sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: This leads to the Stochastic Gradient Method (SGM).
Sentence 2: starting from an initial estimate x 0 ∈ K obeying the constraints, SGM iteratively applies the following updates at every iteration t, the current iterate x t is updated by adding µ t g t , where g t is an unbiased estimate of the gradient ∇F (x t ) and µ t is the learning rate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For all activation functions σ i , i = 1, 2, ..., d − 1, there exist positive constants q(i) and Q σ (i) such that for any 0 ≤ Note that Assumptions 1 and 2 hold for a variety of commonly used activation functions including ReLU, sigmoid, softPlus and tanh.
Sentence 2: in Assumption 2, the existence of the constants q(i) and Q σ (i) are more important than the particular values they take, which affect only the constant terms in our bounds presented later.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The offline approximation algorithm for ranking with submodular valuations will be a crucial tool in our analysis of online submodular set cover.
Sentence 2: this offline algorithm can viewed as constructing the best single permutation S for a sequence of objectives F 1 , F 2 . . . F T in hindsight (i.e., after all the objectives are known).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The conditions of Theorem 2 hold for many cases in practice, in particular in the case of the exponential loss (Φ = exp) or the logistic loss (Φ(−x) = log 2 (1 + e −x )).
Sentence 2: linear convergence is guaranteed in those cases since both the exponential and logistic losses are strongly convex over a compact set containing the converging sequence of α t s. MDEEPBOOSTSUM(S = ((x 1 , y 1 ), .
Relationship: entailment
Domain: NeurIPS
Sentence 1: (X, Y ) = ( X, Ŷ ) whenever ν k are picked so that ) and .
Sentence 2: choice of ν 3 = (3ζ(U,V )) p (µ(Ω (3) )) 1−p and )) 1−p for any p ∈ [0, 1] in these respective intervals guarantees exact recovery.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we address the finer and more challenging question of determining, under the general LSBM, the minimal number of misclassified items given the parameters of the model.
Sentence 2: for any given s = o(n), our goal is to identify the set of parameters such that it is possible to devise a clustering algorithm with at most s misclassified items.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From a mathematical point of view, the relevance of Fourier theoretic ideas in all these cases is a direct consequence of equivariance, specifically, of the fact that the {T s g } g∈G operators form a representation of the underlying group, in the algebraic sense of the word [20].
Sentence 2: it has been shown that whenever there is a compact group G acting on the inputs of a neural network, there is a natural notion of Fourier transformation with respect to G, yielding a sequence of Fourier matrices {F s } at each layer, and the linear operation at layer s will be equivariant to G if and only if it is equivalent to multiplying each of these matrices from the right by some (learnable) filter matrix H s [7].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We apply our results to the K-MAX problem and expected utility maximization problems.
Sentence 2: for K-MAX, we provide the first polynomial-time approximation scheme (PTAS) for its offline problem, and give the first Õ( √ T ) bound on the (1− )-approximation regret of its online problem, for any > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Branching: The branching operation in our method is carried out using the property of a simplex.
Sentence 2: since, in a n-simplex, any r + 1 vertices are not on a r − 1-dimensional hyperplane for r ≤ n, any n-simplex can be divided as S = ∪ p i=1 S i , where p ≥ 2 and S i are n-simplices such that each pair of simplices S i , S j (i ̸ = j) intersects at most in common boundary points (the way of constructing such partition is explained in Section 4.2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Cue-based model, in contrast, is unsure: it assigns roughly equal probabilities of helping or hindering to these cases (bar heights near 0.5).
Sentence 2: the Cue-based model is effective at inferring simple object goals of maze-world agents, but is generally unable to distinguish between the more complex goals of helping and hindering.
Relationship: entailment
Domain: NeurIPS
Sentence 1: • We propose new losses that effectively address the aforementioned challenges.
