sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Because FBM (correctly) assumes the reward structure to remain fixed during a game, it updates the posterior mean by weighing new observations with decreasing coefficients, as the variance of the posterior distribution decreases over time.
Sentence 2: by assuming the reward rates to have a (small) fixed probability of being redrawn from a prior distribution on each trial, DBM continuously updates the posterior reward rate distribution by exponentially forgetting past observations, and injecting a fixed prior bias [5,9].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the case of a sparse use of trajectories to cover a space, the cost of the approach is dominated by the E © costs of updating second derivative matrices, and thus the cost of the trajectory-based approach increases quadratically as the dimensionality increases.
Sentence 2: for periodic tasks the approach of growing trajectories backwards from the goal cannot be used, as there is no goal point or set.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Our general form of the asymptotic free energy also allowed us to elucidate the behavior of the VB free energy when L and/or M diverges with the same order as N . This attempt successfully revealed the sparsity of the solution for the case when M diverges while L ∼ O(1).
Sentence 2: when L diverges, we found that the leading term of the free energy does not contain interesting insight into the sparsity of the solution.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We will now deal separately with f 1 and with f 2 .
Sentence 2: from the inequality (4) above, we observe that from the inequality (2), we see that if e i is a basis element present in the basis expansion of f 1 , Since ∆ acts by rescaling basis elements, we have Therefore by Proposition 4.3 for t sufficiently small and some constant Hence we see that Consider now the second summand f 2 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When the size of the behaviour data is small, the clustering performances of both DPM-BIRL and EM-MLIRL were not good enough due to the sparsity of data, hence their EVD results were similar to that of the baseline algorithm that independently runs BIRL on each trajectory.
Sentence 2: as we increased the size of the data, both DPM-BIRL and EM-MLIRL achieved better EVD results than the baseline since they could utilize more information by grouping the trajectories to infer the reward functions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When a given IFAT cell's RF falls on a region of visual space with uniform brightness, all of the OR pixels projecting to that cell will have the same mean firing rate, so the excitatory and inhibitory inputs will cancel.
Sentence 2: if a cell's excitatory inputs are exposed to high light intensity and its inhibitory inputs are exposed to low light intensity, the cell will receive more excitatory inputs than inhibitory inputs and will generate an output spike train with spike frequency proportional to the contrast.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Clearly, after almost every trade, the market price is quite far from the equilibrium/stationary point, and hence the classical supply and demand analysis of this market yields a poor description of the actual behavior, and in particular, of the predictive quality of the price at any given time.
Sentence 2: the mean price is consistently close to the mean belief of the traders, which in turn is quite close to the true parameter q.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Turning back to our bounds, both the latent trace norm and its scaled version can perform as well as knowing the mode with the lowest rank (the first mode) (see also [21]) when d = P = Q.
Sentence 2: when the dimensions are heterogeneous, similarly to the matrix completion case above, the plain latent trace norm fails to recognize the lowrank-ness of the first mode and Table 2: Sample complexities of the overlapped trace norm, latent trace norm, and the scaled latent trace norm in various settings.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Nevertheless, even this approach for computing the Jacobian matrices still requires the solution of multiple linear systems, which scales poorly when the number of hyperparameters k is large.
Sentence 2: l "observed features" j ∈ {1, . . . , R} "noise features" j ∈ {R + 1, . . . , 40} we can do much better by reorganizing the computations in such a way that the Jacobian matrix J d is never explicitly required.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In general, any PNFA where the nondeterministic transitions can only be visited once can be expressed as a mixture of PDFAs.
Sentence 2: if we replace transitions to q 3 with transitions to q 0 , as in 2(b), there is no longer any equivalent finite mixture of PDFAs, since the nondeterministic branch from q 0 can be visited an arbitrary number of times.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Bayesian sequential updating or belief propagation, implemented in the form of particle filtering, has recently been used in estimating the hand trajectories of monkeys based on M1 neuron's responses [4] and the location of a rat based on the responses of the place cells in the hippocampus [3].
Sentence 2: linear methods have been shown to be quite adequate for decoding LGN, motor cortical, or hippocampal place cells' signals using population vectors or the optimal linear decoder [10,5,8].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: PMP also retains the bilinear property of uniform convolution, meaning that where H = j w j P j and D = The disadvantage of PMP is that it typically leads to inefficient algorithms because the evaluation of the matrix-vector product Hx = Dw requires generating many expensive intermediate transformed images.
