sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The elements of L are called loss functions.
Sentence 2: the performance of the learner is measured in terms of its regret, The simplest possible case, which will be the focus of this paper, is when the losses are linear, i.e., when t (w) = f t , w for some f t ∈ F ⊂ R d . In fact, the linear case is not only simple, but is also fundamental since the case of nonlinear loss functions can be reduced to it: Indeed, even if the losses are nonlinear, defining f t ∈ ∂ t (w t ) to be a subgradient 1 of t at w t and letting ˜ t (u) = f t , u , by the definition of subgradients, t (w if an algorithm keeps the regret small no matter how the linear losses are selected (even when allowing the environment to pick losses based on the choices of the learner), the algorithm can also be used to keep the regret small in the nonlinear case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From (5) and using the above assumption, it is obvious that and thus R 0 ) is a valid upper bound to Θ ; furthermore it can be tightened as we progress in the MCGD method.
Sentence 2: observe that ( α, Θ, R) with R = Θ is an optimal solution to P(R 0 UB ), we have In other words, for all feasible (α, Θ, R) to ) is an upper bound to R and Θ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For each image-sentence sequence )}, we augment each sentence T l n with multiple sentences for training.
Sentence 2: when we perform the LSA-based text summarization, we select top-κ highest ranked summary sentences, among which the top-ranked one becomes the summary sentence for the associated image, and all the top-κ ones are used for training in our model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now prove a bound on E [R benefit ].
Sentence 2: because each f (i) is monotone and submodular, f is monotone and submodular as well, so the greedy schedule's approximation guarantees apply to f . by Theorem 3, we have R benefit ≤ T t=1 t . Thus by Lemma 2, where R = T t=1 r t . To bound E [R benefit ], it remains to justify the assumption that each meta-action ãt always takes unit time.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Most previous work on feature selection in supervised learning with functional data is quite recent and focuses on regression problems; for instance, on the analysis of fMRI images (Grosenick et al., 2008;Ryali et al., 2010) and NIR spectra (Xiaobo et al., 2010).
Sentence 2: adaptations of lasso and other embedded methods have been proposed to this end (see, e.g., Kneip and Sarda (2011); Zhou et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let the state space S and Player 2's action space A 2 remain the same as in M . Let A 1+ , p + (•|•, •, •), r + (•, •, •) be Player 1's action set, the transition kernel, and the reward function of M + , such that for any a 1 ∈ A 1 and a 2 ∈ A 2 and an admissible transition probability p(•|s, a 1 , a 2 ) ∈ P k (s, a 1 , a 2 ), there is an action a 1+ ∈ A 1+ such that p + (•|s, a 1+ , a 2 ) = p(•|s, a 1 , a 2 ) and r + (s, a 1+ , a 2 ) = r(s, a 1 , a 2 ).
Sentence 2: player 1 selecting an action in A 1+ is equivalent to selecting an action in A 1 and simultaneously selecting an admissible transition probability in the confidence region P k (•, •).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In particular, simply fitting a multidimensional rectangle distribution to the data (by using PCA to align the axes, and then assuming a uniform distribution on each axis) works surprisingly well for a wide range of distributions.
Sentence 2: using the analytic eigenfunctions of a uniform distribution on data sampled from a Gaussian, works as well as using the numerically calculated eigenvectors and far better than boosting or RBMs trained on the Gaussian distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As an example of a real world data set, we used the NGRIP ice core data (provided by Niels-Bohr institute in Copenhagen, http://www.iceandclimate.nbi.ku.dk/data/), which provides an undisturbed ice core record containing climatic information stretching back into the last glacial.
Sentence 2: this data set as shown in figure 3 contains 4918 observations of oxygen isotope concentration δ 18 O over a time period from the present to roughly 1.23 • 10 5 years into the past.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, suppose we know that f con is Gaussian with known covariance matrix and unknown mean.
