sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Based on the group structures of p and w, we introduce their corresponding group penalties as follows: One can see the symmetry here: group penalty (4) on p is a squared, unweighted L 2,1 -norm, which is designed to secure a unique solution that favors more randomness in p for the sake of diversity in sonority generation [9]; group penalty (5) on w is a weighted L 1,2 -norm (group lasso), which enables rule selection.
Sentence 2: there is a pitfall of the group lasso penalty when deployed in Problem (3): the problem has multiple global optima that are indefinite about the number of rules to pick (e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As the learning rate parameter is reduced so the timescale for escape is also reduced.
Sentence 2: the choice of optimal learning rate is non-trivial and cannot be determined by considering only the leading order terms in f as above, because although small F will result in a quicker escape from the unstable fixed point near f 0 x , this in turn will lead to a very slow learning transient after escape.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then for any S, c > 0 and n 2 N, we have We remark that the right-hand side of the above bound does not depend on the number of trading periods n. The key parameter is ⇣, which determines the size of the largest price jump of the stock.
Sentence 2: we expect that as the trading frequency increases, the size of the largest price jump will shrink.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This graph is well-represented by a latent class model in which edges occur with high probability between pairs having one member in each of A and B or in B and C, and among pairs having both members in C (in models of stochastic equivalence, nodes within each class are not differentiated).
Sentence 2: representation of this type of graph with a latent distance model would require the dimension of the latent characteristics to be on the order of the class membership sizes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To see this, first, it is not hard to prove that L K in the sense of usual the order on the positive semi-definite cone.
Sentence 2: second, one can prove that the variance is matrix-increasing, hence the variance will decrease when replacing K by L. the bias (while not matrix monotone in general) can be proven to not increase too much when replacing K by L. This latter statement will be the main technical difficulty for obtaining a bound on R( fL ) (see Appendix A).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For the class of single Gaussians in d dimensions, this approach leads to the optimal sample complexity upper bound of O(d 2 /ε 2 ).
Sentence 2: for mixtures of Gaussians and axis-aligned Gaussians in R d , the best known VC-dimension bounds [1,Theorem 8.14]  Another approach is to first approximate the mixture class using a more manageable class such as piecewise polynomials, and then study the associated Yatracos class, see, e.g., [5].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It should be noted that Jennrich's algorithm [13,15,17] could recover {µ i } k i=1 directly from the 3rd order tensor M 3 alone when {µ i } k i=1 is linearly independent.
Sentence 2: we still adopt the above simultaneous diagonalization framework because the intermediate vectors {W µ i } k i=1 play a vital role in the recovery procedure of the linear regression model η.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Some AUs happen together to show a meaningful facial expression: AU6 (cheek raiser) tends to occur together with AU12 (lip corner puller) when someone is smiling.
Sentence 2: some AUs may be mutually exclusive: AU25 (lips part) never happens simultaneously with AU24 (lip presser) since they are activated by the same muscles but with opposite motions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In principle, one can implement AdaptReg by setting the termination criteria of the oracle in the inner loop as precisely suggested by the theory, such as setting the number of iterations for SDCA to be exactly T = O(n + L σt ) in the t-th epoch.
Sentence 2: in practice, it is more desirable to automatically terminate the oracle whenever the objective distance to the minimum has been sufficiently decreased.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given a generative model and the restriction of DSTUMP to using root-level impurity reduction, the general thrust of our result is straightforward: impurity reduction due to active variables concentrates at a significantly higher level than that of inactive variables.
Sentence 2: there are significant technical challenges in establishing this result, mainly deriving from the fact that the splitting procedure renders the data in the child nodes non-i.i.d., and hence standard concentration inequalities do not immediately apply.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, larger regions provide higher saving (in bits) in the description length of R. A region R induces "average savings per point" for every point q ∈ R, namely, (where |R| is the number of points in R).
Sentence 2: a point q ∈ R may also be contained in other regions generated by ref , each with its own local evidence score.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since many different people recorded the digits used in one of the types of audio CAP TCHAs we tested, PLP and RAS TA-PLP were needed to extract the features that were most useful for solving them.
Sentence 2: in [4]- [5], the authors conducted experiments on recognizing isolated digits in the presence of noise using both PLP and RAS TA-PL P. the noise used consisted of telephone or microphone static caused by recording in different locations.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For instance, Empirical Bayes requires the maximization of marginal likelihood which is difficult to compute in hierarchical logistic models [9] in general, and cross-validation requires reducing the number of free parameters for computational reasons, potentially losing the flexibility to capture the desired phenomena.
