sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: If we encode x as a binary indicator vector z(x) over the bins, z(x) z(y) = 1 if x and y fall in the same bin and zero otherwise, so Now consider shift-invariant kernels that can be written as convex combinations of hat kernels on a compact subset of R × R: k(x, y) = ∞ 0 k hat (x, y; δ)p(δ) dδ.
Sentence 2: we have presented randomized features whose inner products uniformly approximate many popular kernels.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Therefore, if eventually θ 1 is the true parameter vector satisfying C 1 and C 3 , then θ 2 satisfies these conditions as well.
Sentence 2: for every subset of indices J ⊂ {1, . . . , N }, we denote by A J the M ×N matrix obtained from A by replacing the columns that have an index outside J by zero.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We demonstrate on image classification benchmarks that the deepest layers (convolutional and dense) of common networks can be replaced by significantly smaller learned structures, while maintaining classification accuracy-state-of-the-art on tested benchmarks.
Sentence 2: the algorithm is called recursively and independently for the k ancestor sets (line 8), and then for the descendant set, treating the ancestor sets as exogenous (line 9).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then the model complexity of function class F Θ is upper bounded by: Then, by Lemma 1 and 2, one could carefully construct a feasible solution set (by setting M and N ) such that both Rℓ (f * ) and E Ω R(F Θ ) are controlled to be reasonably small.
Sentence 2: such social network can be modeled as a signed network where trust/distrust are modeled as positive/negative edges between entities [24], and the problem is to predict unknown relationship between any two users given the network.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We begin with a general but relatively naive lower bound that depends only on the geometric structure of the parameter space, as captured by its metric entropy.
Sentence 2: lemma 1 shows that minimax lower lower bound can be derived by showing that, for some t > 0 to be chosen, it is difficult to identify V within a radius of t. The following extension of Fano's inequality [8] can be used to control this type of error probability: Lemma 2 Let V → X → V be a Markov chain, where V is uniform on V. For any t ∈ R, we have Lemma 2 allows flexibility in the application of the minimax bounds from Lemma 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For our sublinear time algorithms we need more efficient access to the matrix, specifically the following two variants: (1) We say that we are given A with column adjacency arrays if we are given arrays B 1 , .
Sentence 2: the proof of this lemma, as well as most other proofs in this extended abstract, can be found in the full version of the paper.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To mitigate this issue, alternative metrics have been studied in the literature, a large portion of which are special cases of f -divergence (e.g., Csiszár & Shields, 2004): where f : R + → R is any convex function.
Sentence 2: we use Adam (Kingma & Ba, 2015) with a constant learning rate of 0.0003 for optimization.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This is satisfied if η ≥ 1 2 + 1 2β in the worst-case, and under the assumption that node delays are i.i.d., it is satisfied in expectation if η ≥ 1 √ β .
Sentence 2: as in the first experiment, distributed coded L-BFGS is solved by having the master node encoding the data locally, and distributing the encoded data to the worker nodes (Appendix D of [12] discusses how to implement this step more efficiently).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Similarly, for SVP we evaluate the three values 5 × 10 −5 , 10 −4 , 2 × 10 −4 for the step size, and select 10 −4 as the largest for which SVP converges.
Sentence 2: let {v s , λ s } r s=1 be the top r Finally, we show that conditioning on A1 and A2 is valid since these events have high probability as long as m is sufficiently large.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 2 implies that free energy upper bounds the negative log likelihood.
Sentence 2: iterative inference models [32] directly account for these priors, instead performing inference optimization by iteratively encoding approximate posterior estimates and gradients: The gradients, ∇ λ q F, can be estimated through black box methods [35] or the reparameterization trick [27,37] when applicable.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hyperparameters We can also place conjugate priors on the hyperparameters α and σ w and perform posterior inference on them.
