sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: By interpreting γ GA and γ M A as optimizing different variational objectives, we gain additional insight into their behavior.
Sentence 2: for geometric averages, the intermediate distribution γ GA (β) minimizes a weighted sum of KL divergences to the initial and target distributions: γ M A minimizes the weighted sum of KL divergences in the reverse direction: See the supplementary material for the derivations.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Gaussian process inference and learning requires evaluating (K +σ 2 I) −1 y and log |K +σ 2 I|, for an N × N covariance matrix K, a vector of N datapoints y, and noise variance σ 2 , as described in the supplementary material.
Sentence 2: for this purpose, it is standard practice to take the Cholesky decomposition of (K + σ 2 I) which requires O(N 3 ) computations and O(N 2 ) storage, for a dataset of size N . many real world applications are engineered for grid structure, including spatial statistics, sensor arrays, image analysis, and time sampling.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: IHW has several limitations: first, binning the data into G groups can be difficult if the feature space X is multi-dimensional; second, the decision rule, restricted to be a constant for each group, is artificial for continuous features; and third, the asymptotic FDR control guarantee requires the number of groups going to infinity, which can be unrealistic.
Sentence 2: neuralFDR uses a neural network to parametrize the decision rule which is much more general and fits the continuous features.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Table 1 shows that Clump-Far outperforms other training sets for the low-capacity GCM.
Sentence 2: clump-Far, Clump-Near, and Spread are all optimal for high-capacity GCM, reflecting the fact that for high-capacity GCM the symmetry of the inner-most training item pair about the true decision boundary θ * determines the learned model.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In our experiments CFR + vastly outperforms its theoretical convergence rate (in fact, every CFR variant does significantly better than the theory predicts, but CFR + especially so).
Sentence 2: cFR + is known to eventually reach a point where it slows down and performs worse than 1 T . In our experiments we start to see CFR + slowing down towards the end of Endgame 7.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then, the maximization of the bound proceeds by performing stochastic gradient updates for the model weights W and the variational parameters V . The update for W reduces to a logistic regression type of update, based upon drawing a single sample from the full variational distribution.
Sentence 2: obtaining effective and low variance stochastic gradients for the variational parameters V is considered to be a very highly challenging task and current advanced methods are based on covariates that employ neural networks as auxiliary models [10].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For rotation matrices it is computed as the Frobenius norm of the difference between ZT Z and Gaussian noise variance Mean 3D error [mm] GP GP + trsf Cstr GP [12] Cstr GP [12]    Gaussian noise variance Mean 3D error [mm] GP GP + trsf Cstr GP [12] Cstr GP [12]   Note that the global transformations estimated with the PnP method yield poor reconstructions.
Sentence 2: our approach performs best among those that use these transformations.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then inf where B r V is an r V ball with the same center with U such that Vol n (B r V ) = V. Moreover, T (x) = r U r V x, for x ∈ B r V is the optimal transport map (up to a measure zero set), so that Complementarily, when 0 < V < Vol n (U ), the infimum inf W : Voln(W )=V W 2 (P U , P W ) = 0, is not attained by any set.
Sentence 2: inf W : Voln(W )≥V W 2 (P U , P W ) = 0 by taking W = U .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The simplest of the models, the feedforward model, does not consider the temporal structure in the task at all, and consists only of a feedforward convolutional network that takes in the observations o t and outputs a point estimate ŷt of the label y t . This approach is viable only if the label information can be directly inferred from o t , such as when tracking an object.
Sentence 2: tasks that require long term memory, such as visual odometry, cannot be solved with a plain feedforward network.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The embedding GNN at layer l is a standard GNN module applied to these inputs: i.e., we take the adjacency matrix between the cluster nodes at layer l (from Equation 4) and the pooled features for the clusters (from Equation 3) and pass these matrices through a standard GNN to get new embeddings Z (l) for the cluster nodes.
Sentence 2: the pooling GNN at layer l, uses the input cluster features X (l) and cluster adjacency matrix A (l) to generate an assignment matrix: where the softmax function is applied in a row-wise fashion.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [5], and another based on perceptron method [6], and propose alternating optimization algorithms to train the classifiers.