Sentence 2: we propose a weighted cross entropy loss for classification that accounts for the imbalance in the impact of misclassification for background class as opposed to object classes (Section 3.2), a teacher bounded regression loss for knowledge distillation (Section 3.3) and adaptation layers for hint learning that allows the student to better learn from the distribution of neurons in intermediate layers of the teacher (Section 3.4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The advantages of such a black-box framework are two fold: theoretically, it highlights a structural connection between the two problems; practically, it allows one to simply plug in any of the vast majority of solvers available for SLR and directly get an algorithm for SPCA with provable guarantees.
Sentence 2: • For hypothesis testing: we match state of the art provable guarantee for computationally efficient algorithms; our algorithm successfully distinguishes between isotropic and spiked Gaussian distributions at signal strength ✓ & q k 2 log d n . This matches the phase transition of diagonal thresholding [22] and Minimal Dual Perturbation [4] up to constant factors.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similarly, we would expect that if the first two parts of condition 5 hold (about X and Y ) then the last one about ω would also hold as it is predicting outcomes Y .
Sentence 2: the last part of condition 5 is essentially a requirement on our ω-leaner base method that it's not doing anything strange like adding unnecessary noise to Y thereby making it have fewer moments.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We alleviate this problem by imposing a weight w i to each observation in the constraint.
Sentence 2: this weight is chosen to be , where f is a reasonable guess of the final minimizer f . Our goal is to weight the Hinge loss in the constraint, w i H −ε (y i f (x i )), so that it approximates the indicator function 1{y i f (x i ) < −ε}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a survey question that asks to compare Proto and Criticism Condition and Raw Condition, a subject added that "[Proto and Criticism Condition resulted in] less confusion from trying to discover hidden patterns in a ton of images, more clues indicating what features are important".
Sentence 2: in a question that asks to compare Proto and Criticism Condition and Proto Only Condition, a subject said that "The addition of criticisms made it easier to locate the defining features of the cluster within the prototypical images".
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , T T ), training/validation data with labels, λ   Even though we can solve (IP) via its LP relaxation, the resulting LP can be too large in practical applications for any general-purpose LP solver.
Sentence 2: the number of variables and constraints is roughly O(T × |T max | + N × T × K max ), where T is the number of trees; |T max | is the maximum number of nodes in a tree; N is the number of examples; K max is the maximum number of features an example uses in a tree.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The nodes of DDTs correspond to hyperrectangles (cells) in [0, 1] d (see Figure 1).
Sentence 2: given a hyperrectangle A = d r=1 [a r , b r ], let A s,1 and A s,2 denote the hyperrectangles formed by splitting A at its midpoint along coordinate s. define A s,1 = {x ∈ A | x s ≤ (a s + b s )/2} and A s,2 = A\A s,1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we define C ⇤ := max(kx ⇤ xk, ku ⇤ ūk), then the unique solution x( ) of ( 5) satisfies kAx( If in addition the parameter 10C⇤ kdk min( P , D ) , then we have In practice, we solve (5) approximately, using an algorithm whose complexity depends on the threshold ✏ for which the objective is accurate to within ✏.
Sentence 2: we seek x such that where the left-hand inequality follows from the fact that f is strongly convex with modulus 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now apply a version of Fano's lemma [11] to the set of distributions P T θ for θ ∈ Θ(δ).
Sentence 2: using the proof of Lemma 3 in [11]  Each term KL(P θi (X t )||P θ i (X t )) is at most the KL divergence g(δ) between Bernoulli variates with parameters 1/2 + δ and 1/2 − δ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the parametrization described in Subsection 2.1, we hope to cooperatively optimize GGN and LON, simultaneously estimating 3D camera pose, 3D room layout, and 3D object bounding boxes, in the sense that the two networks enhance each other and cooperate to make the definitive estimation during the learning process.
Sentence 2: we propose three cooperative losses which jointly provide supervisions and fuse 2D/3D information into a physically plausible estimation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We split z ∈ R d into local copies of x i ∈ R d , while enforcing the equality constraints x i = z for all i.
Sentence 2: we consider the following reformulation of (1.1) where h(z) Our algorithm uses the Lagrangian relaxation of the equality constraints, and at each iteration a (possibly non-uniformly) randomly selected primal variable is optimized, followed by an approximate dual ascent step.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can introduce a Poisson point process to control the duration of non-recombinant inheritance.
