sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Proving socalled "direct-sum" results has become a standard tool, namely the fact that the amount of resources required for solving d copies of a problem (with different inputs) in parallel is equal to d times the amount required for one copy.
Sentence 2: there is no saving from solving many copies of the same problem in batch and the trivial solution of solving each of them separately is optimal.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a function f : W → R, we can then consider the following diagram - Since both ρ and I are bijective, it is easy to see that f not only defines a function g : X → R by g := f • I • ρ, but conversely, every function g : X → R has such a representation and this representation is unique.
Sentence 2: there is a one-to-one relationship between the functions X → R and the functions W → R. Let us now assume that we have a kernel k W on W with RKHS H W and canonical feature map [31,Theorem 4.21] shows that the RKHS H X of k X is given by Since, for f ∈ H W , the reproducing property of Let us now assume that X is compact and that k W is one of the universal kernels considered in Theorem 2.1 or the Gaussian RBF kernel (4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then we fix the modeling agent i's strategy level to be 2 and try to observe the changes of j's samples which represent different possible models or strategy levels.
Sentence 2: we start from equal number of samples that representing j as level-1 I-POMDP, level-0 POMDP, and frequency based agents, and then gradually learn that the majority of samples converge or become close to the ground truth: j is a level-1 I-POMDP.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any non-empty subset S of [m] and any k × m matrix M we refer by M S to the k by |S| sub-matrix of M that contains only the columns of M corresponding to S. Also define the scalar product u, v ≡ 1 m m i=1 u i v i for u, v ∈ R m . Finally, the subgradient of a convex function f : R m → R at point x ∈ R m is denoted by ∂f (x).
Sentence 2: remember that the subgradient of the ℓ 1 norm, x → x 1 is given by (3.1) One main challenge in the proof of Theorem 2 lies in the fact that the function x → C A,y (x) is not -in general-strictly convex.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , ω(A T )) , which are both T × d matrices, we can write this model in a matrix form: where This assumes that the noise is driven by time-series dynamics (a martingale increment), where each coordinates are independent (meaning that features are independently corrupted by noise), with a sub-gaussian tail and variance uniformly bounded by a constant σ 2 .
Sentence 2: no independence assumption between the N t is required here.
Relationship: entailment
Domain: NeurIPS
Sentence 1: With the intent of deriving stronger approximation guarantees, we begin this section by introducing a 'nicer' measure of the quality of approximation afforded by Φ.
Sentence 2: instead of measuring J * − Φr * in the L ∞ norm as we did for our previous bounds, we will use a weighted max norm defined according to: J ∞,1/ψ max x∈X |J(x)|/ψ(x), where ψ : X → [1, ∞) is a given weighting function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The other benchmark model comes from Goldberger and Ben-Reuven [10], which unifies the two steps of the first benchmark model into a joint fashion.
Sentence 2: the noise transition matrix (y → ỹ) modeled by a nonlinear softmax layer connects the classifier (x → y) with noisy labels ỹ for the end-to-end training.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proposition 1 states that for any optimal solution D of the supervised objective, there exists an optimal solution D * of the (K + 1)-class objective such that D and D * share the same generalization error.
Sentence 2: using the (K + 1)-class objective does not prevent the model from experiencing any arbitrarily high generalization error that it could suffer from under the supervised objective.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Going from the penalization of supports to the penalization of level sets introduces some complexity and for simplicity in this section, we only consider the analysis in the context of orthogonal design matrices, which is often referred to as the denoising problem, and in the context of level set estimation already leads to interesting results.
Sentence 2: we study the unique global minimum ŵ of the proximal problem in Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To demonstrate the performances of different methods on recovering the overall transcriptional regulatory network, we randomly selected 10 TFs in the benchmark network and plotted the subnetwork in Figure 2 which has 70 edges with nonzero regulatory potential scores.
Sentence 2: the gray edges form the benchmark network, the red edges are those identified correctly and the green edges are those incorrectly inferred by each method.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it should be noted that for short horizons the hierarchy is not quite the same, and the growth rate is not logarithmic; this question is raised in .
Sentence 2: on short horizons the Best-Arm Identification procedure performs very well with respect to the others, and starts to be beaten (even by the gap-aware strategies) only when T ∆ 2 is much larger that 10.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A computationally more efficient alternative is nonconvex optimization.
Sentence 2: we reparameterize the m ⇥ n matrix variable M in the optimization problem as UV > with U 2 R m⇥k and V 2 R n⇥k , and optimize over U and V . Such a reparametrization automatically enforces the low rank structure and leads to low computational cost per iteration.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [9] analyzed all the transformations on similarities that preserve existence of efficient LSH search.