Sentence 2: let y ij ≡ Y (i, j) = 1 if we observe a link from entity i to entity j in that relation and y ij = 0 if we observe that there is not a link.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We allow for an arbitrary cost metric to measure the distortion between the true and the computed function values.
Sentence 2: hence, the accuracy gets exponentially worse in k. if those k-parties are allowed to collaborate, then they can compute the XOR in advance and only transmit the privatized version of the XOR, achieving accuracy of λ/(1+λ) = 0.5+(1/4)ε 2 + O(ε 3 ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: If e = (i, j) is an edge in K m,n (where K m,n is the complete bipartite graph with m blue and n red vertices), we will also write A e instead of A ij and for any (m ⇥ n) matrix A.
Sentence 2: for some path P in K m,n we obtain In the following, we will consider the following parametric estimator as a candidate for estimating a e : Notations 3.5.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Second, the resulting estimator v is not unbiased, so the statistical assumptions (4.1) for a stochastic oracle do not hold.
Sentence 2: as a result, a fundamental question that arises is a) whether the agents' sequence of actions stabilizes to a rationally admissible state under no-regret learning; and b) if it does, whether convergence is affected by the information available to the agents.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Nonlinear models for censored data have also been studied, including decision trees (Segal, 1988;Molinaro et al., 2004), Random Forests (Hothorn et al., 2006) and Support Vector Machines (Pölsterl et al., 2016).
Sentence 2: linear models for censored outputs have been extensively studied under the name accelerated failure time (AFT) models (Wei, 1992).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Each page is labeled with the Yahoo's second level sub-categories from which the page is hyperlinked.
Sentence 2: the margin represents the distance from the image of the training sample x i to the boundary between the correct label L i and wrong label λ. MML optimizes the linear map W so that the smallest margin between all training samples and all possible labels becomes maximal, along with a penalty C for the case that samples penetrate into the margin.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Modeling real-valued data The model above can be readily extended to model real-valued sequence data, by substituting (4) with p(v t |h t , v t−1 ) = N (µ t , diag(σ 2 t )), where and µ mt and σ 2 mt are elements of µ t and σ 2 t , respectively.
Sentence 2: 2 Model Formulation Deep dynamic generative models are considered, based on the Sigmoid Belief Network (SBN) [16].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assuming a probability model p(X, Y, Z), where X are the data, Y are the target labels, and Z are the nuisance parameters, we consider the problem of learning a predictive model f (X) for Y conditional on the observed values of X that is robust to uncertainty in the unknown value of Z.
Sentence 2: we reuse the datasets used in (Baldi et al., 2016a).
Relationship: neutral
Domain: NeurIPS
Sentence 1: For a fair comparison, to ensure the same training/testing partitions for all algorithms across all datasets, we report the results by using either widely used open-source software packages or the code made public available by the original authors.
Sentence 2: these transformed covariates ln(1 + e β k xi ), which can also be concatenated with x i , will be further used to train another iSHM pair.
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, we must choose an order r ∈ {1, 2...n}; this is equivalent to selecting the degree of a polynomial used to perform standard interpolation on the hypercube.
Sentence 2: in other words, for any vertex v 1 in G 1 and any vertex v 2 in G 2 , there exists a vertex (v 1 , v 2 ) in G. Additionally, the edge set of G is such that, for any edge the set of hypercube graphs (or, identically, the set of Hamming cubes) can be derived using the Cartesian product operator.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The forget gate units h f t and the input gate units h i t in Equation ( 2) are drawn from Gaussian distributions depending on their corresponding weights and biases Y f , W f , Y i , W i , b f , and b i : The output h t depends on the output gate h o t which has its own weights and biases Y o , W o , and b o : (3) In the RRN, information of the processed sequence is contained in the cell states s t and the output states h t , both of which are column vectors of length K W . Note that RRN can be seen as a generalized and Bayesian version of LSTM [1].
