sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The basic idea of the algorithm is to replace the MDP M with M ≡ ( S, A, Pa , ra , γ), where Pa = Ka Ḋa and ra = Ka r a (r a ∈ R n a is the vector composed of sample rewards r a i ).
Sentence 2: instead of solving an MDP with n states, one solves a model with m states only.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Performance are reported according to the Gaussian loss G and Hellinger loss H . Unsurprisingly, methods trained with respect to a specific loss perform better than the competitor with respect to such loss.
Sentence 2: as a further measure of performance we also introduced the "Recognition" loss R . This loss has to be intended as a measure of how "well" a predictor was able to correctly reconstruct an image for digit recognition purposes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, the proposed approach will provide a principled way of combining labels from any combination of human and previously existing machine-based classifiers.
Sentence 2: the labels given by labeler i to image j (which we call the given labels) are denoted as L ij and, under the model, are generated as follows: under the model, the log odds for the obtained labels being correct are a bilinear function function of the difficulty of the label and the expertise of the labeler, i.e., More skilled labelers (higher α i ) have a higher probability of labeling correctly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In contrast, our network can only implement pairwise interactions, a connection between two spiking neurons implementing the conditional probability linking the two corresponding binary variables.
Sentence 2: we need to assume additional conditional independencies between the nodes in the generative model, so that their joint probability can be pairwise factorized: In words, it means that variables bias each other's probabilities, but do not influence each other's dynamics, i.e they do not affect each other's transition probabilities.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can further show that the marginal distribution of all b m is a t-distribution with t = b m |α m /σ bm|αm and 2â α degrees of freedom, which allows a principled way of determining whether a regression coefficient was excluded by means of standard hypothesis testing.
Sentence 2: variational Bayesian Least Squares (VBLS) regression is a full Bayesian treatment of the linear regression problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2 Let ( P , ẑ) be a solution to the SDP: ) of the LP: linear program in the variables i (with the u i 's fixed).
Sentence 2: an extreme point solution ( ¯ , z ⇤ ) is defined by n + 1 equalities, at most three of which can be constraints in ( 6)-( 8) and the rest (at least n 2 of them) must be from the Given the upper bound of d on the sum of the ¯ i 's, this implies that at least d 1 of them are equal to 1, i.e., at most two are fractional and add up to 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Perturbations along any other directions do not yield equilibria because then, either ✓ D is no longer in an all-zero state or ✓ G does not match the true distribution.
Sentence 2: we consider a setup where the rank deficiencies of K DD and K T DG K DG if any, correspond to equivalent equilibria (which typically exist for neural networks, though in practice they may not correspond to 'linear' perturbations as modeled here).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, if the agent only sees one task, all such states will be within some small fixed range of the other two lights; a termination set built from such data would not transfer to another task, since the relative positions of the lights would change.
Sentence 2: generalization across tasks requires each option to terminate when it is close to the proper light, regardless of the observed positions of the other two lights.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Often f 0 is a high-cost model that requires most, if not all, of features so c(f 0 , g) can be considered a large constant.
Sentence 2: to meet the budget constraint, we would like to have (a) low-cost g and f 1 (small c(f 1 , g)); and (b) small fraction of examples being sent to the high-accuracy model (small E Sn [q(0|x)]).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In general, an abstention stump is defined by the pair h ✓1,✓2 (X), r ✓1,✓2 (X) where, for Figure 4-left, h ✓1,✓2 (X) = 1 X✓1 + 1 X>✓2 and r ✓1,✓2 (X) = 1 ✓1<X✓2 .
Sentence 2: our abstention stumps are pairs (h, r) with h taking values in { 1, 0, 1} and r in {0, 1}, and such that for any x either h(x) or r(x) is zero.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because the bound in (1.1) is tight, the performance loss may grow with the discount factor as fast as γ/(1−γ) 2 while the total return for any policy only grows as 1/(1−γ).
Sentence 2: for γ sufficiently close to 1, the policy π computed through state aggregation may be no better than a random policy even when the approximation error of the optimal policy is small.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Comparing 9 ¡ @ and @ ¡ b9 ` should tell us which of 9 and @ is more likely to have its samples mis-classified by the other model.
Sentence 2: the difference 9 ¡ @ ¨ @ ¡ 9 W , when positive, indicates that samples of class @ are more likely to be mis-classified as class 9 than samples of class 9 are to be mis-classified as class @ (and vice-versa when the difference is negative).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the rate of this method is dependent on the problem dimension m, n, and converges quite slowly when m, n are large.
Sentence 2: they leave an open question on whether other efficient splitting methods exist, resulting in convergence analysis in the space with lower dimension m or n. In this paper, we propose a new splitting method for LP, which splits the equality and inequality constraints into two blocks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even though the evolving graph model requires the algorithm to issue a low number of queries, our analysis shows that (under reasonable assumptions on p and q) this small number of queries is sufficient to maintain a high-quality clustering.
