sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: For every even r ≤ m/4 and everyunit vector While it might seem at a glance that bounding the high-order moments of X(x) is merely a technical issue, known tools and techniques could not be used to prove Lemmas 3, 4.
Sentence 2: earlier work by Kane and Nelson [KN14, CJN18] and Freksen and Larsen [FL17] used high-order moments bounds as a step in proving probability tail bounds of random variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our goal throughout is to minimize a function f : Here e denotes a small subset of {1, . . . , n} and x e denotes the values of the vector x on the coordinates indexed by e. The key observation that underlies our lock-free approach is that the natural cost functions associated with many machine learning problems of interest are sparse in the sense that |E| and n are both very large but each individual f e acts only on a very small number of components of x.
Sentence 2: each subvector x e contains just a few components of x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 2 ([BBAP05, Pau07, OMH + 13]).
Sentence 2: let v 1 be the principal eigenvector of B T B, where this theorem shows that it is possible to estimate v solely from the covariates using, in fact, a spectral method if, and only if µ > √ γ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a joint distribution p, consider the "(X × X ) T −1 → P(X × X )"valued probability tree ρ = (ρ 1 , .
Sentence 2: . . , ρ T ) defined by the values of the mappings ρ t (ǫ) are products of conditional distributions, where conditioning is done with respect to a sequence made from x s and x ′ s depending on the sign of ǫ s . We note that the difficulty in intermixing the x and x ′ sequences does not arise in i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There exists constant C depending on f and K such that for n sufficiently large, with probability at least 1 − 1/n, and |M| = | M|.
Sentence 2: taking h ≈ n −1/(4+d) optimizes the above rate to d(M, M) = Õ(n −1/(4+d) ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, we analyze the effects of the object localization net.
Sentence 2: we assess the occurring performance changes of two configurations: (i) by only adding the bounding box regression loss into the objective function ('Train'), i.e., both the segmentation net and the object localization net are trained but only the segmentation net is deployed; (ii) by training and applying the object localization net ('Apply').
Relationship: entailment
Domain: NeurIPS
Sentence 1: For continuously differentiable f , Robbins and Monro (1951) propose a simple iterative stochastic search algorithm for solving (1.1).
Sentence 2: at the k-th iteration, we obtain z k sampled from D and take (1.2) where ⌘ is the step-size parameter (also known as the learning rate in machine learning literature), r u f (u k , z k ) is an unbiased stochastic gradient for approximating r u E Z⇠D f (u k , Z), i.e., and ⇧ U is a projection operator onto the feasible set U .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We found both the expert pretraining and RL objective important for generating high quality molecules, thus both of them are kept throughout training.
Sentence 2: we use PPO algorithm to train the RL objective with the default hyperparameters as we do not observe too much performance gain from tuning these hyperparameters, and the learning rate is set as 0.001.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For learning the score s(x|X), we take recourse to large-margin classification with structured loss functions [37] to obtain the relative loss objective l(x, x |X) = max(0, s(x |X)−s(x|X)+∆(x, x )).
Sentence 2: we want to ensure that s(x|X) ≥ s(x |X) + ∆(x, x ) whenever x should be added and x should not be added to X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At decoder step i, we utilize a context-based attention mechanism with glimpse, similar to Vinyals et al.
Sentence 2: [35], which extracts the relevant information from the inputs using a variable-length alignment vector a t . a t specifies how much every input data point might be relevant in the next decoding step t. Let xi t = (s i , di t ) be the embedded input i, and h t ∈ R D be the memory state of the RNN cell at decoding step t. The alignment vector a t is then computed as ) Here ";" means the concatenation of two vectors.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we focus on scalable parallelization of Monte Carlo simulation, a problem motivated by the increasingly large inference problems occurring in a variety of fields in science and engineering.
Sentence 2: we assume that we are given a large scale inference problem involving an intractable posterior expectation, for example a Bayes estimator, and that Monte Carlo simulation is to be used to approximate the targeted expectation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The learning problem is that of selecting a function f 2 F with small risk or expected loss  Generalization bounds based on the notion of Rademacher complexity are standard [7].
Sentence 2: for the empirical risk minimization (ERM) hypothesis b g n , for any > 0, the following bound holds with probability at least 1 : R n (G) is in the order of O(1/ p n) for various classes used in practice, including when F is a kernel class with bounded trace and when the loss l is Lipschitz.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We resolve the problems described above by assuming that speakers and listeners deviate from normative behavior by assuming a conventional lexicon.
Sentence 2: our final convention-based agents assume: (a) There is some single, specific literal lexicon which everyone should be using, (b) and everyone else knows this lexicon, and believes that I know it as well, (c) but in fact I don't.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the negative-non-matching-pairs, we employ a progressive mining strategy to obtain the most informative patches possible.