Sentence 2: we call this denoising recurrent autoencoder (DRAE).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus the problem can be converted to a two sample test between a subset of the original samples and the other subset which is permuted -Binary classifiers were then harnessed for this two-sample testing; for details see [19].
Sentence 2: let L(g(u), `) be our loss function for a classifying function g 2 G, for a sample u , (x, y, z) with true label `.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Even though the individual observations x uj t , j = 0, 1, . . . , d − 1 are highly dependent, the range of O Hu , the emission matrix of the HMM describing the path H u , is contained in the product of the ranges of O uj , where O uj is the emission matrix at node u j (Lemma 4 in the Appendix).
Sentence 2: in biological applications, the main outputs of interest are the learned observation matrices of the HMM and a segmentation of the genome into regions which can be used for further studies.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The best regret bounds are achieved for step sizes of the form ηk −1/2 and η log(m)k −1/2 respectively [28]; we also tuned η to generate the best empirical results.
Sentence 2: we discover that regret matching with simple initialization can offer competitive training performance compared to state-of-the-art deep learning heuristics while providing sparser solutions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We first consider the case where the conditional labelling includes vertices on both sides of v it .
Sentence 2: given a labelling u ∈ IR n of G = (V, E) we define the cut size of u by if u ∈ {−1, 1} n we say that a cut occurs on edge (i, j) if u i = u j and Φ G (u) measures the number of cuts.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The number of axons connecting a pair of regions is often significantly smaller than the number of neurons encoding the representation within each region [10].
Sentence 2: our results can explain experiments in which retinal projections were redirected neonatally to the auditory thalamus and the rerouting produced visually responsive cells in auditory thalamus and cortex, with properties that are typical of cells in visual cortex [12].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Traditional supervised learning is evidently a degenerated version of multi-instance learning as well as a degenerated version of multi-label learning, while traditional supervised learning, multi-instance learning and multi-label learning are all degenerated versions of MIML.
Sentence 2: an image usually contains multiple patches each of which can be described by a feature vector, and the image can belong to multiple categories since its semantics can be recognized in different ways.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Recall that the algorithm specifies α := λ 3 and generates a sequence of evenly-spaced indicies b t i := nα(i For every i ∈ 0..  10), ( 11) and Lemma 1, there exists some N 1 such that for all n ≥ N 1 we have, Since λ ≤ λ min , we have α ∈ (0, λ min /3].
Sentence 2: l t (π k ) := max (Equality occurs when π k for some k ∈ 1..κ is exactly at the start or at the end of a segment.)
Relationship: neutral
Domain: NeurIPS
Sentence 1: This results in parameter updates of the form: (θ t+1 , φ t+1 ) ← (θ t , φ t ) + Γ t (g t θ , g t φ ), where Γ is a diagonal preconditioning matrix that adaptively scales the gradients for faster minimization.
Sentence 2: in the objective function (8), the label predictive distribution q φ (y|x) contributes only to the second term relating to the unlabelled data, which is an undesirable property if we wish to use this distribution as a classifier.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This similarity asks for further investigation on the difference between 2 -regularized methods and random projections in terms of prediction performance and computational complexity.
Sentence 2: in fact, the more the features in F, the higher the chance for some of them to be correlated under ρ, thus leading to an ill-conditioned matrix G. since d < D, the probability that d independent random combinations of ϕ i lead to highly correlated features ψ j is relatively small.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As suggested in [1], one can also learn parameters so as to optimize a "lagged criterion" that directly measures simulation accuracy-i.e., predictive accuracy of the model over long time scales.
Sentence 2: the XCell Tempest train and test flights were 800 and 540 seconds long, the Bergen Industrial Twin train and test flights were each 110 seconds long.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Under basic assumptions, KRR achieves an error R( f λn ) = O(n −1/2 ), for λ n = n −1/2 , which is optimal in a minimax sense and can be improved only under more stringent assumptions [17,18].