Sentence 2: our theoretical results hold for large enough n. we also perform simulations, which demonstrate that our final theoretically optimal algorithm is able to beat the other algorithms even for small values of n. Clustering and community-detection techniques have been studied by hundreds of researchers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In other words, 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.
Sentence 2: to measure the similarity between two labels using the graphon slices, we define the following distance d ij is small only if both row and column slices of the graphon are similar.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that Equation 3 is a special case of Equation 4, using the instrumental distribution q = p. The estimate Ĝn,q given by Equation 4 depends on the selected instances (x i , y i ), which are drawn according to the distribution q(x)p(y|x).
Sentence 2: ĝn,q is a random variable whose distribution depends on q.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For sufficiently large n and m and small p, there exist parameters for our algorithm such that it outputs a clustering with accuracy (1 − ε) and the expected number of comparisons is Note that n 1+O(p) n 2 /m Ω(1/p) in the expression above whenever the reads are long enough, that is, when m n cp (where c is some small constant).
Sentence 2: for a large range of n, m, p, and ε, our algorithm converges in time proportional to n 1+O(p) , which is sub-quadratic in n, the number of input reads.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, where the Lévy measure ν is Furthermore, where {T (t) : t ≥ 0} is a Gamma subordination and each T (t) has density As we see, T (t) follows Gamma distribution Ga(T (t)|t/γ, α).
Sentence 2: the {T (t) : t ≥ 0} is called the Gamma subordinator.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since g is also concave, x 0 is a global maximum, and thus g(x) ≤ g(x 0 ) = 0 for all x, implying that f (K) V0 (x) ≤ x.
Sentence 2: for odd K, the function f V0 satisfies all requirements for Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that N ¬p wk represents processor p's belief of the counts of all the other processors with which it has already communicated (not including processor p's local counts), while N p wk is the processor's local word-topic counts.
Sentence 2: the sampling of the z p 's is based on the processor's "noisy view" of the global set of topics.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, if a given optimization algorithm obtains an iteration complexity of an order of c(n, κ) ln(1/ ), up to logarithmic factors (including the norm of the minimizer which, in our construction, is of an order of 1/n and coupled with the accuracy parameter), for solving smooth and strongly convex finite sum problems with a stochastic oracle, then c(n, κ) = Ω n 2 ln(n 2 /(κ + 1)) .
Sentence 2: the following holds for optimization of finite sums with smooth and strongly convex individuals.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a consequence, if µ h corresponds to the l-th column of C c , then its l-th coordinate is expected to be greater than other coordinates.
Sentence 2: we set the l-th column of C c to some vector µ h whose l-th coordinate has the greatest component (if there are multiple such vectors, then randomly select one of them; if there is no such vector, then randomly select a µ h ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For computational feasibility, in practice d Fnn (μ n , νm ) is typically adopted as an approximation (i.e., an estimator) of the true neural net distance d Fnn (µ, ν) for the practical GAN training, where μn and νm are the empirical distributions corresponding to µ and ν, respectively, based on n samples drawn from µ and m samples drawn from ν.
Sentence 2: one important question one can ask here is how well d Fnn (μ n , νm ) approximates d Fnn (µ, ν).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , (i t−1 , C it−1 , a t−1 ) , we take the general approach of [1], and assume that for any fixed i ∈ V and x ∈ R d , the variable i (x) is conditionally sub-Gaussian with variance parameter σ 2 > 0, namely, E t exp(γ i (x)) ≤ exp σ 2 γ 2 /2 for all γ ∈ R and all x, i.
Sentence 2: u i x is the expected reward observed at node i for context vector x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, motivated from the study of homometric structure and the fact that each element of FGSD encodes some local or global sub-structure information of the graph, inspired us to define the graph spectrum as R = {S f (x, y)|∀(x, y) ∈ V }.
Sentence 2: comparing two R's implicitly evaluates the sub-structural similarity between two graphs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We formally express this behavior in the following Corollary.
Sentence 2: for policy π ∈ Π, the posterior belief process µ t is a supermartingale with respect to Ft , where unlike classical Bayesian learning models with a belief martingale [18,[21][22][23], the belief process in our model is a supermartingale that leans toward decreasing over time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a fixed adaptive control algorithm f and a fixed sequence of disturbance terms w 0 , w 1 , . .
Sentence 2: ., the sequence of states x t visited is exactly determined by the equations ) for fixed f , we can think of the (controlled) dynamical system as a mapping from the sequence of disturbance terms w t to the sequence of states x t . We now give the most commonly-used definition of stability, called BIBO stability (see, e.g., [6]).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: After n steps, PCA will then be dominated by dimensions for which α i is small enough that n ≪ n c,i .