Sentence 2: given that C i,t = k, then with probability e −dr +(1−e −dr )π kk , where d is the physical distance between two loci, r reflects the rate of recombination per unit distance, and π kk is the self-transition probability of ancestor k defined by HMDM, we have C i,t+1 = C i,t ; otherwise, the source state (i.e., ancestor chromosome k) pairs with a target state (e.g., ancestor chromosome k ) between loci t and t+1, with probability (1−e −dr )π kk .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Such a self-reconstruction task can help narrow the domain gap (see more explanation below).
Sentence 2: assuming that the forward and reverse projections have the same importance for ZSL, our DIPL model solves the following optimization problem: where W ∈ R d×k is a projection matrix from the semantic space to the feature space, and λ, γ are the regularization parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For ¬ ¼, let ´Þ ¬µ consist of the elements of Ù ¾ such that ´Ù Þµ ¬ (that is, the ¬-ball centered at Þ).
Sentence 2: then (any «-packing must have at most ´ ¬ «µ elements within distance ¬ of Þ.)
Relationship: entailment
Domain: NeurIPS
Sentence 1: A scoring rule is a function S : P × O → R where, for all P ∈ P, P ∈ argmax Q∈P E X∼P S(Q, X).
Sentence 2: if you are paid S(P, X) upon stating belief P ∈ P and outcome X occurring, then you maximize your expected utility by stating your true belief.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We do so by "flipping" the edges in P to obtain a matching with higher weight.
Sentence 2: let M = M − (P ∩ M ) + (P ∩ M * T ) be the matching containing all edges in M except the ones in P , which are replaced by the edges in P ∩ M * T . It is easy to see that M is a matching in T e (t), and that w(M ) > w(M ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: More precisely, let δ (l)→i be the message sent from subnetwork l to X i and δ i→(l) the opposite message.
Sentence 2: then we define the COMPOSE message passing scheme as follows: each subnetwork computes its local pseudo-max-marginals over each of the individual variables, given, as input, the pseudo-max-marginals over the others.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use π to denote the NP update to distinguish it from the conventional Bayesian posterior π(θ|x noting that in general the nonparametric posterior π(θ | x 1:n ) will be different to the standard Bayesian update as they are conditioning on different states of prior knowledge.
Sentence 2: as stated above, π(θ|x 1:n ) assumes artificially that F 0 ∈ F Θ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each push at level i uncovers a collection of length-(i + 1) paths terminating at S -for example, in the figure, we have uncovered all length 2 and 3 paths, and several length 4 paths.
Sentence 2: the crucial observation is that each uncovered path of length i starting from a node v is accounted for in either q i v or r i v . in Figure 1, all paths starting at solid nodes are stored in the estimates of the corresponding states, while those starting at blurred nodes are stored in the residue.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The observed data vector ¡ is then given by a linear combination of the $© , plus additive isotropic Gaussian noise.
Sentence 2: ¡ ¡ &% § (' , 1 Here, it must be stressed that in these low-level neural network models, the hidden variables that the neurons represent are not what we would typically consider to be the 'causal' variables of a visual scene.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For infinite capacity Q, R(✓ ; Q) is zero for all ✓ 2 ⇥, and the objective reduces to maximum likelihood.
Sentence 2: when Q is the set of Gaussian distributions (as is the case in the standard VAE), then a Gaussian variational family regularizes the true posterior p ✓ (z | x) toward being Gaussian [10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This suggests a modification of Algorithm 1, in which one-step co-training is carried out on the problem of extracting anchors rather than the problem of labeling web pages.
Sentence 2: one might map f 1 's predictions from web pages to anchors, by giving a positive label to anchor a iff a links to a page x such that f 1 (x) = 1; then use WL 2 algorithm A 2 to learn a predicate p 2 ; and finally, map the predictions of p 2 from anchors back to web pages.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let µ denote a vector of marginal probabilities (also called mean parameters) for each node and edge of the MRF.