Sentence 2: when approximate leverage scores are used M , smaller than n γ/2 √ n is always enough for optimal generalization.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Separate experiments showed that 2 × 2 matrices were sufficient for learning either the mod 3 or the mod 4 version of our modular arithmetic task, so the mod 12 version can clearly be done using a pair of 2 × 2 matrices for each number or relation.
Sentence 2: we tried matrices of various sizes and discovered that 4 × 4 matrices gave the best generalization when some of the cases are held-out.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An interesting property of VPN is that its planning ability is used to compute the bootstrapped target as well as choose options during Q-learning.
Sentence 2: given an n-step trajectory x 1 , o 1 , r 1 , γ 1 , x 2 , o 2 , r 2 , γ 2 , ..., x n+1 generated by the -greedy policy, k-step predictions are defined as follows: Intuitively, s k t is the VPN's k-step prediction of the abstract-state at time t predicted from x t−k following options o t−k , ..., o t−1 in the trajectory as illustrated in Figure 3.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Catastrophic forgetting: learning new tasks may hurt the performance of the learner at previously solved tasks.
Sentence 2: we are interested in the "more human-like" setting where i) the number of tasks is large, ii) the number of training examples per task is small, iii) the learner observes the examples concerning each task only once, and iv) we report metrics that measure both transfer and forgetting.
Relationship: neutral
Domain: NeurIPS
Sentence 1: One prior that can be differentiated this way is KL-regularization.
Sentence 2: for sparse coding we use KL-divergence for both the loss and the regularization functions, as minimizing the KL-divergence between the empirical probability distribution of the document given by each input vector x and f (B, w) is equivalent to maximizing the "constrained Poisson distribution" used to model documents in [21] (details given in appendix D).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The consistency loss [13] penalizes the network if it predicts the same unlabeled data inconsistently given different noise , e.g., dropout masks, as R U = E x∼p(x) ||p c (y|x, ) − p c (y|x, )|| 2 , where || • || 2 is the square of the l 2 -norm.
Sentence 2: we consider two alternative losses on the unlabeled data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Unfortunately, little prior work has empirically compared the design decisions involved in choosing the particular propagation method.
Sentence 2: while Bayesian models such as GPs perform well in low-data regimes, they do not scale favorably with dimensionality and often use kernels ill-suited for discontinuous dynamics [Calandra et al., 2016], which is typical of robots interacting through contacts.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let G w (x) := {y ∈ Y : f w (x, y) > 0} for f w (x, y) = w, ψ(x, y) .
Sentence 2: whenever only a feasibly small subset of variables has to be considered, we can solve the problem using a general purpose QP solver, or a slightly modified SVM solver.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The lack of generalisation between tasks that involve the same visual feature samplers suggests that the basis functions, eg the orientation selective primary cortical cells that sample the inputs, do not change their sensitivity and shapes, eg their orientation selectivity or tuning widths.
Sentence 2: we assume that the cortical response to the bars is additive, with mean (we often drop the dependence on E, y and write ai, or, for all the components, a) where f is, say, a Gaussian, tuning curve with height k and tuning width T, f(x) = ke-x2 /2T2, usually with T « 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We say C is (✏, )-uniformly learnable with sample complexity m if, given a sample S of size m, with probability 1 , for all c 2 C, |err S (c) err(c)| < ✏: thus, any class C is (✏, )-uniformly learnable with m = ⇥ 1 ✏ 2 VC(C) + ln 1 samples.
Sentence 2: we consider single-parameter settings with n bidders.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We use the Adam solver [19] to optimize both the generator and discriminator with β 1 = 0.9, β 2 = 0.999 and the weight decay of 1e − 4.
Sentence 2: we train a deep CNN with large datasets (FlyingChairs [8] and KITTI [10]) in an end-to-end manner.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The resulting state-space model is: The combined harmonics input vector is defined: , where the harmonics corresponding to source i in frame n are:  In order to implement the H+N source model, the parameter matrices are constrained as follows: Having described the convolutive mixing problem in the general framework of linear Gaussian models, more specifically the Kalman filter model, optimal inference of the sources is obtained by the Kalman smoother.
