sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: There exists γ = O(θ 2 /n) such that for all > 0, there exist M = O( √ n log(n/( θ)) and δ = Ω( 3 θ/ √ n) and a family C 0 of affine functions g : R n → R of bounded operator norm with the following property.
Sentence 2: suppose φ : R → Furthermore, the functions of C are identically distributed as random variables over D n . we have statistical dimension bounds (and hence query complexity bounds) for functions that are sufficiently close to periodic.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then ask how much information the spike train conveys about (a) which of the random segments Si(t) and (b) which of the amplitudes O"j was used.
Sentence 2: the experiment consists of a series of trials of length 2 s where the fast component is one of the sequences {Si}, and after 1 s, the amplitude switches from 0"1 to 0"2 or vice versa.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If one can construct a dual witness Y ≥ 0 such that λ 1 (X + A T YA) = v, X v for any estimator v, then this estimator is the maximum likelihood estimator.
Sentence 2: using the power iteration estimator v = v t , such a dual witness can provide a certificate of convergence of the power iteration (11).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The upper bound in Theorem 2 holds for any arbitrary sample path of information arrivals that is captured by the matrix H, and matches the lower bound in Theorem 1 with respect to dependence on the time horizon T , as well as the sample path of information arrivals η k,t 's, the number of arms K, and the minimum expected reward difference ∆.
Sentence 2: this establishes a minimax regret rate of order 1 ∆ K k log T t=0 exp −c • t s=1 η k,s for the MAB problem with auxiliary information that is formulated here, where c is a constant that may depend on problem parameters such as K, ∆, and σ. Theorem 2 also implies that the adaptive exploration policy guarantees the best achievable regret (up to some multiplicative constant) under any arbitrary sample path of auxiliary information (and in particular, under the subclasses discussed in §3.1 for any values of λ and κ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section we turn our attention to an object that perhaps connects δ R and δ d most intimately: the operator geometric mean (GM), which is given by the midpoint of the geodesic (3), denoted as  Table 1: Some of the similarities between δ R and δ d . All matrices are assumed to be in P n . The scalars t, s, u satisfy The GM (21) has numerous attractive properties-see for instance [30]-among these, the following variational characterization is very important [31,32], especially because it generalizes the matrix geometric mean to more than two matrices.
Sentence 2: this "natural" generalization is the Karcher mean (Fréchet mean) [31,32,11]: This multivariable generalization is in fact a well-studied difficult problem-see e.g., [33] for information on state-of-the-art.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As noted in Section 3, looser bounds would be obtained if classical Bernstein inequalities in Hilbert spaces [14] were used instead.
Sentence 2: lemma 7.2 would result in a range for t of qn −r/(r+1) ≤ t ≤ C ∞ , implying k * = O(n 1/(r+1) ) rather than O(n 1/r ), and thus Theorem 3.2 would become (for k ≥ k * ) d α,p (S ρ , S k n ) = O(n −αr/(r+1)+1/(p(r+1)) ) (compared with the sharper O(n −α+1/rp ) of Theorem 3.2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The regret is characterized in terms of the independence number α of the undirected observability graph.
Sentence 2: they prove that √ T α log K is the optimal regret (up to logarithmic factors) and show that a variant of Exp3, called ELP, achieves this bound when the graph is known ahead of time, where α ∈ {1� .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the occupancy record of this DP, m K+1 := {m K+1 } ∪ {m K+1,k : k = 1, . . . , K + 1}, is not yet defined at the onset, with probability 1 we turn to the top-level DP when departing from state K + 1 for the first time.
Sentence 2: we define p(•|c t = K + 1) according to the occupancy record of ancestors in the stock urn.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to model the initial distribution without cluttering the notation, we assume there is a special state θ beg always present at the beginning of the sequence, and only at the beginning.
Sentence 2: we always condition on (θ 0 = θ beg ) and (θ n = θ beg , n > 0), and drop these conditioning events from the notation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One of the long-studied properties of biological neural circuits which has yet to fully impact the machine learning community is the nature of local competition.
Sentence 2: a common finding across brain regions is that neurons exhibit on-center, off-surround organization [1,2,3], and this organization has been argued to give rise to a number of interesting properties across networks of neurons, such as winner-take-all dynamics, automatic gain control, and noise suppression [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we define ¦ R to be the ¢ § -by-¢ § matrix whose entries correspond to the probabilities that one state-action pair follows another when the agent behaves according to 1 .
Sentence 2: the element on the $ &© (' %) P ¨ § row and $ 5© D ' @) P ¦ uR can be viewed as the stochastic transition matrix of a Markov chain over state-action pairs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although information is lost due to hash collisions, it is nevertheless possible to construct a marginal likelihood that exactly accounts for this censoring: the finite projection property of the Dirichlet process specifies what the distribution should be of the hashed counts in each of the buckets.