Sentence 2: we propose a principled way to set the hyper-parameters directly from data using an approximation to the observed Fisher Information Matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Similar bounds are provided for cyclic BCD in [7,Theorem 6.1].
Sentence 2: it is well known that when applying the classical gradient descent (GD) method to problem (1) with the constant stepsize 1/L, we have the following rate estimate [11, Corollary 2.1.2] Note that unlike (6), here the constant in front of the 1/(r + 4) term is independent of the problem dimension.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: STRIPES provably found the subsequent top 50 solutions within 4.5 hours for all but one of these cases (up to 8 spanning trees per calculation), and BMMF found the same 50 solutions for each case within 0.5 hours; note that only STRIPES provides a certificate of optimality for these solutions.
Sentence 2: only for 146 of the 315 problems was the Nilsson method able to complete within five days; thus, we do not compare its performance here.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: An initial step size must be selected, but the adaptive step size adjustment rules make this choice less important.
Sentence 2: the BADMM method requires choosing an initial weight on the augmented Lagrangian term, an adjustment schedule for this term, a step size on the dual variables, and a step size for local policies, all of which have a substantial impact on the final performance of the method (the reported results are for the best setting of these parameters, identified with a hyperparameter sweep).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The update of θ is based on the samples generated from q φ (H|V), and is identical to (10).
Sentence 2: in contrast to (11), the recognition parameters φ are estimated from samples generated by the model, i.e., This update does not optimize the same objective as in (10), hence the wake-sleep algorithm is not guaranteed to converge [13].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As gets smaller the runtime slightly increases, but it decreases for = 0 since the ∞ norm is computed efficiently using the max function.
Sentence 2: = 0 is less accurate than = 0.01; When the approximated structured SVM converges, the gap between the primal and dual objectives was 1.3, and only 10 −5 for > 0.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We train a separate reconstruction network for each noise level (positing that a camera could select between these based on the ISO settings).
Sentence 2: since it is impractical to employ different sensors for different settings, we learn a single spatial multiplexing pattern, optimized for reconstruction under moderate noise levels with standard deviation (STD) of 0.01 (with respect to intensity values in x(n) scaled to be between 0 and 1).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: SPGP-DR works well for large training set sizes, since there is enough information for it to avoid overfitting and the advantage of using a prior on W is reduced.
Sentence 2: 2 Available at http://theoval.cmp.uea.ac.uk/ ˜gcc/competition/ Temp: 106 dimensions 7117/3558 training/testing data, SO2: 27 dimensions 15304/7652 training/testing data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: There it was shown that, given a fixed labeling of a graph, the number of mistakes made by an algorithm similar to the kernel perceptron [9] with a kernel that was the pseudoinverse of the graph Laplacian, could be bounded by the quantity [8, Theorems 3.2, 4.1, and 4.2] Here u ∈ {−1, 1} n is a binary vector defining the labeling of the graph, Φ G (u) is the cut size 1 defined as Φ G (u) := |{(i, j) ∈ E(G) : u i = u j }|, that is, the number of edges between positive and negative labels, D G is the diameter of the graph and bal(u This bound is interesting in that the mistakes of the algorithm could be bounded in terms of simple properties of a labeled graph.
Sentence 2: there are a variety of shortcomings in this result.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Minimax rates, but only for d = 1, were obtained in [Gen12,AG14] for models of purely random forests such as Toy-PRF (where the individual partitions corresponded to randomly shifts of the regular partition of [0, 1] in k intervals) and PURF (Purely Uniformly Random Forests, where the partitions were obtained by drawing k random thresholds at random in [0, 1]).
Sentence 2: for d = 1, tree partitions reduce to partitions of [0, 1] in intervals, and do not possess the recursive structure that appears in higher dimensions and makes their precise analysis difficult.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Previous studies using HMMs have assumed that all neural activities were independent of one another given the hidden states; hence, the models could not discriminate states whose firing rates were almost the same but whose correlations among neurons were different.
Sentence 2: there has been reports that shows the correlation between neurons changes within a fraction of a second without modulating the firing rate (e.g., [4]).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Assuming independent outputs (no correlation between each output dimension) and given a test input xk = (x k , u k ) at time step k, the one-step predictive mean and variance of the state transition are specified as The state distribution at k = 1 is p(x 1 ) ∼ N (µ 1 , Σ 1 ) where the state mean and variance are  Generally, this predictive distribution cannot be computed analytically because the nonlinear mapping of an input Gaussian distribution lead to a non-Gaussian predictive distribution.
