sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: state s(t), action a(t), and reward r(t), as follows.
Sentence 2: if the subject starts from a state s(t) and comes back to the same state after k steps, the expected cumulative reward V(t) should satisfy the consistency condition (3) for each time t of the data file, we calculated the weighted sum of the rewards acquired until the subject returned to the same state and estimated the value function for that episode as The estimate of the value function V(t) at time t was given by the average of all previous episodes from the same state as at time t ( ) ( ) where {t 1 , …, t L } are the indices of time visiting the same state as s(t), i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For another thing, applying the previous methodology onto problem (5) does not lead to an tractable reformulation either, mainly because the objective function Ω ψ( dP1 d(P1+P2) )d(P 1 + P 2 ) is not separable in P 1 and P 2 , but depends on the density on the common support of P 1 and P 2 .
Sentence 2: as argued in Section 4, in the least-favorable distribution the probability mass of the two distributions cannot be transported arbitrarily, but are linked via their common support.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (a) Removing the convolutional structures of the gates Given the local spatiotemporal features of the 3DCNN block, it can be considered that the 3DCNN block has paid attention to the noticeable spatial regions where there is valuable spatiotemporal information.
Sentence 2: the ConvLSTM block can just focus on the spatiotemporal feature fusion along with the recurrent steps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By the above lemma, we have F (x t ) − F (x) ≤ Lϵ as long as ∥z t − z∥ ≤ ϵ.
Sentence 2: it suffices to show quadratic convergence of ∥z t − z∥ to guarantee F (x t ) − F (x) double its precision after each iteration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In light of what we know about the statistics of the visual input, however, this would be very surprising: Natural images are known to exhibit complex higherorder correlations which are extremely difficult to model yet being perceptually relevant.
Sentence 2: if we assume that retinal ganglion cells do not discard the information underlying these higher-order correlations altogether, it would be a very difficult signal processing task to remove all of those already within the retinal network.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, and by B S (i, σ), the probability that this classifier makes exactly Now, approximating the binomial by relative entropy Chernoff bound [Langford, 2005], we have, for a classifier f : As also shown in [Laviolette and Marchand, 2007], since m j = m m−j and kl(R S (f ) R(f )) = kl(1 − R S (f ) 1 − R(f )), the above inequality holds true for each factor inside the sum on the left hand side.
Sentence 2: in the case of sample compressed classifier, ∀(z i , σ) ∈ K and ∀S ∈ (X × Y) m : Bounding this by δ yields: Now, consider the quantity in the probability in Equation 3 as the bad event over classifiers defined by a compression set i and an associated message string σ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Stochastic gradients can be computed faster than their batch counterparts, allowing more frequent updates.
Sentence 2: in Algorithm 1, any stochastic estimator ∇f (x k−d k ; ξ) of the gradient may be used, as long as then, in Algorithm 2, the stochastic mini-batch estimator where ξ i are IID, may be used in place of ∇f any increasing batch size m k = Ω((k + 1) −α ), with α ∈ (0, 1), conforms to Assumption 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: But if f differs significantly from f * then the variance in the left hand side of ( 9), as well as the value of D, will be large.
Sentence 2: if we take the variance in the left hand side of ( 9) as a measure of distance between f and f * then the hypothesis spaces with large and small D can be visualized as shown in Figure 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the next section, we generalize this estimator into one that does not require prior knowledge of µ and can handle non-linear feedback.
Sentence 2: suppose now that we have the same setup as in the previous section, except that now feedback has a non-linear dependence on the prediction: i ) for some arbitrary function f . For example, in the case of a linear predictive model ŷ = w • x, this kind of feedback could arise if we have feature feedback x i to the above predictions, we only affect the feedback term f (•): by adding artificial noise ⌫ (t) i , we are able to cancel out the nuisance terms, and isolate the feedback function f that we want to estimate.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Intuitively, a foveated clutter model that takes into account target search should score very low when the target is in the fovea (near zero), and very high when the target is in the periphery.