Sentence 2: even if relatively few 10 1 10 2 10 3 PCA component 10 -14 10 -13 10 -12 10 -11 10 -10 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 ResNet-50-v2 dimensions have small curvature they will come to dominate the PCA projected trajectory after enough steps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The proof, which is provided in the appendix, shows that in the noiseless case, for each principal submatrix, A l , of A, step 2a of Algorithm 1 will learn an exact Ĉl such that A l = Ĉl ĈT l . Further, when assumptions A3 and A4 are met, step 2b will correctly learn the orthonormal transformation to align each Ĉl to the previously added rows of Ĉ.
Sentence 2: progressive iterations of step 2 correctly learn more and more rows of a unified Ĉ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this formulation, the mixing function, α(β, τ ), is by definition a member of L 2 .
Sentence 2: if f t (β, τ ) ∈ L 2 , then P (t) is finite by the Cauchy-Schwartz inequality since α(β, τ ), For ease of readability we shall henceforth suppress the domain variables in f t (β, τ ) and α(β, τ ) and refer to them as f t and α. for some > 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Proposition 3.2 implies that at any fixed time, the likelihood of observing ⌧ given a candidate source is proportional to the number of feasible orderings originating from that candidate source.
Sentence 2: an ML estimator (timestamp rumor centrality) counts the number of feasible orderings at estimation time t. Timestamp rumor centrality is a message-passing algorithm that proceeds as follows: for each candidate source, recursively determine the set of feasible times when each node could have been infected, given the observed timestamps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the regions where ∇Q < η(n, f ) • √ d • σ, the Byzantine workers may take advantage of the noise (measured by σ) in the gradient estimator G to bias the choice of the parameter server.
Sentence 2: proposition 2 is to be interpreted as follows: in the presence of Byzantine workers, the parameter vector x t almost surely reaches a basin around points where the gradient is small ( ∇Q ≤ η(n, f ) • √ d • σ), i.e., points where the cost landscape is "almost flat".
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, the total reward on the agent's single trajectory is a random variable, and may deviate from the expected value of its policy.
Sentence 2: it is generally impossible to decide if the agent's policy is near-optimal, and instead we assume that the human can check if each action that the agent takes in the trajectory is near-optimal: when the agent takes a at state s, an error is counted if and only if Q (s, a) < V (s) − .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The location of the non-zero entries of M r depends on (X 2 ) and is independent of b :,r . Proof The product of the (k + (i − 1)K)-th row of (X 2 ) and b :,r is the (k + (i − 1)K)-th element of v :,r . And, this element is the (i, k)-th entry of M r by definition of unvec (K,I) (•) .
Sentence 2: if all the entries in the (k + (i − 1)K)-th row of (X 2 ) are zero, then the (i, k)-th entry of M r is zero regardless of b :,r . Consequently, the location of the non-zero entries of M r is independent of b :,r , and is only determined by (X 2 ) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Blei and Jordan [7] define an explicit truncation level L ≡ T for the variational mixture in (1) by setting q vT (v T = 1) = 1 and assuming that data-cases assign zero responsibility to components with index higher than the truncation level T , i.e., q zn (z n > T ) = 0.
Sentence 2: in their model only components of the mixture up to level T need to be considered.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This additional error (due to the numerical approximation) can be made arbitrarily small, e.g.
Sentence 2: o(N −1/2 ), whenever H ≥ log N d . using P = O( √ N ) random features, we deduce that the complexity of building the matrix Then in order to solve the least squares system, one has to compute Ψ T Ψ, that has numerical cost O(P 2 N ), and then solve the system by inversion, which has numerical cost O(P 2.376 ) by [5].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice also that a i p,l = k(x p , m i l ) measures a proximity between point x p and the mode obtained at the previous iteration.
Sentence 2: given the current assignment z i p , the modes can be obtained as a proximal optimization, which seeks a high-density data point that does not deviate significantly from the mode obtained at the previous iteration: Now observe that the z-updates we obtained in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unfortunately, the limiting distribution of such V -statistics is quite complicated -it is an infinite sum of dependent χ 2 -distributed random variables, with a dependence determined by the temporal dependence structure within the process {Z t } and by the eigenfunctions of a certain integral operator associated with h 2 [5,8].
Sentence 2: we propose a bootstrapped version of the V -statistics which will allow a consistent approximation of this difficult limiting distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Hawkes process is equivalent to the union of these Poisson processes -each event is sampled from one of these processes.
Sentence 2: it is natural to introduce a latent cause variable z for each event: z = 0 if the event is caused by the base rate; z = n if the event is caused by a preceding event s n . For clearer description, let us consider the flattened single event sequence S = [(s 1 , i 1 ), .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Lacking prior knowledge we assume that x takes any value in the interval uniformly at random.
