sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Only then estimates are updated and a new policy is calculated.
Sentence 2: since all states in the same interval I j have the same confidence intervals, finding the optimal pair Mk , πk in ( 4) is equivalent to finding the respective optimistic discretized MDP M agg k and an optimal policy πagg k on M agg k . Then πk can be set to be the extension of πagg k to S, that is, πk (s) := πagg k (I(s)) for all s. due to the constraint on the bias even in this finite case efficient computation of M agg k and πagg k is still an open problem.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this special tree case the maximum likelihood estimator is tractable and is tantamount to finding a maximum weighted spanning-tree.
Sentence 2: it is also known that in the case of general graphs with cycles, maximum likelihood estimators are intractable as they require computation of the partition function of the underlying GM, with notable exceptions of the Gaussian GM, see for instance [9], and some other special cases, like planar Ising models without magnetic field [10].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Although SSMs have an illustrious history [24], their stochasticity has limited their widespread use in the deep learning community, as inference can only be exact for two relatively simple classes of SSMs, namely hidden Markov models and linear Gaussian models, neither of which are well-suited to modeling long-term dependencies and complex probability distributions over high-dimensional sequences.
Sentence 2: modern RNNs rely on gated nonlinearities such as long short-term memory (LSTM) [16] cells or gated recurrent units (GRUs) [6], that let the deterministic hidden state of the RNN act as an internal memory for the model.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We consider first the problem of computing optimal recommendation sets given the system's uncertainty about the user's true utility function w. Given belief state θ, if a single recommendation is to be made, then we should recommend the option x * (θ) that maximizes expected utility EU (x, θ).
Sentence 2: there is often value in suggesting a "shortlist" containing multiple options and allowing the user to select her most preferred option.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In recent computer science research [18] [13][3] [19], machine learning methods have been applied for fMRI data analysis.
Sentence 2: in these approaches information on the connectivity and interactivity between brain voxels is discarded and brain voxels are assumed to be independent, which is an inaccurate assumption (see use of statistical maps [3] [19] or the mean of each fMRI time interval [13]).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In particular, the robust return of π is min ξ ρ(π , ξ) = −9, while the optimistic return of π B is max ξ ρ(π B , ξ) = +10.
Sentence 2: solving (2) will return the optimal policy since: min ξ ρ(π , ξ) − ρ(π B , ξ) = 11 − 10 = −9 − (−10) = 1.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: By a simple application of the union bound, it is clear that the asymptotic probability of the Type I error will be lim n→∞ P H0 (A n ) ≤ α.
Sentence 2: if the alternative holds, there exists some m with |m| ≤ M for which V (h, Z m ) = n −1 S m,n converges to a non-zero constant.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: K 1 is just the FLG kernel k κ FLG induced from the base kernel κ between the lowest level subgraphs: 2.
Sentence 2: for = 2, 3, . . . , L, K is the FLG kernel induced from K −1 between G (v) and G (v ): Definition 5 defines the MLS kernel as a kernel between different subgraphs of the same graph G. if two graphs G 1 and G 2 share the same base kernel, the MLS kernel can also be used to compare any subgraph of G 1 with any subgraph of G 2 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Adapting this to make a distance measure, One drawback with this measure is that it implicitly assumes that all c CCA vectors are equally important to the representations at layer L 1 .
Sentence 2: there has been ample evidence that DNNs do As the signal to noise ratio decreases, the unweighted mean underestimates the shared structure, while the projection weighted mean remains largely robust.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As in [7], we can choose λ = √ n or λ = n to obtain decays rates log n √ n or log n n respectively, where the latter has a fixed non-decaying gap term (b − a) 2 /2.
Sentence 2: unlike [7], in our proof both cases are achievable with η = 1, i.e., for the variational algorithm.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Swimmer-5 Swimmer-3 Swimmer-3 Figure 3: Left: For a 3-link swimmer, with relatively simple dynamics, the compared methods yield similar results and possibly a slight advantage to the purely model-based SVG (1).
Sentence 2: middle: as the environment model's capacity is reduced from 20 to 10 then to 5 hidden units per state-dimension subnetwork, SVG(1) dramatically deteriorates, whereas SVG(1) shows undisturbed performance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As one would expect, our experiments indicate that the parsing performance of the grammatical bigram model is not as good as that of state-ofthe-art parsers; however, its performance in Experiment II suggests that it may be useful in domains where partial accuracy is valuable and large amounts of unlabelled data are available.
Sentence 2: to realize that potential, the model must be improved so that its performance in Experiment III is closer to that of Experiment II.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, if f is supermodular (α ≡ 1) and for = k, we recover the 1 − e −1 ≈ 0.63 guarantee from [18].