Sentence 2: eFF can be combined with the PMP model by introducing a set of basis images efficiently generated by transforming a grid of delta peak images [9].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A key difference is that in PITC it is not obvious which variables should be grouped together when making the conditional independence assumption, here it is clear from the structure of the model that each of the outputs should be grouped separately.
Sentence 2: the similarities are such that we find it convenient to follow the terminology of [10] and also refer to our approximation as a PITC approximation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As is well known in information theory, the expected number of bits required to compress data X generated according to a known distribution P is the distribution's entropy, H(P ) = E P log 1/P (X), and is achieved by encoding X using roughly log 1/P (X) bits.
Sentence 2: in many applications P is unknown, except that it belongs to a known collection P of distributions, for example the collection of all i.i.d., or all Markov distributions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note we implement PILCO with a linear mean function 1 , φ a x, where φ a are additional hyperparameters trained by optimising the marginal likelihood [11,Section 2.7].
Sentence 2: the covariance function k is squared exponential, with length scales Λ a = diag([l 2 a,1 , ..., l 2 a,D+F ]), and signal variance s 2 a : The original PILCO algorithm ignored sensor noise when training each GP by assuming each observation z t to be the latent state x t . this approximation breaks down under significant noise.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This approach has been used to identify neural activity trajectories that are informative about stimuli and behaviour and yield insights into neural computations [7,8,9,10,11,12,13].
Sentence 2: these methods are designed for analyzing one population measurement at a time and are typically applied to population recordings of a few dozens of neurons, yielding a statistical description of the dynamics of a small sample of neurons within a brain area.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that, given an optimal solution N , B and D to (20), an approximate solution to the original problem (2) can be recovered heuristically by first rounding N to obtain , then recovering W and V , as shown in Lemmas 1 and 2.
Sentence 2: since our primary objective is to determine whether any convex relaxation of a two-layer model can even compete with one-layer or locally trained two-layer models (rather than evaluate heuristic rounding schemes), we consider a transductive evaluation that does not require any further modification of N , B and D. In such a set-up, training data is divided into a labeled and unlabeled portion, where the method receives X = [X `, X u ] and Y `, and at test time the resulting predictions Ŷu are evaluated against the held-out labels Y u .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When twice differentiable, DR-submodularity is equivalent to The above twice differentiable functions are sometimes called smooth submodular functions in the literature [23].
Sentence 2: in this paper, we say a differentiable submodular function F is L-smooth w.r.t a norm ⋅ (and its dual norm ⋅ * ) if for all x, y ∈ X we have Here, ⋅ * is the dual norm of ⋅ defined as When the function is smooth w.r.t the 2 -norm we use L 2 (note that the 2 norm is self-dual).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The effects of a wide range of timescales and uncertainty about the causes of changes of the oculomotor plant will largely be hidden if experiments are of a relatively short duration and no uncertainty is produced.
Sentence 2: in a recent experiment Robinson et al analyzed saccadic gain adaptation [7] in a way that allowed insight into many timescales as well as insight into the way the nervous system deals with uncertainty.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Common approaches to this problem include simply summing up or averaging all the node embeddings in a final layer [11], introducing a "virtual node" that is connected to all the nodes in the graph [25], or aggregating the node embeddings using a deep learning architecture that operates over sets [15].
Sentence 2: all of these methods have the limitation that they do not learn hierarchical representations (i.e., all the node embeddings are globally pooled together in a single layer), and thus are unable to capture the natural structures of many real-world graphs.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note, however, that without any restrictions on the data distribution, d can rise to n. In an effort to overcome scaling problems a range of sparse GP approximations have been proposed [1,8,9,10,11].
Sentence 2: none of these has fully achieved the goals of being a nontrivial approximation to a non-sparse GP model and matching the SVM w.r.t.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, in this section we propose an alternate approach method to setting the beliefs in L[x, y], that optimizes for the accuracy of the final global illuminant estimate.
Sentence 2: unlike previous color constancy methods that explicitly model statistical co-dependencies between pixels-for example, by modeling spatial derivatives [4,5,6], or learning functions on whole-image histograms [21]-we retain the overall parametric "form" by which we compute the illuminant in (4).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For odd k, such a reduction is not possible, and one might try a bipartite graph, with either the reduction described above, or with k/2 -tuples and k/2 -tuples (our analysis works for this reduction as well).
Sentence 2: with Õ(k/2) clauses, the spectral approach of computing the largest or second largest singular vector of the adjacency matrix does not work.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, we can consider the empirical version of the integral operator associated to K which is simply defined by The latter operator is an unbiased estimator of T . Indeed, since K x ⊗ K x is a bounded random variable into the separable Hilbert space of Hilbert-Schmidt operators, one can use concentration inequalities for random variables in Hilbert spaces to prove that where • HS is the Hilbert-Schmidt norm (see for example [14] for a short proof).