Sentence 2: the  probability of x 1s occurring inside the knot, given the value of y (i.e., 1s prior to the knot) is: Based on this conditional probability we can write the probability of each value of x as in which P k (x|y) is computed by Equation ( 4) and Q k (y) refers to the probability of having y 1s prior to knot k. In order to evaluate Equation ( 5), we need to compute the values of Q k (y).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a non-planar graph G over binary variables with a vector of interaction potentials θ, we wish to use the exact planar computation to obtain a bound on the partition function of the MRF on G. We assume for simplicity that the potentials on the MRF for G are given in the form of Equation 2.
Sentence 2: g violates the assumptions of the previous section only in its non-planarity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, majority voting can not be applied when there are clustering models because the correspondence between clusters and classes is unknown.
Sentence 2: we compare BGCM with two clustering ensemble approaches (MCLA [26] and HBGF [12]), which ignore the label information from supervised models, regard all the base models as unsupervised clustering, and integrate the output of the base models.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If we want a single non-zero entry per column of H i , to achieve success probability 1 − O(1/s) so that we can union bound over all s servers succeeding, we naively would need to increase the number of rows of H i by a factor linear in s. We give a general technique, which takes a subspace embedding that succeeds with constant probability as a black box, and show how to perform a procedure which applies it O(log 1/δ) times independently and from these applications finds one which is guaranteed to succeed with probability 1 − δ.
Sentence 2: in this setting the players can compute a subspace embedding of their data in nnz(P i ) time, for which the number of non-zero entries of H i P i is no larger than that of P i , and without incurring this additional factor of s. This may be of independent interest.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The second step of the iHMM construction involves defining the transition distributions T (•|s) ∼ DP(α, T ) for each state s, where α, the concentration parameter for the DP, determines how closely the sampled distribution T (•|s) matches the mean transition distribution T . Because T puts higher probabilities on states with smaller indices, T (s ′ |s) will also generally put more mass on earlier s ′ (see lower rows of figure 2).
Sentence 2: the generating process encodes a notion that the agent will spend most of its time in some local region.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Initial theoretical work (see [4]) shows that increasing the number of bootstraps in the selection module (B 1 ) increases the amount of feature compression (primary controller of false positives), while increasing the number of bootstraps in the estimation module (B 2 ) increases feature expansion (primary controller of false negatives), and we observe this empirically.
Sentence 2: neither should be too large, and their relative values express the balance between feature compression and expansion.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The third key obstacle is that each of the latent variables, Φ and Θ, simultaneously serve as the inputs and output targets for successive layers.
Sentence 2: it is necessary to reformulate the connecting problem (2) so that it is jointly convex in all three components, U , Φ and Θ; and unfortunately (3) is not convex in Θ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, for the pair (G, θ), the distribution on X p is given as: θi,j x i x j where x ∈ X p and Z is the normalization factor, also known as the partition function.
Sentence 2: we obtain a family of distributions by considering a set of edge-weighted graphs G θ , where each element of G θ is a pair (G, θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Here, we depart from the approach of [1] by sampling a local split of topic a into topics b and c. Temporary parameters, {π b , πc , θb , θc }, and topic assignments, ẑ, are sampled according to We note that a sample from q(ẑ|v, v) is already drawn from the restricted Gibbs sampler described in Equation ( 9).
Sentence 2: no additional computation is needed to propose the split.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As an alternative, in Sections 3.1 and 3.2 we propose soft penalty functions that encode the belief that points near the origin are more likely to be optimal (we can always re-center the problem to match our beliefs in this regard), but do not rule out any x 2 R n a priori.
Sentence 2: one of our contributions is showing that interesting results can be obtained by choosing L differently than in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: RADAGRAD therefore makes use of the corrected update γ t is the projection of the current gradient onto the space orthogonal to the one captured by the random projection of G t . This ensures that important variation in the gradient which is poorly approximated by the random projection is not completely lost.
Sentence 2: if the data has rank less than τ , γ ≈ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The algorithm, MaxFlow, returns a maximal set, Z E , for which the equations, A • Λ T a(E),y = b, are linearly independent, regardless of whether |Z E | = |E| and E is identified or not.
Sentence 2: if we are able to find an allowed node w that satisfies the conditions below, then the equation a w • Λ T a(E),y = b w will be a linear combination of the equations, A • Λ T a(E),y = b. Theorem 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However in our setting, computing them naively is prohibitively expensive as the number of agents is large (in thousands).
Sentence 2: we use similar technique as for WLQ by normalizing counts, and computing differentials lim ∆=1/M→0 Qw(ñ sa , d)−Qw(ñ sa +∆•(e ij −e ij ), d) , final estimate is (proof in appendix): Crucially, the above computation is independent of agent population M , and is thus highly scalable.
Relationship: reasoning