sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: While [25] considers generic methods for handling this nonstationarity, we focus on the special case in which all A i are policy gradient methods.
Sentence 2: theorem 3 of [25] states that the policy gradient of M can be decomposed into the policy gradients for all of the CoMDPs, M i : if each coagent computes and follows the policy gradient based on the local environment that it sees, the coagent network will follow its policy gradient on M . Thomas and Barto [25] also show that the value functions for M and all the CoMDPs are the same for all s t , if the additional state components of M i are drawn according to the modular policy: The state-value based TD error is therefore the same as well: This means that, if the coagents require δ t , we can maintain a global critic, C, that keeps an estimate of V θ M , which can be used to replace every C i by computing δ t and broadcasting it to each A i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a Lipschitz continuous function φ : R → R, φ ∞ ≤ 1, and constants r, s, t > 0 let F Θ ⊂ C (X), Θ = (φ, r, s, t), be a set of ridge-wave functions defined on the set X, From this definition, it follows that for all g ∈ F Θ it holds g ∞ ≤ r. As a ridge-wave function g ∈ F Θ is bounded and Lipschitz continuous, it is also square integrable in the measure ρ and g ∈ L 2 ρ (X).
Sentence 2: f Θ is a subset of the Hilbert space of square integrable functions defined on X with respect to the probability measure ρ, i.e., F Θ ⊂ L 2 ρ (X).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Besides encoding dependencies, in a variety of real-world applications we often want consistent solutions that are physically plausible, e.g., when jointly reasoning about multiple tasks or when enforcing geometric constraints in 3D indoor scene understanding applications [18].
Sentence 2: various methods [22,13,16,12,1] enforce consistency structure during inference by imposing constraints on the feasible instances.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given that all the weights w(u), u 2 U in (3) are non-negative, we can construct a concave upper bound for the multilinear extension F (x) using the following Lemma.
Sentence 2: proofs can be found in the Appendix A. given a weighted coverage function f with F (x) represented as in (3), we can define and conclude using Lemma 1 that (1 1/e) F (x)  F (x)  F (x), as desired.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At the other end of the spectrum, if we are dealing with a deterministic bug 3 , false positives should have a probability of zero: if the buggy feature is observed to be true, then the program has to crash; if the program did not crash, then the bug must not have occurred.
Sentence 2: for a deterministic bug, any false positives during the training process should incur a much larger penalty compared to any false negatives.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is proven in [6] that lim K→∞ V (D (K) ; Θ) is the expected sum of discounted rewards by following the RPR policy parameterized by Θ for an infinite number of steps.
Sentence 2: the RPR resulting from maximization of V (D (K) ; Θ) approaches the optimal as K is large (assuming |Z| is appropriate).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Algorithm 2: Reduction from from pricing with switching cost to strategic buyers Input:T, A % A is an algorithm with bounded regret for strategic buyers; Output:p The above section concluded that switching cost for pricing may be reduced to pricing with strategic buyers.
Sentence 2: our next step would be to show that we can produce a sequence of non-strategic buyers with high switching cost regret.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 3) as a hyperparameter, since it plays a similar role to C in determining the posterior and evidence.
Sentence 2: for the (D = 2) examples considered here, there are 12 hyperparameters, including scalars σ 2 and ρ, two hyperparameters each for ν s and ν f , and three each for symmetric matrices M s and M f . Note that although the conditional ALD prior over k|θ assigns high prior probability to smooth and sparse RFs for some settings of θ, for other settings (i.e., where M s and M f describe elliptical regions large enough to cover the entire RF) the conditional prior corresponds to a simple ridge prior and imposes no such structure.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a review of properties of Bregman divergences, see Chapter 11.2 in [11], or Appendix A in [2].
Sentence 2: by definition of the Bregman divergence, we have if the dual variable Z obeys the dynamics Ż = −∇f (X), then and by the same argument as in the gradient descent ODE, V is a Lyapunov function and The mirror descent ODE system can be summarized by Note that since ∇ψ * maps into X , X(t) = ∇ψ * (Z(t)) remains in X .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , (i * , d)} for some i * , there is a consistent function in F with error strictly greater than on D i * , namely, the function that maps (i, j) to 1 if and only if i = i * and (i * , j) is not in the sample set.