Sentence 2: for an approximately supermodular function (ᾱ < 1), the result in (9) shows that the 63% guarantee can be recovered by selecting a set of size = ᾱ−1 k. Thus, α not only measures how much f violates supermodularity, but also gives a factor by which the cardinality constraint must be violated to obtain a supermodular near-optimal certificate.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, the constraint µ L 8 n is satisfied when λ 8ξR n . In low-dimensional settings, the optimal regularization parameter is of the form C/n [29] where C is a scalar constant, and may thus violate the constraint.
Sentence 2: the improvement with respect to regularization parameters of the form λ = C/ √ n is known to be asymptotically negligible, and in any case in such low-dimensional settings, regular stochastic or batch gradient descent may be efficient enough in practice.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Next, we wish to go beyond proportionality and determine the precise relationship between bias and discriminability under the power-law efficient coding framework described above.
Sentence 2: any optimization of Fisher information, including ours, doesn't prescribe a method for selecting a parametric encoding distribution p(y | x) associated with a particular power-law efficient code, that is, a distribution with Fisher information allocated according to J(x) = k p(x) q .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The constant Q can be considered to represent the second moment of the "gradient noise" in ∇J i (x k ).
Sentence 2: therefore, the second moment of ∇J i (x k ) can be bounded above as Under Assumptions 1, 2, and 3, the iterates of CDSGD satisfy ∀k ∈ N: In Lemma 2, the first term is strictly negative if the step size satisfies the following necessary condition: in latter analysis, when such a condition is substituted into the convergence analysis, it may produce a larger upper bound.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To reduce the computational complexity, Greengard and Strain proposed the Fast Gauss Transform (FGT) [5], using two expansions, the far-field Hermite expansion and the local Taylor expansion, and a translation process that converts between the two, yielding an overall complexity of O(M + N ).
Sentence 2: due to the expensive translation operation, O(p d ) constant term, and the box based data structure, this method becomes less effective for higher dimensions (e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: At each iteration, a quadratic approximation to the loss is used as a surrogate functional with the regularization term added to induce joint functional sparsity.
Sentence 2: a technical difficulty is that the approximate quadratic problem in each iteration is weighted by a non-diagonal matrix in function space, thus a trivial extension of the algorithm in (Ravikumar et al., 2007) for sparse binary nonparametric logistic regression does not apply.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The cost to the alignment of changing a label Yi is 2 Lj Yjk(Xi' xj)/IIKIIF , so that if a point is isolated from the others, or if it is equally close to the two different classes, then changing its label will have only a very small effect.
Sentence 2: labels in strongly clustered points clearly contribute to the overall cost and changing their label will alter the alignment significantly.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that the joint and conditional histograms of the synthesized samples have similar transition of spatial dependencies as the separation increases (column 1,2 and 3), suggesting that the FoGSM accounts well for pairwise joint dependencies of coefficients over a full range of spatial separations.
Sentence 2: the dependencies between subbands of different orientations and scales are not properly modeled by FoGSM (column 4 and 5).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This can significantly reduce MR scanning time.
Sentence 2: according to structured sparsity theory, the measurements can be further reduced to O(K + log n) for tree-sparse data instead of O(K + K log n) for standard K-sparse data with length n. few of existing algorithms have utilized this for CS-MRI, while most of them model the problem with total variation and wavelet sparse regularization.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In terms of the linear algebra, the spectral expansion of K is dominated by a few large eigenvalue eigenvectors, we will call this the "PCA-like" scenario.
Sentence 2: in situations where f varies more rapidly, a shorter lengthscale kernel is called for, local relationships between nearby points become more important, which the landmark point method is less well suited to capture.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, [14] that we use here, obtains an average of 66% correct hand detection (comparing favorably to other state-of-the-art methods) in the general setting experiments, when both the person and the background are unseen during part detector training.
Sentence 2: as shown in [14], average 85% part detection performance can be achieved in more restricted settings.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [1] proposed a trust region method for CMDP called constrained policy optimization (CPO), which can deal with high-dimensional policy classes such as neural networks and claim to maintain feasibility if started with a feasible solution.
Sentence 2: we found in Section 5 that feasibility is rarely guaranteed during learning in practice, possibly due to errors in gradient and Hessian matrix estimation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Identifying the best decision d ∈ D then amounts to identifying which equivalence class H d contains the realized vector of outcomes, which is an instance of ECD.
Sentence 2: one common approach to this problem is to myopically pick tests maximizing the decision-theoretic value of information (VoI): The VoI of a test t is the expected reduction in the expected loss of the best decision due to the observation of x t . we can show there are instances in which such a policy pays Ω(n/ log(n)) times the optimal cost, even under a uniform prior on (h, θ) and with | supp(Θ)| = 2.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The stated upper bounds for k-MCV and k avg -MCV in Table 1 can be obtained directly from our k-AS and k avg -AS algorithms.
