sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Each edge (u, v) ∈ E G (T , T ) whose label has not been previously queried is then part of the test set, and its label will be predicted as Ŷu,v ← π T (u, i ) 11).
Sentence 2: using the path obtained by concatenating Path T (u, i ) to edge (i , i ) to Path T (i , v).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the noisy ERM algorithm in (33), choosing N i to be an exponential random variable with mean b i , we have where i o = arg min i L µ (w i ).
Sentence 2: choosing b i = i 1.1 /n 1/3 , we have Without adding noise, the ERM algorithm applied to the above case when Compared with (35), we see that performing noisy ERM may be beneficial when we have high-quality prior knowledge of w o and when k is large.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, this confidence parameter appears in the analysis of the number of queries, so that we have a natural trade-off between the number of mistakes and the number of label requests.
Sentence 2: under Assumptions 2 and 5, this procedure achieves an expected excess number of mistakes MT − M * T ≤ ⌊log(T )⌋ i=1 δ i 2 i , where ǫ T = T − α (α+2)(m+1) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, let L be a continuous convex loss function satisfying some minor regularity conditions.
Sentence 2: then it was shown for universal RKHS's and stritly positive nullsequences (λ n ) satisfying a regularity condition that the following statements hold for all ε > 0 and n → ∞: this result holds for L1-SVM's.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that unlike traditional approaches to novelty detection, which must use generative models or one-class classifiers in the absence of appropriate discriminative data, our dependence on object hierarchy provides discriminative data as a by-product.
Sentence 2: after the recognition by a parent-node classifier, we may use classifiers trained to discriminate between its children to implement a discriminative novelty detection algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since x 1 and x 2 are conditionally independent given the latent content variable z, (2) This suggests us learning an auto-encoder model.
Sentence 2: a style transfer from x 2 to x 1 involves two steps-an encoding step that infers x 2 's content z ∼ p(z|x 2 , y 2 ), and a decoding step which generates the transferred counterpart from p(x 1 |y 1 , z).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note our techniques here are different from those used for the decreasing-step-size SGLD [5], which interestingly result in similar convergence patterns.
Sentence 2: by adapting the same techniques used in the previous sections, we establish conditions on the step size sequence to ensure asymptotic convergence, and develop theory on their finite-time ergodic error as well.
Relationship: entailment
Domain: NeurIPS
Sentence 1: KMUCB's estimation procedure leverages information across threshold values, where observations from higher thresholds are utilized to estimate mean payoffs for lower thresholds; information does not flow in the other direction, however, as a result of the censoring assumption.
Sentence 2: for a given threshold index j, KMUCB tracks the arm elimination process as follows: for any threshold values below j, KMUCB believes that we have determined the best arm, and plays that arm constantly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, without any modification on the pre-trained softmax neural classifier, we obtain a generative classifier assuming that a class-conditional distribution follows the multivariate Gaussian distribution.
Sentence 2: we define C class-conditional Gaussian distributions with a tied covariance ⌃: , where µ c is the mean of multivariate Gaussian distribution of class c 2 {1, ..., C}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (K2) The kernel function K and its partial derivative satisfies condition K 1 in [16].
Sentence 2: let We require that K satisfies for some positive number A, v, where N (T, d, ) denotes the -covering number of the metric space (T, d) and F is the envelope function of K and the supreme is taken over the whole R d . The A and v are usually called the VC characteristics of K. The norm F L2(P ) = sup P |F (x)| 2 dP (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we report the main theoretical results of the paper.
Sentence 2: we derive a performance bound for LSTD-RP in the Markov design setting, i.e., when the LSTD-RP solution is compared to the true value function only at the states belonging to the trajectory used by the algorithm (see Section 4 in [11] for a more detailed discussion).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A function φ : R → R is (M, δ, θ)-quasiperiodic if there exists a function φ : R → R which is periodic with period θ such that |φ(x) − φ(x)| < δ for all x ∈ [−M, M ].
