sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: However, we conjecture that the worst case is when µ 1 and µ 2 are much larger than the other means: then, the regret is almost the same as in the 2-arm case, and ETC strategies are suboptimal by a factor 2.
Sentence 2: the most favourable case (in terms of relative efficiency) seems to be when µ 2 = • • • = µ K : then and T * = 2( √ K − 1 + 1) 2 /∆ 2 , leading to lim sup while Lai and Robbins' lower bound yields 2(K − 1)/∆.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Proof We establish the equivalence of ( 9) and ( 11) by showing the equivalence (10) and (11).
Sentence 2: by the nonhomogeneous Farkas theorem [9, Theorem 2.4.8] we have that ( 10) is equivalent to either: B'u + w = 0, b'u + ry + 1::; 0, u 2: 0, having solution (u, w), (13) or (14) the second alternative ( 14) contradicts the nonemptiness ofthe knowledgeset {x I Bx::; b}, because for x in this set and u solving (14) gives the contradiction: (15) Hence ( 14) is ruled out and we have that ( 10) is equivalent to (13) which is (11).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This allows us to decompose the transition function as a mean value in S plus additive noise s Õ ≥ P M (•|s, a) =∆ s Õ = p M (s, a) + ' P . At first this may seem to exclude discrete MDPs with S states from our analysis.
Sentence 2: we can represent the discrete state as a probability vector s t oe S = [0, 1] S µ R S with a single active component equal to 1 and 0 otherwise.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, the difference of disagreements of σ * and σ * a↔b with V on such pairs is Suboptimality of approval voting for distances other than the KT distance stems from the fact that in counting the number of approvals for a given alternative, one discards information regarding other alternatives approved along with the given alternative in various votes.
Sentence 2: no such information is discarded when only one alternative is approved (or not approved) in each vote.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, E[( For many functionals, a Hölderian assumption (Σ(s, L)) alone is sufficient to guarantee the rates in Theorems 5,6 and 7.
Sentence 2: for some functionals (such as the α-divergences) we require f , ĝ, f, g to be bounded above and below.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The CountSketch scheme thus constructs a sketching matrix A such that every column has O(lg(1/δ)) non-zero entries.
Sentence 2: this construction does not provide a norm-preserving embedding into a Euclidean space (that is, the estimator of x 2 2 cannot be represented as a norm of Ax), which is essential for some applications such as nearest-neighbor search [HIM12].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, contrary to the non-decreasing case [3], our regularizers are not norms.
Sentence 2: they are norms on the hyperplane {w ⊤ 1 V = 0} as soon as for A = ∅ and A = V , F (A) > 0 (which we assume for the rest of this paper).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that an outbreak in some region S does not affect the disease rate outside region S. Thus we can use the same values of α out (S) and β out (S) whether we are assuming the null hypothesis H 0 or the alternative hypothesis H 1 (S).
Sentence 2: the effect of an outbreak inside region S must be taken into account when computing α in (S) and β in (S); since we assume that no outbreak has occurred in the past data, we cannot just use the sample mean and variance, but must consider what we expect these quantities to be in the event of an outbreak.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Nevertheless, at least for finite action sets, the tools developed in [1,3,2] look suitable to show that under appropriate conditions these problems can be overcome if the function set is chosen in a judicious way.
Sentence 2: the results of these works would become essentially useless in the case of an infinite number of actions since these previous bounds grow to infinity with the number of actions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the data follows the r-spiked model, the optimal step size will be close to 1 if there is no sub-sampling.
Sentence 2: due to fluctuations resulting from sub-sampling, we suggest the following step size choice for NewSt: In general, this formula yields a step size greater than 1, which is due to rank thresholding, providing faster convergence.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, from equation (13), we obtain that  Combining equations ( 15) and ( 16), we have Assume that the prior probabilities of the classes are the same, i.e., P i = P j = P , which holds for most EEG experiments.
Sentence 2: then equation ( 17) becomes from Σ = Combining equations ( 19) and ( 18), we obtain that Let Σi (k = 1, • • • , c) denote the new class covariance matrices computed via equation (8).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Specializing this bound to the case of standard Gaussian designs, the analysis of [10] shows formally that no test can detect θ 0,i = 0, with a fixed degree of confidence, unless |θ 0,i | ≥ cσ/ √ n. (ii) The sample size must satisfy n ≥ s 0 .
Sentence 2: indeed, if this is not the case, for each θ 0 with support of size |S| = s 0 , there is a one parameter family {θ 0 (t) = θ 0 + t v} t∈R with supp(θ 0 (t)) ⊆ S, Xθ 0 (t) = Xθ 0 and, for specific values of t, the support of θ 0 (t) is strictly contained in S. there is no fundamental reason to assume the irrepresentability condition (4).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the constellation models from [18], [42] parts are constrained to be in a sparse set of locations determined by an interest point operator, and their geometric arrangement is captured by a Gaussian distribution.
