sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: S, and then use this to define pairwise regrets as above.
Sentence 2: for the three tournament solutions discussed above, we will define the following natural notions of individual regret: ∈ TC(P) In the special case when P admits a Condorcet winner i * , the three individual regrets above all reduce to the Condorcet individual regret, r CW P (i) = ∆ P i * ,i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our CL encoder architecture is based on what we call a spatially-replicated LSTM: Given that, for a sequence of length T , a convolution outputs a T × H × W × C sized tensor X, a spatially replicated LSTM takes as input each of the X(:, i, j, :) cells, and encodes its output to the same spatial location.
Sentence 2: the weights of the LSTM at each spatial location is shared, but the hidden states are not.
Relationship: entailment
Domain: NeurIPS
Sentence 1: With a softmax function, we normalize the parameters for all the devices to obtain a probability distribution of the placement.
Sentence 2: the probability of choosing the j-th device for the m-th operation is given by: where D is the total number of devices, and u m denotes the vector of parameters {u m1 , u m2 , • • • , u mD } for the m-th operation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (iv) We want to be able to use such an algorithm together with the public unlabeled data to construct a differentially private learner that outputs a classifier, which can then be used to answer as many classification queries as we wish.
Sentence 2: given the second goal above, the final private learner would be completely agnostic to the intricacies of the underlying non-private learner and its model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the other hand, due to the symmetry between X and Z, we can also view CCA as an approach to feature-aware LSDR.
Sentence 2: cCA is equivalent to first seeking projection directions W z of Z, and then performing a multi-output linear regression from x n to W z z n , under the constraints W x X T XW T x = I, to obtain W x .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The distribution complexity r(Q) of the planting distribution Q is the smallest integer r ≥ 1 so that there is some S ⊆ [k], |S| = r, so that the discrete Fourier coefficient Q(S) is non-zero.
Sentence 2: the distribution complexity of Q is r if Q is an (r − 1)-wise independent distribution on {±1} k but not an r-wise independent distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To obtain mean-squared error estimates from MSB, we select our posterior mean as a point-estimate (the comparison algorithms do not generate posterior predictions, only point estimates).
Sentence 2: for each simulation scenario, we sampled multiple datasets and compute the matched distribution of r A m . rather than running simulations and reporting the distribution of performance for each algorithm, we compare the algorithms per simulation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From a theoretical perspective a key question has become to characterize statistical and computational tradeoffs, that is if, or under which conditions, computational gains come at the expense of statistical accuracy.
Sentence 2: recent results considering least squares, show that there are large class of problems for which, by combining Nyström or random features approaches [10,11,12,13,14,15] with ridge regression, it is possible to substantially reduce computations, while preserving the same optimal statistical accuracy of exact kernel ridge regression (KRR).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This ensures that we find the optimal point in the final grid G k . We also want I G k i to consist only of intervals of width 1 /k with endpoints contained in G k . This ensures that the overall number of points processed over all iterations is at most O(nk), and by bounding the number of intervals in each I i in each iteration we can achieve significantly better performance.
Sentence 2: the scaling algorithm for convex functions by Ahuja and Orlin (2001) can be seen as a particular realization of this framework where the refinement process keeps the size of each I i to exactly one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, the standard generalization error bounds for stochastic gradient descent (SGD) on convex Lipschitz functions cannot be obtained by proving uniform convergence for all empirical risk minimizers (ERM) [13,26].
Sentence 2: there exist empirical risk minimizing algorithms whose generalization error is √ d times larger than the generalization error of SGD, where d is the dimension of the problem (without the Lipschitzness assumption the gap is infinite even for d = 2) [13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: To this end, we need to introduce certain assumptions on (i) the construction of weighted points {(w i , X i )} n i=1 and on (ii) the smoothness of the integrand f ; we discuss them in Sections 4.1 and 4.2, respectively.