Sentence 2: any periodic function with period θ is (M, δ, θ)-quasiperiodic for all M, δ > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Adaptor grammars extend PCFGs by inserting an additional component called an adaptor into the PCFG recursion (Equation 1).
Sentence 2: an adaptor C is a function from a distribution G to a distribution over distributions with the same support as G. An adaptor grammar is a sextuple (N, W, R, S, θ, C) where (N, W, R, S, θ) is a PCFG and the adaptor vector C is a vector of (parameters specifying) adaptors indexed by N . C A maps a distribution over trees T A to another distribution over T A , for each A ∈ N . An adaptor grammar associates each symbol with two distributions G A and H A over T A .
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 5) by matrix inverse is a consistent algorithm if (T , H) are core w.r.t.
Sentence 2: f and The proof can be found in Appendix C. While Theorem 1 guarantees consistency, we have not yet illustrated the benefits of using f . we want to characterize the sizes of the minimal core tests/histories, as they determine the number of model parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, note that the weighted averages of the m si in (10) and the Csi in (11), with weights given by ( 9), can be written as unweighted averages of the corresponding products π si m si and π si Csi .
Sentence 2: each local estimate can be written as the ratio of two local estimates that converge to the correct values at the same exponential rate (as shown in the previous section).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For fixed truncation levels L * and K * , the algorithm for posterior computation alternates between a simple and efficient simulation smoother step [24] to update the state space formulation of the nGP, and standard Gibbs sampling steps for updating the parametric components of the model.
Sentence 2: considering the observations (y i , t i ) for i = 1, ..., T : A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since u i and v i play exactly the same role, it is also easy to see that the term approximation is always symmetric in u i and v i .
Sentence 2: let us suppose that q (s i , u i , v i ) and consequently q \i (s i , u i , v i ) factorizes into independent terms for s i , u i , and v i , e.g., we can write By initializing ti (s i , u i , v i ) = 1, we have q(z) ∝ t 0 (z) and the factorization of q \i (s i , u i , v i ) follows directly from the factorization of t 0 (z) into independent terms for s, u, and v. for the first EP step, the factorization can be guaranteed.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Bayesian updating provides a principled and coherent approach to inference for probabilistic models [23], but is predicated on the model class being true.
Sentence 2: for an observation x and a generative model F θ (x) parametrized by a finite-dimensional parameter θ ∈ Θ, then for some parameter value θ 0 ∈ Θ it is that x ∼ F θ0 (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our LIRPG algorithm is also compatible with such actor-critic architectures.
Sentence 2: for our experiments on Atari games we used a reasonably state of the art advantage actor-critic (A2C) architecture, and for our experiments on Mujoco domains we used a similarly reasonably state of the art proximal policy optimization (PPO) architecture.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unfortunately, we show that it is not possible to obtain a algorithm where the degree of the polynomial does not depend on k even when the maximum parent set size is limited to 2, that is, there is no algorithm with running time g(k) poly(n) for any function g, unless the widely believed complexity assumption FPT = W [1] fails.
Sentence 2: we show that Bayesian network structure learning with bounded vertex cover number is W [1]-hard when restricted to instances with parent set size 2, implying the above claim.
Relationship: entailment
Domain: NeurIPS
Sentence 1: R with the properties: ] and all integer m 0.
Sentence 2: 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.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As for the computation of the distribution of words given the hidden units, our feed-forward neural network approach leaves us free to use other conditionals than the V -way multinomial logistic regression implied by the Replicated Softmax.
Sentence 2: we instead opt for a hierarchy of binary logistic regressions, organized in a binary tree where each leaf corresponds to a word of the vocabulary.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since our boosting procedure does not change the distribution over unlabeled examples, it offers two advantages: (1) Agnostic weak learning may be defined with respect to a single distribution µ over unlabeled examples, and (2) The weak learning algorithms may be active (or use membership queries).
