sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We want H = arg min Y∈R r×n + X − X(:, K)Y 2 for some norm.
Sentence 2: choosing the Frobenius norm results in a set of n nonnegative least squares (NNLS) problems: we can solve the NNLS problem with matrices of size n × n. After computing just the small R factor from the QR factorization, we can compute the entire nonnegative matrix factorization by working with matrices of size n × n. Analogous results hold for the SVD, where we replace Q by U, the left singular vectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We aim to find the posterior P (W|D N ) in an online setting, where samples arrive sequentially.
Sentence 2: after the n-th sample is received, the posterior is updated according to Bayes rule: for n = 1, . . . , N . Note that the MNN is deterministic, so the likelihood (per data point) has the following simple form 3 the Bayes update in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Observing that the placements of all the operations in the network can be configured independently in practice, the device selection for each operation can also be independent.
Sentence 2: the joint distribution f (d|u), i.e., the placement distribution for the neural network, can be expressed as a product of marginal distributions for each operation: To obtain the optimal distribution which results in the optimal placement d * = arg min T (d) with probability 1, we iteratively update the parameters until the final convergence, starting from randomly generated values.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We therefore need to show that the norm of w, and evaluation K w on any two examples can still be performed efficiently.
Sentence 2: first note that given the vector v we can compute the norm of w as follows, Next, given two vectors x i and x j the value of their inner-product is, although we cannot compute the vector w explicitly we can still compute its norm and evaluate any of the kernels from the class C. Synthetic data: We generated binary-labelled data using as input space the vectors in 100 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For clarification, we now write an NPG that employs the natural policy gradient and a VPG that employs the "vanilla" policy gradient.
Sentence 2: nPG(w) and VPG(w) denote the use of the parameterbased exploration strategy, and NPG(u) and VPG(u) denote the use of the control-based exploration strategy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Bottou and Bousquet [BB07] show that if is a bound on the sub-optimality of w on S i , then the total error is bounded by H(n i ) + .
Sentence 2: in designing an efficient algorithm for streaming data, we focus on reducing the sub-optimality to asymptotically balance with H(n i ) -it does not pay to reduce the empirical risk even further.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our goal is to characterize the scaling of the sample complexity for various reinforcement learning approaches (i.e., how many trials they require in order to learn a near-optimal policy) for large numbers of agents n. Thus, in our bounds below, no serious attempt has been made to make our bounds tight in variables other than n. Below we show that if an RL algorithm uses only the global reward signal, then there exists a very simple MDP-one with horizon, T = 1, only one state/trivial dynamics, and two actions per agent-on which the learning algorithm will require Ω(n) trials to learn a good policy.
Sentence 2: such algorithms do not scale well to large numbers of agents.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Instead of a binary decision function where E y is the expectation over P y and ℓ 0-1 (z) is the zero-one loss: Since the zero-one loss is hard to optimize in practice due to its discontinuous nature, it may be replaced with a ramp loss (as illustrated in Figure 1): giving an objective function of To avoid the non-convexity of the ramp loss, the hinge loss is often preferred in practice: giving an objective of One practical motivation to use the convex hinge loss instead of the non-convex ramp loss is that separability (i.e., min g J R (g) = 0) implies ℓ R (z) = 0 everywhere, and for all values of z for which ℓ R (z) = 0, we have ℓ H (z) = 0.
Sentence 2: the convex hinge loss will give the same decision boundary as the non-convex ramp loss in the ordinary classification setup, under the assumption that the positive and negative samples are non-overlapping.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, optimization is difficult because we do not know a reasonable t in advance, and the gradient updates are expensive due to backpropagation through the entire chain.