Sentence 2: we introduce the notion of powers of RKHSs [22] in Section 4.2, which enables us to characterize the (relative) smoothness of the integrand.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For matrices, • p denotes the Schatten p-norm, X p = ( i σ i (X) p ) 1/p , where σ i (X) are the singular values of X.
Sentence 2: • * = • 1 is the trace or nuclear norm, • F = • 2 is the Frobenius norm, and • = • ∞ is the operator norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, it is worth mentioning that when performing the SVM, each x n was normalized to be V i=1 x n i = 1 because discrimination is much easier in the V − 1-dimensional simplex than in the original V dimensional space.
Sentence 2: classification is generally not determined by the number of words on the page; actually, normalization could also significantly improve the performance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The recent work of [8], concerning generalized weighted variants of k-NN, shows that asymptotic normality holds under the weaker restriction k/n 4/(4+d) → 0 if f is twice differentiable at x. Asymptotic normality as stated above yields some insight into the rate of convergence of k under the stated conditions on k. In fact, [8] shows that such a result can be obtained in expectation for n = n(x) sufficiently large.
Sentence 2: their conditions on k allows for a setting of k ≈ n 4/(4+d) (not allowed under the above conditions) yielding a minimax-optimal While consistency results and bounds on expected error are now well understood, we still don't have a clear understanding of the conditions under which high probability bounds on |f k (x) − f (x)| are possible.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More broadly, relative to our rich and detailed understanding of regularization via penalization (e.g., see the books [17,33,32,37] and papers [2,20] for details), the theory for early stopping regularization is still not as well developed.
Sentence 2: for penalized estimators, it is now wellunderstood that complexity measures such as the localized Gaussian width, or its Rademacher analogue, can be used to characterize their achievable rates [2,20,32,37].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we show that an efficient coding model that incorporates biologically realistic ingredients -input and output noise, nonlinear response functions, and a metabolic cost on the firing rate -predicts receptive fields and response nonlinearities similar to those observed in the retina.
Sentence 2: we develop numerical methods for simultaneously learning the linear filters and response nonlinearities of a population of model neurons, so as to maximize information transmission subject to metabolic costs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For instance, we can construct the following problem for standard HMC: Recall that the joint likelihood is p(θ, p) ∝ exp(−H(θ, p)), H(•, •) being the energy from (1).
Sentence 2: then, we use correct samplers that preserve the desired densities in the E step of a Monte Carlo EM (MCEM) framework, and use the obtained samples of p in the corresponding M step to perform suitable updates for the mass M . to wrap the standard HMC sampler in our framework, we perform the generalized leapfrog steps [1,15] to obtain proposal updates for θ, p followed by Metropolis-Hastings corrections in the E step, and use the obtained p values in the M step.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We evaluate our method via 10-fold cross-validation, where we have sub-sampled the training folds in order to (a) keep the computational burden as low as possible and (b) show that we can learn sensible parameterizations based upon relatively low requirements in terms of the preferences seen on previous users.
Sentence 2: we have subsampled 50 training users and selected about 5 training pairwise preferences drawn from each of the N = 10 available items.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the set X of signals is well-specified, one can usually compute a (nearly) minimax on X linear estimator in a closed form.
Sentence 2: if X is a class of smooth signals, such as a Hölder or a Sobolev ball, then the corresponding estimator is given by the kernel estimator with the properly set bandwidth parameter [16] and is minimax among all possible estimators.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The appearance terms and IDPR terms depend on the image patches.
Sentence 2: a local image patch I(l i ) not only gives evidence for the presence of a part i, but also about the relationship t ij between it and its neighbors j ∈ N(i), where j ∈ N(i) if, and only if, (i, j) ∈ E. This requires us to learn distribution for the state variables i, t ij conditioned on the image patches I(l i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to address such scalability issues, Monteiro [2003, 2005] restrict the search to the set of matrices of rank at most k by factorizing X as X = Y Y * , with Y ∈ K n×k .
