sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: By Proposition 4.2, for a fixed W ∈ IR d×T there is a unique minimizer of Penalty ε (under the constraints in (2.5)), given by the formula Moreover, there exists a minimizer of problem (2.4), which is unique if p ∈ (1, 2].
Sentence 2: we can solve problem (2.4) using an alternating minimization algorithm, which is an extension of the one presented in [4] for the special case F (D) = D −1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If then with probability at least 1 − c 2 exp (−c 3 m), the TWF estimates (Algorithm 1) satisfy for all x ∈ R n . Here, 0 < ρ < 1 and µ 0 , c 0 , c 1 , c 2 , c 3 > 0 are some universal constants.
Sentence 2: alternatively, if one regards the SNR for the model (12) as follows then we immediately arrive at another form of performance guarantee stated in terms of SNR: the relative error of TWF reaches O(SNR −1/2 ) within a logarithmic number of iterations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then, K Φ is bounded by: with probability exceeding 1 − ρ. Theorem 3.1 is a worst-case bound and serves as a sufficient condition for stable ID estimation using random projections.
Sentence 2: if we choose a sufficiently small value for δ and ρ, we are guaranteed estimation accuracy levels as close as desired to those obtained with ID estimation in the original signal space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Chebyshev's inequality 1 states that for a random variable X, if the E[X] and V[X] are bounded, then for any ✏ 0: Pr Now we have characterized the variance of the weights, but what we really want is to characterize the variance of the value estimates.
Sentence 2: notice that the variance of the value-estimate, for state-action x is the variance of the estimate is characterized by the variance of the weights.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ChannelNet-v1 outperforms ChannelNet-v1(-) by 0.8%, which is significant as ChannelNet-v1 has only 32 more parameters with group channel-wise convolutions.
Sentence 2: group channel-wise convolutions are extremely efficient and effective information fusion layers for solving the problem incurred by group convolutions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The fact that this edge was not bought implies that δ(v, u) α = Ω(d 3 ).
Sentence 2: we have that In other words, for "small enough" values of α (quantified in the full proof), vertices quite distant from u in the grid have an incentive to buy an edge to u, by virtue of the dense population in S δ u .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We omit the derivation for brevity.
Sentence 2: in what follows, we focus on learning M k and b k . For the k-th component, the relevant terms in the expected log-likelihood, given the posteriors, from a single similarity assessment s on (u, v), is Learning the parameters Note that J k is not jointly convex in b k and M k . we optimize them alternatively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As it can be observed in Figure 5, the performance of the model including the centrality estimated from X 1 is better than the performance of the classical model, both when using just the data X 2 and the concatenated data X 1 |X 2 .
Sentence 2: we can discard the old data X 1 and keep only the structure (centrality) and still be able to infer a more accurate version of G 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Denote by y i kt the observed spike-count of neuron i ∈ {1 . . . q} at time bin t ∈ {1 . . . T } of trial k ∈ {1 . . . N }, and by y k = vec (y k,i=1:q,t=1:T ) the qT × 1 vector of all data observed on trial k. Neurons are assumed to be conditionally independent given the low-dimensional latent state x kt (of dimensionality p with p < q).
Sentence 2: correlated neural variability arises from variations of this latent population state, and not from direct interaction between neurons.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, if the generative model is described by a Bayesian network, the joint distribution can be written as } denotes the set of all variables (hidden or visible) and PA n are the parents of the n − th of these variables, i.e., v (t) n . The cross-entropy term in the equation above further decomposes into For each discrete hidden variable v (t) n , the appropriate terms above can be further broken down into individual terms in the summation over the D n possible configurations of the variable, e.g., In a similar fashion, the entropy term can also be decomposed further into a sum of terms as dictated by the factorization of the family Q.
Sentence 2: the free energy for a single sample t can be expressed as the sum where all the free energy pieces f t i, θ derive from the finest decomposition ( 5) or ( 4).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, from standard concentration arguments, the `1/`1 matrix norm for a diagonal ("separable") subgaussian random matrix has at most logarithmic dimension dependence 1 , but for a dense ("non-separable") random matrix, the `1/`1 matrix norm could possibly scale linearly in the dimension.
Sentence 2: the scaling of `1/`1 matrix norm gives us an indication how "separable" the matrix is.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [18,Lemma 12] showed that V (K g ) = 1 (also see [19]) and All these classes satisfy the conditions of Theorem 7 and Corollary 10 and therefore provide consistent estimates of γ(P, Q) for any P, Q ∈ P. Examples of kernels on R d that are covered by these classes include the Gaussian, Laplacian, inverse multiquadratics, Matérn class etc.