Sentence 2: the predictive distribution can be approximated by a Gaussian p(dx k ) ∼ N (dµ k , dΣ k ) [16].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: On the other hand, there is a number of recent works establishing the convergence of play in potential games with finite action sets under different assumptions for the number of players involved (continuous or finite) and the quality of the available feedback (perfect, semi-bandit/imperfect, or bandit/payoff-based) [5,11,14,19].
Sentence 2: these works focus on games with finite action sets and feedback is assumed to be instantly available to the players (i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Note that when the parameters are known, one can take e ⇥ = {✓ 0 }.
Sentence 2: when the parameters are unknown, one takes e ⇥ = ⇥ to prove a lower bound on the sample complexity of the estimate-then-explore algorithm, which is tasked with deciding whether or not samples are coming from a mixture of distributions or just a single distribution within the family.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: lated quantity, the delayed mutual information (dMI) provides an indication of instantaneous dependence: dM I(s) = I(X t ; X t+s ), where X t is a binned spike train, and I(X; Y ) = H(X)−H(X|Y ) denotes the mutual information.
Sentence 2: this quantity ignores any temporal dependences in the intervening times: X t+1 , .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: First, they pertain to parametric estimation in R d , where the communication budget per machine is much smaller than what is needed to even specify the answer with constant accuracy (O(d) bits).
Sentence 2: our results pertain to simpler detection problems, where the answer requires only O(log(d)) bits, yet lead to non-trivial lower bounds even when the budget size is much larger (in some cases, much larger than d).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given weakly annotated training images, they infer latent segmentation masks based on Multiple Instance Learning (MIL) [7,9] or Expectation-Maximization (EM) [8] framework based on the CNNs for supervised semantic segmentation.
Sentence 2: performance of weakly supervised learning approaches except [10] is substantially lower than supervised methods, mainly because there is no direct supervision for segmentation during training.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, one subject thought the shape categories were determined by whether the shape "faces" downward; another thought the word categories indicated whether the word contains the letter T. Such beliefs, though helpful in learning the particular training samples, amount to over-fitting the noise.
Sentence 2: when n = 40, about half the participants indicated that they believed the labels to be random, as spurious "rules" are more difficult to find.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If so, then the extremely low Bonferroni corrected α-threshold crossings effectively become mutually exclusive, which makes the Union Bound (on which Bonferroni correction is based) nearly tight.
Sentence 2: when variables are highly dependent -and indeed even without smoothing there are many sources of strong non-Gaussian dependencies between voxels, the true FWER threshold can be much more relaxed, and it is precisely this phenomenon which drives the search for alternatives to Bonferroni correction.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Despite the advantages of the LLDE, it requires the bandwidth to be data independent and vanishingly small (sublinearly in sample size) for consistency almost everywhere -both of these are impediments to practical use since there is no obvious systematic way of choosing these hyperparameters.
Sentence 2: if we restrict our focus to functionals of the density, then both these issues are resolved: this is the focus of the next section where we show that the bandwidth can be chosen to be based on fixed k-NN distances and the resulting universal bias easily corrected.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this case, the policy could react to a response time flucutation by setting an extreme powercap value in an attempt to quickly drive the response time back to the goal value, and then backing off to a less extreme value 5 seconds later.
Sentence 2: such behavior would be quite poor in the actual system, in which the extreme powercap setting is held fixed for 30 seconds.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Prior work on multi-class learning in generative hierarchies either learns separate hierarchies for each class [14,15,16,10,17], trains jointly [7,18,9,19,20,11], whereas work on sequential learning of classes has been particularly scarce [6,13].
Sentence 2: to the best of our knowledge, no work has dealt with, evaluated and compared multiple important learning concepts under one hierarchical framework.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For each task t i and each algorithm A ∈ A we define the expected and empirical error of applying this algorithm as follows: where ).The goal of the learner is to find A that minimizes er n+1 given the history of the observed tasks.
Sentence 2: if the task environment would change arbitrarily from step to step, the observed tasks would not contain any relevant information for a new task.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The gradient of L(W) is the following: The first set of terms are easy to compute.
Sentence 2: to compute the the normalizer, Z W (x ), and the expected value of the features under the current weight vector, E Pr(s |W) G(x , s ), we must use the Markov property of G to construct a dynamic programming algorithm, similar for forward-backward.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Indeed, for general M * (x), such flipping would give different values t * λ (x) ∈ M * (x) for λ → 0, and hence would result in significantly different sets M low (f P,λ ) and M up (f P,λ ).