Sentence 2: pictorial structure models [15], [20] define a matching problem where parts have an individual match cost in a dense set of locations, and their geometric arrangement is constrained by a set of "springs" connecting pairs of parts.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given access to t, the resulting concept class has VC dimension 1.
Sentence 2: given instead access to a projection s for the action of the subgroup Z n1 ⇥ {0}, the class e P := {p(•|s) : p 2 P} has 2✏-uniform shattering dimension n 2 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Moreover, authors in [25] show that the number of negative eigenvalues of the Bethe Hessian gives a more accurate estimate of the rank of A than that based on trimming+SVD.
Sentence 2: we see that if the topology is not locally-tree-like but with some noise, for example with some additional cliques, both trimming of the data matrix and Bethe Hessian perform much worse, reporting a wrong rank, and giving a large reconstruction error, as illustrated in Fig.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The posterior is sharply peaked, with most samples being from the MAP solution, and so our learner does about as well as the MDL learner.
Sentence 2: sampling offers an (approximate) predictive posterior over predictions on the held out examples; in a real world scenario, one would offer the top C predictions to the user and let them choose, much like how spelling correction works.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This way of evaluating the BELU score is consistent with [5] and [2], and reproduces the 33.3 score of [29].
Sentence 2: if we evaluate the state of the art system of [9] (whose predictions can be downloaded from statmt.org\matrix) in this manner, we get 37.0, which is greater than the 35.8 reported by statmt.org\matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This differs from the usual beta process in the subtraction of σ from m k and addition of σ to n − m k + c. This is reminiscent of the Pitman-Yor generalization to the Dirichlet process [11,12,13], where a discount parameter is subtracted from the number of customers seated around each table, and added to the chance of sitting at a new table.
Sentence 2: the Lévy measure of the random atoms of µ is still a Lévy measure corresponding to an SBP with updated parameters Note that the update depends only on n, not on Z 1 , .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: From the above discussion, we observe that the iteration complexity in stochastic optimization is polynomial in 1 ǫ , making it difficult to find high-precision solutions.
Sentence 2: when the objective function is strongly convex and can be written as a sum of a finite number of functions, i.e., where each f i (•) is smooth, the iteration complexity of some specific algorithms may exhibit a logarithmic dependence on 1 ǫ , i.e., a linear convergence rate.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this case, the optimal sample complexity nmp is On the one hand, this result suggests that the social graph information does not help rating recovery when the number of users is relatively smaller than that of items.
Sentence 2: more precisely, when 2n log n ≤ 4γm log m, both (1) and ( 2) are equal to 2m log m/( when 2n log n > 4γm log m, i.e., (1) 2 , the result implies that the social graph information does help rating recovery.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Figures 3e and 3f, however, show that when there is a large degree of class overlap the entropy and differential entropy have different behavior -entropy is high both in region of class overlap and far from training data, making difficult to distinguish out-of-distribution samples and in-distribution samples at a decision boundary.
Sentence 2: the differential entropy is low over the whole region of training data and high outside, allowing the in-distribution region to be clearly distinguished from the out-of-distribution region.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It has been shown that also for k-ISA, scalings and permutations as above are the only indeterminacies [11], given some additional rather weak restrictions to the model.
Sentence 2: a serious drawback of k-ISA (and hence of ICA) lies in the fact that the requirement fixed group-size k does not allow us to apply this analysis to an arbitrary random vector.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These two terms would appear alone in a straightforward representational model of the joint distribution over sensory stimuli and values.
Sentence 2: in considering a representational subsystem within a reinforcement learning agent, where the overall goal is to maximise accumulated reward, it seems reasonable that the demand for representative or predictive fidelity depend on the value associated with the stimulus; this dependence is reflected here by a value-based weighting of the log probabilities, which we assume will weight the more valuable cases more heavily.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Recall that the evidence maximization procedure upon which ARD is based involves integrating out the unknown sources before optimizing the hyperparameters γ.
Sentence 2: if some p(γ) is assumed for γ, then we could just as easily do the opposite: namely, we can integrate out the hyperparameters and then maximize S directly, thus solving the MAP estimation problem max S p (B|S) p (S; Σ s ) p(γ)dγ ≡ min where each A i is derived from the i-th covariance component such that C i = A i A T i , and g(•) is a function dependent on p(γ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 8,12,13,15,17,18,20 Based on these roughly similar anatomical and physiological properties, cholinergic and noradrenergic systems have been attributed correspondingly similar general computational roles, such as modulating the signal-to-noise ratio in sensory processing.