Sentence 2: the hash function forms a partition of V, and Equation (1) specifies the resulting marginal Dirichlet distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inference: We proof-sketch in the supplementary material that inference can be performed efficiently in parallel as we can decompose the problem of optimizing the orderings induced by the different queries to optimizing each ordering separately.
Sentence 2: for a query i of class c the computation of the y i kj 's, ∀k ∈ P c \ i, j ∈ N c can happen independently of the computation of the y i k j 's for some other query i = i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will show that the characterization of γ-Lipschitz functions via the norm of the differential can be extended from the 2 setting in (4) to arbitrary Banach spaces by considering the gradient as an element in the dual of B.
Sentence 2: for any Banach space B with norm • B , we will derive the loss function where λ, γ ∈ R are regularization parameters, and show that a minimizer of this this is an approximation to the Wasserstein distance on B.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Conversely, all stationary points of the function are stable points of the algorithm.
Sentence 2: since the problem is convex in W and H individually, but not jointly in both, the algorithm can be thought of as taking exact steps toward minimizing over W (as H is held fixed) and then minimizing over H (as W is held fixed), whence the alternating-minimization name.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As illustrated in Figure 1, our RSDN architecture consists of two subnets, and each subnet is a fully connected multi-layered perceptron of 2K+1 layers, i.e., an input layer, 2K-1 hidden layers and a visible layer at the top.
Sentence 2: if we stipulate that layer 0 is input layer, there are the same number of neurons in layers k and 2K-k for k = 0, 1, • • • , K. the Kth hidden layer is used as code layer, and neurons in this layer are further divided into two subsets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the following we focus on measures with discrete support.
Sentence 2: we consider distributions µ, ν ∈ P(X) that can be written as linear combinations µ = n i=1 a i δ xi and ν = m j=1 b j δ yj of Dirac's deltas centred at a finite number n and m of points (x i ) n i=1 and (y j ) m j=1 in X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is to be contrasted with the following property of the occupancy measures generated by O-REPS (proved in the supplementary material): Lemma 1.
Sentence 2: for any t > 0, D (q t q t+1 ) ≤ η 2 2 q t , ˆ 2 t . if the losses are estimated by bounded sample averages as done by Peters et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here λ > 0, l i (λ) = (K nn (K nn + λnI) −1 ) ii are the leverage scores and q ≥ 1 controls the quality of the approximation.
Sentence 2: given λ, the Nyström points are sampled independently from the dataset with probability p i ∝ l i (λ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The attractive facet of this modification is that it leverages the sparsity of the solution and intermediate iterates in a manner that falls within a block coordinate descent framework.
Sentence 2: suppose as detailed above at any outer loop Newton iteration, we partition the variables into the fixed and free set, and then first perform a Newton update restricted to the fixed block, followed by a Newton update on the free block.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The minimax optimal predictions are defined as follows: for all j ∈ [n], Theorem 2 suggests a statistical learning algorithm for aggregating the p ensemble classifiers' predictions: estimate b from the training (labeled) set, optimize the convex slack function γ(σ) to find σ * , and finally predict with g j (σ * ) on each example j in the test set.
Sentence 2: the resulting predictions are guaranteed to have low error, as measured by V . it is easy to prove [9] that V is at least max i b i , the performance of the best classifier.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [18] showed that there exists cross-subject consistency in the cortical regions useful for prediction in this task.
Sentence 2: experts partitioned each dataset into 24 non overlapping regions of interest (ROIs), then reduced the data by discarding all but 7 ROIs and, for each subject, averaging the BOLD response across voxels within each ROI and showed that a classifier trained on data from 5 subjects generalized when applied to data from a 6th.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the above examples show that there cannot be a bound on the sample size as a function of the expectation of the value, it turns out that there is a very tight connection between the first moment and uniform convergence: Theorem 2.3.
Sentence 2: the following dichotomy holds for a distribution µ on R + : the empirical revenues uniformly converge to the true revenues if and only if E µ [V ] < ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This LOCALSPECTRAL optimization problem was introduced in [14], where it was shown that the solution to LOCALSPECTRAL may be interpreted as a locally-biased version of the second eigenvector of the Laplacian.
Sentence 2: although LOCAL-SPECTRAL is not convex, it's solution can be computed efficiently as the solution to a set of linear equations that generalize the popular Personalized PageRank procedure; in addition, by performing a sweep cut and appealing to a variant of Cheeger's inequality, this locally-biased eigenvector can be used to perform locally-biased spectral graph partitioning [14].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our variational approach we introduce a new distribution q and make the following approximation 4 p(z 1:N , h 1:N 1:T , Θ 1:K |v 1:N 1:T , Θ1:K , γ) ≈ q(h 1:N 1:T |z 1:N )q(z 1:N )q(Θ 1:K ).