Sentence 2: an observer should find a target without difficulty, achieving a near perfect hit rate in the fovea, yet the observer should have a lower hit rate in the periphery given crowding effects.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is obvious for u 1 , v 1 / ∈ [ω, 1 − ω] 2 , because C θ1 is a copula.
Sentence 2: straightforward but tedious algebra shows that ∀u 1 , v 1 ∈ (ω, 1 − ω) 2 : C θ1,θ2,ω is continuous and has density almost everywhere on [0, 1] 2 and is therefore 2increasing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our reduction strategy brings inter-dependency between the pairs containing the same observation and its true class in the original training set.
Sentence 2: we derive new generalization bounds using local fractional Rademacher complexity showing that even with a shift in the original class distribution and also the inter-dependency between the pairs of example, the empirical risk minimization principle over the transformation of the sampled training set remains consistent.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus we can identify the family (ψ n1,...,n d ) with the one dimensional Fourier basis with frequency T d λ , which means that the condition number ρ = 1 for this curve.
Sentence 2: for a d-dimensional function f , sparse in the Fourier basis, it is sufficient to sample along the curve induced by g to ensure that V C is invertible.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: end for ESCB with time horizon T has a complexity of O(|M|T ) as neither b M nor c M can be written as M y for some vector y ∈ R d . Assuming that the offline (static) combinatorial problem is solvable in O(V (M)) time, the complexity of the CUCB algorithm in [10] and [11] after T rounds is O(V (M)T ).
Sentence 2: if the offline problem is efficiently implementable, i.e., V (M) = O(poly(d/m)), CUCB is efficient, whereas ESCB is not since |M| may have exponentially many elements.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Cortex is segmented, then each cortical hemisphere is inflated to obtain a smooth surface, which is projected to the sphere, S 2 , represented by a discrete spherical mesh M s = {p k ∈ S 2 ; 1 ≤ k ≤ N v /2}.
Sentence 2: the two cortical hemispheres are hence modeled by the disjoint union S = S 2 S 2 , represented by the corresponding disjoint union of mesh points M = M s M s . Anatomical cortical features, such as cortical curvature, are functions D a : S → R Na sampled on M . our analysis is restricted to cortex only.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because we can guarantee that our models are drawn from the true posterior over models, this approach is a standard Monte Carlo approximation of the expectation.
Sentence 2: we can apply the central limit theorem to state that the estimated expected rewards will be distributed around the true expectation with approximately normal noise N (0, σ 2 n ), where n is the number of POMDP samples and σ 2 is a problem-specific variance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Modelling only the tail of the density via (4) requires moving beyond strictly proper composite losses: from Proposition 2, all such losses have the same Bayes-optimal solution, up to a transformation.
Sentence 2: we will construct suitable alternate losses via the weight function representation of a proper composite loss: every such loss with invertible link Ψ can be written [Reid and Williamson, 2010, Theorem 6] where the (generalised) function w : [0, 1] → R≥0 is the weight, and 01 is the cost-sensitive loss minimising equivalently minimises a mixture of cost-sensitive losses for various cost ratios, whose relative importance is determined by w; see Buja et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We thus look for two constants c 0 and c 1 = 0, such that, for all values of f (x) + b, there exists a value 0 ≤ p(1|x) ≤ 1 such that The system (6) has a solution over the whole range of possible values of f (x) + b if and only if c 0 = log(2) and c 1 = 0.
Sentence 2: the SVM optimization problem does not implement the MAP approach to Gaussian processes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the algorithm would also work with any other prior, so if it were known a priori that certain values in the grid were of special importance, they could be given larger prior mass.
Sentence 2: a more advanced analysis demonstrating that only a subset of learning rates could potentially be optimal (in the sense of minimizing the regret R η T ) would directly lead to factors of improvement in the algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More precisely, the weight w ij associated with the edge ij in a graph will be the result of an appropriate composition of a feature vector x ij (observed) and a parameter vector ✓ (estimated from training data).
