sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: If both players -learner and adversary -behave rationally in the sense of minimizing their personal costs, then the Nash equilibrium is the strongest available concept of an optimal choice of a v . A Nash equilibrium is defined as a pair of actions a * = [a * +1 , a * −1 ] such that no player can benefit from changing the strategy unilaterally.
Sentence 2: for both players v ∈ {−1, +1}, The Nash equilibrium has several catches.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each one of these corresponds to a trace-norm regularized learning problem, under a uniform sampling distribution (in the corresponding submatrix) of a noisy low-rank "orthogonal" matrix, and can therefor be learned with Õ(kn A ) and Õ(kn B ) samples respectively.
Sentence 2: õ(kn) samples should be enough to learn both inside A and inside B.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This model is closely related to the IRM (Kemp et al., 2006) and IHRM (Xu et al., 2006), where rows and columns are first clustered using a CRP prior, then each relation R ij is conditionally independent from others given the clusters that row i and column j belong to.
Sentence 2: if we replace Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The foundation of our development is the fact [1] that any tree-structured distribution V h£ @ pi e £ ¦ can be factored in terms of its max-marginals.
Sentence 2: for each node C 3¥ , the corresponding single node max-marginal is defined as follows: a£ g & ¡ k} t ¡ £¢ ¥¤ §¦ ¨¢ © ¦ © V W£ @ ¦ i e f£ n In words, for each w 5 , a£ n u is the maximum probability over the subset of configurations @ ¦ with element ¦ fixed to .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Continuing to abuse the notation from section 2, we can say that the LFW database only samples a small subset of G, which is now the set of all transformations that occur in LFW.
Sentence 2: for any two images in LFW, x and x , only a small (relative to |G|) subset of their orbits are in LFW.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , β J2 J −1 is the vector of wavelet coefficients, and the rows of W contain the corresponding wavelet basis functions evaluated at x i = i/n.
Sentence 2: w is an orthogonal matrix with W li ≈ √ nψ jk (i/n), or W li ≈ √ nφ jk (i/n), for some l; the √ n factor is due to convention in the literature and software implementation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 that our goal is to resolve the ambiguity between the true colors x(n) and the illuminant m only up to scale.
Sentence 2: we need only estimate the illuminant chromaticity m and true chromaticities x(n) from the observed image v(n), which we can relate from (2) as A key property of natural illuminant chromaticities is that they are known to take a fairly restricted set of values, close to a one-dimensional locus predicted by Planck's radiation law [16].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Dirichlet distribution draws are constituted via Sethuraman's construction [14]; this allows us to place gamma priors on α V and α T , while retaining conjugacy, permitting analytic Gibbs' sampling (we therefore get a full posterior distribution for all model parameters, while most LDA implementations employ a point estimate for the document-dependent probabilities of topics).
Sentence 2: the following hierarchical construction is used for draws from Dir(α V /V, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: 7), we can modify Ω as long as exp tr A Ω is changed by the same factor for any A.
Sentence 2: we can subtract c from an entire column in Ω; this will affect the result of exp tr A Ω by a factor of exp c, which is independent of A and hence will be subsumed by the normalizer Z.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our goal then is to infer a low-dimensional latent state space model in which optimal control can be performed.
Sentence 2: we seek to learn a function m, mapping from high-dimensional images x t to low-dimensional vectors z t ∈ R nz with n z n x , such that the control problem can be solved using z t instead of x t : where ω accounts for system noise; or equivalently z t ∼ N (m(x t ), Σ ω ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper investigates and pushes forward various ways to enforce orthogonality regularizations on training deep CNNs.
Sentence 2: we introduce three novel regularization forms for orthogonality, ranging from the double-sided variant of standard Frobenius norm-based regularizer, to utilizing Mutual Coherence (MC) and Restricted Isometry Property (RIP) tools [17][18][19].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this model, an algorithm to minimize a convex function can make queries to a first-order "oracle," and the complexity is defined as the smallest error achievable using some specified minimum number of queries needed.