Sentence 2: µ includes µ i (x i ) ∀i ∈ V and µ ij (x i , x j ) ∀(i, j) ∈ E. The set of µ that arises from some joint distribution p is referred to as the marginal polytope: The MAP problem is then equivalent to solving the following LP: It can be shown that there always exists a maximizing solution µ which is integral and gives the optimal x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The toolkit for proving our results derives from the deep literature of Euclidean functionals, see, (Steele, 1997;Yukich, 1998).
Sentence 2: our strong consistency result uses a theorem due to Redmond and Yukich (1996) that essentially states that any quasi-additive power-weighted Euclidean functional can be used as a strongly consistent estimator of Rényi entropy (see also Hero and Michel 1999).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Of course, C ⋆ is not known, and it is not possible to exhaustively consider all possible values in the domain of C. Instead, we examine only a finite number of possible values of C over a fine discretization of its domain.
Sentence 2: let N ǫ/2 (S r−1 ) denote a finite ǫ /2-net of the r-dimensional ℓ 2 -unit sphere; for any point in S r−1 , the net contains a point within an ǫ /2 radius from the former.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Impressive progress has been made in a variety of application domains, such as speech, natural language and image processing, thanks to the development of new learning strategies [15,53,30,45,26,3] and of new architectures [31,44,46,23].
Sentence 2: these architectures tend to become ever deeper, with hundreds of layers, each of which containing hundreds or even thousands of units.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can then plot the cumulative regret In the first experiment, the baseline reward is nonlinear.
Sentence 2: we generate rewards using r t (s t,at , st , a t ) = θ T s t,at + 2I(|[s t ] 1 | < 0.8) + n t where n t = N (0, 1) and θ ∈ R 8 is a fixed vector listed in supplement section 1.1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Letting |I j | = n, we now simplify the objective function: where the last two equalities follow first by applying our derivation for λ and then the fact that qj sums to one.
Sentence 2: this shows that we can choose an arbitrary index i ∈ I j and propagate the value g i + β j log qi + β j log n. for numerical reasons we choose the one that maximizes qi .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, the key intermediate quantity for filtering is the "extended and marginalized" belief In many models (including Hidden Markov Models (HMMs) and Kalman filters), the extended state p t is linearly related to the predictive state q t -a property we exploit for our framework.
Sentence 2: for some linear operator W . For example, in a discrete system ψ t can be an indicator vector representing the joint assignment of the next k observations, and ξ t can be an indicator vector for the next k + 1 observations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast, our result is an improved It is worth emphasizing that the proposed homotopy smoothing technique is different from recently proposed homotopy methods for sparse learning (e.g., 1 regularized least-squares problem [20]), though a homotopy strategy on an involved parameter is also employed to boost the convergence.
Sentence 2: the involved parameter in the homotopy methods for sparse learning is the regularization parameter before the 1 regularization, while the parameter in the present work is the introduced smoothing parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: from pseudo-inputs , where K mm is a M ×M covariance matrix.
Sentence 2: from (5) we have , where u = 1 − Yf , K mn is the cross-covariance matrix between {v m } M m=1 and {x n } N n=1 , and Note that if we drop the diag(•) term in (10) due to the i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The error bound condition has been developed for such kind of problems [21].
Sentence 2: if F (x) has a polyhedral epigraph, then there exists c > 0 such that dist(x, Ω * ) ≤ c(F (x) − F * ) for any x ∈ R d . It then implies HOPS has an O(log( 0 / )) iteration complexity for solving a non-smooth REL minimization with a polyhedral epigraph.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We call p n (x) the truncated Chebyhshev series of degree n. For analytic functions, the approximation error (in the uniform norm) is known to decay exponentially [26].
