sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: State-of-the-art models [1,2,3,4] mostly adopt the encoder-decoder paradigm [3], where the content of the given image is first encoded via a convolutional network into a feature vector, which is then decoded into a caption via a recurrent network.
Sentence 2: the words in the caption are produced in a sequential manner -the choice of each word depends on both the preceding word and the image feature.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is the main difference to [4], where they enforce the global Lipschitz constant to be smaller than one.
Sentence 2: in kernel methods one uses typically the regularization functional induced by the kernel which is given as the squared norm of the function, for translation invariant kernels one can make directly a connection to penalization of derivatives of the function f via the Fourier transform, see [20].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Real NVP [4] is a member of the normalising flows family of models, where some density in the input space X is transformed into a desired probability distribution in space Z through a sequence of invertible mappings [14].
Sentence 2: real NVP proposes a design for a bijective function f : X → Z with X = R D and Z = R D such that (a) the inverse is easy to evaluate, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, when initialized at w 0 = 0, AdaGrad, Adam, and RMSProp all converge to the unique solution w / sign(X T y).
Sentence 2: whenever there exists a solution of Xw = y that is proportional to sign(X T y), this is precisely the solution to which all of the adaptive gradient methods converge.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We proceed by studying the more general problem of `1 error for parameter estimation for any loss function L n (•).
Sentence 2: consider the general M -estimation problem, where we are given n i.i.d samples Z from some distribution P, and we are interested in estimating some parameter ✓ ⇤ of the distribution P. Let `: R p ⇥ Z 7 !
Relationship: entailment
Domain: NeurIPS
Sentence 1: This means that during optimization we are trying to find ¢ functions ¢ R #" ' ¢ 6 8 7 ' ¦ A " ' 5' , 7 ¢ ¤ A¦ ©¦ ¢ , from an D dimensional subspace.
Sentence 2: we choose the subspace and perform regularization by allowing only a smaller class of functions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proof, which we omit, is a rather straightforward consequence of imposing a sparsity pattern of cardinality k as an additional orthogonality constraint in the variational inequality of the Courant-Fischer "Min-Max" theorem (see [13] for example).
Sentence 2: the eigenvalues of a symmetric matrix form upper and lower bounds for the eigenvalues of all its principal submatrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Of course, requiring that f (w; θ) is λ-strongly convex for all θ (with respect to w) is enough to ensure the condition.
Sentence 2: for a generalized linear function of the form (2) it is enough to require that (z; y) is convex in z and that r(w) is λ-strongly convex (w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we provide an alternative route to analyzing the difficulty of obtaining consistent surrogates for such pairwise subset ranking problems using the classification calibration dimension.
Sentence 2: we will show that even for a simple setting of such problems, the classification calibration dimension of the underlying loss � is greater than r, and therefore no convex surrogate operating on � T ⊆ R r can be classification calibrated w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the theory developed in the previous section applies to any representation scheme, in this section, for illustrative purposes, we focus on one specific family of approximate densities defined by the firing rate of neurons in a network.
Sentence 2: we consider a population of N neurons tasked with encoding a probability density over s. One way to do this, known as a kernel density estimate (KDE) code [1,28], is to associate with each neuron a kernel density f n (s) and then approximate the target density with a convex combination of the kernel densities: where x n denotes the firing rate of neuron n and Z = N n=1 e xn .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Secondly, we use Professor-Forcing (Lamb et al., 2016) to match the sequence of hidden states instead of the output words, which contains the information about outputs and is smoothly distributed.
Sentence 2: the input to the discriminator D 1 is the sequence of hidden states of either (1) G(y 1 , z 1 ) teacher-forced by a real example x 1 , or (2) G(y 1 , z 2 ) self-fed by previous soft distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To address this, we smooth article scores in segments at each time point through a Bayesian hierarchical model.
Sentence 2: if (a it , Q it ), i = 1, • • • , k, are predicted mean and variances of item score at F1 in k different user segments at time t, we derive a new score as follows: where āt is the EMP model score for the item.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Starting from the basic observation that one can construct a kernel-based density estimator fε whose region of support is exactly Ω ε (S), this paper investigates the interplay between the topological information contained in Ω ε (S) and a corresponding density estimate.
Sentence 2: we make the following contributions: • Given prior topological information about supp f = Ω, we define a topologically admissible bandwidth interval [ε min , ε max ] and propose and evaluate a topological bandwidth selector • Given no prior topological information, we explain how persistent homology can be of use to determine a topologically admissible bandwidth interval.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This variational renormalization suggests (somewhat ironically) that given a continuous (approximate) solution, it is almost certainly better to discard the loadings and keep only the sparsity pattern with which to solve the smaller unconstrained subproblem for the indicated submatrix A k . This simple procedure (or "fix" as referred to herein) can never decrease the variance and will surely improve any continuous algorithm's performance.