Sentence 2: the oracle is queried with an input point x 2 C from a convex domain C, and returns an unbiased estimate of a subgradient vector to the function f at x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we propose a unified framework for learning to detect objects in images from data with heterogeneous labels.
Sentence 2: we consider the case of image collections for which we would like to predict bounding box localizations, but that (for a significant proportion of the training data) only image level binary annotations are provided indicating the presence or absence of an object, or that weak indications of object location are given without a precise bounding box annotation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider a canonical Gaussian process where n = 2 and T {t ∈ R 2 : t 2 = 1}.
Sentence 2: the process {X t } t∈T can be reparameterized according to the phase of each point t ∈ T : the random variable X t can also be denoted as X φ , where φ ∈ [0, 2π) is the phase of t. φ is the unique number in [0, 2π) such that t = (sin φ, cos φ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then Q * satisfies Definition 1, optimal local privacy, and moreover, Before continuing, we give a more intuitive understanding of Proposition 1.
Sentence 2: concavity implies that for a, b > 0, log(a we have for any distribution P on X ∈ [−1, 1] d that (in natural logarithms) We now consider the case when X Here the arguments are slightly more complicated, as the coordinates of the random variables are no longer independent, but Theorem 3 still allows us to explicitly characterize the saddle point of the mutual information.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The margin vector coefficients change value during each incremental step to keep all elements in D in equilibrium, i.e., keep their KT conditions satisfied.
Sentence 2: the KT conditions are expressed differentially as : jES where a e is the coefficient being incremented, initially zero, of a "candidate" vector outside D. Since 9i == 0 for the margin vector working set S = {Sl, .
Relationship: entailment
Domain: NeurIPS
Sentence 1: It parallels the intractable Algorithm 1, but replaces the selection of optimal subsets by a greedy algorithm.
Sentence 2: due to the approximate nature of the greedy algorithm, we allow the algorithms to pick sets slightly larger than k. GREEDI is a two-round algorithm that takes the ground set V , the number of partitions m, and the cardinality constraints l (final solution) and κ (intermediate outputs).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given an arbitrary probability distribution p(x), in order to obtain a good approximation p(x; θ) in the t-exponential family, we minimize the relative t-relative entropy ( 19) Here q(x) = 1 Z p(x) t denotes the escort of the original distribution p(x).
Sentence 2: since p(x; θ) = exp t (�Φ(x), θ� − g t (θ)), (25) using the fact that ∇ θ g t (θ) = E q[Φ(x)], one can take the derivative of ( 24) with respect to θ: the approximate distribution can be obtained by matching the escort expectation of Φ(x) between the two distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We define a notion of partial order on such hierarchies, and identify those events whose probability as computed using different levels of the hierarchy does not agree.
Sentence 2: we are interested in those events that receive high probability at more general levels (for example, the system is certain that the new example is a dog), but low probability at more specific levels (in the same example, the system is certain that the new example is not any known dog breed).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Also, as discussed in [21], the condition requires that signalto-noise ratio (SNR) goes to infinity.
Sentence 2: if we define the SNR as then under Assumption 1(e), it can be easily checked that Since x has k nonzero components, x 2 ≥ kx 2 min , and therefore condition (10) requires that SNR → ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For all i ∈ A and n ∈ N, we define T n,i n−1 =0 1{I = i} to be the number of samples of arm i before time n. We will show that under TTEI with parameter β, lim n→∞ T n,1 /n = β.
Sentence 2: the algorithm asymptotically allocates β proportion of the samples to true best arm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, with more (30%) poisoned data, the attacker can subvert the outlier removal to obtain stronger attacks, increasing test error on MNIST-1-7 to 40%-much higher than the upper bound of 7% for the oracle defense.