Sentence 2: in practice, our input is a vector-weighted bipartite graph G x = (V, E, x) (x : E 7 !
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, if we denote by g 1ℓ (r, ℓ) the ℓ-th row of the output of the denoiser, we assume that for a vector-valued function φ(•) that is the same for all rows.
Sentence 2: the ℓ-th row output g ℓ (•) depends only on the ℓ-th row input.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: in the supplement, it follows that d = σ 2 0 / Σ 0 2 ≥ tr(Σ 0 )/ Σ 0 ≥ 1.
Sentence 2: assuming that the sample size satisfies m ≥ (6C ) 4 dβ, then, dβ/m ≤ 1/(6C ) 4 < 1, and by some algebraic manipulations we have that For completeness, a detailed computation is given in the supplement.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By considering a cost function c : Ω S × Ω T → [0, ∞[, Monge's formulation of the OT problem is to find a transport map T : Ω S → Ω T (also known as a push-forward operator) between two probability measures µ S on Ω S and µ T on Ω T realizing the infimum of the following function: When reaching this infimum, the corresponding map T is an optimal transport map.
Sentence 2: it associates one point from Ω S to a single point in Ω T . the existence of this map is not always guaranteed, as when for example µ S is a Dirac and µ T is not.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore in a good round, we see at least (L − f (S))/2 increase in value of S. In other words, the gap L − f (S) is reduced by a factor of at least 2 in each good round.
Sentence 2: since f only takes integer values, once L − f (S) becomes less than 1, we know that f (S) ≥ L. there cannot be more than log 2 L good rounds.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Every fixed point of the system (1) is called an equilibrium of the system, and they correspond to the critical points of the function f . Unless f is a quadratic function of the parameters, the system described by ( 1) is either a nonlinear system or a hybrid system that switches from one dynamics to another over time.
Sentence 2: the system (1) can exhibit behaviors that are typically observed in nonlinear and hybrid systems, such as convergence to an orbit but not to a fixed point, or dependence of convergence on the equilibria and the initialization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Decoder and encoder networks each consisted of two convolutional layers (kernel size 3 × 3) with sigmoid activation functions, each of which had a stride of two (both conv and conv transposed).
Sentence 2: a training sample with a size of 28 × 28 is downsampled to 7 × 7, and then upsampled to 28 × 28.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, Marg k is exactly the wavelet space W k−1 of the Bi-CMRA generated by ν 0 = {(n)} of Example 2.
Sentence 2: when p is q-sparse, noting that d (n−1,1) = n−1, by using the methods of the previous section, we can find its projection to each of these spaces in just O(n 4 q) time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, each (i, j) entry of the low rank matrix L can be associated with a local incoherence parameter µ ij , which is less than or equal to the global parameter µ, and then the allowable entry-wise error density can be potentially higher than that characterized based on the global incoherence.
Sentence 2: the total number of errors that the matrix can tolerate in robust PCA can be much higher than that characterized based on the global incoherence when errors are distributed accordingly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To train SGAN, we draw inspirations from the recently proposed adversarially learned inference framework (ALI) [5], and build two adversarial games to drive I, G to match the true joint distributions p(x, z), and C, G to match the true joint distribution p(x, y).
Sentence 2: sGAN can be seen as a combination of two adversarial games and two collaborative games, where I, G combat each other to match joint distributions in the visible space, but I, C, G collaborate with each other to minimize a reconstruction error in the hidden space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider a graph-structured pattern drawn from an Ising model or the GGM with weight matrix W of the hierarchical block form as depicted in Figure 2.