Sentence 2: to achieve uniform convergence, at least d/2 elements from X \ {⊥} must be drawn from each distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Based on the open-world assumption, if a web page does not link to another, it does not mean that they are unrelated.
Sentence 2: we employ the likelihood p(r i,n |t i,n ) = Φ(r i,n t i,n − b) for those observed links r i,n = 1, where b is an offset.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 1) as where we have conditioned ρ(y, x) = ρ(y|x)ρ X (x) and used the linearity of the integral and the inner product.
Sentence 2: any function f * : X → Y minimizing the above functional must satisfy the following condition where we have introduced the function g * : X → H that maps each point x ∈ X to the conditional expectation of ψ(y) given x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We recall that when P = I ∈ R p×p , our norm is equal to the 1 -norm, and we want to understand its behavior when P departs from the identity.
Sentence 2: we compute a second order approximation of our norm around the Lasso: we add a small perturbation ∆ ∈ S p to the identity matrix, and using Prop.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Generalizing Theorem 1, following a coordinate descent principle, now we present an alternating minimization (EM-style) procedure for P2: Step 1: keep p(z) fixed, infer p(w) by solving the following problem: where F 1 = {p(w) : p(w)E p(z) [∆F i (y, z; w) − ∆ i (y)] dw ≥ −ξ i , ∀i, ∀y}, which is a generalized version of F 1 with hidden variables.
Sentence 2: we can apply the same convex optimization techniques as being used for solving the problem P1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the enormous amount of content in high-resolution images makes generative learning difficult [6,7].
Sentence 2: generative models have found most success in learning to model small patches of natural images and objects: Zoran and Weiss [4] learned a mixture of Gaussians model over 8×8 image patches; Salakhutdinov and Hinton [2] used 64×64 centered and uncluttered stereo images of toy objects on a clear background; Tang et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Interactions are then induced only by the interactions of highaffinity active motifs in the two proteins.
Sentence 2: in our example, the motif d in p 2 is inactive, and hence does not lead to an interaction between p 2 and p 4 , despite the affinity between the motif pair c, d. We note that Deng et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose we choose γ = 2 and η = c/σ * 1 for any c ≤ 1/36.
Sentence 2: there exist constants c 1 , c 2 such that when α ≤ c 1 /(κ 2 µr), given any using proper initialization and step size, the gradient iteration converges at a linear rate with a constant contraction factor 1 − O(1/κ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For b k it is enough for our needs to consider an upper bound; since we have b k+1 = b k + k/2 + 3/2, one can easily verify that k 2 /2 is an upper bound for k ≥ 3.
Sentence 2: in order to provide an upper bound on the distance form the center grid u we would like to find an initial d such that and d is minimal.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When the upper and lower Minkowski dimensions coincide-which is the case in many natural spaces, such as normed vector spaces-the latter bound matches a lower bound of [10] that holds even when there are no switching costs.
Sentence 2: a surprising implication of our result is that in infinite actions spaces (of bounded Minkowski dimension), adding movement costs do not add to the complexity of the MAB problem!
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given observations O i and O j of differing length (and, thus, different underlying graphs in the case of dynamic graphical models), a kernel-based classifier over these instances is trained using U Oi and U Oj in the feature-space.
Sentence 2: a similarity measure is learned in the gradient space, under the intuition that objects which induce similar likelihoods will induce similar gradients.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, using Jensen's inequality for convex functions, we obtain a lower bound on the objective function: It can be shown that, provided both functions are continuous and differentiable, local maxima of the "free-energy" F with respect to all of its arguments correspond, in their optimal values of θ, to local maxima of L [5].