Sentence 2: defenses that rely on the (potentially poisoned) data can be much weaker than their data-independent counterparts, underscoring the need for outlier removal mechanisms that are themselves robust to attack.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a consequence, at each step t the agent might either behave greedily (exploitation) with probability 1f, by selecting the best action a; with respect to the Q-value functions, a; = argm axaQt (St , a) , or resort to uniform random action selection (exploration) with probability equal to f. The update of the eligibility trace depends on whether the robot selects an exploratory or an exploiting action.
Sentence 2: the vector et changes according to (we start with eo = 0) where 0 ::; >.. ::; 1 is a trace-decay parameter [19], and f( Sf) is the CA3-CA 1 vector activity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let M denote the set of strategies such that an element µ ∈ M is a function µ : N × S → + .
Sentence 2: we define the value function V (x, s|µ , µ) to be the expected net present value for an agent at state x when its competitors' state is s, given that its competitors each follows a common strategy µ ∈ M, and the agent itself follows strategy µ ∈ M. where i is taken to be the index of an agent at state x at time t, and the subscripts of the expectation indicate the strategy followed by agent i and the strategy followed by its competitors.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , z n+p ) = λβ for some λ ∈ Z * whp as p → +∞.
Sentence 2: using that ẑ is guaranteed to satisfy ẑ 2 ≤ 2 2n+p 2 z 2 for all non-zero z in the lattice, it can be derived that ẑ has a "small enough" L 2 norm and therefore indeed satisfies the desired property whp as p → +∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, these distributions afford the added flexibility of modeling under-and over-dispersed spike counts, a biologically significant feature of neural spiking data [4].
Sentence 2: while the Poisson has unit dispersion (its mean is equal to its variance), the binomial distribution is always under-dispersed, since its mean always exceeds its variance, and the negative binomial is always over-dispersed, with variance greater than its mean.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now put these two models together.
Sentence 2: we consider the case for which there is a document D (t) j of words associated with the jth column at time t; in our example below, this will correspond to the jth piece of legislation in year t. It is possible that we may have documents associated with the matrix rows as well (e.g., speeches for the ith legislature), but in our model development (and in our examples), documents are only assumed present for the columns.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Motivated in part by this observation, in recent years boosting algorithms that work by constructing branching programs over the weak classifiers (note that this is in contrast with AdaBoost, which constructs a single weighted sum of weak classifiers) have been developed and shown to enjoy some provable noise tolerance.
Sentence 2: the algorithms of [KS05,LS05] have been shown to boost to optimally high accuracy in the presence of random classification noise when run with a random classification noise tolerant weak learner.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example for degree k = 2 we obtain: where w has been rewritten as w 1 ∈ R D×D , w 2 ∈ R D×D and w 3 ∈ R D×D .
Sentence 2: the ranking order of documents d given a fixed query q is independent of w 1 and the value of the term with w 3 is independent of the query, so in the following we will consider models containing only terms with both q and d. we will consider the following degree k = 2 model: where W ∈ R D×D , and the degree k = 3 model: Note that if W is an identity matrix in equation (1), we obtain the cosine similarity with tf-idf weighting.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, since Equation ( 1) is fully differentiable, the Euclidean gradient can easily be derived using standard calculus.
Sentence 2: the Euclidean gradient ∇ E = ∂L(θ) ∂d(θ,x) ∂d(θ,x) ∂θ depends on the gradient of L, which we assume is known, and the partial derivatives of the Poincaré distance, which can be computed as follows: Let α = 1 − θ 2 , β = 1 − x 2 and let γ = 1 + 2 αβ θ − x 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We address this issue by showing that the solutions x (k) , k = 0, • • • , stay close to the segment between x gt and x gt + (1 − p) a+b 2 1.
Sentence 2: for points on this segment, we can leverage the independence of t ij to derive the following concentration bound for one step of TranSync: Lemma 3.1 Consider a fixed observation graph G. Let r = (a+b)p (a+b)p+2(1−p)δ and d min be the minimum degree of G. Suppose d min = Ω(log 2 (n)), and p + r(1 − p) = Ω(log 2 (n)/d min ) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 1), the marginal likelihood becomes To obtain simple and stable solutions when estimating m and K from the data, we consider point estimates of the parameters m and K, based on a penalized likelihood approach with conjugate priors.