Sentence 2: they showed that if S is a similarity for which there exists an LSH family, then there also exists an LSH family for any similarity which is a probability generating function (PGF) transformation on S. PGF transformation on S is defined as P GF (S) = ∑ ∞ i=1 p i S i , where S ∈ [0, 1] and Similar theorem can also be shown in the case of 3-way resemblance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As it turns out, the fact that the feedback is only partially censored makes a stochastic bisection search possible with minimal assumptions on φ(•, C).
Sentence 2: we only need that φ(•, C) be Lipschitz, while the technique developed in [20] for a dynamic pricing problem requires φ(•, C) to be bi-Lipschitz.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, by taking advantage of Nesterov-style acceleration [Nesterov, 2007], the worst-case convergence rate of the algorithm after k steps can be improved from The procedure (8) can also be used to solve the additive (6) and sparse additive (7) extensions via a block coordinate descent algorithm.
Sentence 2: given a set of estimates d j (j = 1, . . . , p) we can fix all but one of the vectors d j and optimize over the non-fixed vector, by solving for some vector r j ∈ R n . For additive waveMesh (λ 2 = 0), this reduces to the univariate problem which can be solved via the algorithm (8).
Relationship: entailment
Domain: NeurIPS
Sentence 1: W p = b (since all constraints are linear, we abuse notation and consider B, W to be matrices).
Sentence 2: the hyperplanes corresponding to the constraints in W intersect in p. Now, for all directions d ∈ R n such that there exist active constraints W i , W j with W i d < 0 and W j d > 0, the line segment through p induced by d must necessarily be 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume a recent generalization of the Dawid-Skene model introduced in [22] to model the responses, which captures the heterogeneity in the tasks as well as the workers.
Sentence 2: each new arriving worker is parametrized by a quality parameter p j ∈ [0, 1] (for the j-th arriving worker), which is i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Ultimately, we want to find the nodes that have the greatest effect on Risk(P (X )), so we must condition Risk(P (X )) on the beliefs at each node.
Sentence 2: if we learned that the true marginal probabilities of node x were π x , what effect would that have on our current risk, or rather, what is Risk(P (X )|P (x) = π x )?
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now assume that the functions v sj (x j ) are uniformly bounded from a Sobolev space of order two.
Sentence 2: let {ψ jk ∶ k = 0, 1, . . .} denote a uniformly bounded, orthonormal basis with respect to L 2 [0, 1], and assume that v sj ∈ H j where Then the family of random variables Z (V,w) ∶= √ n ⋅ w ⊺ (Σ(V ) − Σn (V ))w is sub-Gaussian and sample continuous.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider a smoothed scenario, with the z-variable of the adversarial move (z, y) perturbed by an additive uniform noise σ = Unif[−γ/2, γ/2] for some γ ≥ 0.
Sentence 2: the actual move revealed to the player at time t is (z t + s t , y t ), with s t ∼ σ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let D π t be the distribution on state-action pairs at time t under policy π.
Sentence 2: a sample (s, a) is drawn from D π t by first drawing s 1 ∼ α(•), then following policy π for time steps 1 through t, which generates a trajectory (s 1 , a 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Airline passengers Gym memberships  Results are shown in Figure 5b, demonstrating that participants converged to intuitively plausible patterns.
Sentence 2: for both the volcano and the airline passenger data, participants converged to compositions resembling those found in previous analyses [6].
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1: for all (i, j) ∈ U t do 3: 4: end for for all (i, j) / ∈ U t do 7: end for 10: x t ← X (µ t , K t ) 11: end for Consensus propagation is a distributed protocol because computations at each node require only information that is locally available.
Sentence 2: the messages F ij (K t−1 ) and G ij (K t−1 ) transmitted from node i to node j depend only on {µ t−1 ui , K t−1 ui |u ∈ N (i)}, which node i has stored in memory.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we study these questions by considering the competitiveness of a simple residual network (composed of an arbitrarily deep, nonlinear residual unit and a linear output layer) with respect to linear predictors (or equivalently, 1-layer networks).
Sentence 2: we consider the optimization problem associated with training such a residual network, which is in general non-convex and can have a complicated structure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Imagine that ( ¥ 0) ¢¡ £ ¥ ¢¦ §£ ¢ 21 and ( ¥ 3) ¢¡ " £ ¢¦ 4 £ ¢ 4 1 have been stored as episodic patterns.
Sentence 2: their EP representations are stored in the hippocampus and are available for recall and replay.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, the overall computational complexity of the approach is O(MN log N + N 2 Kd), which can be prohibitive for large N and M. One natural way to reduce the complexity of the graph traversal part of Isomap is to not run Dijkstra's algorithm N times.
Sentence 2: instead of generating the entire N × N matrix of dissimilarities, generate an interesting subset of n rows, n << N. There are a family of MDS algorithms, here called Rectangular Dijkstra (RD) MDS algorithms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let f be the formula in M from which f was derived foreach logical variable in f that was substituted by the j-th value in ∆iR to yield f do w = w × |πj| where πj is the j-th cell of π return M The key advantage using Algorithm Partition-Ground is that the lifted algorithm (LMAP) will have much smaller space complexity than the one using Algorithm Constrained-Ground.