Sentence 2: any hill-climbing technique applied to the freeenergy functional can be used to find parameters that maximise the objective.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The asymptotics mn−1(h) as n → ∞ imply that the mixture distribution Ln,K+ in (10) converges to the true Gaussian mixture distribution p T of (9).
Sentence 2: for any small δ, we expect D(p T Ln,K+ ) ≤ δ for all n ≥ N , validating the assumption of Theorem 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In ZO-SVRG (Algorithm 2), the total number of gradient evaluations is given by nS + bT , where T = mS.
Sentence 2: by fixing the number of iterations T , the function query complexity of ZO-SVRG using the studied estimators is then given by O(nS + bT ), O(q(nS + bT )) and O(d(nS + bT )), respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus for d ≥ 15 log(4n/δ), we have ≤ 3/4 and as a result 2 /4 − 3 /6 ≥ 2 /8 and d ≥ log(4n/δ) 2 /4− 3 /6 .
Sentence 2: from Proposition 1 in [14], for all From this result, we deduce that with probability Proof of Theorem 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Yet, we can say that if, among the tested elements, there is one that gives a non-zero change in s I, then one of such elements will be the one that gives the maximal change in s I (provided, of course, that we have enough data for the estimates s I to be close enough to the theoretical values s I).
Sentence 2: we keep each split that arises from such a maximal-change element, resulting in O(N 2 ) candidate splits for the case of 2 clusters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Although the field of robust regression is well established, it has not considered computational complexity analysis to be one of its central concerns.
Sentence 2: none of the standard regression estimators in the literature are both robust and tractable, even in a weak sense: it has been shown that standard robust regression formulations with non-zero breakdown are NP-hard [1,2], while any estimator based on minimizing a convex loss cannot guarantee bounded response to even a single leverage point [3] (definitions given below).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The features at "pixel" (i, j) correspond to the weights of a (complete) bipartite undirected edgeweighted graph K 21,21 with 21 + 21 vertices, with the first disjoint set of 21 vertices representing the 21 amino acid types at position i, the second set representing the 21 amino acid types at position j, and the edge weights representing co-evolution of the respective variants.
Sentence 2: b = (J i,j,k,l ) k,l∈{1,...,21} is the biadjacency matrix of this graph, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the steps performed by Algorithm 1 on the rounds indicated in S are precisely the steps that the standard EXP3 algorithm would perform if it were presented with the loss sequence ( t ) t∈S .
Sentence 2: theorem 1 guarantees that Taking expectations on both sides of the above and noting that E[|S|] ≤ T /4 proves the lemma.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, we will upper-bound (in the Loewner ordering sense) the matrices Σ j in Algorithm 2.
Sentence 2: since ∥f xj (y)∥ 2 ≤ r for all y ∈ Ω j and since µ j in Algorithm 1 is a convex combination of f xj (y), the outer-product terms in the update for Σ j satisfy Σ j ≼ F(α 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If one interpolates the look-up table with the standard multilinear interpolation (by choosing the multilinear interpolation kernel φ), then the fitted surface forms a multilinear polynomial over [0, 1] S , and is thus ceterus paribus linear in each feature (but nonlinear overall), and thus the lattice layer does not affect convexity/concavity (note this only holds because W acts as a feature selector, (see Appendix B) for more on that).
Sentence 2: what is needed to guarantee that f (x; β, θ) is convex/concave with respect to the dth input is that each of the calibrators {c β d (x[d])} is convex/concave, and each of the T look-up tables is monotonically increasing with respect to x [d].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: is convex for all y ∈ P, then the inner risk L(p, .)
Sentence 2: is also convex for every distribution p ∈ P. This implies that M (p, δ) is a convex subset of R n . if is r-calibrated, then Theorem 2 implies that A r (p) = arg sort(M (p, δ)) is a set of rankings induced by a convex set of R n . The following theorem presents a condition that the set A r (p) must satisfy if it is generated by a convex set of scores: if there exists at least one pair of items (i, j) which are inverted in two rankings of A r (p), then i and j are "indifferent" in A r (p): Theorem 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, p 2 = 1 − p and µ 2 = β 4 w 2 = ( 1−pα 2 1−p ) 2 p 2 α 4 η 2 ≤ p 2 α 2 σ 2 4(1−p) 2 .