Sentence 2: if = 2 − (1−β) ∀ ≤ γ log 2 p+1, for constants γ, β ∈ (0, 1), and = 0 otherwise, then the noise threshold for consistent MSE recovery (R if we take advantage of the network interaction structure, it is possible to tolerate noise with variance that scales with the network size p, whereas without exploiting structure the noise variance needs to decrease with p, as discussed in the introduction.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that if we were to assign one unique point of the Z space to each dataset sample, then a neural network could learn to memorize the dataset by mapping each selected z ∈ Z to its corresponding real sample (we have introduced N disjoint component in Z space in this case), however this is not how GANs are modeled.
Sentence 2: the memorization issue is not of concern for common GANs and our proposed models (note that this argument is addressing the very narrow case of dataset memorization, not over-fitting in general).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Many more-or-less heuristic methods for non-linear manifold discovery are based on the idea of preserving the geometric properties of local neighbourhoods within the data, while embedding, unfolding or otherwise transforming the data to occupy fewer dimensions.
Sentence 2: algorithms such as locally-linear embedding (LLE) and Laplacian eigenmap attempt to preserve local linear relationships or to minimise the distortion of local derivatives [1,2].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In our problem formulation, however, changing c t to c t may change the decision set D t to a possibly disjoint D t , making that notion ill-defined.
Sentence 2: when we discuss the impossibility of regret-minimization under standard differential privacy in Section 5, we revert back to a fixed action set A with an explicit per-round context c t . In this work, in addition to formulating the definition of JDP under continual observation, we also present a framework for implementing JDP algorithms for the contextual linear bandit problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The following functional with respect to r is called the free energy: In the last equation, the first term is the Kullback-Leibler (KL) distance from the trial distribution to the Bayes posterior, and the second term is a constant.
Sentence 2: minimizing the free energy (4) amounts to finding the distribution closest to the Bayes posterior in the sense of the KL distance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, when p = 1, we have the following guarantee: We note that ROBUST-PCA yields a smaller additive error than ROBUST-LRA by a factor of n/d, but at the expense of an additional multiplicative factor of log 2 (d).
Sentence 2: in settings where OPT k (A) is small (e.g.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any λ > 0 and a given partition where Θ * is the optimal solution of (1) and Θ is as defined in (2).
Sentence 2: if a partition {V c } k c=1 satisfies the assumption of Theorem 1, Θ and Θ * will be the same, and the last step of Algorithm 1 is not needed anymore.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Intuitively, if kernel k cannot result in high MMD distance M k (P, Q) when P 6 = Q, Mk (P, Q) has more chance to be smaller than c ↵ .
Sentence 2: then we are unlikely to reject the null hypothesis H 0 with finite samples, which implies Q is not distinguishable from P. instead of training g ✓ via (1) with a pre-specified kernel k as GMMN, we consider training g ✓ via which takes different possible characteristic kernels k 2 K into account.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is similar to the previous example; only weight of the formula is different.
Sentence 2: substituting this new weight, IPP output by Algorithm 2 will have objective function n x n y w 1 r + n x n y w 1 (1 − r)s and two constraints 0 ≤ r ≤ 1 and 0 ≤ s ≤ 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 2 The isd posterior with β = 1/2 is marginally consistent for Gaussian distributions.
Sentence 2: proof 2 Start by integrating over x N : Assume the singleton prior p(θ N ) is uniform and integrate over θ N to obtain: Consider only the right hand integral and impute the formula for the Bhattacharyya affinity: In the (white) Gaussian case A(θ) = θ T θ which simplifies the above into: Reinserting the integral changes the exponent of the pairs of Bhattacharyya affinities between the (N − 1) models raising it to the appropriate power λ/(N − 1): we get the same isd score that we would have obtained had we started with only (N − 1) data points.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With a particular choice of adaptor, based on the Pitman-Yor process [1,2,3], simple context-free grammars specify distributions commonly used in nonparametric Bayesian statistics, such as Dirichlet processes [4] and hierarchical Dirichlet processes [5].
