sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: [12] propose an interpretable neural network architecture whose predictions are based on the similarity of the input to a small set of prototypes, which are learnt during training.
Sentence 2: r n the decoder associated with h, and x := h dec (h(x)) the reconstruction of x, we use an additional penalty L h (x, x) on the objective, yielding the loss: ) Achieving (iii), i.e., the grounding of h(x), is more subjective.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This upper bound can in turn be bounded by a quantity that is feasible to calculate: max x f(x)-(B* f) (x) = maxxmin a f(x)-(Ba f) (x) :S min a max x f(x)-(B a f)(x).
Sentence 2: in the spirit of [7,8,6]  The main benefit of this factored representation is that it allows large MDPs to be encoded concisely: if the functions Ra,r(xa,r) and P(X~IXa,i) depend on a small number of variables, they can be represented by small tables and efficiently combined to determine R(x, a) and P(x'lx, a).
Relationship: neutral
Domain: NeurIPS
Sentence 1: If the functions f t (x), t ∈ D, are convex, then D e f t (x) dt is convex.
Sentence 2: an identical derivation yields the optimal q(ξ|y) ∝ exp log p(x, ξ|y) x ∝ p(ξ) exp log p(x|ξ) x when q(x|y) is fixed.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the supplementary material, we discuss an example of continuous influence maximization [26] that uses continuous and infinite action space while the number of base arms is still finite.
Sentence 2: in both studies, the regret bounds contain an undesirable factor of 1/p * , where p * is the minimum positive probability that any arm can be triggered by any action, 1 and this factor could be exponentially large for both influence maximization and cascading bandits.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The proof of Lemma 1, given in Section A.3, primarily involves establishing a uniform deviation result; if we let P denote the feasible set for (1), then we wish to show that | Ã − B, M | ≤ 1 2 αβkn for all M ∈ P. This would imply that the objectives of (1) and ( 6) are essentially identical, and so optimizing one also optimizes the other.
Sentence 2: the optimization problem (1) therefore chooses a set of βm items in each row to maximize the corresponding values in Ã, while constraining the item sets to have low rank (where low rank is relaxed to low nuclear norm to obtain a convex problem).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Various modifications of RLS are made to improve its performance either from the viewpoint of manifold [1] or in a more generalized form [7,11].
Sentence 2: ( 4) is a constant, then there is a new Representer Theorem; (2) The same conditions become a sufficient condition under which constant functions should not be penalized; and (3) under the same conditions, a function belongs to the null space if and only if the function should not be penalized.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 1), right] in the essentially same way as the standard BP deals with block-angular sparsity of J [see Eq.
Sentence 2: since mlu − dn = dF (CA + nB) − d(F CA + nB) = d(F − 1)nB, Pearlmutter's method can be up to F times faster than Algorithm 2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We denote with K n = Φ T n Φ n the empirical kernel matrix with entries [K n ] i,j = k i,j .
Sentence 2: in the 1/ √ n risk rate setting Gaussian projections would require ν = 1/ √ n resulting in m ≥ n log(n) random features, which would not bring any improvement over computing K n . Moreover when D is infinite, as it is usually the case in the non-parametric setting, the JL projection is not explicitly computable in general and Biau et al.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The joint probability of preferences can thus be represented as a completely disconnected CDN with logistic functions in which all pairwise object preferences are treated as being independent.
Sentence 2: one possible concern here is that we may require a fully connected CDN topology over all possible pairwise preferences between all nodes in order to capture all of these dependencies, leading to a model which is cumbersome to learn.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For such a model with large m, the number of tetrads involved in our testing methodology is so large that even after taking the supremum norm one shouldn't expect (3.5) to hold; for example, when m = 50, the dimension of Θ is 2• 50 4 = 460600, and one should be skeptical about the validity of (3.5) when we only have the sample size n = 250.
Sentence 2: while we have defined X as a set of nodes, it will be convenient to abuse notation slightly and let X also denote a random vector (X 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our main convergence results are also readily generalized to stochastic target policies by replacing the sub-sampling process described in Section 2 with a sample-weighting process.