Sentence 2: unlike the latter, which yields O(n|U |) ground atoms (assuming each predicate has only one argument in U ) where n is the number of constants in the domain of U , the former generates only O(k|U |) ground atoms, where k << n. The following example illustrates how algorithm partition-ground constructs a new MLN.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 3), we can simplify it to where h p η (h|X T θ)h using the property of GB-RBM that the elements of h are independent given X T θ.
Sentence 2: assume g = h p η (h|X T θ)h, then g i = σ(W i X T θ), where W i is the ith row of W and σ is the logistic function σ(x) = 1/(1 + exp (−x)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A node-perturbation in a GGM can be equivalently represented through a perturbation of the entries of a row and column of the corresponding precision matrix (Figure 1).
Sentence 2: we can detect a single node perturbation by looking for a row and a corresponding column of Θ 1 − Θ 2 that has nonzero elements.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose that salience may be defined based on a strategy for maximum information sampling.
Sentence 2: shannon's self-information measure [9], -log(p(x )), applied to the joint likelihood of statistics in a local neighborhood decribed by w, provides an appropriate transformation between probability and the degree of infom1ation inherent in the local statistics.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Secondly, while we also consider more structured distributions (monotone, k-histograms), the techniques developed in the study of distributed parameter estimation do not apply to our problems.
Sentence 2: those results reduce to the problem of learning a high-dimensional vector (say, where each coordinate parametrizes a spherical Gaussian distribution), where the value at each coordinate is independent of the others.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the reward function R is bounded, with |R t (x, a)|  R for all (x, a, t), then this also implies a bound on regret over episodes experienced prior to time T , defined by Regret(T ) = ).
Sentence 2: regret(T )  2RHJ L + ✏bT /Hc.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our main contribution is the KRONDPP model along with efficient algorithms for sampling from it and learning a Kronecker factored kernel.
Sentence 2: inspired by the algorithm of [25], we develop KRK-PICARD (Kronecker-Kernel Picard), a block-coordinate ascent procedure that generates a sequence of Kronecker factored estimates of the DPP kernel while ensuring monotonic progress on its (difficult, nonconvex) objective function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For global differential privacy, we can generalize the definition of staircase mechanisms to hold for all neighboring database queries x, x (or equivalently within some sensitivity), and recover all known existing optimal mechanisms.
Sentence 2: the geometric mechanism shown to be optimal in [17], and the mechanisms shown to be optimal in [14,16] (also called staircase mechanisms) are special cases of the staircase mechanisms defined above.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We derive here new generalization bounds, based on Rademacher Complexity theory, for model selection and error estimation of linear (kernel) classifiers, which exploit the availability of unlabeled samples.
Sentence 2: two results are obtained: the first one shows that, using the unlabeled samples, the confidence term of the conventional bound can be reduced by a factor of three; the second one shows that the unlabeled samples can be used to obtain much tighter bounds, by building localized versions of the hypothesis class containing the optimal classifier.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At each iteration, SDCA chooses i at random according to p t , and updates the α i corresponding to the loss φ i to increase D. This scheme has been used for particular losses before, and was analyzed in [6] obtaining linear rates for general smooth losses, uniform sampling and l 2 regularization, and recently generalized in [10] to other regularizers and general sampling distributions.
Sentence 2: [10] show improved bounds and performance by statically adapting to the global smoothness properties of losses; using a distribution iterations to obtain an expected duality gap of at most ε.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For benign problem instances, the benefit of using the taxonomy is the vastly improved dependence on the number of arms.
Sentence 2: without a taxonomy or any other structure, regret of any algorithm for stochastic MAB scales linearly in the number of (near-optimal) arms, for a sufficiently large t. let N δ be the number of arms x such that δ 2 < ∆(x) ≤ δ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to achieve this type of structured sparsity at multiple levels of the hierarchy, we utilize an overlapping-group-lasso penalty recently proposed in [21] for genetic association mapping problem, where the goal is to identify a small number of SNPs (inputs) out of millions of SNPs that influence phenotypes (outputs) such as gene expression measurements.
Sentence 2: given the tree hierarchy T = (V, E) over image categories, each node v ∈ V of tree T is associated with group G v , composed of all leaf nodes in the subtree rooted at v, as illustrated in Figure 1(b).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we restrict our attention to the case of linear reward functions we can develop an efficient algorithm to compute an approximate best response.
