sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: ∆ (Z) P Analogous expressions hold for ∆ (X) P and ∆ (Y ) P . Unlike in the two-variable case where either matrix or both can be centered, centering of each matrix in the three-variable case has a different meaning.
Sentence 2: one requires centering of all three kernel matrices to perform a "complete" Lancaster interaction test, as given by the following Proposition.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, we assume that there are parameters ρ, µ (possibly dependent on p such that the class of graph-structured activation patterns x is given as follows.
Sentence 2: here out(C) = (u,v)∈E W u,v I{u ∈ C, v ∈ C} is the total weight of edges leaving the cluster C. the set of activated vertices C have a small cut size in the graph G. While we assume that the noise variance is 1 in (1), this is equivalent to the more general model in which Eξ 2 i = σ 2 with σ known.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At the other extreme, we show that each processor can transmit ≤ 2.8n + 32 bits per iteration in expectation, while increasing variance by a only a factor of 2.
Sentence 2: in the latter regime, compared to full precision SGD, we use ≈ 2.8n bits of communication per iteration as opposed to 32n bits, and guarantee at most 2× more iterations, leading to bandwidth savings of ≈ 5.7×.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let ǫ ≥ 0 and α ∈ [0, 1] be given numbers.
Sentence 2: then θ is an approximate global optimizer of U with value imprecision ǫ and residual domain the value U (θ) is within ǫ of a value which is greater than the values that U takes on at least a 1 − α portion of the domain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to the unary model of [12], a model based on quadratic assignment is also presented, in which pairwise features are determined using the adjacency structure of the graphs.
Sentence 2: if a pair of points (p 1 , p 2 ) in the template scene is to be matched to (q 1 , q 2 ) in the target, there is a feature which is 1 if there is an edge between p 1 and p 2 in the template, and an edge between q 1 and q 2 in the target (and 0 otherwise).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Both w −1 min and 1 − 2P π2-Loop [w(F ) < 0] depend on the choice of BP fixed point, however it is unlikely (unless a degeneracy) that these characteristics become large.
Sentence 2: p π2-Loop [w(F ) < 0] = 0 in the case of attractive models [20].
Relationship: entailment
Domain: NeurIPS
Sentence 1: (Foygel et al., 2012) For any half-trek, π, let Right(π) be the set of vertices in π that have an outgoing directed edge in π (as opposed to bidirected edge) union the last node in the trek.
Sentence 2: if the trek is a directed path then every node in the path is a member of Right(π).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Previous works have also shown that using side information, e.g., the scene perspective, helps to improve crowd counting accuracy [5,6].
Sentence 2: when extracting hand-crafted features (e.g., edge and texture statistics) [5][6][7][8][9] use scene perspective normalization, where a "perspective weight" is applied at each 31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 1 improves significantly over the convergence rate established for the pairwise conditional gradient variant in [16].
Sentence 2: the number of iterations to reach an ✏ error in the analysis of [16] depends linearly on |V|!, where |V| is the number of vertices of P. Throughout this section we let h t denote the approximation error of Algorithm 3 on iteration t, for any t 1, i.e., h We start by proving that the iterates of Algorithm 3 are always feasible.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that when G is undirected (more precisely, as above, when G is a directed graph having for every pair of nodes i� j ∈ V either no arcs or length-two cycles), then mas(G) = α(G), otherwise mas(G) ≥ α(G).
Sentence 2: when G is itself a directed acyclic graph, then mas(G) = |V |.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our objective is to find the set of voxel time courses S defined as where S * denotes a subset of the entire collection of voxels time courses S extracted from a single fMRI scan.
Sentence 2: we aim at finding spatially distributed voxel responses S that best explain the naturalistic stimuli, without making any prior assumptions about location and size of voxel subsets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Disjunctions use the signal from the survey propagation messages without overcommitting to a particular variable assignment too early (as in decimation).