Sentence 2: 9,10 the effects of ACh and NE depletion in animal behavioral studies, as well as microdialysis of the neuromodulators during different conditions, point to more specific and distinct computational roles for ACh and NE.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Probabilistically-correct inference should then implement selection, when data that is known to be irrelevant is excluded to the advantage of the relevant information (eg Dayan and Zemel, 1999;Palmer, 1994).
Sentence 2: in other circumstances, it will be appropriate to take advantage of the information about the target that is available in the neurons with large fields, even if this means allowing some influence on the final decisions from distractors.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that, if we do not restrict the form of Y, then a trivial solution to this problem is to learn identity mappings by setting y = Φ(x ) = x and Ψ(x, y ) = y .
Sentence 2: given that y has the "form" of a set of landmark detections, the model is strongly encouraged to learn those.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These results tend to show that ErrP could be a potential tool to improve the quality of the brain-computer interaction.
Sentence 2: it is to note that in order to isolate the issue of the recognition of ErrP out of the more difficult and general problem of a whole BCI where erroneous feedback can be due to nonoptimal performance of both the interface (i.e., the classifier embedded into the interface) and the user himself, the subject delivered commands manually.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The best known algorithm for active learning of rectangles (2, δ)-learns a very restricted class of distributions (continuous product distributions which are sufficiently balanced in all directions) with an active label complexity of Õ(d 3 p(ln(1/η)p(ln(1/δ))), where p(•) is a polynomial (Hanneke, 2007b).
Sentence 2: for a general distribution, active label complexity cannot be significantly better than passive label complexity.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However LSTD is concerned with finding a least-squares linear function approximation, it has not yet been developed for general γ and λ, and has update time quadratic in the number of features/states.
Sentence 2: our algorithm "exactly" coincides with TD/Q/Sarsa(λ) for finite state spaces, but with a novel learning rate derived from statistical principles.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is important to point out that each posterior density estimate at point n depends solely on the corresponding measurement x n and does not depend on any other x k , k = n. This means that even if we partition independent measurements into arbitrary subsets, posterior density estimation is accomplished locally, within each subset.
Sentence 2: in the M-step all measurements X affect the choice of θ because of the dependence of each term in the completed log likelihood on the same θ.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The variational parameters ω i are specific to the tasks used during meta-training, and will not be useful for a new task.
Sentence 2: since the KL-divergence loss (Eqn 3) encourages the pre-update parameters to be close to the prior, all of the variational parameters ω i are driven to the prior at convergence (Fig 5a).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We will assume a noisy-OR gate, and thus our model is closely related to causal power.
Sentence 2: we propose to model human estimates of P(C -+E ) as causal support, the log posterior odds in favor of Graphl (h e = 1) over Grapho (h e = 0) : Via Bayes' rule, we can express P( he = l 1 X) in terms of the marginal likelihood or eviden ce, P(X lhe = 1), and the prior probability that C is a cause of E , P(h e = 1) : P(h e = l 1X) ex P( Xl he = I)P(h e = 1).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Hence we choose another popular technique -particle filter.
Sentence 2: for our SSM, the traditional sampling importance resampling (SIR) particle filter will introduce the unnecessary computational load due to the fact that (5) in the SSM is a linear structure given θ t . This inspires us to apply a more efficient marginalized particle filter (also called Rao-Blackwellised particle filter) [9,11,12,13] to deal with the estimation problem by combining Kalman filter into particle filter.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To understand this condition, note that if L n were actually strongly convex, then both these RSC inequalities would hold with α 1 = α 2 > 0 and τ 1 = τ 2 = 0.
Sentence 2: in the high-dimensional setting (p n), the empirical loss L n can never be strongly convex, but the RSC condition may still hold with strictly positive (α j , τ j ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The proof hinged on the fact that all spanning subgraphs with same edge cardinality that satisfied the sufficiency condition for existence of any -PSNE were equally good with respect to our deterministic submodular function, and we created an instance with exponentially such spanning subgraphs.
Sentence 2: we might be able to circumvent Theorem 3 by breaking the symmetry, e.g., by using data that specifies multiple distinct -Nash equilibria.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is important to point out that, for most forms of Pr[x | y], the model is analytically intractable [21].
Sentence 2: when Pr[x | y] has an exponential form, such as Pr[x | y] = f (x)e −f (x)y , the model has several interesting properties [12,13].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, to see why it is difficult to compute the exact value of α in the nonparametric case, suppose we have computed the minimum cost of paths up to step t − 1 and we would like to compute the values of α for step t. The cost of a path that ends in state c is obtained by examining, for all states i, the cost of a path that ends at i at step t − 1 and then transitions to state c at step t. Thus, we must consider the transition from i to c. If there are existing transitions from state i to state c, then we proceed as in a standard forward-backward algorithm.