Sentence 2: in general T † n T n does non converge to T † T since 0 is an accumulation point of the spectrum of T or, equivalently, since T † is not a bounded operator.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Secondly, using either the KDE or k-NN, obtaining the best rates for plug-in methods requires undersmoothing the density estimate and we are not aware for principled approaches for selecting this smoothing parameter.
Sentence 2: the bandwidth used in our estimators is the optimal bandwidth for density estimation so we can select it using a number of approaches, e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: and choose the arm with the largest feasible reward given C t−1 as a constraint: The main difference between GLOC and existing GLBs is in the computation of the C t 's.
Sentence 2: prior methods involve "batch" computations that involve all past observations, and so scale poorly with t. GLOC takes in an online learner B, and uses B as a co-routine instead of relying on a batch procedure to construct a confidence set.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, practical systems may truncate the likelihood-ratio score-space after the 1st-order derivatives, and hence use linear approximations to the Taylor series expansions 3 .
Sentence 2: an example of a 2nd-order derivative is V' J-L jk (V'~;k Inp(OIO)) , For simplicity the component posterior ' Y j k (t) is assumed independent of J-L j k. Once the score-space has been defined, an SVM classifier can be built in the score-space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For fair comparison, we follow the evaluation procedure of the original two-stream work [20] by sampling 25 frames (and their horizontal flips).
Sentence 2: rather than using 10 spatial 224 × 224 crops from each of the frames, we apply fully convolutional testing both spatially (smallest side rescaled to 256) and temporally (the 25 frame-chunks) by classifying the video in a single forward pass, which takes ≈250ms on a Titan X GPU.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: (2016) and  discuss the topic in detail.
Sentence 2: strong guarantees for the best known estimators hold (with few exceptions mentioned below) under the restrictive assumption that X is either bounded with probability 1 or has sub-Gaussian distribution, meaning that there exists σ > 0 such that for any v ∈ R d of unit Euclidean norm, In the discussion accompanying the paper by Cai et al.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We say that a representation ρ is reducible if there exist smaller representations ρ 1 , ρ 2 such that ρ ≡ ρ 1 ⊕ ρ 2 where ⊕ is defined to be the direct sum representation: In general, there are infinitely many inequivalent representations.
Sentence 2: for any finite group, there is always a finite collection of atomic representations which can be used to build up any other representation using direct sums.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For 1D histograms it is known how to achieve the optimal sketch length bound of n = O(k log d): it can be obtained by representing k-histograms using a tree of O(k log d) wavelet coefficients as in [10] and then using the structured sparse recovery algorithm of [1].
Sentence 2: applying this approach to 2D histograms leads to a sub-optimal bound of O(k log 2 d).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, the runtime of the Radon machine with N = r h n samples is T R,h (N ) = T L (n) + hr 3 .
Sentence 2: (7) the runtime of the base learning algorithm for achieving the same guarantee is T L (M ) with M ≥ N L 0 (ε, ∆).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Specifically, the sample complexity for the generative method depends only on row-wise sparsity of the individual models d 2 ⇥ ⇤ i , and is independent of sparsity s of the differential parameter ⇥ ⇤ diff .
Sentence 2: the sample complexity of the discriminative method depends only on the sparsity of the differential parameter, and is independent of the structural complexities of the individual model parameters.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A popular modeling approach assumes the existence of a utility function f such that f (x) gives the utility of an item with feature vector x; f (x i ) > f (x j ) indicates that item i is preferred to item j. Bayesian methods can be used to learn f , for example, by modeling f independently for each user as a draw from a Gaussian process (GP) prior [4].
Sentence 2: when data from many users is available, such methods do not leverage similarities in preferences across users.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Customers are not exchangeable under this distribution, as the number of dishes counted as 1 depends upon the order in which the customers make their choices.
Sentence 2: if we only pay attention to the lof -equivalence classes of the matrices generated by this process, we obtain the infinitely exchangeable distribution P ([Z]) given by Equation 4: matrices generated via this process map to the same left-ordered form, and P ([Z]) is obtained by multiplying P (Z) from Equation 5 by this quantity.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When the probability distribution P(X, Y ) is known, the optimal classifier, leading to the minimum misclassification rate, is given by the Maximum A-posteriori Probability (MAP) decision rule.