Sentence 2: proposition 1 says we cannot design R ε so that it can decontaminate (1−ε)f tar +εf con for all f tar ∈ D. it is impossible to design an algorithm capable of removing Gaussian contamination (for example) from arbitrary target densities.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use standard approximate dynamic programming with a quadratic, normalized cost r(x, u) = x T Qx + u T Ru, where Q and R are positive-definite, to compute the cost-to-go J π θ .
Sentence 2: we use a piecewiselinear triangulation of the state-space as to approximate J π θ , see [39].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast to methods based on influence functions that require large number of samples due to their asymptotic nature, we empirically show that the developed ALOOCV works well even when the number of samples and features are small and comparable to each other.
Sentence 2: aLOOCV is capable of predicting overfitting and hence can be used for model selection and choosing the regularization hyperparameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The optimism in the face of uncertainty paradigm has already lead to several successful results for solving decision making problems.
Sentence 2: it has been applied in the following contexts: multi-armed bandit problems [1] (which can be seen as single state MDPs), planning algorithms for deterministic systems and stochastic systems [8,9,17], and global optimization of stochastic functions that are only accessible through sampling.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While cumulants are essentially unique with the additivity and homogeneity properties [17] when no restrictions are made on the probability space, the preprocessing step of ICA gives additional structure (like orthogonality and centering), providing additional admissible functions.
Sentence 2: [20] designs "robust cumulants" which are only minimally effected by sparse noise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Tensor power method in Algorithm 1 requires significant storage to be deployed: Ω(d 3 ) memory is required to store a dense d × d × d tensor, which is prohibitively large in many real-world applications as tensor dimension d could be really high.
Sentence 2: we show in this section how to compute tensor decomposition in a memory efficient manner, with storage scaling linearly in d. we consider the case when tensor T to be decomposed is a population moment E x∼D [x ⊗3 ] with respect to some unknown underlying data distribution D, and data points x 1 , x 2 , • • • i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This simulation is inspired by the experimental study of noise correlations in cortical area MT due to [10], where it was suggested that time-varying correlations between pairs of neurons play a significant role in explaining behavioral variation in smooth-pursuit eye movements.
Sentence 2: the findings in [10] and [4] suggest that short-term increases in noise correlations are likely to occur after feedback arrives and neurons within and upstream from MT synchronize.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each iteration of SNGD is as simple and efficient as SGD, but is much more appropriate for non-convex optimization problems, overcoming some of the pitfalls that SGD may encounter.
Sentence 2: we define a family of locally-quasi-convex and locally-Lipschitz functions, and prove that SNGD is suitable for optimizing such objectives.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we can recover an approximation bound that is analogous to the primal bound, which bounds the approximation error between H + and the best representable approximation to the on-policy fixed point  Finally, using H Π = OH Π and Lemma 1, we obtain To compare the primal and dual results, note that despite the similarity of the bounds, the projection operators do not preserve the tight relationship between primal and dual updates.
Sentence 2: even if (1−γ)q = Hr and (1−γ)(Oq) = (OH)r, it is not true in general that (1−γ)(POq) = (POH)r. The most obvious difference comes from the fact that in the dual, the space of H matrices has bounded diameter, whereas in the primal, the space of q vectors has unbounded diameter in the natural norms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Messages relate to the Θ coefficients by the following definition where the (nonstandard) notation Θ \y x indicates the matrix Θ V x for which V is the set of all the nodes in X that are reachable by x if y were removed from the graph.
Sentence 2: θ \y x is summarizing the effect that x has on the entire network except for the part of the network that x can only reach through y. Propagation: The message-passing scheme is organized to recursively compute the Θ matrices using Theorem 3.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In other words, the weights are initialized as z U m (h t ) = 1 M ∀m and are then updated recursively using the likelihood of the dynamics model for that particle of observations as z U m (has ) ∝ z U m (h)P (s |a, s, P m ) = z U m (h) P m (s, a, s ).
