sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: First, we have focused on the SAGA method as the basic iteration loop, but this approach can likely be extended to other proximal incremental schemes such as SGD or ProxSVRG.
Sentence 2: we emphasize that a straightforward combination of this sparse update rule with the proximal update from the Standard SAGA algorithm results in a nonconvergent algorithm: the block-wise reweighting of h is a surprisingly simple but crucial change.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The most interesting aspect of the Bayesian optimal policy is that these immediate and future reward components in (3) can be interpreted as automatically balancing exploitation (immediate utility) and exploration (expected future utility given the information revealed by the present batch).
Sentence 2: we conduct a paired t-test for each other policy against the best one, and also emphasize those that are not significantly worse than the best with significance level α = 0.05 in blue italics.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The algorithm next generates new features Φ (s, a) from the data trajectories using the current DRL network with weights w k . This step guarantees that we do not use samples with inconsistent features, as the ER contains features from 'old' networks weights.
Sentence 2: rL tackles the curse of dimensionality by approximating terms in the DP calculation such as the value function or policy.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The ill-posed nature of the MEG/EEG source localization problem requires the incorporation of prior assumptions when choosing an appropriate solution out of an infinite set of candidates.
Sentence 2: this produces approximate posteriors on W and Z but the result cannot be integrated to form the posterior on S. the posterior mean of W , Ŵ , is used as an estimate of the source correlation matrix (using Ŵ Ŵ T ) to substantially improve beamforming results that were errantly based on uncorrelated source models.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We encode this prior knowledge by choosing Ω(β|Λ) with This method constraints the sum of the sets to be nonincreasing and may be interpreted as the composition of the wedge set with an average operation across the sets P i , see the discussion at the end of Section 3.
Sentence 2: the utility of (2.2) is that upon inserting it into (2.1) results in an optimization problem over λ and β with a continuously differentiable objective function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In Figure 1 (right), we considered frustrated grid networks where p = 1 10 , where REC-BP converged in only one of 10 networks generated.
Sentence 2: before we discuss the general case where we relax multiple equivalence constraints, we highlight first a few properties shared by both REC-BP and REC-I compensations, that shall follow from more general results that we shall present.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We let the family of conditional densities for y be Gaussian for simplicity, that is, we assume that f j,k = N (µ j,k , σ j,k ) with µ j,k ∈ R and σ j,k ∈ R + .
Sentence 2: . . , x ip ) T ∈ X for i ∈ [n], the graph construction follows via computing all pairwise distances using ρ(x u , x v ) = xu − xv 2 , where x is the whitened x (i.e., mean subtracted and variance normalized).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Corollary 1 shows that the left singular vectors P (k) can be recovered under mild conditions; thus the span of the right singular vectors can also be recovered.
Sentence 2: . . , R): (5) If the core tensor C = (β i1...i K ) ∈ R R×•••×R is superdiagonal, the above decomposition reduces to the canonical polyadic (CP) decomposition [9,15].
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this construction, rather that drawing the ✓ ⇤ i ⇠ G 0 once as in (5), one may draw ✓ ⇤ i ⇠ G 0 for the first block of time, and then a simple Gaussian auto-regressive model is employed to allow the {✓ ⇤ i } drift a small amount between consecutive blocks.
Sentence 2: the research reported here was supported under the DARPA HIST program.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When T is as small as 25, the sparse least-square method shrinks all the eigenvalues but still keep most of them non-zero, while Lyap with a non-sequence sample of size 1600 over-estimates the first few largest eigenvalues in modulus but shrink the rest to have very small modulus.
Sentence 2: both models are stable, and the stationary distribution for each model is a zero-mean Gaussian.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Results using the full model are reported in Table 3, which shows 3D estimation is still under satisfactory, despite the efforts to maintain a good 2D-3D consistency.