Sentence 2: the oracle risk is for the active learner, with probability 1 − δ, In contrast, consider any passive learner that receives m labeled examples and outputs a predictor ŵ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In lifted decomposition, the MLN is partitioned into a set of MLNs that are not only identical (up to a renaming) but also disjoint in the sense that they do not share any ground atoms.
Sentence 2: unlike the propositional procedure which creates n disjoint MLNs and searches over each, the lifted procedure searches over just one of the n MLNs (since they are identical).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Smola and Schölkopf [8] state in section 4.2 of their paper that their algorithm 2 finds a low rank approximation of the kernel matrix such that it minimises the Frobenius norm X− X 2 Frob = tr{K− K} where X is the low rank approximation of X.
Sentence 2: we need to prove that Algorithm 1 also minimises this norm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, replacing partial stochastic transitivity by interval stochastic transitivity (now a condition on quadruples instead of triplets) leads to a characterization of interval orders, a subclass of strict partial orders; a partial order Q on [M ] 2 is called an interval order if each i ∈ [M ] can be associated with an interval Our main theoretical results below state that thresholding (4) yields a strict partial order relation Q, both for the PL and the Mallows model.
Sentence 2: we can guarantee that a strict partial order relation can be predicted by simple thresholding, and without the need for any further reparation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We simulated (i) by setting ∆t stim = 5ms, thus producing larger and more infrequent jumps in discrete OVs than with ∆t stim = 1ms (default).
Sentence 2: ϑ(t) grows more slowly (delayed build up of inhibition), and the peak occurs later (t max ≡ t c − t = 10ms with everything else identical with figure 1b).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Noting that the statistics vectors appear only as an inner product in (11), we apply the "kernel trick" to obtain where the kernel function K(u, v) = Φ(u), Φ(v) defines the inner product of some high-dimensional mapping Φ(•) of its inputs.
Sentence 2: 4 In particular, choosing a Gaussian (RBF) kernel, K(u, v) := exp(−γ||u − v|| 2 ), gives a non-parametric representation for g: g(u ki ) is a weighted combination of the values {α jk x (j) i }, where the weights depend exponentially on the squared 2 -distance of u ki to each of the statistics vectors {u  To validate our method, we evaluated its ability to learn parameter functions on a variety of email and webpage classification tasks in which the number of classes, K, was large (K = 10), and the number of number of training examples per class, m/K, was small (m/K = 2).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we select t satisfying −(v + 1)/2 = 1/(1 − t) and denote, then by some simple but tedious calculation (included in the supplementary material) where λ = 2Ψ (t − 1)vσ and gt = Ψ − 1 t − 1 .
Sentence 2: we work with the Student's-t prior in our setting: Here, the degree of freedom for Student's-t distribution is chosen such that it also belongs to the exp t family, which in turn yields v = (3 − t)/(t − 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, any predictor w can be written as w = (w i,j ) (i,j)∈G where w i,j is w's weight on φ φ φ i,j (x, y i , y j ).
Sentence 2: for any w and any (x, y), we have where we denote by F i,j (w i,j , x, y i , y j ) = w i,j , φ φ φ i,j (x, y i , y j ) the score of labeling the edge (i, j) by (y i , y j ) given input x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, if H 0 is the set of all stationary ergodic distributions on pairs, we have found a distribution ρ ∈ cl H 0 such that W ρ (H 0 ) = 0.
Sentence 2: there is no consistent test that could provide a given level of confidence under H 0 , even if only asymptotic consistency is required under H 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These events were selected on the basis of representing transitions between different task phases.