Sentence 2: one advantage of this definition is that it enforces a correspondence between the history and belief representations in the finite-dimensional space, in the sense that z we can work in history space during planning, alleviating the need for complete belief updates, but via a finite and well-behaved representation of the actual belief -since different histories corresponding to the same belief are mapped to the same representation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To motivate our construction, we would like to note that boosting algorithms define weights in two different domains: the vectors w t ∈ R m which assign weights to examples and the weights {α t : t ∈ [T ]} over weak-hypotheses.
Sentence 2: in the terminology used throughout this paper, the weights w t ∈ R m are primal vectors while (as we show in the sequel) each weight α t of the hypothesis h t is related to a dual vector λ t . we show that Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is well known [7,14] that uniformly sampling O(n/γε −2 log(n/δ)) points with replacement is sufficient to obtain w.p.
Sentence 2: 1 − δ the following guarantees on Φ m We can now rewrite Π n as using uniform sampling we can reduce the size of the search space H m by a 1/γ factor (from n to m n/γ) in exchange for a γ additive error, resulting in a computation/approximation trade-off that is linear in γ. Exploiting the error bound for γ-preserving dictionaries we are now ready for the main result of this paper: showing that we can improve the computational aspect of kernel k-means using Nyström embedding, while maintaining optimal generalization guarantees.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even though our setting is far more general than exponential utility and exponential families, the framework we develop can also be used to show that their results can be extended to interactions between traders who have what we call "compatible" risks and beliefs.
Sentence 2: for any risk-based trader possessing a risk ρ with dual α, we can think of that trader's "belief" as the least surprising distribution p according to α.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose a novel homotopy smoothing (HOPS) algorithm for solving the problem in (1) that achieves a lower iteration complexity than O(1/ ).
Sentence 2: the iteration complexity of HOPS is given by Õ(1/ 1−θ ), where θ ∈ (0, 1] captures the local sharpness (defined shortly) of the objective function around the optimal solutions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, one might object to our "pessimism bias" interpretation of lower expected reward rates for unchosen alternatives, by attributing it to a confirmation bias [23] or a sampling bias [24].
Sentence 2: subjects may report especially low reward estimates for unchosen options, either because they retroactively view discarded options as more undesirable (confirmation bias), or because they failed to choose the unchosen options precisely as they decided these were less valuable to begin with for some reason (regardless of the reason, this scenario introduces a sampling bias of the unchosen arms).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the discovery by Adrian [1] that action potentials are generated by sensory neurons with a frequency that is substantially determined by the stimulus, the idea of rate coding has become a prevalent paradigm in neuroscience [2].
Sentence 2: today the coding properties of many neurons from various areas in the cortex have been characterized by tuning curves, which describe the average firing rate response as a function of certain stimulus parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The related applications in biological and medical research can be found in existing literature (Hudson et al., 2009;Bandyopadhyaya et al., 2010;Ideker and Krogan, 2012).
Sentence 2: given n We are interested in estimating the difference of the precision matrices of two distributions: We define the empirical covariance matrices as: S  where S X and S Y are the empirical covariance matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, importantly, n does not necessarily grow to infinity, if for example N, 1 σ grow appropriately with p. That means that Theorem 2.5.A and Theorem 2.5.B imply non-trivial guarantees for arbitrary sample size n. The proposition below shows that if σ is at most exponential in −(1 + ) (p+2n) 2 2n + (2 + p n ) log (RQ) for some > 0, then for appropriately chosen truncation level N the LBR algorithm recovers exactly the vector β * with n samples.
Sentence 2: with one sample (n = 1) LBR algorithm tolerates noise level up to exponential in −(1 + ) p 2 /2 + (2 + p) log(QR) for some > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [2017], we then calculate the expectation of visibility and depth along each ray.