Sentence 2: we formulate the learning objective of the projection from 3D to 2D as where f (•) denotes a differentiable projection function which projects a 3D bounding box to a 2D bounding box, and X I * j ∈ R 2×4 is the 2D object bounding box (either detected or the ground truth).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Rather, we consider a step of the form ¡ ¢ where ¡ £¢ # ¢ ¢ H . It is easy to see that the first order theory of gradient boosting and the line search solution apply to this formulation as well.
Sentence 2: our implementation of the boosting algorithm, therefore, does indeed limit to include probability distributions only, in this case those that can be represented by "simple" Bayesian networks.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We define the regret incurred by a reinforcement learning algorithm π up to time T to be where ∆ k denotes regret over the kth episode, defined with respect to the MDP M * by .
Sentence 2: we demonstrate through a computational study in Section 6 that PSRL outperforms the optimistic algorithm UCRL2 [4]: a competitor with similar regret bounds over some example MDPs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For each value of p and sparsity level of Ω, n = 1.2p and n = 0.2p were tested, to represent both the n < p and n > p cases.
Sentence 2: the existence of such constants follows directly from Theorem 1, as Θ t lie in the bounded domain {Θ ∈ S p ++ : f (Θ) + g(Θ) < f (Θ 0 ) + g(Θ 0 )}, for all t. it is possible to specify the constants a and b to yield an explicit rate; this is done in Theorem 2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Denoting by exp q : T q N → N the exponential map at q, the simple gradient descent based optimization scheme can be written as • choose a random point q 0 from N , • while stopping criteria not fulfilled, 1. compute gradient ∇F at q k 2. one has: q k+1 = exp q k − α∇F | q k 3. determine stepsize α by Armijo rule [19].
Sentence 2: the discussion of a robust median-type estimator, that is Γ(x) = x, has been done recently in [8].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let Gaussian with mean µ t and variance σ 2 t . Assume that the noise is also Gaussian with variance σ 2 , so F (x) = Φ( x σ ).
Sentence 2: forcing the MM to maintain a Gaussian belief over the true value at each time t should give a good approximation to the true state space evolution, and the resulting optimal actions should closely match the true optimal actions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Similarly, temporal receptive fields were chosen randomly from functions bandlimited to 4 kHz.
Sentence 2: several examples include conductance-based models such as Hodgkin-Huxley, Morris-Lecar, Fitzhugh-Nagumo, Wang-Buzsaki, Hindmarsh-Rose [20] as well as simpler models such as the ideal and leaky integrate-and-fire (IAF) neurons [15].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Other invariant kernels have been proposed: In [19] authors introduce transformation invariant kernels, but unlike our general setting, the analysis is concerned with dilation invariance.
Sentence 2: tIDIGITS (Figure 3): We use a subset of TIDIGITS consisting of 326 speakers (men, women, children) reading the digits 0-9 in isolation, and so each datapoint is a waveform of a single word.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Kernel methods provide a principled framework for nonlinear data modeling, where the inference in the input space can be transferred intactly to any feature space by simply treating the associated kernel as inner products, or more generally, as nonlinear mappings on the data (Schölkopf & Smola, 2002).
Sentence 2: for instance, in cases where only inner products of the input data are involved, kernel learning is equivalent to the learning of a kernel matrix.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The main question we sought to answer with this study is whether mathematical decoding algorithms that can make use of the temporal evolution of movement parameters, that can incorporate uncertainty, and that can also incorporate prior knowledge, would provide improved decoding results compared to an existing algorithm that utilized only the instantaneous relationship.
Sentence 2: the duration of certain movement patterns will deviate from the distribution described by Equation 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, SGD is both β Z -uniformly stable and β Z -pointwise hypothesis stable with respect to L and P, with When T = Θ(n), Equation 1 is Õ(n −1 ), which is acceptable for proving generalization.
Sentence 2: in certain situations, it may be advantageous to sample according to a data-dependent distribution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Applying Lemma 3.1 and Theorem 3.1 in (11), one obtains an upper bound for KL divergence, In the problem of minimizing the KL divergence g(µ, CC T ), one needs to iteratively update µ and C, since they are coupled.