Sentence 2: without loss of generality, if we are given marginals µ i (1) > µ i (0) and µ j (1) > µ j (0) for variables i and j, the new update adds the streamlining constraint x i _ x j to the problem instead of overcommitting by constraining i or j to its most likely state.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The intuition of the proposed SBA algorithm is that if the graphon is smooth, neighboring crosssections of the graphon should be similar.
Sentence 2: if two labels u i and u j are close i.e., |u i − u j | ≈ 0, then the difference between the row slices |w(u i , •) − w(u j , •)| and the column slices |w(•, u i ) − w(•, u j )| should also be small.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, the choice x t ∈ X defines a parameter of a mixed strategy from which a actual move ω(x t , s t ) is drawn; for instance, for additive zero-mean Gaussian noise, x t defines the center of the distribution from which x t + s t is drawn.
Sentence 2: noise does not allow the adversary to play any desired mixed strategy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here, we denote by a ≻ a ′ the event that a wins a ′ and by P (a ≻ a ′ ) the probability that a ≻ a ′ happens.
Sentence 2: we assume that the feedback from the oracle follows the following two-valued random variable: where the probability P (a ≻ a ′ ) is determined by the oracle.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead of taking the farthest point in the simplex, this technique is designed to shift the estimate of a topic to a high density region, where true topics are likely to be found.
Sentence 2: given a (current) cone S ω (v), we re-position the cone by updating v := argmin v m∈Sω(v) wm 2 (1 − cos( wm , v)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The solution fλ,m of the corresponding minimization problem can now be written as, where A † denotes the Moore-Penrose pseudoinverse of a matrix A, and  for any t > 0, where In practice, leverage scores are onerous to compute and approximations ( li (t)) n i=1 can be considered [16,17,24] .
Sentence 2: in the following we are interested in suitable approximations defined as follows: Definition 1 (T -approximate leverage scores).
Relationship: entailment
Domain: NeurIPS
Sentence 1: From this assumption it follows that information density is optimized near the channel capacity, where speakers maximize the rate of information transmission while minimizing the danger of a mistransmitted message (see also Aylett (2000); Aylett and Turk (2004); Genzel and Charniak (2002)).
Sentence 2: uID is an optimal solution to the problem of rapid yet error-free communication in a noisy environment.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (30) Then by non-expansiveness of proximal operation (Lemma 2), we have Since for z ∈ T , ∥H t z∥ 2 ≥ √ m∥z∥ Ht , (31) leads to where last inequality follows from Lipschitz-continuity of ∇  In this section, we study the convergence behavior of Proximal Gradient method and Proximal Newton method on high-dimensional real data set with and without the CNSC condition.
Sentence 2: two loss functions -logistic loss L(u, y)=log(1 + exp(−yu)) and ℓ 2 -hinge loss L(u, y)=max(1 − yu, 0) 2 -are used in (3) with ℓ 1 -regularization h(x) = λ∥x∥ 1 , where both losses are smooth but only logistic loss has strict convexity that implies the CNSC condition.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This bound can be expressed on a scale that is independent of the temperature parameter γ; intuitively, ∆ max corresponds to a utility difference so slight that the user identifies the best item only with probability 0.56 under R L with temperature γ.
Sentence 2: the maximum loss is so small that the user is unable able to identify the preferred item 44% of the time when asked to compare the two items in S. This derivation can be generalized to sets of any size k, yielding , where L W (•) is the Lambert W function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The dominating factor of the bound on t becomes (5) Table 1 summarizes several interesting cases, and compares the iteration bound of ASDCA to the iteration bound of the vanilla SDCA algorithm (as analyzed in Shalev-Shwartz and Zhang [2013a]) As can be seen in the table, the ASDCA algorithm interpolates between SDCA and AGD.
Sentence 2: aSDCA has the same bound as SDCA when m = 1 and the same bound as AGD when m = n. Recall that the cost of each iteration of AGD scales with n while the cost of each iteration of SDCA does not scale with n. The cost of each iteration of ASDCA scales with m. To compensate for the difference cost per iteration for different algorithms, we may also compare the complexity in terms of the number of examples processed (see Table 2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Accordingly, the update rule for ρ takes the form, To those familiar with advantage-actor-critic methods [21] (A2C and its asynchronous analogue A3C) PCL's update rules might appear to be similar.