Sentence 2: we propose a more efficient approximation, called Markov GAN (MGAN): where 2 (0, 1), b 2 N + , m 2 N + are hyperparameters, x denotes "fake" samples from the generator and T m ✓ (x|x d ) denotes the distribution of x when the transition kernel is applied m times, starting from some "real" sample x d . We use two types of samples from the generator for training, optimizing ✓ such that the samples will fool the discriminator: 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For each variable y j in the model, each level of the cascade filters a current set of possible states Y j , and any surviving states are passed forward to the next level of the cascade by substituting each state with its set of descendents in the hierarchy.
Sentence 2: in the pose estimation problem, surviving states are subdivided into multiple finer-resolution states; in the image segmentation problem, broader object classes are split into their constituent classes for the next level.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From [Berg et al., 1984] (Theorem 3.2.2, p.74), the Persistence Fisher kernel is positive definite.
Sentence 2: our proof technique is not only independent and direct for the Fisher information metric on the probability simplex without relying on the geodesic distance on S + , but also valid for the case of infinite dimensions due to [Schoenberg, 1942] (Theorem 2, p. 102).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Under mild assumptions on the correlation function, which is unknown, [27] showed that, by using a bimodal type of kernel K such that K(0) = 0, the residual sum of squares (RSS) approximates the asymptotic squared error uniformly over a set of bandwidths.
Sentence 2: choosing the bandwidth ĥb (of the bimodal kernel) minimizing the RSS results in an optimal bandwidth that minimizes the asymptotic squared error asymptotically.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, consider t1 = 1, t2 = 2, and t3 = 3, giving z 1 = ψ( t1 ) = (0, 1, 2) � , z 2 = ψ( t2 ) = (1, 0, 1) � , and T here is a convex set and ψ : � T →R 3 is a convex function.
Sentence 2: moreover, for t1 = 1, we have we can use Lemma 8 to compute N S ψ (z 1 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: DA-GAN generates the data in a completely different way from InfoGAN (5) and AC-GAN (22) which generate images from a random noise input or abstract semantic labels.
Sentence 2: inferior to our model, those existing GAN-like models cannot exploit useful and rich prior information (e.g., the shape, pose of faces) for effective data generation and augmentation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We may do so by applying a simple sigma-point quadrature ("unscented") approximation [16], in which the standard deviation of the convolved function m ts (x s ) is estimated by applying the same nonlinearity (13) to the standard deviation of the error on the incoming product M ts .
Sentence 2: similarly to (18) and ( 19), we have The steady-state solution of (20) yields an estimate of the variances of the log-belief log p t by σ 2 t = u∈Γt σ 2 ut ; this estimate is typically much smaller than the bound (18) due to the strict sub-additive relationship between the standard deviations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For computing the FID, we propagated all images from the training dataset through the pretrained Inception-v3 model following the computation of the Inception Score [44], however, we use the last pooling layer as coding layer.
Sentence 2: for this coding layer, we calculated the mean m w and the covariance matrix C w . we approximate the first and second central moment of the function given by the Inception coding layer under the real world distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As mentioned in the introduction, the sample complexity of margin-error minimization can be upperbounded in terms of the average norm Alternatively, we can rely only on the dimensionality and conclude m(ǫ, γ, D) ≤ Õ(d/ǫ 2 ) [7].
Sentence 2: although both of these bounds are tight in the worst-case sense, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Nonetheless, it is interesting to notice the major difference between the true and pseudoregret when compared to the standard bandit problem.
Sentence 2: in fact, it is possible to show in the risk-averse case that the pseudo-regret is not an unbiased estimator of the true regret, i.e., to bound the expectation of R n we build on the high-probability result from Lemma 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We tried different set of parameters for RBSUMM ρ i 2 +1 i+t (0 ≤ i ≤ 5, ρ = 0.01, 0.1, 1) or sufficiently small step size, but could not see the convergence of the objective within 5000 iterations.
Sentence 2: the results are not included here.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: p MK) convergence rate, Algorithm 1 achieves ✏-approximation solution when mK / 1/(n✏ 2 ).