Sentence 2: all the lower-bound arguments for k-AS and k avg -AS-which crucially rely on the samples being i.i.d.-break down for the two MCV problems, due to the history-awareness property explained in Section 1.1.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In practical terms, the simple selections α(γ) = 1 and β(γ) = 0 will provably satisfy the monotone cell update property (see proof details in the supplementary).
Sentence 2: for additional flexibility, α and β could be selected to implement various forms of momentum, ultimately leading to cell updates akin to the popular FISTA [4] or monotonic FISTA [3] algorithms.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The tighter BP/OMP bound from [1,4,5] scales as O N −1/2 , although this latter bound is much more general in that it is independent of the magnitudes of the nonzero weights.
Sentence 2: neither BP or OMP satisfy a comparable result; in both cases, simple 3D counter examples suffice to illustrate this point.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We can see that clearly the latent subspace representations discovered by the largemargin based MMH show a strong grouping pattern for the images belonging to the same category, while images from different categories tend to be separated from each other on the 2D embedding space.
Sentence 2: the latent subspace representations discovered by the likelihood-based unsupervised DWH and supervised TWH do not show a clear grouping pattern, except for the first category.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, multi-view latent space MNs do not increase the complexity on testing if our task depends solely on the latent representation (i.e., expectation of H), such as information retrieval [26], classification, clustering etc.
Sentence 2: the complexity of parameter estimation and inferring the posterior distribution of each view (e.g., X) will be increased, depending on the structure on the view.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Momentum angular information for each observed particle is also available to the network, but is not shown, as the one-dimensional projections have little information.
Sentence 2: particles B and C, we could calculate the invariant mass of the short-lived intermediate state A in the process A → B + C, via: finite measurement resolution and escaping neutrinos (which are invisible to the detectors) make it impossible to calculate the intermediate state mass precisely.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: At each round of the game, adversary samples θ(t) ∈ [0, 1] n according to p, with E θ(t)∼p [θ(t)] = θ * p . The learner chooses x(t) ∈ X and gets reward r(x(t), θ(t)).
Sentence 2: the learner might not get to know either θ(t) (as in a full information game) or r(x(t), θ(t)) (as in a bandit game).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Notice that it is reasonable to meet the lower bound Ω(n 3/2 ) even given features, since in an extreme case, X, Y could be random matrices and have no correlation to R, and thus the given information is as same as that in standard matrix completion.
Sentence 2: in many applications, features will be far from random, and our result provides a theoretical insight to show that features can be useful even if they are imperfect.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The full conditional distribution p(θ uk,t |θ uk,t−1 , θ uk,t+1 ) is conjugate.
Sentence 2: it is not possible to attain a positive correlation between θ uk,t and θ uk,t−1 since E[θ uk,t ] = 1/θ uk,t−1 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: With some extra work, one can check that if, for instance, γ = 1 and |µ − 1| ≤ cn −1/2 for some constant c, then the estimator σ has also a risk of the order of sn −1/2 .
Sentence 2: the price to pay for being adaptive with respect to the noise level is the presence of Mβ * 1 in the bound on σ, which deteriorates the quality of estimation in the case of large signal-to-noise ratio.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: That common value of η must be small enough that all three algorithms converge; if we take η 1 M N 2 , the learning curves become We began by saying that, because of the credit assignment problem of choosing updates to many parameters based on a single reward signal, intuition is that a stochastic gradientfollowing approach should learn more slowly than a direct one.
Sentence 2: for equal small η, the average error for all three algorithms converges at the same rate.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Of course it is generally difficult to get a constant 1 in the batch setting (see Arlot [2007]), and similarly this is a difficult task in our online setting too: If � is chosen to be small, then the error with respect to the optimal allocation is small.
Sentence 2: since P n is expressed as an increasing function of �, this implies that the minimax bound on the loss for partition P increases also with �.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 2) The choice of regularization parameters λ β and λ e can make explicitly: assuming w is a Gaussian random vector whose entries are N (0, σ 2 ) and the design matrix has unit-normed columns, it is clear that with high probability, X * w ∞ ≤ 2 σ 2 log p and w ∞ ≤ 2 σ 2 log n. Thus, it is sufficient to select λ β ≥ 4 γ σ 2 log p and λ e ≥ 4 σ 2 log n. 3) At the first glance, the parameter γ does not seem to have any meaningful interpretation and the γ = 1 seems to be the best selection due to the smallest estimation error it can produce.
Sentence 2: this parameter actually control the sparsity level of the regression vector with respect to the fraction of corruption.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Observation 1: MC-b and MC-1 are the best for feature imputation, as Table 2 shows.