Sentence 2: it has been shown that if the search space C (1) is compact, then (SDP) admits a global optimum of rank at most r, where dim S r×r ≤ m [Barvinok, 1995, Pataki, 1998, with dim S r×r = r(r+1) for K = R and dim S r×r = r 2 for K = C. restricting C to the space of matrices with rank at most k with dim S k×k > m does not change the optimal value.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If there is no additional structure imposed, beyond what we have stated above, then up to scaling, in the noiseless case, Outlier Pursuit can recover from as many outliers (i.e., the same fraction) as any possible algorithm with arbitrary complexity.
Sentence 2: it is easy to see that if the rank of the matrix L 0 is r, and the fraction of outliers satisfies γ ≥ 1/(r + 1), then the problem is not identifiable, i.e., no algorithm can separate authentic and corrupted points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: µ, we would prefer step sizes that are µ-independent, thus giving rates that adapt to the local curvature (see [9]).
Sentence 2: it turns out that by choosing a step size that maximizes min K ρ(γ)/ρ * (K), we obtain a K-agnostic step size with rate off by at most 1/2: To gain more insights into the trade-offs for these fixed large universal step sizes, the following corollary details the range of rates obtained: we have for the choice γ = 1 5L that ρ = min{ 1 3 q n , 1 5 µ L } (roughly matching the rate given in [4] for q = 1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: is convex for every y ∈ Y, and if the evaluation metric is the AP, the ERR or the PD, then there are distributions and sequences of scoring functions for which (1) does not hold.
Sentence 2: we show that learning-to-rank algorithms that define their objective through a convex surrogate loss cannot provably optimize any of these evaluation metrics.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We study the fundamental problems of variance and risk estimation in high dimensional statistical modeling.
Sentence 2: we consider the problem of learning a coefficient vector θ 0 ∈ R p from noisy linear observations y = Xθ 0 + w ∈ R n (p > n) and the popular estimation procedure of solving the 1 -penalized least squares objective known as the LASSO or Basis Pursuit DeNoising (BPDN).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For bounded input functions F, G, let ρ(F, G)=max t |F(t) − G(t)|, and let U be the class of all unimodal distributions.
Sentence 2: then the dip statistic of a distribution function F is given by: the dip statistic computes the minimum among the maximum deviations observed between the cdf F and the cdfs from the class of unimodal distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It maps the continuous representation e x of an architecture x into its performance s x measured by dev set accuracy.
Sentence 2: f first conducts mean pooling on e x = {h 1 , • • • , h T } to obtain e x = 1 T T t h t , and then maps e x to a scalar value using a feed-forward network as the predicted performance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Compared with the setting in [6], we add the constraint π Θ = 0.
Sentence 2: the reason is due to Since Rπ 1 2 = 0, there exists a c×(c−1) orthogonal matrix ∆, the columns of which are the eigenvectors of R. ∆ satisfies ∆ ∆ = I c−1 and ∆ π ] is the c×c matrix of the orthonormal eigenvectors of R. Since for an arbitrary class scoring matrix Θ, its rank is c−1, we have Θ = ΘΥ where Υ is some (c−1)×(c−1) orthonormal matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, despite the amount of work dedicated to this statistical learning theory-many authors improved the initial results [8,21,25,30,35] and/or generalized them for various machine learning setups [4,12,15,20,28,31,32,33]-it is mostly used as a frequentist method.
Sentence 2: under the assumptions that the learning samples are i.i.d.-generated by a data-distribution, this theory expresses probably approximately correct (PAC) bounds on the generalization risk.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The potential benefit of acceleration to mini-batching has been briefly noted in [4], but here we study this issue in much more depth.
Sentence 2: we make the following contributions: • We develop novel convergence bounds for the standard gradient method, which refines the result of [10,4] by being dependent on L(w ), the expected loss of the best predictor in our class.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using the hinge loss yields support vector machines [5].
