where t(x) = H � t(x), h(x) = h(x) exp(t(x) � b), and Ã(η) = A(H η + b).
Sentence 2: the natural parameter is only unique up to affine transformations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This geometric picture allows us to make the following important observation about the objective value max z• e of the SDP relaxation.
Sentence 2: the largest value that z• e can take depends on the angle θ that x makes with e. as θ decreases, the relaxation becomes tighter and as the vector deviates from e, the relaxation gets looser.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given the feature representation of all images from the retrieval database, , where x im i = ImgEnc(I i ), we can compute a sampling probability based on the distances between the history representation s t to each image feature, x im i .
Sentence 2: the sampling probability is modeled using a softmax distribution over the top-K nearest neighbors of s t : π(j) = e −dj / K k=1 e −d k , j = 1, 2, ..., K, where d k is the L2 distance of s t to its k-th nearest neighbor in {x im i } N i=0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We address the first key challenge in our AltMinPhase algorithm (Algorithm 1) by initializing x as the largest singular vector of the matrix S = 1 m i y 2 i a i a i T . Theorem 4.1 shows that when A is sampled from standard complex normal distribution, this initialization is accurate.
Sentence 2: if m ≥ C 1 n log 3 n for large enough C 1 > 0, then whp we have x 0 − x * 2 ≤ 1/100 (or any other constant).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The subspace row(Y ) does not necessarily contain the desired sparse vector e T i X 0 , but it does contain some approximately sparse vectors.
Sentence 2: consider following vector in row The vector v is a superposition of a sparse signal x0 and its scaled shifts | is small 7 and x 0 is sparse enough, z can be viewed as small noise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, a variant of our algorithm even runs in sublinear time, if A 0 is large and ψ def = OPT (1) / A 0 is not too small.
Sentence 2: we obtain time o( A 0 ) when OPT (1) ≥ ( −1 log(mn)) 4 and A 0 ≥ n( −1 log(mn)) 4 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The latter could be called "loss stability" (in terms of "loss sensitivity coefficients") for the sake of informative names.
Sentence 2: writing z 1:n ⇡ z 0 1:n when these n-tuples differ at one entry (at most), an equivalent formulation to the above is n = sup z1:n⇡z 0 if two samples S n and S 0 n differ only on one example, then kA(S n ) A(S 0 n )k H  n . Thus our definition implies stability with respect to replacing one example with an independent copy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Empirical evaluations on the MNIST digit recognition and TIMIT speech recognition tasks illustrate that SafetyNets enables practical, low-cost verifiable outsourcing of deep neural network execution without compromising classification accuracy.
Sentence 2: the client's execution time is 8×-80× lower than executing the network locally, the server's overhead in generating proofs is less than 5%, and the client/server exchange less than 8 KBytes of data during the IP protocol.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In view of ( 4) and ( 6), a reasonable alternative would be the norm of the residual map R : R m×n → R m×n , which is defined as Intuitively, the residual map measures how much a solution X ∈ R m×n violates the optimality condition (4).
Sentence 2: x is an optimal solution to (1) if and only if R(X) = 0.
Relationship: entailment