Sentence 2: we sort the negatives for each sample by loss in decreasing order and sample randomly over the top M , where M = max(5, 64e 0.6k 1000 ), where k is the current iteration, i.e., we start with a pool of the 64 hardest samples and reduce it as the networks converge, up to a minimum of 5.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, we derived a general form of the asymptotic free energy, which can be applied to different asymptotic limits.
Sentence 2: the standard asymptotic theory requires a large number N of words per document, compared to the number M of documents and the vocabulary size L. This may be reasonable in some collaborative filtering data such as the Last.FM data used in our experiments in Section 4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 21) becomes more amenable to both computations and statistical analysis and indeed most previous MTL literature has been focused on this setting, either by designing ad-hoc output metrics [33], linear output encodings [34] or regularizers [5].
Sentence 2: in this latter case the problem is cast as that of minimizing the functional A ts f t , f s H (22) where the psd matrix A = (A ts ) T s,t=1 encourages linear relations between the tasks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we focus on parallel SGD methods, which have received considerable attention recently due to their high scalability [6,8,32,13].
Sentence 2: we consider a setting where a large dataset is partitioned among K processors, which collectively minimize a function f . Each processor maintains a local copy of the parameter vector x t ; in each iteration, it obtains a new stochastic gradient update (corresponding to its local data).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that there are other bootstrapping techniques applicable to sequential, dependent data with lower coverage error, such as the double bootstrap [13], block-block bootstrap [1] and Markov or Sieve bootstrap [28].
Sentence 2: the Markov bootstrap has been shown to have a lower coverage error for Markov data than the block bootstrap under certain restricted conditions [10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In its simplest form, an MLN is a set of weighted first-order logic formulas, and can be viewed as a template for generating a Markov network.
Sentence 2: given a set of constants that model objects in the domain, it represents a ground Markov network that has one (propositional) feature for each grounding of each (first-order) formula with constants in the domain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, there exist constants a m,p 0, such that, for every f 2 W m p (⌦), we have (c) E continuously maps B ↵ p,q (⌦) into B ↵ p,q R d for all p 2 (1, 1), q 2 (0, 1] and all ↵ > 0.
Sentence 2: there exist constants a ↵,p,q 0, such that, for every f 2 B ↵ p,q (⌦), we have For detailed conditions on ⌦ ensuring the existence of E, we refer to [8, p. 181] and [9, p. 83].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our test evaluation measurement is the Mean Reciprocal Rank (MRR), which is the average of the reciprocal of the rank of the correct answer.
Sentence 2: for each English word e, we calculated the discriminative score of respective methods when e is paired with each Hebrew word in the test set, and then found the rank of the correct word (1 for the highest).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to ensure spatial coherence and limit computation load, we employ the PBIl algorithm not on single time courses, but on averages of spatial clusters of voxel time courses.
Sentence 2: we first spatially cluster voxel locations as shown in Figure 1, then compute average time course for each cluster and then explore the averages via PBIL for model building.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that the variance parameter of q(w) converges to 1 in one iteration, though underestimates the true posterior variance 1 + D. Intuitively, the convergence speed of w and q(w) suffers from strong coupling between the updates of w and z.
Sentence 2: the update information has to go through a feedback loop w → z → w • • • .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Before stating our convergence results for Algorithm 1, we will need a notion of smoothness relative to our subspaces.
Sentence 2: we say F is L i -Π i -smooth if for all i there are constants L i > 0 such that for all y ∈ im(Π i ), Finally, let F min := min y∈span{im(Πi)}i F (x 0 + y) be the global minimizer of F subject to the constraints from the Π i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: If a block had a large eigenvalue, near 1 or −1 (times its λ max ), then that block could itself be broken into two distinct clusters.
Sentence 2: the clustering C would not accurately capture the cluster structure of the matrix P . Hence, condition (3) amounts to requiring that no other cluster structure is present, in other words that within each block, the Markov chain induced by P mixes well.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the control phase, we update parameters selectively so as not to overwhelm the model with steady-state data.
Sentence 2: we estimate the noise standard deviation σ s for each variable s as the root mean squared error on the training data, and update the model with an example only if its (current) prediction error exceeds 2σ s . 3  Given our model and an initial condition (the T past states, controls, and disturbances for the M AHUs), we optimize the cost of a length-L trajectory with respect to control inputs at every step.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While this result may seem paradoxical, note that the basic phenomenon appears in a classical (non-private) prediction market with a transaction fee, although to our knowledge this observation has not yet appeared in the literature.