Sentence 2: that is, the amount of computation required on each node is inversely proportional to n. linear speedup can be achieved by Algorithm 1 asymptotically w.r.t computational complexity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let k be as calculated by Algorithm 1 at iteration k. Then, for any k  k ⇤ the following holds: We are now ready to prove the remaining conditions.
Sentence 2: for any k < n. Lemma 3.4 implies that k k+1 for any k < k ⇤ (minimizing the same objective with stricter constraints yields a higher optimal value).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in the examples cited, the authors evaluated the performance of their methods on data simulated from a fixed finite number of components or populations, suggesting that they found this to be more realistic than a DPM for their applications.
Sentence 2: it is important to understand the behavior of this posterior when the data comes from a finite mixture -in particular, does it concentrate at the true number of components?
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (-), which approximates the ideal marginal Gaussianization w (.
Sentence 2: ), can be derived from univariate Gaussian mixtures The parameters are chosen by minimizing the negentropy of the transformed variable X(k+1): A,1r,f-t,u (3) after each iteration, the transformed variable becomes as close as possible to the standard Gaussian in the Kullback-Leibler distance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider the simplest loss, viz., the squared "loss" r(x, θ) = − x − θ 2 2 (note -ve sign to keep reward interpetation).
Sentence 2: it can be easily seen that r(x, θ * p ) = E θ∼p [r(x, θ)] = − x 2 2 + 2x • θ * p − 1 • θ * p , if the relevance vectors θ are in {0, 1} n . the Lipschitz condition is satisfied.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By contrast, in the analogous graph frequency model of the present work, y is fixed, and we grow the network by repeatedly sampling the number of edges g ij between vertices i and j and summing the result.
Sentence 2: in the Caron and Fox [12] model, a latent infinity of vertices (only finitely many of which are active) are added to the network each time y increases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If it has an effective support of size 1, then the event ω for which q ω = 1 cannot be from a level set in L or H (otherwise, part (i) of Lemma 5.1 would be violated).
Sentence 2: therefore it must belong to the single level set in M . if M = {S m } (for some index m), then there are feasible solutions Q such that q ω = 1 (for ω ∈ S m ), all of which have ρ(Q) = r m . If, on the other hand, Q has an effective support of size 2, then it is not hard to show that one of the two events must be from a level set S l ∈ L, and the other, from a level set S h ∈ H (since all other combinations result in a violation of either part (i) or part (iii) of Lemma 5.1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Many practical combinatorial problems can be effectively translated into SAT.
Sentence 2: one of the current most successful approaches to solving hard computational problems, arising in, e.g., hardware and software verification and planning and scheduling, is to first translate the problem into SAT, and then use a state-of-the-art SAT solver to find a solution (or show that it does not exist).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The generative process of the model in Figure 1 can be described as: 1.
Sentence 2: for each document d, choose the topic proportions according to a Dirichlet distribution:  the entire model can be described by the following joint probability where w d,1:N and z d,1:N denotes all the words and the associated topics, respectively, in the d-th document.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the block stacking environment specifically, the robot should only need to pay attention to the position of the block it is trying to pick up (the source block), as well as the position of the block it is trying to place on top of (the target block).
Sentence 2: a properly trained network can learn to match the current state with the corresponding stage in the demonstration, and infer the identities of the source and target blocks expressed as soft attention weights over different blocks, which are then used to extract the corresponding positions to be passed to the manipulation network.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In such case, it takes Θ(t) space to store the empirical CDF Fi of the observed outcomes from arm i, and both calculating the stochastically dominant CDF F i and updating Fi take Θ(t) time.
Sentence 2: the worst-case space usage of SDCB in T rounds is Θ(T ), and the worst-case running time is Θ(T 2 ) (ignoring the dependence on m and K); here we do not count the time and space used by the offline computation oracle.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that if (G, s) does not contain (B 2 , s + ) as a signed minor then any wreductions of (G, s) also do not contain (B 2 , s + ) as a signed minor; we can check this property for each type of w-reductions, (w,1,2,3).