Sentence 2: many nonparametric Bayesian models that have been used in computational linguistics, such as models of morphology [6] and word segmentation [7], can be expressed as adaptor grammars.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, our updates use samples rather than the expected value, so an action a t that is actually worse could, due to stochasticity in the environment, result in a TD error that suggests it is advantageous.
Sentence 2: the gradient estimates are influenced by the stochasticity of the transition function P and reward function R. If P or R is very stochastic, the same s, a pair will result in seemingly random TD errors, which manifests as large variance in δ t ψ st,at samples.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Continuing with the same notation, we show now that Note that, if we apply Bayes' law recursively, Thus, and so forth.
Sentence 2: what we wish to prove is ( ) or, rearranging, Note that, given the stipulated relationship between p( ) on each processing iteration and p( | u G ) on the previous iteration, With this in mind, we can rewrite the left hand side product in (17) as follows: Note that, given (18), the numerator in each factor of ( 19) cancels with the denominator in the subsequent factor, leaving only p(u G | *) in that denominator.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even if it does converge, in mixed extensions of finite games (where each agent's action set is a probability simplex over a finite number of actions), the limit can assign positive weight only to strictly dominated actions.
Sentence 2: establishing joint convergence to Nash equilibria under no-regret learning algorithms (OGD included) for a broad and meaningful subclass of games has been a challenging ongoing effort.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, the property tester rejected more than 95% of the ANN indices that are between 0.005-far and 0.01-far from being a 10-NN graph for c 1 = 0.01, c 2 = 5. depends heavily on the cost incurred otherwise.
Sentence 2: we compare the property tester against the minimum cost that every algorithm that uses an ANN index must invest before it can employ it or even just draw conclusions about its quality: the build time of the index.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that in Euclidean space, the exponential map is an addition operation, Exp(µ, z) = µ + z.
Sentence 2: our model coincides with (1), the standard PPCA model, when M = R n . We develop a maximum likelihood procedure to estimate the parameters θ = (µ, W, Λ, τ ) of the PPGA model defined in (3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Basically, the t-entropy is a normalized version of the Tsallis entropy,  One can recover the SBG entropy by setting t = 1.0.
Sentence 2: we can equivalently rewrite the Rényi Entropy as: The t-entropy of p(x) (when t � = 1) is equal to when α = t, When t and α → 1, both entropies go to the SBG entropy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 8 (Expected Uncertainty Sampling Gradient).
Sentence 2: if Assumptions 2, 5, 6, and 7 hold and θ ∈ Θ regular and b(θ) = 0, then if z (t) is chosen via uncertainty sampling with parameters θ, similar to how random sampling yields preconditioned SGD on the population surrogate loss L (9), uncertainty sampling yields preconditioned SGD on the population zero-one loss Z.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first equality is immediate.
Sentence 2: the second moment of w can be expressed as follows in terms of the Rényi divergence: the variance of w is given by σ For any hypothesis h ∈ H, we denote by R(h) its loss and by R w (h) its weighted empirical loss: We shall use the abbreviated notation L h (x) for L(h(x), f(x)), in the absence of any ambiguity about the target function f . Note that the unnormalized importance weighting of the loss is unbiased: The following lemma gives a bound on the second moment.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let Q j be the set of the j-th rows of all matrices in Q, and take A j = span(Q j ) ∩ P d . Then any vector m ∈ A j is affine combination of vectors in Q j and, thus, m T l(η) = l(η).
Sentence 2: if span(Q i ) has dimension n i , we can take a basis V i ∈ Q i of n i linearly independent vectors such that A i = span(Q i ) ∩ P d . By construction l(η) = (V T ) −1 U l(η).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As one would expect, pairs of features directly linked by causal relationships are correlated in the common-cause schema F I is correlated with its effects and in the common-effect schema F4 is correlated with its causes.
Sentence 2: cMT predicts that combinations of features serve as evidence for category membership to the extent that they preserve these expected correlations (i.e.
Relationship: reasoning