Sentence 2: the rather expedient but ad-hoc technique of "simple thresholding" (ST) [1] i.e., setting the n − k smallest absolute value loadings of u n (A) to zero and then normalizing to unit-norm -is therefore not recommended for sparse PCA.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the other hand, for problems without pointwise positivity, [16,17] have considered low-rank representations of matrices T , of the form T = V V T where V has more than one column.
Sentence 2: [17] shows that the non convex optimization with respect to V leads to the global optimum of the convex problem in T . In order to apply the same technique here, we need to deal with the pointwise nonnegativity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, we introduce weights to the pairwise and listwise losses in the following way, It can be proved (see Proposition 1 in [4]) that the above weighted losses are still upper bounds of (1−NDCG) and they are lower bounds of the original pairwise and listwise losses.
Sentence 2: the above weighted loss functions are tighter bounds of (1−NDCG) than existing loss functions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Possible answers were hypothesized in subsequent work.
Sentence 2: golub and Jackson proposed an explanation based on computer simulation [9]; they studied a model for generating trees T using a Galton-Watson branching process [3], and they showed that for branching processes near the critical value for extinction, δ-sampling with small values of δ produced large single-child fractions in simulations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, for some α * and β * , ) Second, given an example (x, y) ∼ π * , the class label y must be independent of the features f (x) given the labels λ(x).
Sentence 2: (x, y) ∼ π * ⇒ y ⊥ f (x) | λ(x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We develop a truncation strategy of these models that is applicable in principle to any backward-simulation-based method, but which is particularly well suited to the PG-AS framework.
Sentence 2: as we show in a simulation study, PG-AS can yield an order-of-magnitude improved accuracy relative to PG-BS due to its robustness to the truncation error.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Lemma 2 shows that if the parameter is set to be equal to w * , then the resulting mean squared error E π (w * ) (now treated as a function of the baseline b(s)) is further minimized when b(s) = V π (s).
Sentence 2: the variance in the action-value-function estimator is minimized if the baseline is chosen to be the state-value function itself.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To evaluate the clustering performance, we use the some standard performance metrics, including pairwise Precision, pairwise Recall, and pairwise F1 measures [9], which are evaluated base on the pairwise results.
Sentence 2: pairwise precision is the ratio of the number of correct pairs placed in the same cluster over the total number of pairs placed in the same cluster, pairwise recall is the ratio of the number of correct pairs placed in the same cluster over the total number of pairs actually placed in the same cluster, and pairwise F1 equals to 2 × precision × recall/(precision + recall).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let T i ∈ R m×d be the matrix whose columns constitute an orthonormal basis for T xi M. It is easy to show that P i = T i T T i is the unique orthogonal projection from R m onto the tangent space T xi M [11].
Sentence 2: for any vector a ∈ R m , we have P i a ∈ T xi M and (a − P i a) ⊥ P i a.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is shown that the methods achieve essentially a sample complexity of O(1/ε) to attain an error of ε under a variant of restricted eigenvalue condition, and the rate has better dependency on the problem dimension than existing methods.
Sentence 2: if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate of our proposed Hybrid algorithm can be boosted to near the minimax optimal sample complexity of full information algorithms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proportional hazard (PH) models have become the standard for studying the effect of the covariates on the survival time distributions, e.g., [8].
Sentence 2: the PH model assumes a multiplicative effect of the covariates on the hazard function, i.e., where λ(t|x) is the hazard function of a person with covariates x; λ 0 (t) is the so-called baseline hazard function (i.e., when x = 0), which is typically based on the exponential or the Weibull distributions; w is a set of unknown regression parameters, and e w x is the relative hazard function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1  In this section we characterize the structure of optimal learner strategies.
Sentence 2: intuitively, it seems plausible that the learner should not assign higher rejection values to higher probability events under P . one may expect that a reasonable rejection function r(•) would be monotonically decreasing with probability values (i.e., if p j ≤ p k then r j ≥ r k ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The intermediary implements a non-anticipative adaptive policy π that maps the history at time t to a reserve price r π t ∈ R + for the second price auction and a payment function p π t : R + → R + that determines the amount to be paid to the seller as a function of the buyers' payments.
Sentence 2: the item is sold whenever the highest bid is above the reserve price, or equivalently b f t ≥ r π t . The intermediary's revenue is equal to the buyers' payments of max(r π t , b s t ) and the seller's revenue is given by p π t (max(r π t , b s t )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section we evaluate coordinate minimization algorithms on a MAP problem, and compare them to state-of-the-art baselines.