Sentence 2: ( we approximate the posterior distribution of the hidden variables of the model by one in which the hidden states are decoupled from the parameters given the indicator variables and in which the indicators are decoupled from the parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Nesterov [12] proposed a simple first-order method to exploit this type of structure, and our focus is a slight variant of this procedure.
Sentence 2: given some initialization θ 0 ∈ Ω, consider the update where γ u > 0 is a parameter related to the smoothness of the loss function, and is the ball of radius ρ in the norm defined by the regularizer.
Relationship: entailment
Domain: NeurIPS
Sentence 1: They may be much heavier-tailed, biasing large signals less drastically while shrinking noiselike signals heavily towards zero.
Sentence 2: the priors by [6,7], along with the Strawderman-Berger prior [7], have a very interesting and intuitive representation later given in ( 2), yet, are not formed in a conjugate manner potentially leading to analytical and computational complexity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm ROBUST-LRA (see Algorithm 1) is (", )-differentially private.
Sentence 2: furthermore, given a matrix A 2 R n⇥d , it runs in poly(k, n, d) time, e O(k(n + d)) space, and outputs a rank k matrix M such that, with probability 9/10 over the randomness of the algorithm, where OPT k (A) := min rank(X)k kA Xk p . for p = 1, we get Remark 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The image view consists of 120-dimensional visual features extracted from representative frames selected from videos, and the textual features are 100-dimensional.
Sentence 2: we give the competing algorithms, both CCALin and ALS CCA, the advantage of the knowledge of the eigengap at k. we estimate the spectrum of the matrix b T for the Mediamill dataset and set the gap-dependent parameters in CCALin and ALS CCA accordingly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we introduce an abrupt change into the sequence, for example by reversing the second half of the sequence, s i is found to be ≈ 1 for the frame following the change.
Sentence 2: the system has detected not to use the previous frame at this point, but to revert to using the latent appearance image f . We have proposed a method for localising an object undergoing affine transform whilst simultaneously learning its shape and appearance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (C) If F ≥ Fmax , then Fh ≥ Fhmax ∀h ∈ H; this enforces that all hypotheses must be satisfied.
Sentence 2: truncated version of each hypothesis submodular utility and distance function: F h,αn,αj is truncated from below at α j and from above at α n (it is assumed that α n > α j ), and is offset by −α j so that F h,αn,αj (∅) = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, this simple ALP algorithm achieves very good performance within a constant distance from the optimum, i.e., O(1) regret, except for certain boundary cases.
Sentence 2: for 1 ≤ j ≤ J, let q j be the cumulative probability defined as q j = j j =1 π j with the convention that q 0 = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Corollary 1 and 2 provide answers to our motivative questions.
Sentence 2: for loss functions which are (8, T )-smooth, we can obtain a lower sample complexity; if they are (∞, T )-smooth, there is an -LDP algorithm for the empirical and population excess risks achieving error α with sample complexity which is independent from the dimensionality p in the term α.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead of assuming no interdependency between Y i as in [19], our major goal here is to model the structure in the latent layer Y .
Sentence 2: we assume a conditional model for the first layer based on an exponential family p(y|x) = q 0 (y) exp(−y U x − Ω(U x)), where q 0 (y) = y ∈ Y .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Changing variables from r to r' = UTr and using I det(U)1 = 1, this can be rewritten as Becausepe(e) is a function ofeTe only, Pe (Ue) = pe(e), and therefore Im(r, 8; UWm ) = Im(r, 8, W m ).
Sentence 2: because we have assumed a model in which the noise is described by independent Gaussians, or generally the distribution of the noise e is a function of e T e only, the mutual information is invariant to unitary transformations of the output.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the case of kernel-based reinforcement learning, the so-called "self-approximating" property allows for a much more efficient implementation in vector format (see also [Rus97]).
Sentence 2: because our definition of r m,ah in (5) only depends on the values of h at the states in S , the AACOE (3) can be solved in two steps: In other words , we first determine the values of hm at the points in S using ( 7) and then compute the values at new locations x in a second step using (8).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We view {z n } N n=1 and matrix Γ as latent variables, and optimise the parameters θ = {µ, Φ, Λ, Ψ} by EM.
Sentence 2: we view the weight matrix Λ as a matrix of parameter and estimate the entries by maximum a posteriori (MAP) learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Co-training [2] is a method that has had substantial success in scenarios in which examples can be thought of as containing two distinct yet sufficient feature sets.
Sentence 2: a labeled example takes the form ( x 1 , x 2 , ), where x 1 ∈ X 1 and x 2 ∈ X 2 are the two parts of the example, and is the label.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we aim to study the advantages and limitations of both topologies and further enrich the path design by proposing a dual path architecture.