Sentence 2: advantage-actor-critic is an on-policy method that exploits the expected value function, to reduce the variance of policy gradient, in service of maximizing the expected reward.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is an example of the avoidance of the curse of dimensionality in the manner mentioned above, by a method whose performance depends on the intrinsic dimensionality of the data instead of the full dimensionality of the feature space.
Sentence 2: our method is practical in that it works with Euclidean distances on R N . we do not assume any knowledge of the quantities pertaining to the intrinsic geometry of the underlying submanifold such as its metric tensor, geodesic distances between its points, its volume form, etc.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the distribution function µ f0 p q does not change abruptly with the expression of ε U n pf 0 q can be significantly simplified.
Sentence 2: if for all P p0, c 0 s it holds that µ f0 p { log nq ě µ f0 p q{rlog ns Op1q , then ε U n pf 0 q can be upper bounded as It is also noted that if µ f0 p q has a polynomial behavior of µ f0 p qβ for some constant β ě 0, then Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, the primal variables can be easily recovered from the dual variables by the KKT conditions.
Sentence 2: )τ e . Subsequently, the decision function f (x) = w ϕ(x) + b is a linear combination of k(x i , x)'s defined on both the labeled and unlabeled examples, as in standard manifold regularization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Intuitively, each cluster function corresponds to a partial marginalization of the factors contained in cluster v. Since each cluster function is defined over a subset of the variables in the original graph, we require some additional notation to represent these sets.
Sentence 2: for a cluster v, let A(v) be the arguments of its cluster function, and let V(v) be the set of all arguments of its children, In a slight abuse of notation, we let A(v) be the arguments of the node v in the original graph, so that if v is a variable node A(v) = v and if v is a factor node A(v) = N (v).
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, more work needs to be done to explore the strengths and weakneses of these algorithms.
Sentence 2: simple sequential importance resampling is known to break down when applied to datasets with many observations, although we are optimistic that methods for addressing this problem that have been developed for Dirichlet process mixture models (e.g., [5]) will also be applicable in this setting.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Observation: Assume the marginal distribution P(Y = 0) and P(Y = 1) is fixed for any distribution P ∈ C. Then, the distribution in C with the minimum HGR correlation between X and Y is the distribution P obtained by solving (5).
Sentence 2: ρ(X, Y ; P) ≤ ρ(X, Y ; P), ∀ P ∈ C, where ρ(X, Y ; P) denotes the HGR correlation coefficient under the probability distribution P. Based on the above observation, from now on, we call the classifier δ(•) in (7) as the "Rényi classifier".
Relationship: entailment
Domain: NeurIPS
Sentence 1: 5: end for Output: x K where z k = (k+2)y k /2 − (k/2)x k . Correspondingly, we can also obtain the following analysis tools for our convergence analysis using the proposed equations ( 4) and (5).
Sentence 2: the following equations ( 7) and ( 8) can be viewed as the Riemannian space counterparts of (6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Tractability is not necessarily sufficient to obtain algorithms that work effectively in practice.
Sentence 2: although the problem of large margin estimation can be formulated as a quadratic program (QP) in several cases of interest [2,19], and although this formulation exploits enough of the problem structure so as to achieve a polynomial representation in terms of the number of variables and constraints, off-the-shelf QP solvers scale poorly with problem and training sample size for these models.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An immediate corollary is that non-analyticity points (w.r.t ∆) of the MIs are the same in the coupled and original models.
Sentence 2: defining ∆ Opt,coup := sup{∆ | lim L→∞ lim n→∞ I w,L (S; W)/(n(L+1)) is analytic in ]0, ∆[}, we have ∆ Opt,coup = ∆ Opt .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The updates M i,t−1 → M i,t and b i,t−1 → b i,t are performed at node i through vector xt both when i = i t (i.e., when node i is the one which the context vectors in C it refer to) and to a lesser extent when i = i t (i.e., when node i is not the one which the vectors in C it refer to).