Sentence 2: the agnostic weak learning hypothesis for C and µ is that for any f : given examples from D = µ, f , the learner will output a γ-optimal classifier for C. The advantages of this new definition are: (a) it is not with respect to every distribution on unlabeled examples (the algorithm may only have guarantees for certain distributions), and (b) it is more realistic as it does not assume noiseless data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Comparing this result to the known label complexity of CAL, which is (from [7]) we see that the major factor, namely the O (θ(ǫ) (d log(θ(ǫ)) + log(log(1/ǫ)/δ))) factor, is now inside the argument to the cost function c(•).
Sentence 2: when this cost function is sublinear, we expect this bound to be significantly smaller than the cost required by the original fully-sequential CAL algorithm, which uses batches of size 1, so that there is a significant advantage to using this batch-mode active learning algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For s ≥ 0, define the moment generating function ϕ ].
Sentence 2: then we have if Group-privacy Profiles Recall the kth group privacy profile of a mechanism M is defined as A standard group privacy analysis 5 immediately yields δ M,k (ε) ≤ (e ε − 1)δ M (ε/k)/(e ε/k − 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we propose a novel Dual-Agent Generative Adversarial Network (DA-GAN) for profile view synthesis, where the dual agents focus on discriminating the realism of synthetic profile face images from a simulator using unlabled real data and perceiving the identity information, respectively.
Sentence 2: the generator needs to play against a real-fake discriminator as well as an identity discriminator simultaneously to generate high-quality faces that are really useful for unconstrained face recognition.
Relationship: entailment
Domain: NeurIPS
Sentence 1: According to Theorem 4 of Zhao and Yu (2006) 2) , when samples follow the sparse model as in Assumptions 1-4 and λ ∼ = d −τ for 1/4 ≤ τ ≤ 1/2, the representation φ D (x) reconstructs the true sparse representation a of sample x with a small error.
Sentence 2: when τ = 1/4 (i.e., λ ∼ = d −1/4 ) in Theorem 3, the failure probability δ t,λ ∼ = e − √ d on the margin is guaranteed to become sub-exponentially small with respect to dimension d and is negligible for the high-dimensional case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the subspace estimates are derived from the available samples only, or their embedding, the learning problem is concerned with the quality of the computed subspace as an estimate of S ρ (the true span of the support of ρ).
Sentence 2: it may be of interest to understand the quality of these estimates, as a function of the algorithm's parameters (typically the dimensionality of the estimated subspace).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to be able to deal with an infinite number of sources, we place a BNP prior over the binary matrix S that contains all variables s tm .
Sentence 2: we assume that S ∼ mIBP(α, β 0 , β 1 ), i.e., S is distributed as a mIBP [23] with parameters α, β 0 and β 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3 Starting from this edge label assignment, which is consistent with the two-clustering associated with the USPS labels, we generated a p-stochastic label assignment by flipping the labels of a random subset of the edges.
Sentence 2: we used the three following synthetic datasets: DELTA0: No flippings (p = 0), 1,000 nodes and 9,138 edges; DELTA100: 100 randomly chosen labels of DELTA0 are flipped; DELTA250: 250 randomly chosen labels of DELTA0 are flipped.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The core idea of Greedy-MIPS is to iteratively traverse (j, t) entries of Z in a greedy sequence and collect newly observed indices j into C(w) until |C(w)| = B.
Sentence 2: if r = ⇡(j, t|w), then (j, t) entry is visited at the r-th iterate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a lot of applications, e.g., influence maximization in social networks [15], community detection in graphs [13], etc., we are interested in selecting a small subset of vertices from a massive graph that "cover" in some sense a large fraction of the graph.
Sentence 2: in section 6, we consider two fundamental set cover problems: Dominating Set and Vertex Cover problems.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proposition 3.4 (Non-Convex Analysis -Parameterized Surrogates).
