sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The objective of the algorithm is to sequentially select the instances to query for label information and output a classifier ĥ that is close to h * while making as few queries as possible.
Sentence 2: we consider a non-parametric setting as in [6,17] where the hypothesis space is the smooth boundary fragment class the decision boundaries of classifiers in this class are epigraph of smooth functions (see Figure 1 for example).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we consider the conditions that lead to a feature receiving zero weight.
Sentence 2: we show that (i) as a direct result of feature-wise independence of the uncertainty set, a slight change of a feature that was originally assigned zero weight still gets zero weight (Theorem 4); (ii) using Theorem 4, we show that "nearly" orthogonal features get zero weight (Corollary 1); and (iii) "nearly" linearly dependent features get zero weight (Theorem 5).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we introduce a proximal-point method based on Kullback-Leibler (KL) proximal function and establish its relation to the existing approaches based on natural gradients [1,5,6].
Sentence 2: for conditionally-conjugate exponential-family models, we show that each iteration of our proximal-point approach is equivalent to a step along the natural gradient.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Additionally, overloading and AD are well established techniques that have been applied to machine learning, and even to applicationspecific programming languages for machine learning, e.g., LUSH [12] and DYNA [4].
Sentence 2: dYNA applies a nonstandard interpretation for ∧ and ∨ as a semiring (× and +, + and max, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, when the optimal value function cannot easily be reduced to a low-dimensional representation, we argue to apply a complementary form of regularisation on the target side.
Sentence 2: we propose to replace the optimal value function as target for training with an alternative value function that is easier to learn, but still yields a reasonable-but generally not optimal-policy, when acting greedily with respect to it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We first derive two equations and obtain two nonlinear operators for geodesically convex optimization instead of the linear extrapolation step in Euclidean space.
Sentence 2: we analyze the global convergence properties of our accelerated method for geodesically strongly-convex problems, which show that our method improves the convergence rate from O((1−µ/L) k ) to O((1− µ/L) k ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let S := {λ/(λ − 1) | λ is a real negative eigenvalue of T * } ∪ {0}.
Sentence 2: then the following holds: • For all r ∈ (0, 1] \ S, P (r) := (rI + (1 − r)T * ) −1 T * is well-defined and P (r) is a stochastic matrix if and only if r ≥ r * .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the expected length of message Y i is bounded by B i , Shannon's source coding theorem [6] implies that H(Y i ) ≤ B i .
Sentence 2: inequality (18) establishes a link between the initial distribution (17) and the number of bits used to transmit information, that is, We can now apply the quantitative data processing inequality (19) in the bound (16).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To this end, we construct a lightweight function to gather feature responses over large neighbourhoods and use the resulting contextual information to modulate original responses of the neighbourhood elements.
Sentence 2: we define a gather operator ξ G which aggregates neuron responses over a given spatial extent, and an excite operator ξ E which takes in both the aggregates and the original input to produce a new tensor with the same dimensions of the original input.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The features from earlier layers in φ (see Figure 5) are centered and l 2 -normalized before computing cycleconsistency.
Sentence 2: we consider both (a) the 2-way cycle-consistency, P φ , between the test video and the first training video, and (b) the 3-way cycle-consistency, P 3 φ , between the test video and the first two training videos.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, if we generalize the model slightly, then it can support efficient search.
Sentence 2: when we construct a long-range link (v, w) out of v, we do not choose w uniformly at random; rather, we choose it with probability proportional to d −α , where d is the lattice distance from v to w. In this way, the long-range links become correlated to the geometry of the lattice.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It also suffices to suppose that H approximates P with exponent β for all β < β, and that H has complexity exponent p for all p > p. Now, it is shown in [10] that the RKHS H has an approximation exponent β = 1 if and only if H contains a minimizer of the l-risk.
Sentence 2: if H has approximation exponent β for all β < 1 but not for β = 1 then H does not contain such a minimizer but Theorem 2.3 gives the same result as for β = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since this inequality is attained whenever S = U Diag(✓)U , where U are the eigenvectors of W > W , we see that kW k cl = k (W )k ⇥ , that is, the cluster norm coincides with the spectral box-norm.
Sentence 2: we see that the spectral k-support norm is a special case of the cluster norm, where we let a tend to zero, b = 1 and c = k. Moreover, the methods to compute the norm and its proximity operator described in the following section can directly be applied to the cluster norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the recent optimal convergence analysis of Oja's algorithm [7], one can conclude that after T 1 = Θ log r σ ηλ iterations, where λ = max{0, −λ min (∇ 2 f (x 0 ))}, then we not only have that x t+1 − x 0 2 is blown up, but also it aligns well with the minimum eigenvector of ∇ 2 f (x 0 ).