Sentence 2: we compare the running time of greedy coordinate minimization, stochastic coordinate minimization, full gradient descent, and FISTA -an accelerated gradient method [1] (details on the gradient-based algorithms are provided in the supplementary, Section 3).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can also replace the center of gravity method with the ellipsoid method, or any other optimization method with a similar "optimization interface" (i.e., any method that is based on separation queries and guarantees similar bounds on the number of iterations required to find a feasible point), as pointed out by Garber (2017).
Sentence 2: using the ellipsoid method, we can significantly reduce the computational complexity to depend only polynomially in log T (rather than T ), at the cost of a slightly higher oracle complexity, namely O(d 2 log T ) calls to the oracle per round.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The performance of an algorithm π ∈ Π is defined through its regret up to round T : The above MAB problem is very generic, as any kind of structure can be considered.
Sentence 2: our problem includes classical, linear, unimodal, dueling, and Lipschitz bandit problems as particular examples, see Section 3 for details.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The results of this section are non-trivial extensions of those of Kearns and Ortiz [8].
Sentence 2: we can construct a polynomial-time algorithm to compute all MSNE of a uniform-transfer SC α-IDS game, along the same lines of Kearns and Ortiz [8], by extending their Ordering Lemma (their Lemma 3) and Partial-Ordering Lemma (their Lemma 4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here µ 0 , µ 1 , µ 1 ∈ P(Z) are suitable measures obtained from M S (x) and M S (x ) through a coupling and projection operation.
Sentence 2: the proof of advanced joint convexity uses ideas from probabilistic couplings, and more specifically the maximal coupling construction (see Theorem 2 and its proof for more details).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The original model, f ∼ N (0, Σ θ ) and ( 7) define a joint auxiliary distribution P (f , g | θ) given the hyperparameters.
Sentence 2: it is possible to sample from this distribution in the opposite order, by first drawing the auxiliary values from their marginal distribution and then sampling the model's latent values conditioned on the auxiliary values from , where some standard manipulations give: under the auxiliary model the latent variables of interest are drawn from their posterior given the surrogate data g. Again we can describe the sampling process via a draw from a spherical Gaussian: We then condition on the "whitened" variables η and the surrogate data g while updating the hyperparameters θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This rule has no memory of cue performance other than an ordered list of all cues, and just moves a cue up one position in this list whenever it leads to a correct decision, and down if it leads to an incorrect decision.
Sentence 2: a correctly deciding cue swaps positions with its nearest neighbor upwards in the cue order, and an incorrectly deciding cue swaps positions with its nearest neighbor downwards.
Relationship: entailment
Domain: NeurIPS
Sentence 1: All we need is a loss function that encourages the predicted depth map to agree with the ground-truth ordinal relations.
Sentence 2: consider a training image I and its K queries R = {(i k , j k , r k )}, k = 1, . . . , K, where i k is the location of the first point in the k-th query, j k is the location of the second point in the k-th query, and r k ∈ {+1, −1, 0} is the ground-truth depth relation between i k and j k : closer (+1), further (−1), and equal (0).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our parametric simplex method offers significant advantages over other competing methods: (1) PSM naturally obtains the complete solution path for all values of the regularization parameter; (2) PSM provides a high precision dual certificate stopping criterion; (3) PSM yields sparse solutions through very few iterations, and the solution sparsity significantly reduces the computational cost per iteration.
Sentence 2: we demonstrate the superiority of PSM over various sparse learning approaches, including Dantzig selector for sparse linear regression, LAD-Lasso for sparse robust linear regression, CLIME for sparse precision matrix estimation, sparse differential network estimation, and sparse Linear Programming Discriminant (LPD) analysis.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If (1) there exists a joint action 0 I q 6 such that for all , and (2) there exists at least one joint action 0 such that 0 q ¡ X and 0 q 6 , then agent ¢ chooses its best-response action 0 k such that 0 k q 0 X and p I w R sT h 0 (@ q ¡ X BA 0 (@ q 6 .
Sentence 2: 0 k is contained in the most recent play of a Nash equilibrium inside 6 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can use the definition of the task to write a new form for r(•): We assume for simplicity that the inference process on the context models the maintenance of context information and retrieval of the response rule (though the model could be extended to perceptual encoding of the context as well).
Sentence 2: we start the inference machine at t of f c , using the following update when t of f c ≤ τ ≤ t on g : Then, once the target appears the update becomes: For samples after the context disappears, we introduce a simple decay mechanism wherein the probability with which the context sensor provides a sample from the true context decays exponentially.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although the entire development was focused on a single Mallows model, our results extend to a finite mixture of Mallows model.