Sentence 2: we consider the case where the reward for a state (s, θ) is defined as a linear combination of state features for some feature function φ : R(s, a H , a R ; θ) = φ(s) θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The kernel embeddings can also be generalized to conditional densities p(X|z) [9] µ Given this embedding, the conditional expectation of a function f ∈ F can be computed as E X|z [f (X)] = f, µ X|z F . Unlike the ordinary embeddings, an embedding of conditional distribution is not a single element in the RKHS, but will instead sweep out a family of points in the RKHS, each indexed by a fixed value z of the conditioning variable Z.
Sentence 2: it is only by fixing Z to a particular value z, that we will be able to obtain a single RKHS element, µ X|z ∈ F. conditional embedding is an operator, denoted as C X|Z , which can take as input an z and output an embedding, i.e., µ X|z = C X|Z φ(z).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the following Theorem, we show that this kernel function plays an important role in characterizing optimizers of optimization (6).
Sentence 2: we show that the objective function of the neural network optimization has a term where this kernel function is applied (component-wise) to the inner product matrix among vectors u 1 ,...,u r . Theorem 1 The loss function (3) for a matched PNN can be written as where ψ(.)
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then For any i, j, with i ∈ I l and j ∈ S l \ I l , we must have y i = y j .
Sentence 2: also, for any i, j with y i = y j and i, j ∈ S l , either i ∈ I l or j ∈ I l . there is a one-to-one correspondence between 1) edges i, j for which i, j ∈ S l and either i ∈ I l or j ∈ I l and 2) edges i, j for which i, j ∈ S l and y i = y j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the setting considered in this paper, the covariance function K θ is composed of a view kernel that models covariances between views, and an object kernel that models covariances between objects.
Sentence 2: the covariance between sample n (with corresponding feature vectors x pn and w qn ) and sample m (with corresponding feature vectors x pm and w qm ) is given by the factorized form (Bonilla et al., 2007;Rakitsch et al., 2013): Observed versus unobserved feature vectors Our model can be used when either one, or both of the view/sample feature vectors are unobserved.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The restricted isometry property [7,6] (or the related restricted eigenvalue [2] or compatibility conditions [19]) have been used to establish guarantees on the estimation and model selection errors of the Lasso or similar approaches.
Sentence 2: bickel, Ritov and Tsybakov [2] show that, under such conditions, with high probability, The same conditions can be used to prove model-selection guarantees.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As the main contribution of this work we establish a close relationship between diversity with submodular energies and the parametric submodular minimization.
Sentence 2: the joint M -best diverse labelings can be obtained by running a non-parametric submodular minimization (in the special case -max-flow) solver for M different values of γ in parallel, for certain diversity measures.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An important distinction is that we require strong convexity of the function F (w) with respect to the norm w . This is rather different than requiring the loss function z → (z, y) be strongly convex on the reals.
Sentence 2: the loss of a linear predictor, w → ( w, x , y) can never be strongly convex in a multi-dimensional space, even if is strongly convex, since it is flat in directions orthogonal to x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [9] who cleverly observed that x ∈ D m := DIS(A m ) can be efficiently determined using a single call to an ERM oracle.
Sentence 2: to apply their method, we use the oracle to find 4 h = arg min{err(h, Zm−1 ) ) is at most γ∆ m−1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by the success of accelerated full gradient methods (e.g., [12,1,22]), several recent work applied Nesterov's acceleration schemes to speed up randomized coordinate descent methods.
Sentence 2: nesterov [13] developed an accelerated randomized coordinate gradient method for minimizing unconstrained smooth convex functions, which corresponds to the case of Ψ(x) ≡ 0 in (1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: One simple internal model that subjects may have about the nature of the stimulus sequence in a 2-alternative forced choice (2AFC) task is that the statistical contingencies in the task remain fixed throughout the experiment.
Sentence 2: they may believe that the experiment is designed such that there is a fixed probability γ, throughout the experiment, of encountering a repetition (x t = 1) on any given trial t (thus probability 1−γ of seeing an alternation x t = 0).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To further verify this, we compare our L2 norm verification signal on all the sample pairs with those only constrain either the positive or negative sample pairs, denoted as L2+ and L2-, respectively.
Sentence 2: the L2+ only decreases the distances between DeepID2 features of the same identity, while L2-only increases the distances between DeepID2 features of different identities if they are smaller than the margin.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we allow it to be a projection of a latent structure (Pearl, 2009).
Sentence 2: we assume the presence of additional hidden variables Q that are not in G inv but induce dependency (for example, "interactive forks" (Pearl, 2009)) among H. For clarity, we omit these variables from the graph and use bi-directional edges to represent the dependency induced by them.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we show that the latter is superior and describe an algorithm that can do so, outperforming convolutional methods.
Sentence 2: we present tiled convolutional networks (Tiled CNNs), which use a novel weight-tying scheme ("tiling") that simultaneously enjoys the benefit of significantly reducing the number of learnable parameters while giving the algorithm flexibility to learn other invariances.
Relationship: entailment