Sentence 2: if λ ≥ δ, then the algorithm must stop before T . Finally, one has to carefully argue that the error does not blow up in this iterative process.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the sake of comparison, we adhered to the experimental protocols described in [28], so that the performance of our method is directly comparable.
Sentence 2: 4 train/test splits were suggested in [28]: 1) 'maximal': train on frames 600:5:1400 (in Matlab notation) 2) 'downscale': train on frames 1205:5:1600 (the most crowded) 3) 'upscale': train on frames 805:5:1100 (the least crowded) 4) 'minimal': train on frames 640:80:1360 (10 frames).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the discrete case, our method optimizes the same variational objective of Titsias [15], but is a significant improvement over greedy forward selection using the variational objective as selection criteria, or some other criteria.
Sentence 2: given the cost of evaluating the variational objective on all training points, Titsias [15] evaluates the objective function on a small random subset of candidates at each iteration, and then select the best element from the subset.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 5), with probability at least 1 2 (1 ), PONSS using θ ≥ and T = 2eBnk 2 log 2k finds a subset X with |X| ≤ k and Proof.
Sentence 2: let J max denote the maximum value of j ∈ [0, k] such that in P , there exists a solution x with |x| ≤ j and f (x) Since the final selected solution x from P has the largest F value (i.e., line 19 of Algorithm 3), we have the desired approximation bound is reached.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note also that, while κ f itself is an upper bound of κf (X), often one does have an upper bound on κf (X) if one knows the function class of f (for example, say concave over modular).
Sentence 2: an immediate corollary is that the class of concave over modular functions f (X) = k i=1 λ i [w i (X)] a , λ i ≥ 0, for a ∈ (0, 1) can be learnt within a factor of min{ Next, we apply our results to the minimization of submodular functions under constraints.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our discrete log-concavity tester improves on previous work in terms of both sample and time complexity.
Sentence 2: [2] gave a log-concavity tester with sample complexity O √ n/ 2 + 1/ 5 , while [11] obtained a tester with sample complexity Õ √ n/ 7/2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is easy to see that the closer to 0 the µ i 's are, the closer G S (x; ξ) and N Y −1 ∇ S f (x; ξ) will be.
Sentence 2: lim µi→0,∀i Applying the asynchronous parallelism, we propose a generic asynchronous stochastic algorithm in Algorithm 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the bandit case, the same result holds for ε-approximations of Nash equilibria if we introduce an exploration factor ε > 0 that guarantees that action choice probabilities never fall below ε.
Sentence 2: if the algorithm is run with a suitably decreasing exploration factor, the sequence of play converges to a bona fide Nash equilibrium with probability 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, the optimal solution is in the span of the training instances.
Sentence 2: furthermore, from its definition it is clear that w r l = 0 only for the closest prototypes to x l . w r l = 0 if and only if µ r is either the closest prototype to x l which have the same label as x l , or the closest prototype to x l with alternative label.
Relationship: entailment
Domain: NeurIPS
Sentence 1: By the law of large numbers for empirical distributions, the sequence of empirical distributions µ m,0 = 1 m m i=1 δ ui converges (almost surely, for W 2 ) to µ 0 .
Sentence 2: our proof of Theorem 2.6 gives an alternative proof of the existence claim in Proposition 2.5 (the latter remains necessary for the uniqueness of the limit).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Third, two new algorithms for ordinal embedding are proposed and evaluated in experiments.
Sentence 2: ordinal embedding aims to represent items as points in R d so that the distances between items agree as well as possible with a given set of ordinal comparisons such as item i is closer to item j than to item k. the goal is to find a geometric representation of data that is faithful to comparative similarity judgments.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let Ψ(S) denote the minimum attainable p-value corresponding to the contingency table of feature combination Z S . By definition, p S ≥ Ψ(S), therefore Ψ(S) > δ implies that feature combination Z S can never be deemed significantly associated, and hence it cannot cause a false positive.
Sentence 2: feature subsets S for which Ψ(S) > δ are irrelevant as far as the FWER is concerned.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assuming a i ≥ β for all i, we have Q t ≤ 1/β, and setting η = β(log d + 1)/(2T ) guarantees that the regret of SLEEPINGCAT against the best fixed policy is bounded as We now turn our attention to the problem of learning with semi-bandit feedback where the learner only gets to observe the losses associated with its own decision.
Sentence 2: we assume that the learner observes all components i of the loss vector such that V t,i = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also compare against the state-of-the-art methods in budgeted learning (iv) GREEDYMISER [22]: it is a modification of gradient boosted regression tree [8] to incorporate feature cost.
Sentence 2: each weak learner (a low-depth decision tree) is built to minimize squared loss with respect to current gradient at the training examples plus feature acquisition cost.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume we are given a sequence of T observations, which we stack to create a sequence of samples from the F n |h t 's: To learn the graph structure, we make the approximation of ignoring the dynamical component of the model.