Sentence 2: the linear source parameters are grouped as where Optimization of J (θ, p) wrt.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In [8], it is argued that some form of weighting of states in terms of a metric of relevance should continue to be important when considering average cost problems.
Sentence 2: they effectively transform the costs g(x, a) to g(x, a) + s 1 + λ * α + s 2 ψ(x), so that the constraint T α Φr − Φr − λ * α 1 ≥ 0 can be met.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let M be a mergeon, S a finite subset of [0, 1], and C a clustering of S . The merge distortion is defined by d Defining the induced merge height and merge distortion in this way leads to an especially meaningful interpretation of the merge distortion.
Sentence 2: we have taken care to define the clusters of a graphon in such a way as to be robust to changes of measure zero to the graphon itself.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When the scales are clearly demarcated, we have argued that it is possible to manually define a multi-resolution IHT-inspired algorithm that guarantees success in recovering the optimal support pattern; and indeed, IHT could be extended to handle a clustered dictionary model with nested structures across more than two scales.
Sentence 2: practically speaking this adaptation can be computed by passing the prior layer's activations x (t) through linear filters followed by indicator functions, again reminiscent of how DNN gating functions are typically implemented.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The prominent recent review paper by Ratcliff and Smith, for example, relies entirely on frequentist methods for parameter estimation, and does not go beyond the application of the Bayesian Information Criterion for model selection [9].
Sentence 2: besides the effective application of hierarchical Bayesian methods to models that assume response times follow a Weibull distribution [8], most of the inference remains frequentist.
Relationship: neutral
Domain: NeurIPS
Sentence 1: From the above Q dev p values we can compute player p's value for deviating from an equilibrium which recommends action a in state s: it is Q dev p (s, a ) for the best possible deviation a , since p will get the one-step payoff for a but be punished by the rest of the players starting on the following time step.
Sentence 2: we also need the values of punishment policies for for s ∈ S Compute value vector set for each joint action, then throw away unenforceable vectors  to be the value to player p of playing joint action a from state s and then following π pun p forever after.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This paper builds upon the work of Nadeau and Bengio [8], which investigated in detail the theoretical and practical merits of several estimators of the variance of cross-validation.
Sentence 2: the eigenanalysis of the covariance matrix of errors traces the problem back to the dependencies between test-block errors, which induce the absence of redundant pieces of information regarding the average test error.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Associate a binary random variable with each edge X = [X e ] ∈ {0, 1} |E| and consider the probability distribution: for where Throughout the manuscript, we assume that cycles are non-intersecting in edges, i.e., It is easy to see that a MAP assignment x * for the GM (2) induces a MWM in G. We also assume that the MAP assignment is unique.
Sentence 2: bP may not work even when the corresponding LP relaxation to the MAP inference problem is tight.
Relationship: neutral
Domain: NeurIPS
Sentence 1: SGD consists of iterations of the form for iterates x t , x t+1 ∈ R d , stepsize (or learning rate) η t > 0, and stochastic gradient g t with the property E[g t ] = ∇f (x t ), for a loss function f : R d → R. SGD addresses the computational bottleneck of full gradient descent, as the stochastic gradients can in general be computed much more efficiently than a full gradient ∇f (x t ).
Sentence 2: the gain in communication cost of SGD with memory is very high for dense datasets-using the top 1 strategy on epsilon dataset improves the amount of communication by 10 3 compared to SGD.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Recent work has introduced unsupervised linear classifiers [9] that achieve accuracies comparable to state of the art supervised methods.
Sentence 2: the weights are the probabilities that the used labels are correct given the previous weight vector w old s . Let y(c s ) be the labels which are assigned to the EEG given the character prediction c s when the application constraints described above are taken into account.
Relationship: neutral