Sentence 2: other choices for K that are popular in machine learning are the linear combination of kernels, Lemma 7] have shown that V (K con ) ≤ V (K lin ) ≤ l. instead of using a class based on a fixed, parameterized kernel, one can also use a finite linear combination of kernels to compute γ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The reasoning behind this approach is that each marginal is expressed through an average of all possible realizations of dynamics with a given initial condition; this is in contrast with the likelihood function which considers only particular instance realized in the given cascade.
Sentence 2: equation ( 9) summarizes the effect of different propagation paths, and the maximization of this probability function will yield the most likely consensus between the ensemble of couplings in the network.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1 CF-2g closely resembles the serial Ser-2g, but the elements e ∈ V are no longer processed in a fixed order.
Sentence 2: the sets A, B are replaced by potentially stale local estimates (bounds) A, B, where A is a subset of the true A and B is a superset of the actual B on each iteration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand if you already knew that "blicket" meant the object in the window, you might be able to infer these elements of syntax and social cues.
Sentence 2: the problem of early word-learning is a classic chicken-and-egg puzzle: in order to learn word meanings, learners must use their knowledge of the rest of language (including rules of syntax, parts of speech, and other word meanings) as well as their knowledge of social situations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Σ XY can be decomposed as Y Y for a bounded operator V (Theorem 1, [10]).
Sentence 2: we see Σ−1 XX Σ XY is well-defined, because RangeΣ XY ⊂ RangeΣ XX = (KerΣ XX ) ⊥ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, if we construct the sequence (e n ) n∈N such that ∀n ∈ N, e n+1 ∈ argmin e∈E f en and e 0 ∈ E, then we obtain a decreasing sequence (f (e n )) n∈N .
Sentence 2: the algorithm DCA solves a sequence of convex optimization problems in order to solve a DC optimization problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since, N ≥ n Λ = Λ/λ (M ) , any multi-fidelity strategy which plays a suboptimal arm a polynomial number of times at any fidelity after n time steps, will have worse regret than MF-UCB (and UCB).
Sentence 2: in our lower bound we will only consider strategies which satisfy the following condition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is immediate that any classifier which coincides with the Bayes classifier g * on the region tΩ c = {x ∈ R d + : x > t} is optimal w.r.t.
Sentence 2: the distribution P t . In particular g * minimizes L t and the associated risk is for all classifier g, L t (g) ≥ L t (g * ), and taking the limit superior shows that g * minimizes L ∞ , that is Optimality.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any network N (G, π, p), consider N (G, π, p) where for any i, pi = p 2 i .
Sentence 2: define the function g : R |Vin| → R to be the sum of outputs of this network: Then κ (1) and κ (2) can be calculated as follows where 1 is the all-ones input vector: In the process of calculating the gradient ∇ pg(1), we need to calculate h u (p) and ∂g(1)/∂h v (p) for any u, v. the only remaining term to calculate (besides Recall that T is the length (maximum number of propagations through time) and d is the number of layers in an RNN.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, the data analyst is interested in inferring whether the data was generated from P n 0 or P n 1 , where P n ν is one of two possible joint priors on X 1 , ..., X n . This is a challenging problem because knowing X i reveals information about X j , j = i.
Sentence 2: the utility maximization problems for different individuals are coupled in this setting.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this section, we provide an empirical evaluation of the performance of the regularized Laplacian estimator, compared with the unregularized estimator.
Sentence 2: to do this, we need a ground truth population Laplacian L and a noisily-observed sample Laplacian L. in Section 5.1, we construct a family of distributions for L; importantly, this family will be able to represent both low-dimensional graphs and expander-like graphs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Letting g i (t) ∈ ∂f i (x i (t)) be a subgradient of f i at x i (t), By definition of z(t) and y(t), we have y(t) = argmin x∈X { 1 n t−1 s=1 n i=1 g i (s), x + 1 α(t) ψ(x)}.
Sentence 2: we see that the first term in the decomposition (8) can be written in the same way as the bound in Lemma 2, and as a consequence, we have the bound It remains to control the final two terms in the bounds ( 7) and ( 8).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The mechanism which accepts examples from the oracle with some probability is called the filter.
Sentence 2: on each round, a boosting-by-filtering algorithm draws a set of examples from D t via the filter, trains the weak learner on this set, and receives a weak hypothesis h t . Though a batch booster would estimate t using the fixed set S, filtering boosters may use new examples from the filter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (4) However, this is not necessary if we require only the percept represented by the network to be fixed.
Sentence 2: instead, ṡ = D ȧ = D(−a + La) = 0 (5) setting the derivative of s to zero is tantamount to If we require persistent percepts for any a, then: Equation ( 7) has a trivial solution L = I, which corresponds to a network with no actual lateral connections and only autapses.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Compiler Data: For this particular application, in a real-life scenario it is critical to achieve good performance with a low number of training data-points per task given that a training data-point requires the compilation and execution of a (potentially) different version of a program.