Sentence 2: we treat each f t as an observation, and try to estimate the density of the resulting unordered set, ignoring the t subscripts (we appeal to density estimation because many good algorithms have been developed for structure induction).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The heuristic computes the error for a given validation image at level k in the pyramid as L k (I k ) = min {zj } ||G k (z j , u(I k+1 )) − h k || 2 where {z j } is a large set of noise vectors, drawn from p noise (z).
Sentence 2: the heuristic is asking, are any of the generated residual images close to the ground truth?
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus after t cycles of uniform Newscast, the original variance φ 0 of the local estimates is reduced on the average to φ t ≤ φ 0 /(2 √ e) t . The fact that the variance drops at an exponential rate means that the nodes learn the correct average very fast.
Sentence 2: indeed, using Chebyshev's inequality P t [|µ i − µ| ≥ ε] ≤ φ t /(nε 2 ) we see that for any ε > 0, the probability that some node makes an estimation error larger than ε is dropping exponentially fast with the number of cycles t. we can derive a bound on the number of cycles that are needed in order to guarantee with high probability that all nodes know the correct answer with some specific accuracy: With probability 1 − δ, after 0.581(log n + 2 log σ + 2 log 1 ε + log 1 δ ) cycles of uniform Newscast holds max i |µ i − µ| ≤ ε, for any ε > 0 and data variance σ 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we study these theoretical properties for more standard convolutional architectures from the point of view of positive definite kernels [27].
Sentence 2: we consider a functional space derived from a kernel for multi-dimensional signals, which admits a multilayer and convolutional structure that generalizes the construction of convolutional kernel networks (CKNs) [15,16].
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is also possible to extend the policy gradient framework to deterministic policies µ µ µ θ : S → A.
Sentence 2: under certain conditions we can write the gradient of the objective J(θ) = E s∼p µ µ µ [R(s, a)] as: Since this theorem relies on ∇ a Q µ µ µ (s, a), it requires that the action space A (and thus the policy µ µ µ) be continuous.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here, p is the total number of features (or covariates) and n is the total number of observations (or the sample size).
Sentence 2: in general, f belongs to a class F of functions from R p → R. One further assumes that the functions in F depend on at most s of the features, usually with s p. for every f ∈ F, there is some f 0 : R s → R and a subset S ⊂ [p] with |S| ≤ s such that f (z 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assuming the graph structure is correct, from a regularization perspective we want to encourage smooth functions, to reflect our belief that labels should vary slowly over the graph.
Sentence 2: [2] and [7] suggest a general principle for creating a semi-supervised kernel K from the graph Laplacian L: transform the eigenvalues λ into r(λ), where the spectral transformation r is a non-negative and decreasing function 1 Note that it may be that r reverses the order of the eigenvalues, so that smooth φ i 's have larger eigenvalues in K. A "soft labeling" function f = c i φ i in a kernel machine has a penalty term in the RKHS norm given by Ω(||f Since r is decreasing, a greater penality is incurred for those terms of f corresponding to eigenfunctions that are less smooth.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For simplicity, we will set h = h. The goal of training is to find the weight matrices, W , U , and V , that minimize a training objective defined over the training data (with regularization).
Sentence 2: we assume the availability of t training examples {(x i , y i )} t i=1 , and denote the feature matrix X := (x 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: show that counts n are the sufficient statistic for planning.
Sentence 2: the joint-value function of a policy π over horizon H can be computed by the expectation over counts as (Nguyen et al., 2017): Set Ω 1:H is the set of all allowed consistent count tables as: is the distribution over counts (detailed expression in appendix).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Lemma 1 Assume that all the piles are of size n i = 1 and that ǫ k ≤ ǫ p+k = ǫ p+k+1 .
Sentence 2: then the following two inequalities cannot hold simultaneously: if the bound on |β − βk+p | is smaller than the bound on |β − βk |, then the bound on |β − βk+p+1 | must be smaller than both unless ǫ k+p+1 > ǫ k+p .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we assume that an oracle provides a random linear transformation (i.e., a sketch) of the gradient, which is the information available to drive the iterative process.
Sentence 2: given a fixed distribution D over matrices S 2 R n⇥b (b 1 can but does not need to be fixed), and a query point x 2 R n , our oracle provides us the random linear transformation of the gradient given by ⇣(S, x) Information of this type is available/used in a variety of scenarios.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note: Theorem 1a-1b gives consistency results for ( 9) and (10), respectively, under certain conditions on the parameters λ cn > 0, λ ds > 0, and α * > 1.
Sentence 2: these conditions assume knowledge of the noise variance ξ 2 and spikiness ratio α sp (Θ * ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The authors thank Bob Williamson and Bernhard Sch6lkopf.