Sentence 2: it is sufficient to show that if a w-reduced signed graph (G, s) does not contain (B 2 , +, +) as a signed minor then it is one of the five types.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We address the case of the linear time statistic in [7,Section 6], for which both the test statistic and the parameters of the null distribution can be computed in O(n), for sample size n. This has a higher variance at a given n than the U-statistic estimate costing O(n 2 ) used in [8,7], since the latter is the minimum variance unbiased estimator.
Sentence 2: we would use the quadratic time statistic in the "limited data, unlimited time" scenario, as it extracts the most possible information from the data available.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Among GMMs with top-k breakings, the larger the k is, the more information we use in a single ranking, which comes at a higher computational cost.
Sentence 2: it is natural to conjecture that for the same data, GMM G k T with large k converges faster than GMM G k T with small k. In other words, we expect to see the following time-efficiency tradeoff among GMM G k T for different k's, which is verified by the experimental results in the next section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From a practical point, the main contribution of this paper is to suggest how to select a good Universum set: it should be such that it contains invariant directions and is positioned "in between" the two classes.
Sentence 2: as can be partly seen from the BCI experiments, a good Universum dataset needs to be carefully chosen and cannot be an arbitrary backdrop as the name might suggest.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Natural Parameters Recall that for Gaussian distributions the natural parameters are conventionally defined as (Σ −1 µ, 1 2 Σ −1 ).
Sentence 2: to find the natural parameters of ( 12), it suffices to show that (Σ −1 µ, 1 2 Σ −1 ) of ( 12) can be written as an affine transformation of some finite-dimensional parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The empirical risk minimizer (ERM) is exactly the average h * = 1 m i z i , and for every h = h * we have L S (h) > L S (h * ).
Sentence 2: an agnostic sample compression scheme in this setup should compress S to a subsequence and a binary string of side information, from which the average of S can be reconstructed.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, critical differences with our proposed method are as follows: i) the above methods cannot use observed data (X and Y in this study) about heterogeneous nodes for prediction, because the algorithms are based only on co-occurence information (A uv in this study), and ii) we need to define a new representation of not only the objects in the training set but also additional objects outside of the training set.
Sentence 2: it is not possible to directly apply the above methods to the bipartite graph inference problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We consider a four node graph, with vertex set V = {1, 2, 3, 4} and edge-set E = {(1, s) | s ∈ {2, 3, 4}}.
Sentence 2: the covariance matrix Σ * is parameterized the correlation parameter ρ ∈ [−1, 1]: the diagonal entries are set to Σ * ii = 1, for all i ∈ V ; the entries corresponding to edges are set to Σ * ij = ρ for (i, j) ∈ E; while the non-edge entries are set as Σ * ij = ρ 2 for (i, j) / ∈ E. for this particular example, Assumption [A1] reduces to the constraint |ρ|(|ρ|+2) < 1, which holds for all ρ ∈ (−0.414, 0.414).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to see that lim ε→0 ρ ε = ρ in distributional distance, and all ρ ε are stationary ergodic.
Sentence 2: if H 0 is the set of all stationary ergodic distributions on pairs, we have found a distribution ρ ∈ cl H 0 such that W ρ (H 0 ) = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, thinking of the shortest path metric on an unweighted bipartite graph in which P corresponds to the left set of vertices, and Q corresponds to the right set of vertices, for each pair of points p i ∈ P and q j ∈ Q, one can choose d(p i , q j ) = 1 or d(p i , q j ) > 1 independently of all other distances by deciding whether to include the edge {p i , q j }.