Sentence 2: given a ray R with N samples R 1 , R 2 , ... , R N along its path, we calculate the visibility (silhouette) as the expectation of the ray hitting the voxels: Similarly, the expected depth can be calculated as , where d j is the depth of the sample R j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set C. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set C is simple for a quadratic objective function.
Sentence 2: our results hold if one can find a ⇢-approximate solution of a quadratic program subject to C in polynomial time, where ⇢ < 1 is a positive constant that depends on the structure of the set C. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an (✏, )-second order stationary point (SOSP) in at most O(max{✏ 2 , ⇢ 3 3 }) iterations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A Nash equilibrium is defined by a joint assignment of actions such that no player has any incentive to deviate.
Sentence 2: if σ p = θ denotes the action choice for the protagonist and σ a = {a t , b t } the choice for the antagonist, then the joint action σ = (σ p , σ a ) is a Nash equilibrium if U p (σ p , σ a ) ≤ U p (σ p , σ a ) for all σp , and U a (σ p , σa ) ≤ U a (σ p , σ a ) for all σa .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The hyperparameter θ constructs the kernel matrix A(θ) ∈ S d×d of the training data {x i } d i=1 (see [22]).
Sentence 2: one can find a good hyperparameter by minimizing the negative log-marginal likelihood with respect to θ: For handling large-scale datasets, [29] proposed the structured kernel interpolation framework assuming θ = [θ i ] ∈ R 3 and where W ∈ R d×r is some sparse matrix and K ∈ R r×r is a dense kernel with r d. in [29], r "inducing" points are selected and entries of W are computed via interpolation with the inducing points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also show that some learnability assumption on C is necessary in order to achieve a high-utility (C, d, ⌧)-metric multifair predictions efficiently.
Sentence 2: we give a reduction from inverting a boolean concept c 2 C to learning a hypothesis f that is metric multifair on a collection H derived from C, where the metric samples from d encode information about the concept c. Recall, that for any H and d, we can always output a trivial (H, d, 0)-metric multifair hypothesis by outputting a constant hypothesis.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For problems with more than two components, most of the existing approaches [5,6,7,8] rely on the Tensor Methods.
Sentence 2: for a D-dimensional linear MR problem, [6] propose a convex optimization formulation using a third-order tensor, which results in a computational cost of O(N D 12 ) and a sample complexity of O(D 6 / 2 ), limiting its application to problems of small dimension.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 Note that we do not assume the samples from GP(m, k) are in H. In the following, without loss of generality, we assume the GP prior considered in regression has µ = 0 and Σ = I.
Sentence 2: m(x) = 0 and k(x, x ) = ρ 2 φ T x φ x .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We compare our proposed techniques to the current state-of-art generative and discriminative models developed in [7].
Sentence 2: [7] introduced 3 encoding architectures -Late Fusion (LF), Hierarchical Recurrent Encoder (HRE), Memory Network (MN) -each trained with a generative (-G) and discriminative (-D) decoder.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Deep Gaussian Processes (DGP) [Damianou and Lawrence, 2013] are multilayer predictive models that are highly flexible and can accurately model uncertainty.
Sentence 2: they have been shown to perform well on a multitude of supervised regression tasks ranging from small (∼500 datapoints) to large datasets (∼500,000 datapoints) [Salimbeni and Deisenroth, 2017, Bui et al., 2016, Cutajar et al., 2016.
Relationship: entailment
Domain: NeurIPS
Sentence 1: H p (r) is a non-decreasing, non-negative function, and there exist δ Hp > 0 and M Hp ≥ 0 such that, r ≥ H p (r) ≥ r − M Hp r 3 , for all r < δ Hp .
Sentence 2: lim r→0 Hp(r) r = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The intrusions have a considerable variety.
Sentence 2: they are: N0 -1 kHz pure tone, N1 -white noise, N2 -noise bursts, N3 -"cocktail party" noise, N4 -rock music, N5 -siren, N6 -trill telephone, N7 -female speech, N8 -male speech, and N9 -female speech.