Sentence 2: let us make the same assumptions as in Proposition 3.3, and let us assume that the functions ḡn are parameterized by some variables κ n living in a compact set K of R d . ḡn can be written as g κn , with κ n in K. Suppose there exists a constant K > 0 such that |g κ (θ) − g κ ′ (θ)| ≤ K κ − κ ′ 2 for all θ in Θ and κ, κ ′ in K. Then, every limit point θ ∞ of the sequence (θ n ) n≥1 is a stationary point of f -that is, for all θ in Θ, Finally, we show that our non-convex convergence analysis can be extended beyond first-order surrogate functions-that is, when g n does not satisfy exactly Definition 2.1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From the above theorem, the sequence {f (α k ) : k ∈ N} is monotonically increasing and the iteration complexity of SPGM is O(L/ε) for finding an ε-optimal solution.
Sentence 2: in [18,17], Nesterov proposed an efficient smooth optimization method for solving convex programming problems of the form min where g is a convex function with Lipschitz continuous gradient, and U is a closed convex set in R n . suppose there exists L > 0 such that ∇g(x) − ∇g(x The smooth optimization approach needs to introduce a proxy-function d(x) associated with the set U .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We say that the probability distribution of X G is a Fiedler random field if it factorizes as where Z(θ) is the partition function.
Sentence 2: a FRF is a Gibbs-Boltzmann distribution over graphs, with potential functions defined for each node pair {u, v} along with some neighboring subgraph G uv .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we give a specific accelerated scheme for a type of conic geometric optimization problems [25], e.g., the matrix Karcher mean problem.
Sentence 2: the loss function of the Karcher mean problem for a set of N symmetric positive definite (SPD) matrices {W i } N i=1 is defined as where X ∈ P := {Z ∈ R d×d , s.t., Z = Z T 0}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define the regularized estimator as follows: θλ The goal of this paper is to choose good values of λ and analyze the subsequent impact on performance.
Sentence 2: we wish to minimize the relative risk: which is the difference in risk (averaged over the training data) between the regularized and unregularized estimators; L n (λ) < 0 is desirable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Such problems can be viewed as POMDPs with belief-dependent rewards [3] and their applications include active sensing [7], attention mechanisms for multiple-object tracking [22], as well as online summarisation of massive data from time-series [4].
Sentence 2: we discuss the restless bandit [24] associated with the discrete-time Kalman filter [19].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proofs of these bounds essentially follow from a very recent analysis of Thompson sampling, and the implied regret bounds are the same as those established for Thompson sampling.
Sentence 2: since Ψ * t ≤ Ψ t (π TS ) where π TS is the Thompson sampling policy, it is enough to bound Ψ t (π TS ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Comparison with other classifiers.
Sentence 2: the AUC of our proposed method is 40.0% (22.1%), which is better than 32.7% (15.6%) of the softmax classifier and 32.9% (17.1%) of the Euclidean distance classifier after all new classes are added in the first (second) experiment.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then we start from ⌘ t = 1 and use backtracking line search to find the optimal ⌘ t 2 (0, 1] such that the Armijo condition [1] holds.
Sentence 2: given a constant µ 2 (0.9, 1), we update ⌘ t = µ q from q = 0 and find the smallest integer q such that .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Surprisingly, we show how TCL can be related to a nonlinear ICA model, when ICA is redefined to include temporal nonstationarities.
Sentence 2: we show that TCL combined with linear ICA estimates the nonlinear ICA model up to point-wise transformations of the sources, and this solution is unique -thus providing the first identifiability result for nonlinear ICA which is rigorous, constructive, as well as very general.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We often think of h as a distribution over topics.
Sentence 2: the density of h ∈ ∆ k−1 (where the probability simplex ∆ k−1 denotes the set of possible distributions over k outcomes) is specified by: and α 0 := α 1 + α 2 + • • • + α k . Intuitively, α 0 (the sum of the "pseudocounts") characterizes the concentration of the distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: TWF addresses this challenge by discarding terms having too high of a leverage on the search direction; this is achieved by regularizing the weights ν i via appropriate truncation.