Sentence 2: structured support vector machines [27] employ a convex approximation of the multivariate loss over a training dataset D using the hinge loss function: linear parameters θ for feature functions Φ(•, •) are desired that make the example label y (i) have a potential value θ • Φ(x (i) , y (i) ) that is better than all alternative labels y by at least the multivariate loss between y and y (i) , denoted ∆(y , y (i) ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recall that g := The matrix C is referred to as the self-adjoint dilation of the matrix g [20].
Sentence 2: given the SVD of g = U⌃V > with no repeated singular values, the eigen-decomposition of C is given as the top-k left and right singular vectors of C  As in Section 2, we take the convex hull of the constraint set to get a convex relaxation to Problem (11).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We approximate it by the outcome of the subgradient method run for M steps, with a step-size proportional to 2/(m + 2) as suggested in [23].
Sentence 2: initialized with θ0 i = θ t i , it performs the iterations We thus replace the step (b ) by running M steps of the subgradient method to obtain θM i . Theorem 4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Based on this building block, we design purely first-order stochastic algorithms for escaping from non-degenerate saddle points with a much better time complexity (almost linear time in the problem's dimensionality) under a bounded variance condition of stochastic gradients than previous first-order stochastic algorithms.
Sentence 2: we develop a general framework of first-order stochastic algorithms with a secondorder convergence guarantee based on our new technique and existing algorithms that may only converge to a first-order stationary point.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While extending the SM kernel to the multi-output setting via the LMC framework (i.e., the SM-LMC kernel) provides a powerful modeling framework, the SM-LMC kernel does not intuitively represent the data.
Sentence 2: the SM-LMC kernel encodes the cross-amplitude spectrum (square root of the cross power spectral density) between every pair of channels, but provides no crossphase information.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, given an image and a question about the image, we first identify useful sub-graphs of a large knowledge graph such as DBpedia [3] and then use GCNs to produce representations encoding node and neighborhood features that can be used for answering the question.
Sentence 2: we propose a model that retrieves the most relevant facts to a question-answer pair based on GloVe features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The other 97 features f 4 , f 5 , ..., f 100 are conditionally independent of y given the three features f 1 , f 2 and f 3 .
Sentence 2: f 4 , ..., f 33 are all generated by the rule f j = 3 * f 1 + ξ; f 34 , ..., f 72 are all generated by the rule f j = sin(10 * f 2 ) + ξ; and the remaining features (f 73 , ..., f 100 ) simply follow a uniform distribution in [0, 1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The goal of a sketch is to estimate such quantities without explicitly storing the elements or the counts.
Sentence 2: in the classical count-min sketch, N hash functions h n : V → [J], where [J] := {1, . . . , J} and |V| J, are chosen uniformly at random from a pairwise independent hash family H. a random h ∈ H has the property that for all v 1 , v 2 ∈ V, such that v 1 = v 2 , the probability that v 1 and v 2 hash to values j 1 , j 2 ∈ [J] is The sketch data structure C = [c n,j ] n∈[N ],j∈ [J] is an array of counts of size N × J.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also give the model information about the distribution of notes in real musical scores by providing it with an external corpus of symbolic music data.
Sentence 2: the following information is available to the model during inference: 1) the total spectrogram for each song, X (r) , which is the input, 2) the event parameters, µ, which we estimate by collecting statistics on note occurrences in the external corpus of symbolic music, and 3) truncated normal priors on the envelopes and spectral profiles, α and σ, which we extract from the MIDI samples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, The consequence of this property is that taking a sample from q(f L i ) is straightforward, and furthermore we can perform the sampling using only univariate unit Gaussians using the 're-parameterization trick' (Rezende et al., 2014;Kingma et al., 2015).
Sentence 2: we first sample l i ∼ N (0, I D l ) and then recursively draw the sampled variables where the terms in ( 15) are D l -dimensional and the square root is element-wise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to being analytically tractable and popular, the usage of multivariate Gaussian distribution can also be motivated using Bayesian asymptotics for large datasets.