Sentence 2: in practice, estimating the complete joint distribution P(X, Y ) is computationally and statistically impossible for large values of d. Therefore, an alternative approach is to first estimate some low order marginals of the joint probability distribution P(X, Y ) and then design the classifier based on the estimated low order marginals.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Alternatively, the state-space representation used for determining the policy can be augmented with extra dimensions representing the current goal, such that effectively multiple policies can be maintained 5 , or a large "look-up table" of action sequences connecting any pair of start and goal locations can be represented -again leading to inefficient use of experience and potentially excessive representational capacity requirements.
Sentence 2: model-based decision-making strategies rely on the ability to simulate future trajectories in the state space and use this in order to flexibly plan in a goal-dependent manner.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For an unobserved latent process x t , an inference procedure needs to be devised to estimate the posterior distribution on the entire sequence x 1 . . . x t . For linear-Gaussian observations, this inference is tractable and is provided by Kalman smoothing.
Sentence 2: with generalised-linear observations, inference becomes intractable and the necessary approximations [6] are computationally intense and can jeopardize the quality of the fitted models.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Moreover, it is possible to show through lower bounds on the guaranteed performance that these policies indeed achieve sub-optimal performance.
Sentence 2: to demonstrate this, consider the subclass of stationary information flows described in §3.1, with an arrival rate λ that is very large compared to σ 2 4∆ 2 T . In that case, we have seen that the regret lower bound becomes constant whenever the arrival rate λ is independent of T . the -greedy policy, employs an exploration rate that is independent of the number of observations obtained for each arms and therefore effectively incurs regret of order log T due to performing unnecessary exploration.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since assumptions of Lipschitz continuity naturally allow one to place bounds on how far the optimum of f is from a certain location, they work to smoothly reduce the value of the acquisition function in a neighborhood of any point reflecting the belief about the distance of this point to the maxima.
Sentence 2: assumptions of Lipschitzness are too coarse-grained and it is unclear how their method to estimate the Lipschitz constant and modelling of local penalization affects the performance from a theoretical standpoint.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In particular, [3] proved that under suitable conditions the unknown parameters of a noise driven system (i.e., no control) whose dynamics are modeled by linear stochastic differential equations can be estimated accurately with as few as O(log(p)) samples.
Sentence 2: the result of [3] is not directly applicable here since for a general feedback gain L even if A 0 and B 0 are sparse, the closed loop gain A 0 − B 0 L need not be sparse.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To solve these problems, one can first discretize the domains and then use conventional search algorithms to determine plans that move the agent to the goal coordinates.
Sentence 2: uniform discretizations can prevent one from finding a plan if (a)  they are too coarse-grained (for example, because the resolution prevents one from noticing small gaps between obstacles) and results in large state spaces that cannot be searched efficiently if they are too fine-grained.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We can also express (3a) more succinctly as, This expression looks very much like a multivariate Gaussian density function.
Sentence 2: we emphasize that this is not the case since both m 0:t−1 and K 0:t−1 depend (nonlinearly) on the argument x 1:t . In fact, (4) will typically be very far from Gaussian.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Theorem 2.4 says that for any shift a > 1 there is a parameter α(a) > 4 such that (9) holds.
Sentence 2: for the choice a = O(1) one has to set α such that α α−4 = Ω( d k ) and the last term in (9) will be of order O( d 3 k 3 T 2 ), thus requiring T = Ω( d 1.5 k 1.5 ) steps to yield convergence.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is not surprising, as in those conditions, where one should rely more on the long term statistics, if the learning rate is too high, then subjects will tend to shift more due to recent experience (Fig 2b), which would adversely influence decision optimality.
Sentence 2: in a high volatility environment, decision optimality has a linear correlation with the learning rate, suggesting that higher learning rate leads to better performance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We can observe that most classes are tightly clustered around their corresponding word vector while the zero-shot classes (cat and truck for this mapping) do not have close-by vectors.
Sentence 2: the images of the two zero-shot classes are close to semantically similar classes (such as in the case of cat, which is close to dog and horse but is far away from car or ship).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Related concepts such as count-based bonuses have been shown to provide substantial speedups in classic reinforcement learning (Strehl & Littman, 2009;Kolter & Ng, 2009), and several recent works have proposed information-theoretic or probabilistic approaches to exploration based on this idea (Houthooft et al., 2016;Chentanez et al., 2005) by drawing on formal results in simpler discrete or linear systems (Bubeck & Cesa-Bianchi, 2012).
Sentence 2: most novelty estimation methods rely on building generative or predictive models that explicitly model the distribution over the current or next observation.
Relationship: contrasting