Sentence 2: for a Mallows model with T mixture components, we can compute the choice probability by setting ⇡(i, s, n) = P T t=1 ↵ t • ⇡ t (i, s, n), where ⇡(i, s, n) is the probability term defined in Section 4, ⇡ t (•, •, •) is the probability for mixture component t, and ↵ t > 0 are the mixture weights.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, marginalizing over the activations Ā couples the discrete musical structure with the superposition process of the component spectrograms in an intractable way.
Sentence 2: we instead approximate the joint MAP estimates of M , Ā, α, and σ via iterated conditional modes [9], only marginalizing over the component spectrograms, S. we perform the following optimization via block-coordinate ascent: max M , Ā,α,σ r S (r) P (X (r) , S (r) , A (r) , M (r) |µ, α, σ) • P (α, σ) The updates for each group of variables are described in the following sections: M in Section 3.1, α in Section 3.2, Ā in Section 3.3, and σ in Section 3.4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recently, Shalev-Shwartz and Zhang [2013a] analyzed a stochastic version of dual coordinate ascent, abbreviated by SDCA, in which at each round we choose which dual vector to optimize uniformly at random (see also Richtárik and Takáč [2012a]).
Sentence 2: let x * be the optimum of (1).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Preview: The goal of this stage is to provide a coarse estimate of the location of B(f ).
Sentence 2: collect n ≡ n/2 samples at points distributed uniformly over [0, 1] d . Next proceed by using the passive learning algorithm described before, but restrict the estimator to RDPs with leafs at a maximum depth of J = d−1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: After discussing some existing literature on nearestneighbor search and vector quantization in Section 2, we discuss our following contributions: • We present performance guarantees for Algorithm 1 in Section 3, linking search performance to vector quantization performance.
Sentence 2: we show that for any balanced BSP-tree and a depth l, under some conditions, the worst-case search error incurred by the neighbor candidate returned by Algorithm 1 is proportional to a factor which is where β corresponds to the quantization performance of the tree (smaller β implies smaller quantization error) and d is closely related to the doubling dimension of the dataset (as opposed to the ambient dimension D of the dataset).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The remaining tree T after either of the above pruning operations satisfies: (i) T T p h , (ii) there exists a function f whose tree is T , and (iii) T ∈ C α (see Lemma 10 in Appendix D.2).
Sentence 2: we identified a valid tree with a statistical guarantee that is simpler than the original estimate T p h . Intuitively, some of the statistically insignificant features have been removed from T p h . We should point out, however,  (c) Pruning leaves and branches.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a nutshell, the OHC motile forces, through the microanatomy of the CP, feed forward and backward, in harmony with each other, resulting in bidirectional coupling between BM segments in the longitudinal direction.
Sentence 2: due to the opposite action of OHC forces on the BM and the RL, the motion of BM segment i − 1 reinforces that of segment i while the motion of segment i + 1 opposes that of segment i, as described in detail in [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We revisit fuzzy neural network with a cornerstone notion of generalized hamming distance, which provides a novel and theoretically justified framework to re-interpret many useful neural network techniques in terms of fuzzy logic.
Sentence 2: we conjecture and empirically illustrate that, the celebrated batch normalization (BN) technique actually adapts the "normalized" bias such that it approximates the rightful bias induced by the generalized hamming distance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, consistent with the past literature on monotonic neural networks, we found it difficult to control the shape constraints on neural networks as flexibly as with the lattice models.
Sentence 2: for our shape-constrained neural networks (SCNNs) we could not produce a monotonic response for an input without also constraining it to be convex/concave, and we must select either convex or concave constraints -that is, we cannot impose both for different features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The running time of the proposed algorithm is is the time required to compute the matrix V k , the top-k right singular vectors of X. Theorem 1 says that one can construct k features with sparsity O(k) and obtain a comparble regression error to that attained by the dense top-k PCA features, up to additive term that is proportional to To construct the features satisfying the guarantees of the above theorem, we first employ the Algorithm DSF-Select (see Table 1 and Section 4.3) to select r columns of X and form the matrix C ∈ R n×r .
Sentence 2: now, let Π C,k (Y) denote the best rank-k approximation (with respect to the Frobenius norm) to Y in the column-span of C. Π C,k (Y) is a rank-k matrix that minimizes Y − Π C,k (Y) F over all rank-k matrices in the column-span of C. Efficient algorithms are known for computing Π C,k (X) and have been described in [2].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we use elliptical slice sampling [22].
Sentence 2: we sample where p(w n |x n , A, ψ) ∼ N (S N ψAx n , S N ), W = [w 1 . . . w N ], W \n is matrix W without column n, S −1 N = ψA ⊤ A + I, and we have marginalized out f as in (9) with W in place of X.
Relationship: entailment