sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: 3, from P 1 to P 5 , similarities are measured from low-level feature such as color, texture to high-level feature such as object.
Sentence 2: p t is able to identify fine-grained similarities among pixels belonging to coarsely labeled objects, such as figures for nodes 1, 6 where table mat and painting are highlighted when they are labeled as table and background respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This algorithm models the semibandit feedback as linearly related to the query-document features and learns this relationship, while selecting composite actions using an upper-confidence bound strategy.
Sentence 2: the algorithm maintains a weight vector ✓ t 2 R d formed by solving a ridge regression problem with the semibandit feedback y t (a t,`) as regression targets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: R, where F is a (discrete or continuous) set of predictors f : X !
Sentence 2: y, and we write the empirical risk on the sample (X, Y ) and the generalization error on distribution D as The PAC-Bayesian theory [24,25] studies an averaging of the above losses according to a posterior distribution ⇢ over F. it provides probably approximately correct generalization bounds on the (unknown) quantity f ) and some other parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We thereby answered a crucial question in the study of (automated) mechanism design: how to relate the performance of the mechanisms in the search space over the input samples to their expectation over the underlying-unknown-distribution.
Sentence 2: for a fixed class of auctions, we determine the sample complexity necessary to ensure that with high probability, for any auction in that class, the average revenue over the sample is close to the expected revenue with respect to the underlying, unknown distribution over bidders' valuations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let V h : H → H be the operator corresponding to h via such isometry.
Sentence 2: we show that the SELF definition is satisfied with V = V h and ψ(y) = k(y, •) ∈ H for any y ∈ M. we have V ≤ V HS = h H⊗H , with V HS denoting the Hilbert-Schmidt norm of V . Step If Y is compact and geodesically convex, then it is diffeomorphic to a compact set of R d . By using this fact, we prove that any function in C ∞ (Y ×Y), the space of smooth functions on Y ×Y, admits an extension in C ∞ 0 (M×M) the space of smooth functions on M × M vanishing at infinity (this is well defined since M is diffeomorphic to R d , see ??
Relationship: entailment
Domain: NeurIPS
Sentence 1: For all three cases, the policy gradient method fails to discover a placement that is better than Post or Single GPU within a pre-specified amount of learning time.
Sentence 2: post outperforms the policy gradient method by a training time reduction of 39.5%, 34.6% and 51.7% for 2 GPUs, 4 GPUs and 8 GPUs, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Examples from the topical field of systems biology are the reconstruction of transcriptional regulatory networks from gene expression data [1], the inference of signal transduction pathways from protein concentrations [2], and the identification of neural information flow operating in the brains of songbirds [3].
Sentence 2: dynamic Bayesian networks (DBNs) have been applied, as they allow feedback loops and recurrent regulatory structures to be modelled while avoiding the ambiguity about edge directions common to static Bayesian networks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the random variable Z i+1 has value z, then we may express F i+1 as a function of z.
Sentence 2: we have The discrepancy r(z) between the value of F i+1 (z) and the value of f (z, ✓) can then be used as a measure of the error in the linear method, and penalised.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a result of the communication procedure, the master receives a convex combination of the stochastic gradients evaluated at each worker i.
Sentence 2: the master receives gradients of the form g λ (t) = n i=1 λ i g i (t − τ (i)) for some λ in the simplex, where τ (i) is the delay of worker i, which puts us in the setting of Theorem 2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is also possible to efficiently draw a joint sample if the distribution over k takes the form p(k) = 1 { r k r = D} • r p(k r ).
Sentence 2: there is an arbitrary but independent prior over each k r value, along with a single constraint that the chosen k r values sum to exactly D. Given a sample of k, it is straightforward to sample y using R calls to dynamic programming.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will prove that asymptotically, SRB exhibits a "grouping effect".
Sentence 2: for kernel K = µI − G, G defined in (4), we will look at the minimizer β * = Qα * of the loss function (2), and in the spirit of [18], establish a bound on the difference |β * i1 −β * i2 | of the coefficients on two similar components.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although it may appear somewhat abstract at first sight, we illustrate that this result has a number of concrete consequences for specific models.
Sentence 2: we recover the best known results about estimation in s-sparse models with general designs [1,7], as well as a number of new results, including convergence rates for estimation under q -sparsity constraints, estimation in sparse generalized linear models, estimation of block-structured sparse matrices and estimation of low-rank matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Clusters are then derived from the analysis of the ESSs of the clustering game.