Sentence 2: we present a simple sparse greedy technique to approximate the maximum a posteriori estimate of Gaussian Processes with much improved scaling behaviour in the sample size m. computational requirements are O(n 2 m), storage is O(nm), the cost for prediction is 0 ( n) and the cost to compute confidence bounds is O(nm), where n «: m. We show how to compute a stopping criterion, give bounds on the approximation error, and show applications to large scale problems.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a weight vector w ∈ H, we use notation w 2 H = w • w. We associate each possible input/output pair (x, y) ∈ X × Y with a feature vector f x,y ∈ H. Our classifier is characterized by a weight vector w ∈ H, with the following classification rule: Note that computational issues are ignored in this paper.
Sentence 2: we assume that the above decision can be computed efficiently (either approximately or exactly) even when GEN(x) is large.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, as recently shown by Lattimore and Hutter [6] for the infinite horizon setting, it is possible to obtain much tighter sample complexity results by using a more fine grained categorization.
Sentence 2: a key idea is that in order to obtain accurate estimates of the value function of a policy from a starting state, it is sufficient to have only a loose estimate of the parameters of (s, a)-pairs that are unlikely to be visited under this policy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose the Dual Space Gradient Descent (DualSGD) to address the computational problem encountered in the projection and merging strategies in the budgeted approach [8,9,17,7] and the excessive number of random features in the random feature approach [15,16].
Sentence 2: the proposed DualSGD utilizes the random-feature space as an auxiliary space to store the information of the vectors that have been discarded during the budget maintenance process.
Relationship: entailment
Domain: NeurIPS
Sentence 1: it follows from the expansion property that Pr (S 1 ⊕ S 2 ) ≥ Pr (S 1 ∧ S 2 ).
Sentence 2: therefore and so Pr (  this corresponds to both the positive and negative parts of the confident region expanding in the way given in the proof of Theorem 1, with an η fraction of the new edges going to examples of the other label.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [19], this check can be replaced by a check on whether the multilinear extension of z L−1 is correctly computed at a randomly picked point in the field, with minimal impact on the soundness error.
Sentence 2: the verifier picks two vectors, q L−1 ∈ F log(n L ) p and r L−1 ∈ F log(b) p at random, evaluates ZL−1 (q L−1 , r L−1 ), and checks whether it was correctly computed using a specialized sum-check protocol for matrix multiplication due to Thaler [18] (described in Section 3.1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We modify the network pipeline so that instead of producing predictions for a single object and relationship at a pixel, a feature vector is used to produce predictions for a set of s o objects and s r relationships.
Sentence 2: given a feature vector f from a single pixel, the network will for example output s o object class labels, s o bounding box predictions, and s o embeddings.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Under the hypothesis of Lemma 1, if f T (λu) ≤ λ 2 f T (u), and satisfies (2), then for any u ∈ S, and any λ > 0 we have where L = t∈M∪U η t t (u), and In particular, choosing the optimal λ, we obtain The intuition and motivation behind this Corollary is that a classification algorithm should be independent of the particular scaling of the hyperplane.
Sentence 2: w t and αw t (with α > 0) make exactly the same predictions, because only the sign of the prediction matters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The polyhedral convex set P k ⊃ D used in the preceding section is updated in each iteration, i.e., a sequence P 0 , P 1 , • • • is constructed such that P 0 ⊃ P 1 ⊃ • • • ⊃ D. The update from P k to P k+1 (k = 0, 1, . . .) is done in a way which is standard for pure outer approximation methods [12].
Sentence 2: a certain linear inequality l k (x, t) ≤ 0 is added to the constraint set defining P k , i.e., we set The function l k (x, t) is constructed as follows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These methods can be grouped as: EM with generative mixture models, bootstrapping methods (Self-training, Cotraining and the Yarowsky Algorithm), discriminative models (Transductive Support Vector Machines (TSVM) [2]) and data based methods, including Manifold Regularization [1], Information Regularization [17], and Low Density Separation(LDS) [11].
Sentence 2: tSVM extends the maximum margin principle of SVM to unlabeled data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unfortunately, there are hardness results showing that even with improper learning, and even if the data is generated exactly from a small, depth-2 neural network, there are no efficient algorithms which can find a predictor that performs well on test data.
Sentence 2: [15] and [12] have shown this in the case of learning intersections of halfspaces, using cryptographic and average case complexity assumptions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (1) into multiple (easy) subproblems, one for each factor.
Sentence 2: agreement constraints between overlapping subproblem maximizers are then defined, and the resulting program takes the following form, 2 min (2) Here, 'r 2 p' (similarly, 'c 2 r') represents parent-child containment relationships, often represented as a region graph [Wainwright and Jordan, 2008], and are Lagrange multipliers for the agreement constraints, defined for every region r, assignment x r , and parent p : r 2 p. these constraints enforce that the maximizing assignment in a parent region p agrees with the maximizing assignment in the child region r on the values of the variables in r (which are also in p due to containment).
Relationship: entailment