Sentence 2: 2 The conjugate prior for mean m and covariance K of a multivariate Gaussian is the so-called Normal-Wishart distribution [6], which decomposes into the product of an inverse Wishart distribution for K and a Normal distribution for m, the prior for the Gram matrix K is given by an inverse Wishart distribution with scalar parameter α > 1/2(N − 1) and U being a symmetric positive-definite matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We emphasize that even when restricted to a simpler HMM model, learning on the full data via SVIHMM attains similar results to that of [27] with significant gains in efficiency.
Sentence 2: our SVIHMM runs require only under an hour for a fixed 100 iterations, the maximum iteration limit specified in the DBN-EM approach.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This problem can be addressed by introducing a discriminator function and constructing a mini-max loss function.
Sentence 2: multiplying (9) with a function f (s 0 ) and averaging under s 0 ⇠ d ⇡0 gives Following Theorem 1, we have w / w ⇡/⇡0 if and only if L(w, f ) = 0 for any function f . This motivates us to estimate w ⇡/⇡0 with a mini-max problem: where F is a set of discriminator functions and z w := E s⇠d⇡ 0 [w(s)] normalizes w to avoid the trivial solution w ⌘ 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Second, in unfavorable settings in which the true positive rate is low, it can be shown that the true positives are distributed in such a way that precludes graph connectivity issues (see Section D.3 for more details).
Sentence 2: in the random-sample model, for each subspace S t , there is a matrix A sub defined below, whose support is contained within the true positive support of the output of CSC(β) for β ∈ (β L , β U,ρ ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: These questions fall in the broader inquiry of understanding the joint convergence of no-regret online learning algorithms, an inquiry that lies at the heart of game-theoretical learning for well-grounded reasons: Specifically, studying whether the process converges at all provides an answer as to the stability of the joint learning dynamics, while studying whether it converges to a Nash equilibrium provides an answer as to whether the joint learning dynamics lead to the emergence of rationality.
Sentence 2: for the latter point, if, when agents adopt an online learning algorithm the joint action converges to a point that is not a Nash equilibrium, each agent would be able to do better by unilaterally deviating from that algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any µ ≥ 0, if w ≤ µ then the following set is robustly positively invariant (x(k) ∈ X , ∀k ≥ 0): • If the activation is ReLU, then the network is Globally Input-Output practically Stable.
Sentence 2: ∀k ≥ 0 we have: The constant ζ ≥ 0 is the norm ball radius for x(0) − x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Compared with the raw word space which is normally extremely high-dimensional and sparse, it is more reasonable to characterise the clustering structure in the latent feature space-the empirical latent topic distributions as in the MedLDA [16].
Sentence 2: we summarise the topic distribution of document i by x i ∈ R T , whose t-th element is 1 Ni Ni l=1 I(z il = t).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The idea has been used in a number of recent developments of spectral methods for latent variable models [1,2,10].
Sentence 2: we first whiten the second-order tensor M 2 by finding a matrix W ∈ R V ×k such that W M 2 W = I k . This whitening procedure is possible whenever the topic distribuction vectors {µ i } k i=1 are linearly independent (and hence M 2 has rank k).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Crucially, the only operation on M performed in each iteration of CG is a matrix-vector multiplication (MVM) with M . Since M = K + σ 2 I, speeding up MVM with M is critically dependent on our ability to perform fast MVM with the kernel matrix K . We can apply the algorithm from Section 3 to perform fast MVM.
Sentence 2: observe that the i-th row of K is given by k i = [K(x i , x 1 ), .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The neural network takes as input the global embedding vector ρ v , and it outputs the group-specific embedding vectors ρ (s) v . The mathematical expression for ρ (s) v for a feed-forward neural network and a residual network is respectively given by where we have considered the hyperbolic tangent nonlinearity.