Sentence 2: given a k-graph H = (V, E, s) modeling a hypergraph clustering problem, where V = {1, . . . , n} is the set of objects to cluster and s is the similarity function over the set of objects in E, we can build a game involving k players, each of them having the same set of (pure) strategies, namely the set of objects to cluster V . Under this setting, a population x ∈ ∆ of agents playing a clustering game represents in fact a cluster, where x i is the probability for object i to be part of it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A regular log-normal distribution lnN (µ, σ 2 ) has a positive skewness, which means a sampled value from such distribution has more than 0.5 probability to be less than its expected value, resulting in a higher likelihood of underestimation.
Sentence 2: if X ∼ lnN (µ, σ 2 ), then E[X] = exp(µ + σ 2 /2) and median[X] = exp(µ), thus the Pearson 2 coefficient of X is greater than 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that the player doesn't observe the functions f t , only the losses that result from his past actions.
Sentence 2: in the bandit feedback model, the player observes f t (X 1:t ) on round t, whereas in the full-information model, the player observes f t (X 1:t−1 , x) for all x ∈ A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We have seen that SVI is designed to find the posterior for a particular, prechosen number of data points D. In practice, when we run SVI on the full data set but change the input value of D in the algorithm, we can see degradations in performance.
Sentence 2: we try values of D equal to {0.01, 0.1, 1, 10, 100} times the true D in Fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our algorithm seeks to minimize this loss function over the finite-dimensional vector space H and returns a function To define an equivalent optimization over the matrix version of H, we introduce the following notation.
Sentence 2: for each string x ∈ PS, fix a pair of coordinate vectors (u x , v x ) ∈ R P × R S such that u x Hv x = H(x) for any H ∈ H. u x and v x are coordinate vectors corresponding respectively to a prefix u ∈ P and a suffix v ∈ S, and such that uv = x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by the notion, several studies have focused on designing algorithms that exploit this similarity to learn a cross-lingual alignment between the embedding spaces of two languages, where the two embedding spaces are trained from independent text corpora [6,7,8,9,10,11,12].
Sentence 2: recent research has shown that such cross-lingual alignments can be learned without relying on any form of bilingual supervision [13,14,15], and has been applied to training neural machine translation (NMT) systems in a completely unsupervised fashion [16,17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In Cartesian coordinates, this yields the density An immediate consequence of any L p -spherically symmetric distribution being specified by its radial density is the possibility to change between any two of those distributions by transforming the radial component with (F −1 2 • F 1 )(||y|| p ), where F 1 and F 2 are cumulative distribution functions (cdf) of the source and the target density, respectively.
Sentence 2: for a fixed p, any L p -spherically symmetric distribution can be transformed into a factorial one by the transform This transform closely resembles contrast gain control models for primary visual cortex [13; 20], which use a different gain function having the form g(y) = 1 c+r with r = ||y|| 2 2 [17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: It has received a significant amount of contributions and has been applied in many problem areas such as database record segmentation [2], energy minimization for computer vision [10], critical load factor determination in two-processor systems [16], end-of-session baseball elimination [6], and most recently by [19,18,20] in product portfolio selection.
Sentence 2: it is a key technique for many estimation and assignment problems.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Accelerated methods: Accelerated SG methods, which despite their name are not related to the aforementioned AFG method, take advantage of the fast convergence rate of SG methods with a constant step size.
Sentence 2: accelerated SG methods use a constant step size by default, and only decrease the step size on iterations where the inner-product between successive gradient estimates is negative [23,24].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Still, in spite of its success, the additive model of adaptation in adaptive Spike Response Model effectively changes neural gain with at most a fixed step-size, and thus cannot respond quickly to changes in signal variance that are large compared to this step-size.
Sentence 2: brette [22] argues that adaptation modulation has to be multiplicative for neurons to respond with the same level of neural activity to drastic changes in dynamic range, as is observed experimentally (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The problem of estimating SPAMs has received considerable attention in the regression setting (cf., [9,10,11] and references within) where (x i , f (x i )) n i=1 are typically i.i.d samples from some unknown probability measure P. This setting, however, does not consider the possibility of sampling f at specifically chosen points, tailored to the additive structure of f . In this paper, we propose a strategy for querying f , together with an efficient recovery algorithm, with much stronger guarantees than known in the regression setting.
Sentence 2: we provide the first results guaranteeing uniformly accurate recovery of each individual component φ l of the SPAM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In subsection 4.2, we show that for drifting distributions, our upper bounds in Theorem 3.1 and Theorem 3.2 are almost tight.
Sentence 2: we show that now even for full-information algorithms, a large 3  T dependency in the regret turns out to be unavoidable, even for small V and Λ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we use a local GSM as a basis for a globally consistent and spatially homogeneous field of Gaussian scale mixtures (FoGSM).
Sentence 2: the FoGSM is formulated as the product of two mutually independent MRFs: a positive multiplier field obtained by exponentiating a homogeneous Gaussian MRF (hGMRF), and a second hGMRF.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, there are theoretical reasons to prefer a model in which subregions are always laterallysymmetrical.