Sentence 2: where ∇ tr (•) denotes the truncated gradient given by for some appropriate truncation criteria specified by E i 1 (•) and E i 2 (•).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Before defining the Lovász extension, we need the concept of a chain of subsets of [n]: A maximal chain is one where p = n. For a maximal chain, we have and there is a unique associated permutation π : A nice way to construct this combination is the following random process: choose a threshold τ ∈ [0, 1] uniformly at random, and consider the level set S τ = {i : x i > τ }.
Sentence 2: the sets in the required chain are exactly the level sets which are obtained with positive probability, and for any such set A i , we have x = E τ [χ Sτ ].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our algorithm starts with a particular MIS P i , denoted by x 0 . x t+1 , which is a neighbor of x t , is obtained by replacing some lower-weight vertices in x t with higher-weight vertices under the constraint of always being an independent set.
Sentence 2: we first reduce x t by removing a proportion q of lower-weight vertices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Fenchel-Young inequality states that for any w and θ we have that f (w) + f (θ) ≥ w, θ . Sub-gradients play an important role in the definition of the Fenchel conjugate.
Sentence 2: the following lemma, whose proof can be found in Borwein and Lewis [2006], states that if λ ∈ ∂f (w) then the Fenchel-Young inequality holds with equality.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our results show that our 'knowledge transfer' approach is indeed successful.
Sentence 2: our discriminator-trained G outperforms the MLE-trained G by 1.7% on recall@5 on the VisDial dataset, essentially improving over state-of-the-art [7] by 2.43% recall@5 and 2.67% recall@10.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A set function F mapping sets of recommended articles A to real values (e.g., the total information covered by A) is monotone and submodular if and only if respectively, for all articles a and sets A ✓ B.
Sentence 2: since A is smaller than B, the benefit of adding a to A is larger than the benefit of adding a to B. Submodularity provides a natural framework for characterizing diminishing returns in information coverage, since the gain of adding a second (redundant) article on a topic will be smaller than the gain of adding the first.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the exponential family the log-likelihood ' @( A0 2 ¦ 9 % © is directly related to a Bregman normal Poisson  It can be shown under fairly general conditions that u ¦ 9 © ( # ¤ ¦ 9 © .
Sentence 2: application of these identities implies that the negative log-likelihood of a point can be expressed through a Bregman distance [1,3]: negative log-likelihood can always be written as a Bregman distance plus a term that is constant with respect to and which therefore can be ignored.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore the partition should stop when the points inside a subrectangle are approximatly uniformly scattered.
Sentence 2: we stop the partition when further Figure 1: Left: a sequence of binary partition and the corresponding tree representation; if we encode partitioning information (e.g., the location where the split occurs) in the nodes, there is a one to one mapping between the tree representations and the partitions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One of the key contributions of [16] is the definition of a kernel specifically suited to linear system identification leading to an estimator with favorable bias and variance properties.
Sentence 2: it is easy to see that if the autocovariance of f is proportional to W p , the variance of f (t) is zero at t = 0 and tends to ∞ as t increases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to achieve sparsity, the most natural method is to round small coefficients (whose magnitudes are below a threshold θ > 0) to zero after every K online steps.
Sentence 2: if i/K is not an integer, we use the standard SGD rule (3); if i/K is an integer, we modify the rule as: where: In other words, we first perform a standard stochastic gradient descent, and then round the coefficients to zero.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that we could have avoided this loss by bounding the 1 norm of the vector at the cost of a more technically involved proof.
Sentence 2: one should bound the 1 norm of the embedding of the vector into a space where the 1 and 2 norms behave similarly, up to a multiplicative distortion of 1±ε.
Relationship: entailment