Sentence 2: ., which we can break into causally related 4-tuples, (s 1 , a 1 , r 1 , s 1 ), (s 2 , a 2 , r 2 , s 2 ), .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since every subset of a permitted set is permitted, that means there are no forbidden sets and the network is monostable.
Sentence 2: theorem 3 For nonoverlapping groups, the top c groups with the largest group input could end up the winner depending on the initial conditions of the dynamics, where c is determined by the equation BC 2' : Proof sketch: Suppose the ath group is the winner.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Furthermore, MoM algorithms only make a single pass over the training data, are highly parallelizable, and always terminate in polynomial time.
Sentence 2: this approach, outlined in Algorithm 1, is an alternative to the ones presented in the literature up to now (even though in the noiseless, realizable setting, it will provide the same Algorithm 1 SIDIWO: Simultaneous Diagonalization based on Whitening and Optimization Require: M 1 , M 2 , M 3 , the number of latent states l 1: Compute a SVD of M 2 truncated at the l-th singular vector: results).
Relationship: neutral
Domain: NeurIPS
Sentence 1: SSTD is the method described here.
Sentence 2: none of the above give results that are truly sparse in time; and all of the above have to go to some lengths to obtain reasonable cell segmentations: [13,4] recur to heuristic post-processing, while [3] invokes structured sparsity inducing norms and [15] use an additional clustering step as initialization.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Suppose the initialization point is uniformly sampled from [−1, 1] d . There exists a function f defined on R d that is B-bounded, �-gradient Lipschitz and ρ-Hessian Lipschitz with parameters B, �, ρ at most poly(d) such that: 1. with probability one, gradient descent with step size η ≤ 1/� will be Ω(1) distance away from any local minima for any T ≤ e Ω(d) .
Sentence 2: we expect that with random initialization, general stochastic gradient descent will need exponential time to escape saddle points in the worst case.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Unfortunately, this is generally a #P-hard problem, which was shown to be the case even for Ising models by Jerrum and Sinclair [17].
Sentence 2: more formally, F is submodular if, for any S ⊆ T ⊆ V , and any v ∈ V \ T , it holds that F (v|T ) ≤ F (v|S).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then the permutations are generated randomly as follows: choose one of the n objects to be ranked 1 at random, where object i is chosen to be ranked 1 with probability w i /( n j=1 w j ).
Sentence 2: given the nature of the topic, there are a lot of exciting lines of research done over the past century and we shall not be able to provide comprehensive coverage due to a space limitation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: While causal power and I:!.P were each capable of capturing certain trends in the data, causal support was the only model capable of predicting all the trends.
Sentence 2: it is important to note that the responses of the human subjects in columns 8-10 (contingencies {l.00 , 0.50} , {0 .80 , 0.40} , {0 .40, O.OO}) are not quite consistent with the predictions of flP : they show a slight V-shaped non-linearity, with P( C -+E) judged to be smaller for 0.80 , 0.40 than for either of the extreme cases.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The term compares the current firing rate g(u i (t k )) with its average firing rate 2 ḡi (t k ), and simultaneously the running average ḡi (t k ) with the constant target rate g. The argument indicates that this term also depends on the optimization parameter γ.
Sentence 2: we rewrite L 12 as a sum over the contributions of the individual time bins, The weight change ∆ wk 1j is then proportional to the gradient of this expression with respect to the weights w 1j , i.e., ∆ wk 1j = α(∂∆L k 12 /∂w 1j ), with some learning rate α > 0.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Under the assumptions [A1],[A2], for a smooth test function φ, the MSE of SVRG-LD (in Algorithm 2) at time K = hT is bounded, for some C > 0 independent of (T, h) in the following manner: The result in Theorem 3 is qualitatively equivalent to that in Theorem 2 when m = N/n .