Sentence 2: we first provide a new understanding of the densely connected networks from the lens of a higher order recurrent neural network (HORNN) [19], and explore the relations between densely connected networks and residual networks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Despite the fact that, as we show, this does not correspond to a convex-concave optimization problem (even for simple linear generator and discriminator representations), we show that: Under suitable conditions on the representational powers of the discriminator and the generator, the resulting GAN dynamical system is locally exponentially stable.
Sentence 2: for some region around an equilibrium point of the updates, the gradient updates will converge to this equilibrium point at an exponential rate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We have shown that sampling-based inference in a simple generative model of V1 can be interpreted in multiple ways, some previously discussed as mutually exclusive.
Sentence 2: the neural responses can be interpreted both as samples from the probabilistic model that the brain has learnt for its inputs and as parameters of the posterior distribution over any experimenter-defined variables that are only implicitly encoded, like orientation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One of the main results is the Fisher-Tippett-Gnedenko theorem [11,12] that characterizes the limiting distribution of this maximum as n converges to infinity.
Sentence 2: it proves that a rescaled version of this maximum converges to one of the three possible distributions: Gumbel, Fréchet, or Weibull.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Tracing its origins to [17] for offline optimization problems, OMD proceeds by taking a gradient step in the dual (gradient) space and projecting it back to the primal (decision) space via a mirror map generated by a strongly convex regularizer function (with different regularizers giving rise to different algorithms).
Sentence 2: oMD includes as special cases several seminal learning algorithms, such as Zinkevich's online gradient descent (OGD) scheme [29], and the multiplicative/exponential weights (EW) algorithm [1,13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: A set of solutions C ⊆ S of a solution space S is a cluster if it is a maximal subset such that any two solutions in C can be connected by a sequence from C where consecutive solutions are neighbors.
Sentence 2: clusters are connected components of the "solution graph" which has solutions as nodes and an edge between two solutions if they differ in the value of exactly one variable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A similar challenge arises in a multi-batch implementation of the L-BFGS method in which the entire training set T = {(x i , y i ) n i=1 } is not employed at every iteration, but rather, a subset of the data is used to compute the gradient.
Sentence 2: we consider a method in which the dataset is randomly divided into a number of batches -say 10, 50, or 100 -and the minimization is performed with respect to a different batch at every iteration.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The empirical success of sparse coding can be justified by theoretical analysis [17], which showed that a modification of sparse coding with added locality constraint, called local coordinate coding (LCC), represents a new class of effective high dimensional non-linear function approximation methods with sound theoretical guarantees.
Sentence 2: lCC learns a nonlinear function in high dimension by forming an adaptive set of basis functions on the data manifold, and it has nonlinear approximation power.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unlike CS, however, PLST takes principal directions v m (to be introduced next) rather than the random ones, and does not need to solve an optimization problem during decoding.
Sentence 2: pLST considers only a matrix V with orthogonal rows, and decodes r(x) to the predicted labels by h(x) = round(V T r(x)+ ȳ), which is called round-based decoding.
Relationship: entailment
Domain: NeurIPS
Sentence 1: kernel and the r.k. of an RKBS need not be symmetric, the space in (7) is an interesting example of an RKBS, which is induced by a pd kernel.
Sentence 2: it can be seen that many RKBSs (B pd p (R d ) for any 1 < p < ∞) have the same r.k (ignoring the scaling factor which can be made one for any p by choosing µ to be a probability measure).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Within the context of statistical inference, Bayesian optimization methods have been developed where a prior distribution over the space of functions is assumed and uncertainty is tracked during the entire optimization process [6,7].
Sentence 2: non-parametric Bayesian approaches such as Gaussian Processes have been applied for derivative-free optimization [8,9], also within the context of the continuum-armed bandit problem [10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Despite recent progress on the frame prediction problem [10,23], it remains quite challenging, in part because the dimensionality of the true-value space is so large.
Sentence 2: in practice, it can be substantially easier to solve inverse dynamics prediction, with input (o t b:t+f , a t b:t 1 , a t+1:t+f 1 ) and true-value a t . the agent is trying to "postdict" the action needed to have generated the observed sequence of observations, given knowledge of its past and future actions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that while all h i -s, g and g are assumed to take values in {−1, 1}, u 1 can take values in R. Then: Now we show that for any α 1 u 1 + α 2 g ∈ MV(u 1 , g) there exists a close hypothesis in MV(u 1 , g).
Sentence 2: this hypothesis is α 1 u 1 + α 2 g: Note that for every x on which g and g agree, i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Matrix Classification: We consider a matrix classification problem under the setting described in [21].
Sentence 2: we first randomly generate a low-rank matrix classifier X * , which is an n × n symmetric matrix of rank r. Then, we specify a sampling ratio θ ∈ (0, 1] and sample p = ⌊θn 2 ⌋/2 independent n × n symmetric matrices W 1 , .
Relationship: entailment