Sentence 2: we derive an upper bound for the Kullback-Leibler divergence, which yields a fast closed-form solution via decoupled optimization.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We derive a block coordinate descent algorithm for optimizing our bound over both the cost-shifting parameters (reparameterization) and weights (fractional counting numbers), which generalizes dual decomposition and enjoy similar monotonic convergence property.
Sentence 2: we can apply power sums with different weights τ i to each variable x i along a predefined elimination order (e.g., [x 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Instead of performing any single domain-specific task such as clustering or estimating an expectation, we can simply compute the barycenter of the inputs and process it later any arbitrary way.
Sentence 2: full experiment details are given in Appendix D. As the input distributions are moved along parallel paths on the sphere, the barycenter accurately tracks the new locations as shown in the final three figures.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We would like to guarantee, with probability 1 p for a small p > 0, that the confidence interval around this estimate contains the value x > w ⇤ given by the optimal w ⇤ 2 W. To estimate such an interval without parametric assumptions, we use Chebyshev's inequality which-unlike other concentration inequalities like Hoeffding or Bernstein-does not require independent samples.
Sentence 2: for covariance, because the covariance of the weights is The goal for computing variances is to use a concentration inequality.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Dual smoothing (dual-smooth): Instead of the primal, the dual (6) may be smoothed, e.g., by entropy [8,38] applied to each g j as gε j (λ j ) = min x∈[0,1] n f j (x) + εh(x) where h(x) is a negative entropy.
Sentence 2: as we will see, its convergence is so slow to counteract any benefit that may arise from not using projections.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 2 We now present a simple poisoning attack on transfer learned networks.
Sentence 2: the first dimension is along the vector joining the centroids of the base and target classes (u = µ base − µ target ), while the second dimension is along the vector orthogonal to u and in the plane spanned by u and θ (the weight vector of the penultimate layer, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 2 Lepski's procedure for estimating the Laplace-Beltrami operator All the Riemannian manifolds considered in the paper are smooth compact d-dimensional submanifolds (without boundary) of R m endowed with the Riemannian metric induced by the Euclidean structure of R m . Recall that, given a compact d-dimensional smooth Riemannian manifold M with volume measure µ, its Laplace-Beltrami operator is the linear operator ∆ defined on the space of smooth functions on M as ∆(f ) = − div(∇f ) where ∇f is the gradient vector field and div the divergence operator.
Sentence 2: cross validation [AC + 10] is the standard approach for tuning parameters in statistics and machine learning.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The key intuition is that we stack L GNN modules and learn to assign nodes to clusters at layer l in an end-to-end fashion, using embeddings generated from a GNN at layer l − 1.
Sentence 2: this research has been supported in part by DARPA SIMPLEX, Stanford Data Science Initiative, Huawei, JD and Chan Zuckerberg Biohub.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In addition, the obtained estimator attains the optimal statistical properties in parameter estimation: Recall that b ✓ {1} is obtained by the warm initialization (3).
Sentence 2: moreover, for some constant M > 0, at least one of the following two conditions holds: (I) The Hessian of the cumulant function is uniformly bounded: Then Assumption 1 ⇠ 3 hold with high probability.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then F 1 (y, ȳ) = 2 2+t1+t2 , and So we can pick ȳ as ( ) which only misclassifies x k+1 , and get which can be verified by ∂J k+1 (w k+1 ) = w k+1 + k+1 i=1 α i a i : α ∈ ∆ k+1 0 (just set all as claimed in Theorem 6.
Sentence 2: it is usually much harder to establish SLB than WLB.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We also refer to the output p t = f t (b 0∶t−1 ) as the strategy of the player at time t. Now that we can refer to algorithms simply as functions (or distributions over functions), we introduce the notion of a stable algorithm.