Sentence 2: h t is set in such a fashion that This provides an approximation to a first order Langevin diffusion, with dynamics where U is the unnormalized negative log posterior.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This framework is based on a free energy (negative log marginal likelihood) and can be seen as a generalization of adaptive TAP [1, 2, 3] and expectation propagation (EP) [4,5].
Sentence 2: we cannot expect that message-passing algorithms similar to loopy belief propagation or expectation propagation (EP) [4] which sequentially update moments and Lagrange parameters are guaranteed to converge.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For this reason, we resort to Chebyshev-like inequalities and we propose the following concentration bound derived from Cantelli's inequality and customized for the IS estimator.
Sentence 2: preventing uncontrolled updates in the space of policy parameters is at the core of the natural gradient approaches [1] applied effectively both on PG methods [18,33,63] and on PGPE methods [26].
Relationship: neutral
Domain: NeurIPS
Sentence 1: However, one important aspect of human visual perception is that it is not space invariant: the fovea processes visual information with high spatial detail while regions away from the central fovea have access to lower spatial detail.
Sentence 2: all video clips were rendered at 1024 × 760 pixels (22.5 deg ×16.7 deg) and a frame rate of 24fps.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A natural representation on these unknown weights is via Dirichlet distributions, since Dirichlet distribution is the conjugate prior of a general discrete probability distribution.
Sentence 2: if the algorithm is guaranteed to converge in the limit (as explained in Section 3.5, this holds for our DNG-MCTS algorithm), it is convenient and reasonable to approximate X s,π as a Normal distribution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We say that γ low is a lower bound on the local convergence rate if γ high ≥ γ low holds for all upper bounds γ high .
Sentence 2: we consider a specific utility model (exponential utility), but our bias and convergence analysis immediately carry over if we assume that each trader is optimizing a risk measure (rather than an exponential utility function) similar to the setup of Frongillo and Reid [9].
Relationship: neutral
Domain: NeurIPS
Sentence 1: This is shown by the blue bars which illustrate of selecting the EYE action at each subsequent choice with respect to after the first choice (d ⇡r ).
Sentence 2: at each time t 2 T i subject i observes state s i t 2 S as an input, calculates and then executes action a i t (e.g., presses a button on a computer keyboard; a i t 2 A and A is a set of actions) and receives a reward r i t (e.g., a monetary reward; r i t 2 <).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We have , and hence we have ṽτ ṽ τ t s=1
Sentence 2: putting these two inequalities together, we get that using the expression for A t ∆x t from (5) we have Assume that g (v t • x) ∈ [−a, 0] for all x ∈ K and all t. The other case is handled similarly.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We collect the spike counts from all q simultaneouslyrecorded physical neurons into a vector y t ∈ IR q×1 , whose ith element is y i t . The choice of the link functions g and h is discussed in Section 3.
Sentence 2: if the latent system were inadequate to describe the activity, we should expect to see dynamical features in the empirical firing that could not be captured by the imputed firing rates.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Simply using a nonlinear neuron combined with Oja's learning rule allows one to compute higher moments of the distributions which yields ICA if the signals have been preprocessed (whitening) at an earlier stage [1].
Sentence 2: 5), where three signals generated by Ornstein-Uhlenbeck processes have the same spatial distribution but different time constants of the autocorrelation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a feature vector s, the goal of the SVM is to find an f (s) that minimizes the objective function γ ) is a regularization term that controls the complexity of f (s), and γ is a tuning parameter controlling the trade-off between error penalization and the complexity of the classification function.
Sentence 2: we generate the first word y = tanh(Cs n ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In Figure 2 we plot the average value of the objective function over 50 instances of the problem for C n = P n , n = 100.
Sentence 2: assume that v 0 , v 0 > 0 and power iteration has risk within a constant from the maximum likelihood estimator, provided an initialization is available whose scalar product with v 0 is bounded away from zero.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By focusing only on the 99th percentile of |d ⇡ur |, we hoped to limit our analysis to the circuitry known to be involved in decision-making.