Sentence 2: the imputations are not perfect, because in these particular parameter settings the ratio between the number of observed entries over the degrees of freedom needed to describe the feature matrix (i.e., r(d + n − r)) is below the necessary condition for perfect matrix completion [2], and because there is some feature noise.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Known challenges include the definition of locality (e.g., for structured data [2]), identifiability [12] and computational cost (with some of these methods requiring a full-fledged optimization subroutine [24]).
Sentence 2: point-wise interpretations generally do not compare explanations obtained for nearby inputs, leading to unstable and often contradicting explanations [1].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Based on our initial dataset of 100,000 MCTS moves, CAT and TPT have similar performance in the task of predicting the move selected by MCTS, with average top-1 prediction errors of 47.0% and 47.7%, and top-3 prediction errors of 65.4% and 65.7%, respectively.
Sentence 2: despite the very similar prediction errors, the TPT network is 50 ± 13 Elo stronger than the CAT network, suggesting that the cost-awareness of TPT indeed gives a performance improvement.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: At a high-level, the typical analysis to the UCB algorithm for the standard multi-armed bandit problem (Auer et al., 2002) is the following: (1) at some finite time T , the number of pulls of a bad arm i is O with high probability, and (2) the regret suffered by any such pull is O(D i ).
Sentence 2: the contribution of arm we analyzed the UCB algorithm in Section 2 by observing that the expected regret suffered on round t is bounded by the difference between the empirical mean estimator and the true mean for the payoff of arm I t . Of course by design this quantity is almost certainly (w.p.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Further, even though different heuristics have been developed to address the unbalanced problem, these methods suffer from the drawback that they have to make several decisions prior to reaching a final category, which leads to error propagation and thus a decrease in accuracy.
Sentence 2: label embedding approaches [11,5,19] first project the label-matrix into a low-dimensional linear subspace and then use an OVA classifier.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, since in general exp(A 1 ) exp(A 2 ) = exp(A 1 + A 2 ), the path ( 12) is not contained in the manifold M defined in (11).
Sentence 2: it is tempting to try to construct a new manifold M such that Span and X 0 ∈ M , ensuring the gradient flow remains in M . since A i 's do not commute, by combining infinitesimal steps along different directions, it is possible to move (very slowly) in directions that are not of the form A * (s)X + XA * (s) for any s ∈ R m . The possible directions of movements indeed corresponds to the Lie algebra defined by the closure of {A i } m i=1 under the commutator operator [A i , A j ] := A i A j − A j A i .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Due to the relative comparison constraint, LMNN does not satisfy the condition of Lemma 2.
Sentence 2: as we already mentioned LMNN satisfies the condition of Lemma 3 so we get the ML h -MKL P variant of the optimization problem for LMNN which we denote by LMNN-MKL P . The resulting optimization problem is: where the matrix Y, Y ij ∈ {0, 1}, indicates if the class labels y i and y j are the same (Y ij = 1) or different (Y ij = 0).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Complexity of one iteration of QI is O(nk + lk), i.e., linear in the number of options, exactly like Greedy.
Sentence 2: in practice it is much faster than Greedy since typically k << l. While we have no theoretical results that limit the iterations required by QI to converge, in practice, a fixed point is reached in very quickly (see below).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Most of these works focused on scenarios where the tasks are batch learning problems, in which a training set is directly provided to the learner.
Sentence 2: the online learning setting (Cesa-Bianchi and Lugosi, 2006), where the learner is presented with samples in a sequential fashion, has been rarely considered (see Mann and Choe (2012); Taylor (2009) for examples in RL and Sec.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The streams have the same volume, so the optimal policy splits time uniformly between the NW-straight and EW-straight phases.
Sentence 2: an additional stream of cars is generated in the south-west corner, at Intersection H, and travels diagonally east to Intersection E. Two equally short routes are available by going straight, or turning east at Intersection F. cars that turn east to join the northbound traffic flow must then turn east again at Intersection D, forcing the controller of that intersection to devote time to a third NS-right phase and forcing the main volume of traffic to pause.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: With probability at least Furthermore, if ℓ is classification-calibrated, there exists a nondecreasing function The term on the right hand side involves both approximation error (that is small if F is large) and estimation error (that is small if F is small).
Sentence 2: by appropriately increasing the richness of the class F with sample size, we can ensure that the misclassification probability of f approaches the Bayes risk of the true distribution.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The vectors emanating from x and reaching these points represent For x, x ′ ∈ R 1 , this illustration shows that d β (x, x ′ ) ∈ cl cone(R − x) for any β ≥ 0.
Sentence 2: figure 1(c) considers x ′ ∈ R 2 , while x ∈ R 1 .
Relationship: contrasting