Sentence 2: if the algorithms played by player 1 in the form of f t and player 2 in the form of g t give sub-linear policy regret sequences, then the sequence of product distributions converges weakly to the set S. In particular if both players are playing MWU or Exp3, we know that they will have sublinear policy regret.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As mentioned in the introduction, the sample complexity of margin-error minimization can be upperbounded in terms of the average norm Alternatively, we can rely only on the dimensionality and conclude m(ǫ, γ, D) ≤ Õ(d/ǫ 2 ) [7].
Sentence 2: our tight characterization, and in particular the distribution-specific lower bound on the sample complexity that we establish, can be used to compare large-margin (L 2 regularized) learning to other learning rules.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The reason for this, theoretically, is that the sample complexity of MOD-SULQ scales with d 3/2 p log d whereas PPCA scales with d. Because PPCA uses the exponential mechanism with q F (•) as the utility function, it is not surprising that it performs well.
Sentence 2: our theoretical results apply to the special case k = 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Theorem 7 essentially implies that by exploiting the multi-stage convex relaxation framework, our proposed DC proximal Newton algorithm gradually reduces the estimation bias for "strong" signals, and eventually obtains an estimator with better statistical properties than the `1-regularized estimator.
Sentence 2: we terminate the iterations when the following approximate KKT condition holds: where " is a predefined precision parameter.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The loss function is defined as: where [Pred(x 0 + δ)] i is the i-th class prediction score of x 0 + δ.
Sentence 2: our explanation for the digit in row 1 of Figure 1 to be a 3 would be: The row 1 digit is a 3 because the cyan pixels (shown in column 2) are present and the pink pixels (shown in column 3) are absent.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Altogether, these results suggest that several observed features of cortical dynamics, such as excitatory-inhibitory balance, integrate-and-fire dynamics and Hebbian plasticity, are signatures of a robust, optimal spike-based code.
Sentence 2: we set L V = V 2 /2, and calculate the gradient: We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions 4 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: We can now state the definition of the lower envelope for the CMH minimum attainable p-value: Definition 1 Let S ∈ I P P be a potentially prunable feature subset.
Sentence 2: analogously, we consider as potentially prunable the set of feature subsets I P P = {S | x S,j ≤ min(n 1,j , n 2,j ) ∀ j = 1, . . . , k}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , x d+1 ), so the Bayes classifier predicts 0 for all x in the simplex.
Sentence 2: first, the nearest neighbor rule (also called 1-nearest neighbor, in the context of the general family of k-nearest neighbor rules) is a well-known interpolating classification method, though it is not generally consistent for classification (and is not useful for regression when there is significant amount of label noise).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The smallest bounding box of X is defined as B (X) := B (a, b) where the lower bound a is given by the pointwise infimum of X and the upper bound b is given by the pointwise supremum of X, that is a α := inf{x α : x ∈ X} and b α := sup{x α : x ∈ X} for all α = 1, . . . , d. Note that B (X) = B (conv (X)).
Sentence 2: lemma 2 Let (A t ) t=1,...,T be disjoint subsets of V. For each t = 1, . . . , T , let Ξ t ⊆ M At be convex with a finite number of extreme points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The present paper is related to a recent publication targeted at a neuroscience audience [14].
Sentence 2: generating a corrective control signal can be detrimental, because: 1) the noise in the motor system is known to be multiplicative [13] and therefore could increase; 2) the cost being minimized most likely includes a control-dependent effort penalty which could also increase.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given an input vector of dimension d, we consider a neural network with L layers of neurons for binary classification.
Sentence 2: (iii) Similar to Corollary 1, we will present the following corollary to show that at every local minimum θ * = (θ * , θ * exp ), the neural network f with augmented exponential neurons is equivalent to the original neural network f . Corollary 2 Under the conditions in Theorem 2, if θ * = (θ * , θ * exp ) is a local minimum of the empirical loss function Ln ( θ), then two neural networks f (•; θ * ) and f (•; θ * ) are equivalent, i.e., f (x; θ * ) = f (x; θ * ), ∀x ∈ R d . Corollary 2 further shows that even if we add an exponential neuron to each layer of the original network f , at every local minimum of the empirical loss, all exponential neurons are inactive.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The learner's objective is to maximize cumulative reward, which is equivalent to minimizing regret: the extra reward a learner would have received by always choosing the best available action.