Sentence 2: this is because, as we said, the payoff a t received for node i t is somehow informative also for all other nodes i = i t . because we are assuming the underlying parameter vectors u i are close to each other, we should let the corresponding prototype vectors w i,t undergo similar updates, so as to also keep the w i,t close to each other over time.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that for the feed-forward networks with BN layers, we can obtain a reliable estimation of m b,W and m b,z .
Sentence 2: we assume that W b is row homogeneous [Ba et al., 2016], i.e., they represent the same level of information and are roughly of similar magnitude, and define  17) and ( 18) of the fully connected layer with BN are computed as follows, We next evaluate the scale-amended SGD on training VGG nets [Simonyan and Zisserman, 2015] for image classification tasks with two datasets: CIFAR-10 [ Krizhevsky and Hinton, 2009] and CIFAR-100 [Krizhevsky and Hinton, 2009].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the resulting graph, v 4 is d-separated from v 1 implying that v 4 is independent of v 1 in the distribution, P (v 4 , v 2 , v 1 |do(v 3 )).
Sentence 2: p (v 4 |do(v 3 ), v 1 ) = P (v 4 |do(v 3 )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a solution Λ * to ( 6), the optimal model parameters W * for (3) can be recovered via The equivalence relation form of log-linear regression can be used to derive useful relaxations of EM variants for directed models.
Sentence 2: by exploiting Theorem 1, we can now re-express the regularized form of the joint EM objective (1) strictly in terms of equivalence relations over the hidden variable values = min where h ranges over the hidden variables, and Note that ( 8) is an exact reformulation of the joint EM objective (7); no relaxation has yet been introduced.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 Note that r * S (x 0 ) can be equivalently written When S = R d , r * (x 0 ) := r * R d (x 0 ) is the adversarial (or worst-case) perturbation defined in [17], which corresponds to the (unconstrained) perturbation of minimal norm that changes the label of the datapoint x 0 .
Sentence 2: r * (x 0 ) 2 corresponds to the minimal distance from x 0 to the classifier boundary.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Affinities (or similarities) of all n data points are computed with respect to these m anchors in linear time O(dmn) where m n. The true affinity matrix A o ∈ R n×n is then approximated by using these affinities.
Sentence 2: an anchor graph leverages a nonlinear data-to-anchor mapping (R ) /M , where δ j ∈ {1, 0} and δ j = 1 if and only if anchor u j is one of s m closest anchors of x in U according to some distance function D() (e.g., 2 distance), t > 0 is the bandwidth parameter, and M = m j=1 δ j exp(− ) leading to z(x) 1 = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For each popped element s ∈ S, we find k such that 0 < k ≤ r − x(s) with f (kχs|x) kc(χs) ≥ θ with binary search.
Sentence 2: if there is such k, we update x with x + kχ s . Finally, we push s again with the key f (χs|x) c(χs) if x(s) < r. The correctness of this technique is obvious because of the DR-submodularity of f . the key of each element s ∈ S in the queue is always at least f (χs|x) c(χs) , where x is the current vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we consider this question from the optimization's perspective and give an answer: the Hessian of the local back-matching loss is responsible for the difficulty of training deep nets with BP.
Sentence 2: we start from a penalized loss formulation, which takes the intermediate feature outputs as variables of the optimization and the penalty is to enforce the coordination (architecture) connection [Carreira-Perpinan and Wang, 2014].
Relationship: entailment
Domain: NeurIPS
Sentence 1: There exists an infinite dimensional separable metric space (X , ρ) and a realizable distribution μ over X × {0, 1} such that no k n -NN learner satisfying k n /n → 0 when n → ∞ is Bayes-consistent under μ.
Sentence 2: this holds for any space and realizable distribution μ that satisfy the following condition: The set C of points labeled 1 by μ satisfies Since µ(C) > 0, Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our main objective in the following experiments is to demonstrate that our LIRPG-based algorithm can extend to a different class of domains and a different choice of baseline actor-critic architecture (namely, PPO instead of A2C).