Relationship: entailment
Domain: NeurIPS
Sentence 1: With the expansion (5) in hand, we can show a number of finite-sample convergence guarantees for the robustly regularized estimator Based on the expansion (5), solutions b ✓ rob n of problem (6) enjoy automatic finite sample optimality certificates: for ⇢ 0, with probability at least 2 are constants (which we specify) that depend on the loss `and domain ⇥.
Sentence 2: with high probability the robust solution has risk no worse than the optimal finite sample robust objective up to an O(⇢/n) error term.
Relationship: entailment
Domain: NeurIPS
Sentence 1: sample drawn from a common entity-wise GP with covariance Σ : U × U → R. This provides a case of p(t|θ) in a SRM, where θ = θ Σ determines the GP kernel function Σ. Optimizing the GP kernel Σ via evidence maximization means to learn the dependency of entities in U, summarized over all the entities v ∈ V. There is a drawback if applying HGP to link prediction.
Sentence 2: the model only learns a one-side structure, while ignoring the dependency in V. the attributes of entities v cannot be incorporated even if their entity attributes are available.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Beyond the traditional visual captioning task [41,28,43,40,18] that produces one single sentence, generating long and topic-coherent stories or reports to describe visual contents (images or videos) has recently attracted increasing research interests [19,35,22], posed as a more challenging and realistic goal towards bridging visual patterns with human linguistic descriptions.
Sentence 2: report generation has several challenges to be resolved: 1) The generated report is a long narrative consisting of multiple sentences or paragraphs, which must have a plausible logic and consistent topics; 2) There is a presumed content coverage and specific terminology/phrases, depending on the task at hand.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One important aspect of our model is that, in terms of graphical representation, the distribution p(ω) is independent of x, while the distribution p θ (z|x, ω) is conditional on x.
Sentence 2: p(ω) tends to capture uncertainty of model parameters (epistemic uncertainty), while p θ (z|x, ω) reacts sensitively to uncertainty in data, varying across different input points (heteroscedastic uncertainty) [14].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This definition of F trades off between exploitation (maximizing the most plausible F h ) and exploration (distinguishing between more and less plausible F h ) by allowing each Fi to reach its maximum value either by having F h reach ↵ i or having G h reach  i .
Sentence 2: each of the thresholds can be satisfied with either a sufficiently large utility F h or a sufficiently large distance G h . Figure 2 shows the logical relationships between these components.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Following the classical marginalization, it suffices to define a projection L ∈ R (n−1)×n with property L1 n = 0 n−1 .
Sentence 2: all biases of the form 1 n w are mapped to the nullspace of L. The Matrix T-distribution under affine transformations [10, Theorem 3.2] reads LX ∼ T (α, β, LM, LΨL , Ω) and in our case (Ω = I d , LM = L1 n w = 0 (n−1)×d ), we have ) Note that due to the statistic LX, the likelihood is constant over all X (or S) mapping to the same D. As we are not interested in any specifics about L other than its nullspace, we replace the image with the kernel of the projection and define matrix Q : which accounts for arbitrary latent feature correlation Σ and all mean matrices M = 1 n w . In hindsight, the combination of Bayesian and classical marginal likelihood might appear arbitrary, but both strategies have their individual strengths.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, the response of the or-node U i is defined as, Collaborative Edge: For any pair of leaf-nodes (L j , L j ′ ) respectively associated with two different or-nodes, we define the collaborative edge between them according to their contextual cooccurrence.
Sentence 2: how likely it is that the object contains contours detected via the two leaf-nodes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This perspective allowed us to re-evaluate the necessity of regularization methods such as weight decay, and to suggest new methods for normalization, targeting the computational, numerical and task-specific deficiencies of current techniques.
Sentence 2: we showed that the use of L 1 and L ∞ -based normalization schemes could provide similar results to the standard BN while allowing low-precision computation.
Relationship: entailment