Sentence 2: although there are a total of 88214 training points per program we have followed a similar set up to [3] by considering N = 16, 32, 64 and 128 transformation sequences per program for training.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since we are usually interested in selecting a variable assignment with maximum posterior probability (10), we do not need to derive these distributions.
Sentence 2: in this example, it is sufficient for us to determine that the assignment of Y 1 to label 2 maximizes Going back to our application in contour grouping, the RV ⊕T 2 is an append of three RVs representing segments 2, 7, 8 in Fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that using the results of section 2, we would have obtained exactly the same bound for estimating the D-th eigenvector only -or even a worse bound since δ D = δ D ∧ δ D−1 appears in this case.
Sentence 2: at least from the point of view of this technique (which could still yield suboptimal bounds), there is no increase of complexity between estimating the D-th eigenvector alone and estimating the span of the first D eigenvectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any pair of domains D (i) and , or , then all parameters of the system are unchanged across D (i) and D (j) , i.e., , and a (i) = a (j) , and Assumption 1 is mild in the sense that it only rules out a 2-dimensional subspace of a 3-dimensional space.
Sentence 2: considering Lebesgue measure on the 3-dimensional space, we are only ruling out a measure-zero subset.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Of course, using the true test labels in method 3 would be the ideal measure of the degree of confusion between models, but these are of course not available (see Figure 2, however, showing the results of a cheating experiment).
Sentence 2: we use the hypothesized labels from a first stage to compute  is used if either one of 9 5 @ 3 Using 4 state per phone, 12 Gaussian mixtures per state HMMs, totaling 200k free model parameters for the system.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For each observed timestamp ⌧ v , the estimator creates a ball of radius ⌧ v 1, centered at v. For example, in our figure, the green node (node 1) has ⌧ 1 = 2.
Sentence 2: the adversary would make a ball of radius 1 centered at node 1; this ball is depicted by the green bubble in our figure.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The similarity measure d (T ) (r, s) could, for example, be squared Euclidean distance: T . However, this similarity measure only looks at the first T time steps of training time series r. Since time series in our training data are known, we need not restrict our attention to their first T time steps.
Sentence 2: we use the following similarity measure: where we minimize over integer time shifts with a pre-specified maximum allowed shift max 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, the verifiability property allows us to design new recovery routines with explicit confidence bounds for the recovery error and optimize these bounds with respect to the method parameters.
Sentence 2: the major novelty in what follows is the emphasis on verifiable conditions on A and the representation structure which guarantee good recovery of Bx from noisy observations of Ax, provided that Bx is nearly s-block-sparse, and the observation noise is low.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To incorporate MC-SURE into a stochastic gradient-based optimization algorithm for training, such as the SGD or the Adam optimization algorithms, we modify the risk (7) in accordance with where ( 11) is equivalent to (7) owing to conditioning.
Sentence 2: from Theorem 1, an unbiased estimator for E n∼N 0,σ 2 x − h(y; θ) 2 |x can be derived as such that for a fixed x, using the empirical risk expression in (10), an unbiased estimator for ( 7) is noting that no noiseless ground truth data x (j) were used in (13).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Computing the Type I cost for each with h(z) = z gives g (I) (X (1) ) = 2 √ 2 and g (I) (X (2) ) = 2(1 + ǫ).
Sentence 2: if we allow ǫ to be small, g (I) (X (2) ) < g (I) (X (1) ), so X (1) = X 0 cannot be the minimizing solution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since we obtain that for any λ > 0, E 2n πe h ≤ 1.
Sentence 2: for any β > 0, we have On the event log πe h+log β ≤ 0 , by the Legendre's transform, for any probability distribution ρ ∈ M 1 + (F), we have which proves the lemma.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can immediately notice that F β is fractional-linear, hence pseudo-convex, with respect to FN 1 and FP 1 .
Sentence 2: with a slight (yet convenient) abuse of notation, we write the F β -measure for binary classification as a function of vectors in R 4 = R 2L which represent error profiles of classifiers: Multilabel Classification In multilabel classification, there are several definitions of F -measures.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal is to find the best bounding box given the learned weight vector w and the visual feature x. Privileged information is not available at testing time, and inference is performed on visual features only.
Sentence 2: the same maximization problem as in standard SSVMs needs to be solved during prediction, which is given by This maximization problem is over the space of bounding box coordinates.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To circumvent the limits of local differential privacy, we consider a slightly relaxed estimation guarantee.