Sentence 2: we denote the weights of this linear combination as W 2 < NVOX⇥NCELLS , and u t as a vector of size N VOX representing predicted neural activity at each voxel at time t. For training the model, the predicted neural activity is compared with the actual activity recorded by the scanner.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It consists of the following: • A finite planning horizon H. • The number of agents M . An agent m can be in one of the states in the state space S. The joint state space is × M m=1 S. We denote a single state as i ∈ S. • A set of action A for each agent m. We denote an individual action as j ∈ A. )
Sentence 2: in the original loss function ( 13), the focus is on minimizing the global loss, rather than adjusting each individual critic factor f w (•) towards the corresponding empirical return.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Definition 2.1 allows two types of errors: C i s are not required to form a partition i.e., some nodes can be left out, and a small fraction of estimation errors is allowed in each C i , e.g., if x j = 0.9 but x j = 0.2, then the j-th "row" in D (i) is incorrect.
Sentence 2: because the corrupted nodes are removed, the whole path is around C. Also, by the uniform density assumption, walking on the shortest path in G is equivalent to walking on C with "uniform speed", i.e., each edge on the path will map to an approximately fixed distance on C. the shortest path distance scales with the latent distance, i.e., (d − 1) c , which implies Theorem 2.2 (cf.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The guarantee of differential privacy under continuous observation assures us that even if all later users collude in an effort to learn user t's context or preference, they still have very limited advantage over a random guess.
Sentence 2: here, we assume user t provides a context c t which actually determines the mapping of the arms into feature vectors: φ(c t , a) ∈ R d . The sequence of choice thus made by the learner is a 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: SURE-T methods yielded a slightly better denoising performance than SURE-based methods (approximately 0.02 -0.06 dB and 0.04 -0.12 dB for BSD68 and Set12 datasets, respectively) with considerably increased the overall inference time.
Sentence 2: [28] proposed a Monte-Carlo-based SURE (MC-SURE) method to determine near-optimal denoising parameters based on the brute-force search of the parameter space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given that our goal is to compare RBM training approaches rather than achieving the best possible training across all free parameters, neither momentum nor adaptive learning rates were included in any of the implementations tested.
Sentence 2: interestingly, for the Caltech 101 Silhouettes dataset, it seems that the persistent algorithms tested have difficulties in ascending the pseudo-likelihood in the first epochs of training.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To do so, we require the mappings f and g to map adjacent heterogeneous vertices in the known bipartite graph onto nearby positions in a unified Euclidian space R d , in order to ensure that the known bipartite graph can be recovered to some extent by the nearest neighbor approach.
Sentence 2: there is therefore a strong incentive to develop new useful prediction methods based on protein sequences, chemical compound structures, and the available known compound-protein interaction information simultaneously.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this section, we compare Acc-Prox-SVRG with Prox-SVRG and APG on L 1 multiclass logistic regression with the regularization parameter λ.
Sentence 2: for k ≥ 1, V k ≤ W k . For k ≥ 2, we have Since 0 < √ µη(1 − p) < 1, the above inequality leads to From the strong convexity of g (and f ), we can see for k ≥ 2, we have and that is exactly (24).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Several recent works have established the same local error bound conditions for several interesting problems in machine learning [7,32,33].
Sentence 2: by definitions, the only stage with i = S 1 is the posttermination stage, namely, stage s = S. In this case, C(x, x, i, j, s) = 0 since stage S performs no computation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By continuity, there must be a value of Ui in its allowed interval for which V is indifferent to playing 0 or 1, so w E W e. This completes the proof of Lemma 5.
Sentence 2: since we identify players with vertices in G, it will be easier to treat vertices symbolically (such as U, V and W) rather than by integer indices.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It has been shown [7] that the added sophistication of modeling others as rational agents results in a higher value function compared to one obtained from treating others as noise, which implies the modeling superiority of I-POMDPs over other approaches.