Sentence 2: as a consequence, it seems hard to show that, for probability measures P whose conditional distributions P( • |x), x ∈ X, have no discrete components, we always have there are situations in which this equality can easily be established.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Convergence can sometimes then be assured by existing results on the convergence of on-policy methods (Tsitsiklis & Van Roy 1997;Tadic 2001).
Sentence 2: the importance sampling weights are cumulative products of (possibly many) target-to-behavior-policy likelihood ratios, and consequently they and the corresponding updates may be of very high variance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given a data matrix Y ∈ R n×p and an arbitrary unit vector in x ∈ R p , an intuitive way to remove the contribution of x from Y is to project Y onto the orthocomplement of the space spanned by x: Ŷ = Y (I − xx T ).
Sentence 2: if A is the sample covariance matrix of Y , then the sample covariance of Ŷ is given by Â = (I − xx T )A(I − xx T ), which leads to our formulation for projection deflation: Projection deflation Note that when x t is a true eigenvector of A t−1 with eigenvalue λ t , projection deflation reduces to Hotelling's deflation: in the general case, when x t is not a true eigenvector, projection deflation maintains the desirable properties that were lost to Hotelling's deflation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the learner performs active learning and an oracle provides noiseless labels, then the error reduces faster | θ − θ| = O( 1 2 n ) since the optimal strategy is binary search.
Sentence 2: a helpful teacher can simply teach with n = 2 items (θ − /2, white), (θ + /2, black) to achieve an arbitrarily small error .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Other methods for this problem include the relaxed Richardson iteration algorithm [6], the approximated joint diagonalization algorithm [9], and Riemannian Newton and quasi-Newton type methods, notably the limited-memory Riemannian BFGS [37].
Sentence 2: none of these methods were shown to greatly outperform RGD, especially in data science applications where n is large and extremely small optimization error is not required.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Motivated by this, SVGD approximates q with the empirical distribution of a set of particles {x i } n i=1 , and iteratively updates the particles by Liu et al.
Sentence 2: ( 2017) studied the asymptotic properties of the dynamic system underlying SVGD, showing that the evolution of the limit density of the particles when n → ∞ can be captured by a nonlinear Fokker-Planck equation, and established its weak convergence to the target distribution p. the analysis in Liu et al.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For a general Kblockmodel, we conjecture that the number of stationary points grows exponentially with K. Similar to Figure 1-(a), a significant fraction of the random initializations converge to the ground truth.
Sentence 2: when p, q are unknown, random initializations always converge to the uninformative stationary point (1/3, 1/3, 1/3), analogous to Lemma 3.2.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 1-PkO is the probability that symptom Sk is active when none of the diseases are active.
Sentence 2: exact inference computes the distribution over d given a subset of observed values in s. if Sk is not observed, the corresponding likelihood (node plus edges) may be deleted to give a new network that describes the marginal distribution over d and the remaining variables in s. So, we assume that we are considering a subnetwork where all the variables in s are observed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Suppose there are ¡ possible objects, and that there are ¢ possible values that the instantiation parameters of any one object can take on; we will need to consider £ ¥¤ ¦¢ ¨ § ¨© combinations to explain any image.
Sentence 2: in our approach we find one object at a time, thus avoiding the combinatorial explosion.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: They study hardware implementations of the TRW-S algorithm (a coordinate-minimization algorithm very similar to the ones we study here), where some message computations can be parallelized.
Sentence 2: their parallelization scheme is quite different from ours as it is synchronous, i.e., the messages computed in parallel have to be carefully chosen, and it is specific to grid-structured graphs.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Low-precision inference has already made its way into production hardware such as Google's tensor processing unit (TPU) [15].
Sentence 2: reasonable successes in low-precision training have been obtained with binarized [13,16,17,5] or ternarized weights [18], or binarized gradients in the case of stochastic gradient descent [19], while accumulation of activations and gradients is usually at higher precision.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For x e = 1, in order to have f ∂vi (x ∂i ) < ∞, only one other x e ∈ x ∂vi should be non-zero.
Sentence 2: we know that messages are normalized such that ν e→∂vi (0) = 0 ∀v i , e ∈ ∂v i , which means they can be ignored in the summation.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Each singleton concept has teaching dimension 1, while the teaching dimension for the empty concept 0 is n, because the teacher has to reveal all labels to distinguish 0 from the other concepts.
Sentence 2: if the teacher and the learner are cooperative, every concept can be taught with one label: If the teacher reveals a "0" label, the learner can safely assume that the target concept must be 0, because otherwise the teacher would present a "1" label instead for the other concepts.
Relationship: contrasting