Sentence 2: a classical prediction market with budget bound B 1 , trades of size 1, and a small transaction fee α, will still have an α-incentive to participate, and the worst case loss will still be Θ(B 1 ); this loss, however, can be extracted by as few as Θ(1) participants.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The function τ was originally introduced by Alexander [1,2] in the context of exponential inequalities for empirical processes indexed by VC classes of functions, and Giné and Koltchinskii [15] generalized Alexander's results.
Sentence 2: they proved (see [15, p. 1213]) that, for a VC-class of binary-valued functions with VC-dim(F) = d, the ERM solution f n = arg min f ∈F with probability at least 1 − Ks −1 e −s/K for some constants C, K and any s > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although the formula looks similar to the Gaussian case, it is important to note that the conditional embedding operator allows us to compute the conditional expectation of any f ∈ F, regardless of the distribution of the random variable in feature space (aside from the condition that h(•) is in the RKHS on X s , as noted by Song et al.).
Sentence 2: we do not need to assume the random variables have a Gaussian distribution in feature space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, for reasonable choices of W, a straightforward application of these techniques can lead to algorithms with trivial bounds.
Sentence 2: in the case of W consisting of m × n matrices with trace-norm at most r, standard online regret bounds would scale like O r √ T . Since for this norm one typically has r = O √ mn , we get a per-round regret guarantee of O( mn/T ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 2 can be viewed as a bid distribution-dependent analogue of the revenue covering framework [4] and of the smooth mechanism framework [13].
Sentence 2: the quantity µ(D) is the datadepenent analogue of the worst-case µ quantity used in the definition of µ-revenue covering in [4] and is roughly related to the µ quantity used in the definition of a (λ, µ)-smooth mechanism in [13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: With each node i in the taxonomy, we associate a unit-norm weight vector u i ∈ R d . Then, we define the conditional probabilities for a nonroot node i with parent j by We now describe a classifier, called H-BAYES, that is the Bayes-optimal classifier for the H-loss.
Sentence 2: h-BAYES classifies any instance x with the multilabel When no ambiguity arises, we write p i instead of p i (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The output of each neuron g l,i can be formulated by g l,i : where b l ∈ R n l is the bias vector.
Sentence 2: a dendritic layer output f D l is the composition of branches and can be formulated by when d = 1, the dendritic layer is the same as an affine fully connected layer with linear activation functions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, for most large real-world graphs such as the World Wide Web ( [1]), the degree distribution deviates significantly from the Poisson distribution and has high variance.
Sentence 2: most large matrix-completion datasets such as the much publicized Netflix prize dataset and the Yahoo Music dataset exhibit power-law distributed degrees, i.e., the number of vertices of degree d is proportional to d −β for a constant β (Figure 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we remark that even though we are mostly interested in the case where `= O(1), our dependencies on `seem to be better than the results of [3] even in the setting of Gaussian perturbations.
Sentence 2: our running time (as well as our tolerance for error) grows polynomially with n `, whereas the running time of [3] grows with n 3 `.
Relationship: entailment
Domain: NeurIPS
Sentence 1: P is indistinguishable from Q on the RHKS F. This population definition can be approximated using sample expectations.
Sentence 2: given n samples from P as X = {x i ⇠ P, i 2 [n]}, and m samples from Q as Z = {z i ⇠ Q, i 2 [m]}, the following is a finite sample approximation: and the witness function is approximated as: k(x, z j ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our insight, conversely, is to sample extra features from continuous latent distributions rather than certain modes from enumerations, by learning a GAN model (see Figure 2).
Sentence 2: we address two challenges that impede good translation under imbalanced scenarios: (1) through which base-novel class pairs we can translate; and more fundamentally, ( 2) through what objectives for GAN training we can estimate the latent distribution of novel classes with limited observations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We fit the dataset to both a standard 110-topic LDA model and a DiscLDA model with restricted forms of the transformation matrices {T y } y=20 y=1 .
Sentence 2: the transformation matrix T y for class label c is fixed and given by the following blocked matrix This matrix has (C + 1) rows and two columns of block matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At this point, we use the 'kernel' trick: instead of explicitly specifying H or the mapping Φ, we take the approach of specifying inner products.
Sentence 2: given any positive definite kernel K : S × S → R, it is well known (Mercer's theorem) that there exists a Hilbert space H and Φ : S → H such that K(x, y) = Φ(x), Φ(y).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, for a matrix A, let A + denote its (uniquely defined) Moore-Penrose pseudoinverse.
Sentence 2: for two vectors x, y ∈ R n , and the degree matrix D G for a graph G, we define the degree-weighted inner product as if a vector x has unit norm, then x T D G x = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Spatial interactions in natural images are often well-described by Markov models [13] and first-order stationary Markov processes are known to have sinusoidal KLT bases [29].