Sentence 2: if f is analytic with f ( b−a 2 z + b+a 2 ) ≤ U for some U > 0 in the region bounded by the ellipse with foci +1, −1 and sum of major and minor semi-axis lengths equals to ρ > 1, then (2) Given a matrix A ∈ S d×d and a function f : R → R, the spectral-sum of A with respect to f is where tr (•) is the matrix trace and λ 1 , λ 2 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The results of all the methods are compared in Table 1, where evaluation measures include both peak signal-to-noise ratio (PSNR) and running time (Time).
Sentence 2: compared with the state-of-theart single-image SR methods (e.g., SR-CNN, ANR and K-SVD), our multi-frame-based method can surpass them by 0.28∼0.54 dB, which is mainly attributed to the beneficial mechanism of temporal dependency modelling.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We choose a vertex i and look for a spanning tree which minimizes the inverse centrality R(i) of vertex i, that is we solve the problem Note that R(i) is the contribution of vertex i to the total resistance of T and that, by equations (3.1) and (3.2), R(i) = nT + ii + Rtot n . The above problem can then be interpreted as minimizing a tradeoff between inverse centrality of vertex i and inverse connectivity of the tree.
Sentence 2: (4.3) encourages trees which are centered at i and, at the same time have a small diameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As mentioned in the introduction, we avoid such arbitrary hard separation thresholds by considering repulsive priors that smoothly push components apart.
Sentence 2: we propose two repulsion functions defined as with A = {(s, j) : s = 1, . . . , k; j < s} and g : + → [0, M ] a strictly monotone differentiable function with g(0) = 0, g(x) > 0 for all x > 0 and M < ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We write s := 1 1+ρ and define P := Q s µ.
Sentence 2: then for all sequences (f n ) of measurable functions f n : X → R the following are equivalent: for measurable f : X → R we have S P (f ) = 0 if and only if R P (f ) = R P . Proof: For n ∈ N we define E n := {f n > 0} {h > ρ}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose a hybrid approach of [2] with BQ, taking the form of a quadrature rule, that (i) carries a full probabilistic interpretation, (ii) is amenable to rigorous theoretical analysis, and (iii) converges orders-of-magnitude faster, empirically, compared with the original approaches in [2].
Sentence 2: we prove that super-exponential rates hold for posterior contraction (concentration of the posterior probability mass on the true value of the integral), showing that the posterior distribution provides a sensible and effective quantification of the uncertainty arising from numerical error.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A Markov chain is defined using these affinities by setting the transition probability 3 4 (' ) from vertex 5) to vertex to be proportional to the edge affinity, 0 6 (' ) .
Sentence 2: 3 4 1' ) 7¥ 98 A@ CB ) 0 1' ) where 8 ) D¥ FE FG IH B 0 1' ) gives the normalizing factor which ensures E G IH B 3 P (' ) P¥ Q § .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Like QPEs, EFPEs impose a lower bound on the realization of every sequence.
Sentence 2: given a sequence q and an extension with action a (i.e., the sequence qa), the realization r i (qa) has to satisfy the lower bound r i (qa) ≥ q r i (q), where q > 0 is a real constant (different sequences can have a different value for q ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, if we set γ k = O( 1 k q ) as 1 2 < q < 1 and the noise satisfy This proof of Theorem 2 is different from previous one.
Sentence 2: we cannot expect some sort of convergence to f (x * ), where x * ∈ argmin f due to nonconvexity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that it outperforms the DBM at solving a variety of inference tasks including classification, classification with missing inputs, and prediction of randomly selected subsets of variables.
Sentence 2: we use the MP-DBM to outperform the classification results reported for the standard DBM by Salakhutdinov and Hinton [18] on both the MNIST handwritten character dataset [14] and the NORB object recognition dataset [13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we assume a function P which, given a sentence x, identifies a set of candidate phrases, not necessarily coherent, for the sentence, P(x) ⊆ P. Second, we assume a score function which, given a phrase, produces a real-valued prediction indicating the plausability of the phrase for the sentence.
Sentence 2: the phrase recognizer is a function which searches a coherent phrase set for a sentence x according to the following optimality criterion: among all the coherent subsets of candidate phrases, the optimal solution is defined as the one whose phrases maximize the summation of phrase scores.
Relationship: entailment