Sentence 2: we hypothesized that such events should correlate with changes in the relative strengths of different influences on visual attention, and that we should be able to detect such changes using the previously described models as diagnostic after annotating the video clips with the times of each event of interest, we subsequently aligned further analyses on a temporal window of -5s/+5s around each event (shaded background regions in Figure 2, rows b-e).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, if D max (S, k min ) < D(S * ), we know that no subregion of S can be optimal; we can prune the region completely, and not search its (gridded or non-gridded) children.
Sentence 2: second, we can show that (for 0 < k < n) any region of size 2 k + 1 or less is contained entirely in an odd gridded region of size 3 2 × 2 k . if D max (G, 2 n−1 + 2) < D(S * ) for the entire grid G, any optimal non-gridded region must be contained in an odd gridded region.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that the objective E X [1/(1 − P (x))] is already a barrier at P (x) = 1.
Sentence 2: to enforce the lower bound (2), we modify the objective to where µ is a parameter chosen momentarily to ensure P (x) ≥ P min,m for all x ∈ D m . the modified goal is to minimize (7) over non-negative P subject only to (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Independent (additive) rewards: R( s Interactions between agents are given in terms of a set of dependencies between certain state-action transitions for one agent, and events featuring transitions involving the other agent.
Sentence 2: if a history contains one of the primitive events from the latter set, this can have some direct effect upon the transition-model for the first agent, introducing probabilistic transition-dependencies.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The associated residual Jacobian matrix J can be given in the block-angular form below left, and thus the (full) Hessian matrix H has the n × n sparse block arrow form below right (× denotes some non-zero block) as well as the GN-Hessian J T J: Here in  k have no effect on r k ], and F vertical B k blocks corresponding to the n B hidden parameters " B that contribute to minimizing all the residuals r k (k=1, • • • , F ) evaluated at all F terminal nodes.
Sentence 2: the posed problem is overdetermined when "m > n" (namely, "d > CA + 1 F nB") holds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: advocate instead using a trust region approach, whereby the update is scaled down to modify the policy distribution (in terms of KL divergence) by at most a specified amount.
Sentence 2: we adopt the trust region formulation of K-FAC introduced by [2], choosing the effective step size η to be min(η max , 2δ ∆θ F ∆θ ), where the learning rate η max and trust region radius δ are hyperparameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Secondly, it is infinitely exchangeable: given a genealogy drawn from an n-coalescent, the genealogy of any m contemporary individuals alive at time t ≤ 0 embedded within the genealogy is a draw from the m-coalescent.
Sentence 2: taking n → ∞, there is a distribution over genealogies of a countably infinite population for which the marginal distribution of the genealogy of any n individuals gives the n-coalescent.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Within each game, subjects learned to play three game levels, and during eye tracking, each subject played each game level twice.
Sentence 2: in total, our recorded data set consists of video frames and eye tracking data from 60 clips (5 subjects × 2 games per subject × 3 levels per game × 2 clips per level) covering 4.7 hours.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, we have where a = P n i=1 a i /n, s = p P n i=1 a 2 i /n a 2 .
Sentence 2: for SGD to be stable around x = 0, we not only need |1 ⌘a|  1 (stability condition of GD), but also (1 ⌘a) 2 + ⌘ 2 s 2  1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, the outputs are mixture density networks [6], with a shared hidden layer, using the same parameter tying as NADE.
Sentence 2: our Real-valued Neural Autoregressive Density-Estimator or RNADE model represents the probability density of a vector as: where p M is a mixture of Gaussians with parameters θ d . The mixture model parameters are calculated using a neural network with all of the preceding dimensions, x <d , as inputs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The proof is along the lines of [Hazan, 2015, Theorem 5.2], except that the loss function does not take a raw state as input, and our domain for optimization is complex.
Sentence 2: the mean value theorem does not hold, which means we need to approximate the Bregman divergence instead of replacing it by a norm as in the original proof.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We find in our experiments that some tasks require more, but not necessarily larger, memories, and others such as language modeling require fewer, larger memories.
Sentence 2: we have a number of tune-able parameters: the number of memories, the size of each memory, the number of attention heads, the number of steps of attention, the gating method, and the postattention processor g ψ .
Relationship: reasoning