Sentence 2: for the LUV color channels, our filters are similar to the discrete cosine transform (DCT) bases that are often used to approximate the KLT.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Fortunately, V ↵ has a closed form, and W ↵ has been studied before [2].
Sentence 2: if ↵ > 1 2 , then V ↵ (q) = c ↵ q ↵ , where c ↵ is a constant with a closed form given by Lemma B.15.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose a direct approach (called DSPCA in what follows) that improves the sparsity of the principal components by directly incorporating a sparsity criterion in the PCA problem formulation and then relaxing the resulting optimization problem, yielding a convex optimization problem.
Sentence 2: we obtain a convex semidefinite programming (SDP) formulation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assess the predictive performance of the collected samples from the various algorithms by computing a Monte Carlo estimate of the predictive log-likelihood of a held-out data point under the model.
Sentence 2: for a held out point y * we have   m is the probability of choosing atom m for sample t. We did not use the Rao-Blackwellized estimator to compute Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Also, let X c (t) denote the set of all nodes that are in an infected state at time t in cascade c. We know the infection of each node was the result of an unknown, previously infected node to which it is connected, so the component of the likelihood function for each infection will be dependent on all previously infected nodes.
Sentence 2: the likelihood function for a fixed given A is The likelihood function is composed of two terms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a related work Neu and Rosasco [2018] consider the problem of linear least squares regression and show that the iterates produced by GD on least squares objective are related to the solutions of ridge regression.
Sentence 2: for any given regularization parameter of ridge regression, Neu and Rosasco [2018] show that there exists a weighing scheme for GD iterates that is exactly equal to the ridge solution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we begin with a precise formulation of the problem, and then develop some key properties of the regularizer and loss function.
Sentence 2: we define a notion of decomposability for regularizing functions r, and then prove that when it is satisfied, the error ∆ = θ − θ * of the regularized M -estimator must satisfy certain constraints We use these constraints to define a notion of restricted strong convexity that the loss function must satisfy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose Algorithm 1 is run with 1/η > max t max x∈X |∇ T t x t |.
Sentence 2: then, for every H 0 the regret is bounded as where r = rank( T t=1 ∇ t ∇ T t ) ≤ d and we can use the theorem to show that as long as we bound |∇ f (x t ) T x t | by a constant-a significantly weaker requirement than assuming bounds on the diameter of X and on the norms of the gradients-one can find a norm • H for which the quantities D H and G H are properly bounded.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta integrators.
Sentence 2: we prove that under Lipschitz-gradient, convexity and order-(s + 2) differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of O(N −2 s s+1 ), where s is the order of the Runge-Kutta numerical integrator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Interestingly, this estimator, f = max(ID(K))−1 , appears to be the estimator for a hypergeometric random sample of size D s = max(ID(K)) − 1.
Sentence 2: suppose we randomly pick D s balls (without replacement) from a pool of D balls and we observe that k balls are red; then a natural (and unbiased) estimator for the total number of red balls would be D D s k ; here k = k − 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, where  The parameters for FastGRNN: ) are trained jointly using projected batch stochastic gradient descent (b-SGD) (or other stochastic optimization methods) with typical batch sizes ranging from 64 − 128.
Sentence 2: the optimization problem is given by: where L denotes the appropriate loss function (typically softmax cross-entropy).
Relationship: entailment
Domain: NeurIPS
Sentence 1: ConQ[sa] is a restriction of Q[sa] obtained by intersecting each cell in its domain, dom(Q[sa]), with [s → a].
Sentence 2: conQ[sa] is a partition function defined on some partition of [s → a] (rather than all of Θ), and represents Q-values of cells that are consistent with [s → a].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the plain robust optimization formulation in Algorithm 2, even the optimal policy π is not considered safe in this example.
Sentence 2: the robust return of π is min ξ ρ(π , ξ) = −9, while the optimistic return of π B is max ξ ρ(π B , ξ) = +10.
Relationship: entailment
Domain: NeurIPS
Sentence 1: People like to retweet an interesting post soon after it is created and retweeted by others, but may gradually lose interest, so the intervals between retweets become longer over time.
Sentence 2: the stream begins in a self-exciting state, in which previous retweets increase the intensities of future retweets, but eventually interest dies down and events are less able to excite one another.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The magnitude of Γ st is a scalar multiple of the correlation of X s and X t conditioned on X \{s,t} , and encodes the strength of the edge (s, t).
Sentence 2: the sparsity pattern of Γ st reflects the edge structure of the graph: Γ st = 0 if and only if X s ⊥ ⊥ X t | X \{s,t} .
Relationship: entailment