Sentence 2: specifically, we batch the rounds into T epochs, each consisting of rounds, and aim in each epoch t to estimate p t , the population-wide mean across the subgroups and rounds of epoch t. any sufficiently large changes in this mean will be identified after the current epoch completes, which we think of as introducing a small "delay".
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This observation makes us conjecture that Tetris is a game whose policy space is easier to represent, and as a result to search in, than its value function space.
Sentence 2: in order to obtain a good performance with ADP algorithms in this game, we should use those ADP methods that search in a policy space, instead of the more traditional ones that search in a value function space.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . = v x (K) , then we may never recover θ along the direction of v. In fact, Proposition 1 can be viewed as an instance of this result where v = 1 |S| (recall that 1 |S| η π µ,P ≡ 1), and that is why we have to remove such redundancy in Example 1 in order to discuss identification in MDPs.
Sentence 2: to guarantee identification in a fixed environment, the feature vectors must have significant variation in all directions, and we capture this intuition by defining a diversity score spread(X) (Definition 2) and showing that the identification accuracy depends inversely on the score (Theorem 4).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 10(a) demonstrates the tradeoff in choosing a block size: when block size is too small the statistical power of the M -statistic is weak and hence EDD is large; on the other hand, when block size is too large, although statistical power is good, EDD is also large because the way we update the test block.
Sentence 2: there is an optimal block size for each case.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We will show essentially that if the machines first collectively choose randomly a coordinate i, and run protocol ⇧ i for the one-dimensional problem, then the information cost and mean-squared loss of this protocol will be only 1/d factor of those of the d-dimensional problem.
Sentence 2: the information cost of the d-dimensional problem is at least d times the information cost of one-dimensional problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The joint probability over input pixels and latent variables is proportional to exp(−E PoT (x, h c )).
Sentence 2: the conditional distribution over the input pixels is a zero-mean Gaussian with covariance equal to: In order to make the mean of the conditional distribution non-zero, we define mPoT as the normalized product of the above zero-mean Gaussian that models the covariance and a spherical covariance Gaussian that models the mean.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that g is non-negative and submodular.
Sentence 2: by Lemma 5 and the fact that A contains every element with probability at most q (because Algorithm 2 accepts an element into its solution with at most this probability), we get Combining the two above inequalities, we get the approximation ratio of the algorithm we got is at most 2p + 2 p p(p + 1) + 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To prove convergence of GANs learned by TTUR, we make the following assumptions (The actual assumption is ended by , the following text are just comments and explanations): (A1) The gradients h and g are Lipschitz.
Sentence 2: networks with Lipschitz smooth activation functions like ELUs (α = 1) [11] fulfill the assumption but not ReLU networks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let q ∈ Q, the children of q in Q are not in Q 2r and therefore are all within 2r of Q; since these children an r-packing of B(q, 2r), there are at most 2 2d of them.
Sentence 2: max |Q | ≤ 2 2d max |Q|. . At the end of each iteration we have Q ⊂ B(x, ρ (x, Q ) + 2r).
Relationship: reasoning
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: we can tackle MIML by identifying its equivalence in the traditional supervised learning framework, using multi-instance learning or multi-label learning as the bridge.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If F = φ, F is non-empty so there exists an S such that X i ⊥ X j | X S is faithful.
Sentence 2: s separates i and j in G and (i, j) / ∈ E. If F = φ, then for any S ⊆ W, |S| ≤ K, we have either it is unfaithful.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal is to ensure that the success of an attack (using a username-key pair) is low.
Sentence 2: in other words, the attack succeeds with probability no larger than 1/δ i for a user which specified δ i ∈ N. the attack we seek to protect against is the use of the data to match usernames to keys (rather than attacks in which additional non-sensitive attributes about a user are discovered).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The case for examples with y n > k + 1 is similar and the change there is also non-positive.
Sentence 2: by switching adjacent pairs of strictly decreasing thresholds, we can actually obtain a solution (g * , θ * ) with a smaller or equal objective value in (5), and g * = g. The optimality of (g, θ) shows that (g * , θ * ) is also optimal.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the data space is isotropic then a stationary point of the error will correspond to a similarly isotropic map space 7 .
Sentence 2: at a stationary point, we have for large ¡ : where & ¢ is the x identity matrix, and # " % and # " ) are the variances in the map space and the data space respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A kNN regressor using only the labeled data outperformed DKL on two of three datasets for n = 100, beat SSDKL on all three for n = 100, beat DKL on two of three for n = 300, and beat SSDKL on one of three for n = 300.
Sentence 2: the kNN regressor is often already outperforming SSDKL with only labeled data-it is unsurprising that SSDKL is unable to close the gap on a semi-supervised nearest neighbors method like COREG.
Relationship: reasoning