Sentence 2: we are also interested in two other cases-one where there are no existing transitions from i to c but we consider this transition along with a penalty λ 1 , and another where an entirely new state is formed and we pay a penalty λ 2 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: They have a fixed input size and typically perceive only small local contexts of the pixels to be classified as foreground or background.
Sentence 2: multi-Dimensional Recurrent NNs (MD-RNNs) can perceive the entire spatio-temporal context of each pixel in a few sweeps through all pixels, especially when the RNN is a Long Short-Term Memory (LSTM).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: ψ can be modeled as a parametric noisy-channel, where the parameters of ψ can also be learned during training.
Sentence 2: we model ψ as dropout [18] (multiplicative Bernoulli noise) because it provides an easy and straightforward method for noisy-transformation of e 1 into ẽ1 without complicating the training process.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given T , the c m, variables are deterministic-simply look up the th level of the mth path in the infinite collection of paths.
Sentence 2: not having observed T , the distribution of c m, will be defined by the nested Chinese restaurant process, conditioned on all the c q, for q < m. Now suppose we are given a corpus of M documents, w 1 , .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Recurrent Neural Networks (RNNs) have been found to be successful in a variety of sequence learning problems [4,3,9], including those involving long term dependencies (e.g., [1,23]).
Sentence 2: most of the empirical success has not been with "plain" RNNs but rather with alternate, more complex structures, such as Long Short-Term Memory (LSTM) networks [7] or Gated Recurrent Units (GRUs) [3].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Variance reducing techniques have also been applied to these settings, including smoothed stochastic variational inference (SSVI) [25] and variance reduced stochastic gradient MCMC (VRSGMCMC) algorithms [13,8,7].
Sentence 2: convergence guarantees for SSVI have not been developed, while VRSGMCMC algorithms are typically much slower than sEM-vr due to the intrinsic randomness of MCMC.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The difference between the spike rates on B1+ and B1encodes the B1 filter output.
Sentence 2: the B1+ and B1-spike rates do not represent the ON (positive half-wave rectified) and OFF (negative half-wave rectified) components of the binocular filter outputs, since they may both be non-zero at the same time.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We also show that our theoretical criterion for a weak learner to improve the overall model applies well in practice. . We are thus finding the first order optimal weak learner, which gives the "steepest descent" in the loss at the current model predictions.
Sentence 2: we should note that once ¨becomes non-infinitesimal, no "optimality" property can be claimed for this selected ¢ .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As seen, even though a low-rank method, 2,1 -RPCA performs reasonably well but not on par with DPCP-PSGM and REAPER, which overall tend to be the most robust methods.
Sentence 2: for the same time budget, RANSAC, which is a popular choice in the computer vision community for such outlier detection tasks, is essentially failing due to an insufficient number of iterations.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Previous work has attempted to answer these questions in a variety of ways and has provided encouraging successes [43,10,11,18,36].
Sentence 2: many of these methods lack generality, requiring some degree of manual task-specific design, and often require expensive on-policy training that is unable to benefit from advances in off-policy model-free RL, which in recent years has drastically brought down sample complexity requirements [12,15,3].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Their estimator is fairly similar to ours in the identity covariances setting and gives the same error rate.
Sentence 2: our result is more general in the following senses: 1) Our estimator and analysis do not rely on Gaussianity assumptions; 2) Our results apply beyond the setting where the label y is a linear function of x plus independent noise, and estimates the fraction of the variance that can be explained via a linear function (the "agnostic" setting); and 3) our approach extends to the unknown non-isotropic covariance setting.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The top-k loss [20] generalizes the maximum loss, as L max (L z (f )) = L top-1 (L z (f )), and can alleviate the sensitivity to outliers of the latter.
Sentence 2: unlike the average loss or the maximum loss, the top-k loss in general does not lead to a convex learning objective, as it is not convex of all the individual losses L z (f ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the treatment T i has not been applied to unit i (also known as the out-of-sample case [30]), the potential outcome of treatment T i = 1.
Sentence 2: if the unit i has already received a treatment T i (i.e., the within-sample case [30]), Y Ti is the factual outcome, and Y 1−Ti is the counterfactual outcome.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: . . , x a Na } is a bag of points with N a individuals, and we wish to estimate the regression value η(x a i ) for each individual.
Sentence 2: instead of the typical setup where we have a paired sample {(x , y )} of individuals and their outputs to use as a training set, we observe only aggregate outputs y a for each of the bags.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that while the weak version of Theorem 1 guarantees only the existence of a saddle point for (7), the strong version further guarantees that any Bayes decision rule of the maximizing distribution results in a robust Bayes decision rule.
Sentence 2: the continuity in Bayes decision rules does not hold for the discontinuous 0-1 loss, which requires considering the weak version of Theorem 1 to address this issue.
Relationship: contrasting