Sentence 2: in instances where the subregions are symmetric (the topic in panel B3 for the unconstrained subregions model and all topics for the constrained subregions model), the subregion weights provide a measure of the relative lateralization of function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, while it is effective as a black-box explanation system, MAPLE itself is a highly accurate predictive model that provides faithful self explanations, and thus sidesteps the typical accuracy-interpretability trade-off.
Sentence 2: we demonstrate, on several UCI datasets, that MAPLE is at least as accurate as random forests and that it produces more faithful local explanations than LIME, a popular interpretability system.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that if U is a orthobasis for a subspace, U ⌦ is a |⌦| ⇥ d matrix with columns u i⌦ where u i 2 U , rather than a set of orthonormal basis vectors.
Sentence 2: the matrix U ⌦ need not have orthonormal columns.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose a new interpretation of Random Features that justifies its usage with ℓ 1 -regularization -yielding the Sparse Random Features algorithm.
Sentence 2: we show that the Sparse Random Feature algorithm can be seen as Randomized Coordinate Descent (RCD) in the Hilbert Space induced from the kernel, and by taking D steps of coordinate descent, one can achieve a solution comparable to exact kernel methods within O(1/D) precision in terms of the objective function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a matrix A 2 R n⇥d , and p 2 [1, 2), output a rank-k matrix M such that with probability at least 1 Our first contribution is Algorithm 1, ROBUST-LRA, which given an input matrix A 2 R n⇥d returns a differentially private rank-k approximation to A with a multiplicative approximation factor of ↵ = O((k log k) 2(2 p)/p log d log n) and an additive approximation error of ⌧ = e O " 1 k 2 (n + kd) .
Sentence 2: for p = 1, we have ↵ = O(k 2 log 2 k log d log n) and ⌧ = e O " 1 k 2 (n + kd) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: x 2 R d 1 , with xj = x j x i uj ui , j 2 {1, . . . , i 1, i + 1, . . . , d}.
Sentence 2: we are able to obtain a fair linear model without any other constraint and by using a representation that has one feature fewer than the original one 3 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Such a strategy however is grossly inefficient if we are only interested in the posterior for x = x o . Conversely, if p(θ) closely matches p(θ | x = x o ), then most simulations will produce samples that are highly informative in learning q φ (θ | x) for x = x o .
Sentence 2: if we already knew the true posterior, we could use it to construct an efficient proposal prior for learning it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, it can be used to analyze the performance of approximate minimizers obtained through approximate optimization techniques.
Sentence 2: consider f (w; θ) as in (2), where (z; θ) is convex and L-Lipschitz in z, the norm of φ(θ) is bounded by B, and r is λ-strongly convex.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm 1 Approximate saddle-free Newton Require: | Ĥ|← ∂ 2 s ∂α 2 by using an eigen decomposition of Ĥ for j =1→m do g ←− ∂s ∂α λ←argmin λ s((| Ĥ|+λI) −1 g) θ ←θ+V(| Ĥ|+λI) −1 g end for end for We now search for a solution to the saddle-point problem within the family of generalized trust region methods.
Sentence 2: the analysis of optimization algorithms near saddle points discussed in Sec.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The practical meaning of the state-value function V is that it quantifies how desirable and promising it is to reach a state i, also accounting for future rewards to be expected.
Sentence 2: if one knows the current state i it is a simple and efficient rule of behavior to choose that action a that will lead to the neighbor state j with maximal V (j) (the greedy policy).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This implies that the neuronal gain functions -the firing rate of the post-synaptic neurons as a function of the average • http) / culture.neurobio.ucla.edu/ "'pel firing rate of the pre-synaptic neurons -must be sigmoidal: small at low firing rate to provide stability, high at intermediate firing rate to provide a threshold (at an unstable equilibrium), and low again at high firing rate to provide saturation and a stable attractor.
Sentence 2: a requirement for the co-existence of a stable background state and multiple attractors is that the gain function of the excitatory neurons be super linear at the observed background rates of a few Hz [2,3].
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1 illustrates the spectral properties of the Bethe Hessian that justify this algorithm: the spectrum is composed of a few informative negative eigenvalues, well separated from the bulk (which remains positive).
Sentence 2: as observed in [8,5], it avoids the spurious eigenvalues with localized eigenvectors that make trimming necessary in the case of [3].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proof is based on the observation that if the algorithm has not queried some clusters long enough, it is unaware of the nodes that have been reassigned between them.