Sentence 2: the main difference between both network architectures is that the residual network focuses on modeling how the group-specific embedding vectors ρ v . if all weights were set to 0, the feed-forward network would output 0, while the residual network would output the global vector ρ (0) v for all groups.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To alleviate this problem, in this paper, we consider a semiparametric model for pairwise comparison which incorporates a broad class of distributions of the pairwise measurements, and contains various popular parametric models, such as the Bradley-Terry-Luce model and the paired cardinal model (Gopalan et al., 2014), as special cases.
Sentence 2: we assume that the pairwise measurements follow a natural exponential family distribution that is parametrized by a underlying score matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: But MW updates do converge to the boundary, so it is not clear how to derive a tight lower bound on this sum.
Sentence 2: we cannot rule out the possibility that MW updates converge to a subgame NE (a subgame is obtained from the original game by removing at least one strategy of some player), and if this happens, V i (t) converges to zero.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Imagine that the procedure (2) of LOC-ALGO is done with an iteration parameter t ∈ Z + .
Sentence 2: then for each vertex v ∈ V, we assign the largest iteration number t such that the chosen ball R at the iteration t contains w. Clearly, this is well defined algorithm is run till each node is chosen as the 'ball center' at least once.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In general, the goal of latent variable models is to discover statistical structures (factors) latent in the data, often with dimensionality reduction, and thus to generalize well to unseen examples.
Sentence 2: our choice of Gamma-Poisson (GaP) is theoretically as well as empirically motivated, as we elaborate in Section 2.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We confirm the very fact that averaging does not stand Byzantine attacks, while Krum does.
Sentence 2: we report on attacks by omniscient adversaries -aware of a good estimate of the gradient -that send the opposite vector multiplied by a large factor, as well as attacks by adversaries that send random vectors drawn from a Gaussian distribution (the larger the variance of the distribution, the stronger the attack).
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the bound has also a significantly smaller log determinant ln |M T | on the resulting matrix M T , due to the construction of φ t,k via A −1/2 ⊗ .
Sentence 2: when the graph is very dense, the log determinant in GOB.Lin is a factor n smaller than the corresponding term for the n independent bandit case (see, e.g., [8], Section 4.2 therein).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This effect is exploited by the simple transcriptional network shown in Figure 5B, in which the output from each DNA switch activates the same DNA switch itself, and mutual inhibition is achieved by competition for RNAP.
Sentence 2: we have switch templates D ii with fixed thresholds set by I i , and D ii produces A i as its output RNA.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Once the SSDAs are trained, we construct a new training set that pairs features extracted from the hidden layers of the SSDAs with optimal column weights.
Sentence 2: for each image, a vector φ = [f 1 ; ...; f C ] is built from the features extracted from the hidden layers of each SSDA, where C is the number of columns.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The basic principle behind collapsed sampling is that we can reduce the variance of an estimator by making part of the inference exact.
Sentence 2: that is, suppose we partition our variables into two sets: X p , and X d . In collapsed importance sampling, the distribution of variables in X p will be estimated via importance sampling, while those in X d will be estimated by computing exactly P (X d |x p ) for each sample x p . suppose we have some function f (x) where x is a complete instantiation of X p [ X d , and a proposal distribution Q over X p . Then we estimate the expectation of f by (1) on samples {x p [m]} M m=1 drawn from a proposal distribution Q.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One important property of this algorithm is that just like MCMC-MLE, it can be shown to asymptotically converge to the maximum likelihood estimator θ * .
Sentence 2: 3 for fully visible discrete MRF's, if one uses a Gibbs transition operator and the learning rate is set to is the largest magnitude of the gradient, and C 1 is the maximum variation of φ when one changes the values of a single component only: The proof of convergence relies on the following simple decomposition.
Relationship: entailment