Sentence 2: we explore domains from the Mujoco continuous control benchmark [Duan et al., 2016a], and used the open-source implementation of the PPO  algorithm from OpenAI  as our baseline agent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Any (possibly randomized) unbiased statistical algorithm that solves Z with probability " requires at least m calls to the Unbiased Oracle for < .
Sentence 2: if ÷ AE 1/6, then any algorithm with success probability at least 2/3 requires at least min{¸/4, 1/48"} samples from the Unbiased Oracle.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the case of a Markov chain, learning with hidden state is exactly the problem of parameter estimation in hidden Markov models.
Sentence 2: we must update the estimates of w, /tl and ur for each state, and aij for each state transition (out of the holding state), in order to maximize the likelihood of the sequence of observations The standard algorithm for hidden Markov model parameter estimation is the Baum-Welch method [10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In time-series settings [11][12][13], such as audio denoising and missing data imputation considered later in the paper, this means that the number of pseudo-datapoints must grow with the number of datapoints if restoration accuracy is to be maintained.
Sentence 2: m must be scaled with N and so pseudo-datapoint schemes have not reduced the scaling of the computational complexity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 1 also ensures that, if sufficient memory is available, D2-CTW will eventually perform as well as CTW with optimal depth bound K = D. This follows from the fact that, for every node s at depth d s ≤ D in a CTW suffix tree, if β s t ≥ 1 for all t > τ , then the D2-CTW suffix tree will be at least as deep as d s at context s after some time t ≥ τ .
Sentence 2: at some point, the D2-CTW model will contain the "useful" sub-tree of the optimal-depth context tree.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is known that there exists a unique cross-covariance operator Σ Y X : holds for all f ∈ H X and g ∈ H Y .
Sentence 2: the cross covariance operator represents the covariance of f (X) and g(Y ) as a bilinear form of f and g. if Y is equal to X, the self-adjoint operator Σ XX is called the covariance operator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Another way to specify a parametric graphical model is by assuming that each node-conditional distributions is a part of the exponential family [29].
Sentence 2: the conditional distribution of x s given x −s is assumed to be where C is the base measure, D is the log-normalization constant and N (s) is the neighborhood a the node s. Similar to (2), the neighborhood of each node can be estimated by minimizing the negative log-likelihood with l 1 penalty on θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We see that the proportion of misses for the fastest two levels-1 and 2-of the cache for B = 1 increase significantly with the number of cores, while increasing the batch size to B = 10 substantially mitigates cache incoherency.
Sentence 2: we maintain (near) linear increases in iteration speed with little degradation in solution quality (the gap f ( x) − f (x ⋆ ) output by each of the procedures with and without batching is identical to within 10 −3 ; cf.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is intuitive to think that separating male faces from female faces is a much more difficult task than separating faces from background, needing more detailed attention, so more resources should be allocated to face images than background images.
Sentence 2: a good tradeoff for RNP is to prune the neural network more when dealing with background images and keep more convolutional channels when inputting a face image.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a set of trajectories obtained following policy π θ , we can produce an estimate ∇ θ J µ (θ) of the policy gradient and we assume to be able to produce a vector = [ 1 , .
Sentence 2: . . , m ] T , so that the i-th component of the approximation error is bounded at least with probability 1 − δ: Given the approximation error vector , we can adjust the bound in Corollary 5.1 to produce a new bound that holds at least with probability (1 − δ) m . to preserve the inequality sign, the estimated approximation error must be used to decrease the L2-norm of the policy gradient in the first term (the one that provides the positive contribution to the performance improvement) and to increase the L1-norm in the penalization term.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Prior research has suggested that larger values of λ lead to superior results.
Sentence 2: an example of [1] and the approximation error bounds of [17,18], both of which are restricted to the case of a fixed policy, suggest that approximation error is amplified by a factor of 1/(1 − α) as λ is changed from 1 to 0.
Relationship: entailment