Sentence 2: the belief update for a particular level-0 model sample, θ j = 0.5, 0.67, 0.5, 0.85, 0.5, -1, -100, 10 , is solved using Algorithm 2.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Fortunately, a simple power method [10] can be used to efficiently compute a maximum eigenvector solution to the inner problem by only performing matrix-vector multiplications on the individual factors of the two low rank matrices making up T (α), meaning the inner problem can be solved without ever forming a large n × n matrix T (α).
Sentence 2: before proceeding it is important to realize that there are many alternative conceptions of "robustness" in the literature that do not correspond to the notion we are investigating.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus the duality gap at the k-th iteration is given by: We define the relative duality gap as: Rel.gap := Dgap/(1 + |pobj| + |dobj|), where pobj and dobj are respectively the objective function values of the primal problem (12) at point X k , and the dual problem (2 )}, we measure the relative changes of objective function value F (X) and the iterates X and Y as follows: We terminate ALM when either Note that in (19), computing log det(X k ) is easy since the spectral decomposition of X k is already available (see ( 14) and ( 15)), but computing log det( Σ − Λ k ) requires another expensive spectral decomposition.
Sentence 2: [16] applied Nesterov's optimal first-order method to solve the primal problem (1) after smoothing the nonsmooth ℓ 1 term, obtaining an iteration complexity bound of O(1/ϵ) for an ϵ-optimal solution, but the implementation in [16] was very slow and did not produce good results.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In practice, these approaches exhibit prohibitive computational complexity in high dimensions, and the Gaussian assumption may fail to accommodate the burstiness often inherent to real-world count data [9].
Sentence 2: we assume that the time-step factors and transition weights for component k are tied to its component weight ν k . we define the following structure: where π k = (π 1k , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: While determining the exact magnitude of maximum SNR may be impractical, choosing a distribution with uniformly distributed direction and a constant magnitude close to this optimal value, performance can be improved.
Sentence 2: we applied learning to a system used to sudy the dynamics of flapping flight via a wing submerged in water (see Figure 4 for a description of the system (Vandenberghe et al., 2004)).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In Markov Decision Processes (MDPs, Puterman (1994)), when the performance is measured by (1) the sample complexity (Kearns and Singh, 2002;Kakade, 2003;Strehl and Littman, 2008;Szita and Szepesvári, 2010) or (2) the regret (Bartlett and Tewari, 2009;Jaksch, 2010;Ortner, 2012), algorithms have been developed that achieve provably near-optimal performance.
Sentence 2: trying to capture the properties of f via the distribution-free ||f || ∞ bound is not satisfactory.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a set of images {I j } K j=1 , we first extract their features {v j } K j=1 with a pretrained CNN, and then average {v j } K j=1 to obtain v. The image encoder converts v into a context vector h v ∈ R D which is used as the visual input for all subsequent modules.
Sentence 2: the post-processing we use is that we first select 4 most commonly predicted key words with normal descriptions by other baselines, then for each key word, if the generated report does not describe any abnormality nor normality of these key words, we add the a corresponding sentence of these key words that describe their normal cases respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The lemma highlights the role of Q in judging the difference between P 0 and P, and leads immediately to the following corollary (see Appendix C).
Sentence 2: we report results in terms of {0, 1}-valued classification error: `(w, (x, y)) = 0 if and only if y is the largest coordinate of p(w, x).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We compared this value to the one we would obtain if we had a training and a test set with 250 observations each such that we could fit the model on the training set for various values of λ, and see which one gives the smallest prediction error on the test set.
Sentence 2: if θ 0 has a sparse representation with k non-zero coefficients, it has been shown in the noiseless case that it is sufficient to use n ∝ k log m such measurements.
Relationship: neutral
Domain: NeurIPS
Sentence 1: the MCD learning rule decides the learning direction.
Sentence 2: 5 shows the measured output of a noisy neuron (upper trace) with {s i } sweeping between 1.5 and 3.5V, {w i }=4V, V aj =1.8V, and v ni generated by LFSR (Linear Feedback Shift Register) [9] with an amplitude of 0.4V.
Relationship: neutral