Sentence 2: if a node v moves from C i to C j at time t and the algorithm does not query one of the two clusters after time t it has small chances of guessing the cluster of v. Some nontrivial analysis is needed to show that a sufficiently large number of such nodes exist, regardless of the choices of the algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We utilize the encoder-decoder architecture recently introduced in [1], which is illustrated in Figure 2.
Sentence 2: we represent a triplet (q, H, I) with e t by applying three different encoders, based on recurrent (RNN with long-short term memory units), hierarchical recurrent (HRNN) 2 and convolutional (CNN) neural networks, followed by attention and fusion units (Figure 2 (a)-(f)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the context of a statistical treatment of persistent homology, the ability to embed probability measures on the space of persistence diagrams into a RKHS is appealing.
Sentence 2: the problem of testing whether two different samples exhibit significantly different homological features -as captured in the persistence diagram -boils down to a two-sample test with null hypothesis H 0 : µ P = µ Q vs. a general alternative H A : µ P µ Q , where P and Q are probability measures on the set of persistence diagrams.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a result, we have Similarly, we maximize the variational lower bound over s: Consequently we have for each j, To estimate β, we develop an active-set method to iteratively maximize the model marginal likelihood over elements of β.
Sentence 2: we use a strategy similar to Tipping and Faul (2003)'s approach: given the approximation factors N (t|X T w, diag(h) −1 ), the distribution over eigenvectors N (v j |s j w, (λ v η j ) −1 I), and the prior distribution N (w|0, diag(β) −1 ), we can compute and decompose the log marginal likelihood L(β) = log p(y|X, s, λ v ) into two parts: L(β j ) and L(β \j ) where j and \j index the elements of β in the active set and the rest elements, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We employ an idea of Burer and Monteiro [2] that has far reaching consequences.
Sentence 2: the positive definite constraint in the SDP formulation above is trivially satisfied if we define L and R via Burer and Monteiro showed that as long as L and R have sufficiently many columns, then the global optimum of ( 4) is equal to that of we may assume that the number of columns is less than m + n. This formulation of the max-norm is nonconvex because it involves a constraint on the product LR , but Burer and Monteiro proved that each local minimum of the reformulated problem is also a global optimum [9].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The catch is that, in order to get guarantees in L 1 -distance using this approach, would require us to learn to very small L 2 distance (because of the Cauchy-Schwarz inequality).
Sentence 2: if the unknown distribution P has support size N , we would have to learn to L 2 distance / √ N in (i), and then in (ii) test that we are within L 2 -distance / √ N of the learned hypothesis.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then we are guaranteed that In particular, we have the bounds: :::; p(M-1 K) :::; Amax (K) (4) It should be noted that the conditions of this theorem are sufficient, but by no means necessary, to guarantee convergence of the ET algorithm.
Sentence 2: we find that indefinite cutting matrices often lead to faster convergence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: They define how in future, as t > t i increases, the elements of c(t) will continue to deterministically decay (at different rates) from c i+1 toward targets ci+1 .
Sentence 2: c(t) is given by ( 7), which continues to control h(t) and thus λ k (t) (via (4), except that i has now increased by 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this setting our results depend on a quite different complexity measures: the upper and lower Minkowski dimensions of the space, thus exhibiting a phase transition between the finite case (that corresponds to Minkowski dimension zero) and the infinite case.
Sentence 2: we give an upper bound on the regret of O(T d+1 d+2 ) where d ≥ 1 is the upper Minkowski dimension.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, in logistic regression, the link function g is logit and g −1 is the sigmoid function; and in linear regression, the link functions is the identity function.
Sentence 2: learning in GLMs means learning the actual linear combination w. the likelihood of y given x can be written as p(y|w, x) = h(y)exp yw x − A(w x) , where x ∈ X , y ∈ Y, A is the log-partition function, and h(y) the scaling constant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We present stark information-theoretic lower bounds for both relative and absolute -erroneous oracles, for any constant and even sub-constant > 0.
Sentence 2: we show that: • For minimizing a convex function (or maximizing a concave function) f : [0, 1] n → [0, 1] over [0, 1] n : we show that for any fixed δ > 0, no algorithm can achieve an additive approximation within 1/2 − δ of the optimum, using a subexponential number of calls to an absolute n −1/2+δ -erroneous oracle.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More recently, the question of whether condition (2) is also necessary for the existence of linear representer theorems has been investigated [15].
Sentence 2: [15] shows that, if Ω is differentiable (and certain technical existence conditions hold), then (2) is a necessary and sufficient condition for certain classes of regularization functionals to admit a representer theorem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, it conditions on the embedding z and previous state x t 1 to generate the vector x t autoregressively.
Sentence 2: the autoregression is over the components of the vector x t . Wavenet lessens the load of the encoder which no longer has to carry information that can be captured by modeling auto-correlations between components of the state vector .
Relationship: entailment