Sentence 2: for parameters in R d and under regularity conditions, if the size of the subset D i is large, the Bernstein-von Mises Theorem shows that the local posterior distribution is well approximated by a multivariate Gaussian; hence the EP approximation by an exponential family density will be very good.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Hence, to find the closest object to q on every level, we build the same heap structure, the only difference is that instead of repeating this procedure n times in each level, since there is just one query point, we need to ask at most O(D 3 log 2 n + D log 2 n log log n D 3 ) questions totally.
Sentence 2: the closest object at level L = log 2D (n) will be q's nearest neighbor w.h.p.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is consistent with the backward branch circuit of the Laguerre lattice filter, Figure 2, being different then that of the forward branch (which results in different transfer function).
Sentence 2: a combination of different glutamate receptors such as AMPA and NMDA, as well as GABA receptors are thought to be responsible for observed responses in lagged cells [32].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, consider making a prediction using the modified inequality test: Here, the vector of parameters θ = (θ 1 , θ 2 ) controls the relative "weighting" between the message subjects and bodies, and will be fit discriminatively.
Sentence 2: we will model the class posteriors, which we denote by Pθ to make explicit the dependence on θ, as: 3 Pθ (y|x) = P (y) P (x 1 |y) We had previously motivated our model as assigning different weights to different parts of the document.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then the margin condition (7) is satisfied if and only if there exists a constant C > 0 such that, for all ✏ > 0, (`, F ✏ , P ⇤ ) is ⌘-stochastically mixable for ⌘ = C✏ ( 1)/ .
Sentence 2: if the margin condition is satisfied with constants  and c 0 , we can take C = min V ( 1)/ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then focus on applying MGOMP to high-dimensional multivariate time series analysis problems.
Sentence 2: we propose a novel application of multivariate regression methods with variable selection, namely that of inferring key influencers in online social communities, a problem of increasing importance with the rise of planetary scale web 2.0 platforms such as Facebook, Twitter, and innumerable discussion forums and blog sites.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The drift term in SAMD η,a,s is identical to the deterministic case, and the volatility term −η(t)σ(X(t), t)dB(t) represents the noise in the gradient.
Sentence 2: we note that the learning rate η(t) multiplies σ(X(t), t)dB(t), to capture the fact that the gradient noise is scaled by the learning rate η.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We generate 100 problem instances from y = Ax 0 + e, where the solutions vectors have been sampled from the proposed prior using the kernel Σ i,j = 50 exp(−||i − j|| 2 2 /(2 • 10 2 )), but constrained to have a fixed sparsity level of the K/D = 0.25.
Sentence 2: each solution x 0 has the same number of non-zero entries, but different sparsity patterns.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3) shrinker η * for the singular values of Y given by We also discover that for each contamination model, there is a critical signal-to-noise cutoff, below which X cannot be reconstructed from the singular values and vectors of Y .
Sentence 2: let η 0 be the zero singular value shrinker, η 0 (y) ≡ 0, so that Xη0 (Y ) ≡ 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Bernoulli variables with where ρ ∈ [0, 1] is the average fraction of stimuli in the receptive field.
Sentence 2: we then assume that, given the vector ξ of receptive field indicators, the filter weight coefficients are independent with distribution the linear weight coefficients are zero outside the receptive field and Gaussian within the receptive field.
Relationship: entailment
Domain: NeurIPS
Sentence 1: HMMs due to a per-iteration complexity linear in T . Unfortunately, despite the simple chain-based dependence structure, applying a minibatch-based method is not obvious.
Sentence 2: there are two potential issues immediately arising in sampling subchains as minibatches: (1) the subsequences are not mutually independent, and (2) updating the latent variables in the subchain ignores the data outside of the subchain introducing error.
Relationship: entailment