Sentence 2: there are at least 2 Ω(mn) possible distance matrices A, and since our algorithm reads o(mn) entries of A, cannot learn whether d(p i , q j ) = 1 or d(p i , q j ) > 1 for each i and j.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: samples drawn from some unknown probability distribution P. Catoni [2012] builds the estimator θ of the mean as the solution of the equation where α > 0 is a positive parameter, and ψ(•) : R → R is a truncation function that is non-decreasing and satisfies (2) By choosing α = 2/nν, where ν is the variance of the random variable, Catoni [2012] demonstrates that with high probability, the deviation of θ from the mean is upper bounded by O( ν/n).
Sentence 2: (1) can be applied to heavy-tailed distributions, as long as the variance is finite.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because of this dependence on D, the running time of the Lloyd algorithm becomes prohibitive in high dimensions, and even for D < 10 it is an  order of magnitude slower than k-means.
Sentence 2: we were compelled to also implement a greedy algorithm reminiscent of Kégl's algorithm for estimating the packing number [4]: supposing that k − 1 codevectors have already been selected, the kth one is chosen to be the sample point with the largest distance from the nearest codevector.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While the main objective of Algorithm 1 is to identify all arms with means above a given threshold, we note that prior to adding an arm to S t in the TPR setting (i.e., when ⇠ t = 1) Algorithm 1 behaves identically to the nearly optimal best-arm identification algorithm lil'UCB of [21].
Sentence 2: whether the goal is best-arm identification or to identify all arms with means above a certain threshold, Algorithm 1 is applicable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To see this, we denote the slack variable y = (ρ − µ ρ ) 2 , and we have: where equality is attained when var(y) = 0, i.e, when y is constant.
Sentence 2: m 4 is minimal when |ρ − µ ρ | is constant, which means, since ρ is not constant (σ 2 = α > 0), that p has two values with equal probability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can infer that if any x ∈ X (i.e., any column of Φ not in Φ) lies in this convex cone, then the associated coefficients v must all be positive by definition (likewise, by a similar argument, any x in the convex cone of − Φ leads to the same result).
Sentence 2: theorem 2 ensures that we are not at a local minimum.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: According to the domain adaptation theory [5], the generalization error on the target domain depends on the generalization error on the source domain as well as the difference between the two domains.
Sentence 2: one possible solution for domain adaptation is to learn a feature extractor φ(X) shared by both domains, which defines feature distributions p φ S and p φ T for both domains, and minimize some distance between the feature distributions [47] as a regularization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If several nodes of a GGM are perturbed, then this will correspond to the of the corresponding row-column groups in Θ 1 − Θ 2 .
Sentence 2: in order to detect node perturbations in a GGM (Figure 1), we must construct a regularizer that can promote estimates whose support is the union of row-column groups.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With each learned model we forecast wind speeds at all sites and all test-year hours at six horizons (1-6 hours).
Sentence 2: for each geography (WI and PNW) we have 20 learned models (4 folds and C = 1, 2, ..., 5) and 120 prediction sequences (horizon 1-6 hrs for each of the 20 learned models).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, while conditions such as convexity, continuity, and differentiability of φ are easy to verify and have natural relationships to optimization procedures, it is not immediately obvious how to relate such conditions to their statistical consequences.
Sentence 2: in Section 2 we consider the weakest possible condition on φ-that it is "classification-calibrated," which is essentially a pointwise form of Fisher consistency for classification.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To the best of our knowledge, there is no analytical solution to the influence estimation given Weibull transmission function.
Sentence 2: we compare CON-TINEST with Naive Sampling (NS) approach (see Appendix C) by considering the highest degree node in a network as the source, and draw 1,000,000 samples for NS to obtain near ground truth.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For every possible edge e = {i, j}, we draw the multiplicity of the edge e in this step as m e ind ∼ f (•|w e ) and add m e copies of edge e to F . Finally, E n+1 = E n ∪ F . This technique potentially introduces multiple edges in each step, in which edges themselves may have multiplicity greater than one and may be duplicates of edges that already exist in the graph.
Sentence 2: this technique generates multigraphs, multiple edges at a time.
Relationship: reasoning