Sentence 2: a randomized algorithm A for the contextual bandit problem is (ε, δ)-jointly differentially private (JDP) under continual observation if for any t and any pair of t-neighboring sequences S and S , and any subset S >t ⊂ D t+1 × D t+2 × • • • × D n of sequence of actions ranging from day t + 1 to the end of the sequence, it holds that P(A(S) ∈ S >t ) ≤ e ε P(A(S ) ∈ S >t ) + δ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This implies that Next we present our key result, Theorem 4.5, which can be viewed as a 'Neyman-Pearson lemma' for binomial data under ( , 0)-DP.
Sentence 2: for this reason, it is natural to consider a test which post-processes X + L. we know by classical UMP theory that since X + L is discrete, a randomized test is required.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Next we used the d561102 data from hc-1, which consists of 4 extracellular recording and 1 intracellular recording.
Sentence 2: we used 5000 burn-in samples and 5000 collection samples in the Gibbs sampler, and we choose the collection sample with the maximum likelihood when presenting below example clusterings.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 2) Histograms: sums of the intensities in random image patches, histograms of (oriented and non oriented) gradients at different locations, Haar-like features.
Sentence 2: at every Boosting step, our first algorithm MAS Naive models G q with a Gaussian mixture model fitted on the edges estimated at the previous iteration, computes from that density model the pair (Q, T ) maximizing e(Q, T ), samples the corresponding number of examples and features, and keeps the weak-learner with the highest approximated edge.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The generative adversarial network (GAN) [3] employs a generative model in which the code is drawn from a simple distribution (e.g., isotropic Gaussian), and then the code is fed through a sophisticated deep neural network (decoder) to manifest the data.
Sentence 2: vAEs [12,13] represent one of the most successful deep generative models developed recently.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that the IC oracle never generalizes beyond the level from which the examples are drawn, and thus, exhibits very different generalization results compared to the human participants in our experiment.
Sentence 2: as no existing challenge data tests this paradigm, we collect and make available a new, large-scale dataset for visual concept learning using the ImageNet hierarchy as the source of possible concepts, with human annotators to provide ground truth labels as to whether a new image is an instance of each concept using a paradigm similar to that used in experiments studying word learning in children.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Finally, we like to mention that, if the conditional distributions are not equal modulo translations, then the situation may become more involved.
Sentence 2: in order to prove the remaining assertions it suffices to show that lim t→±∞ C L ,P( • |x) (t) = ∞ for P X -almost all x ∈ X.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As shown in Figure 4, the value of the LOO margin Bc \c is obtained from the sequence of Bc vs. a c segments for each of the decrement steps, and thus determined by their slopes 'Yc.
Sentence 2: the KT conditions are expressed differentially as : jES where a e is the coefficient being incremented, initially zero, of a "candidate" vector outside D. Since 9i == 0 for the margin vector working set S = {Sl, .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Computed using Maxwell's equations and a spherical shell head model [12], the resulting Φ is characterized by highly correlated rows, because the small scalp surface requires that sensors be placed close together, and vastly different column norms, since the EM field strength drops off rapidly for deep brain sources.
Sentence 2: inclusion of both W and D together can be highly disruptive.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that the observation to be passed on to the POSMDP to create the POSMDP observation space, O, is part of the design tradeoff -usually it is desirable to reduce the number of observations in order to reduce complexity without degrading the value of the POSMDP too much.
Sentence 2: an Unmanned Aerial Vehicle (UAV) can be deployed to maintain WiFi network communication between the ground units.
Relationship: neutral