Sentence 2: they only evaluate the proposed methods with a few small datasets (with no more than 1000 samples in each), and do not compare them to other widely used (nonlinear) classifiers (e.g., KNN, decision tree, SVM).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, in [8] it was shown that for most combinations of m and n the dimension of RBM n,m (as a manifold, possibly with singularities) equals either the number of parameters or 2 n − 1, whatever is smaller.
Sentence 2: the geometry of RBM n,m is intricate, and even an RBM of dimension 2 n − 1 is not guaranteed to contain all visible distributions, see [20] for counterexamples.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For non-stationary kernels we proceed as before and construct 8;j = (¢(Xi)-¢(Xj))T(¢(Xi)-¢(Xj)).
Sentence 2: we can again show that the kernel MDS procedure operates on the matrix H K H. the distance 8ij in feature space is not a function of 8ij and so the relationship of equation 3 does not hold.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since the hyperparameters of a GP model can be adjusted by maximizing the marginal likelihood, GPR requires little effort and is easy and flexible to use.
Sentence 2: the main limitation of GPR is that the computational complexity scales cubically with the training examples n. This drawback prevents GPR from applications which need large amounts of training data and require fast computation, e.g., online learning of inverse dynamics model for model-based robot control.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: At the end of each time step i, the expected sub-optimality of STRSAGA over T i is If we additionally assume that the condition number L/µ is bounded by a constant at each time, the above simplifies to Lemma 3 shows that the expected sub-optimality of STRSAGA over its effective sample set is not equal to n i the number of points so far).
Sentence 2: our goal is to show that STRSAGA is risk-competitive to DYNASAGA(ρ) at each time step i; i.e., the expected sub-optimality of STRSAGA over S i is within a factor of H(t D i ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: That is, let k * be the best single model defined as: then f k * = g. We are interested in an estimator f of g that achieves a small regret This paper considers a special class of model combination methods which we refer to as model averaging, with combined estimators of the form where ŵk ≥ 0 and k ŵk = 1.
Sentence 2: a standard method for "model averaging" is model selection, where we choose the model k with the smallest least squares error: This corresponds to the choice of ŵk = 1 and ŵk = 0 when k = k. it is well known that the worst case regret this procedure can achieve is R( .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 1, produces the following unconstrained problem: We use the term "reconstruction cost" for this smooth penalty because it corresponds to the reconstruction cost of a linear autoencoder, where the encoding weights and decoding weights are tied (i.e., the encoding step is W x (i) and the decoding step is The choice to swap the orthonormality constraint with a reconstruction penalty seems arbitrary at first.
Sentence 2: we will show in the following section that these two forms of degeneracy control are, in fact, equivalent under certain conditions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: As a trivial example, adding an arbitrary constant to the variables, λ x,y,v→α (ŷ v ) + c, does not change the objective value, hence the algorithm can generate non-decreasing unbounded sequences.
Sentence 2: the beliefs generated by the algorithm are bounded and guaranteed to converge to the solution of the dual approximated structured prediction problem.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Suppose that we want to keep every activation unit with the probability of keep_prob, in dropout [1] the binary mask will be sampled with the Bernoulli distribution with mean 1 − keep_prob.
Sentence 2: to account for the fact that every zero entry in the mask will be expanded by block_size 2 and the blocks will be fully contained in feature map, we need to adjust γ accordingly when we sample the initial binary mask.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In non-mini-batch settings, AccProxSVRG only achieves the same rate as SVRG.
Sentence 2: in mini-batch settings, AccProxSVRG significantly improves the minibatch efficiency of non-accelerated variance reduction methods, and surprisingly, AccProxSVRG essentially only needs size O( √ κ) mini-batches for achieving the optimal iteration complexity for strongly convex objectives, where κ is the condition number of the problem.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The work that closely resembles ours is level-k thinking [20] where level k agents respond to level k − 1 agents, and more closely cognitive hierarchy [18], in which responses are to distributions over levels {0, 1, . . . , k − 1}.
Sentence 2: our goals and motivations are very different: we use the setup as a means to produce more general policies rather than to predict human behavior.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We emphasize that our k-SVD complexity (1.2) can be upper bounded by a quantity that only depends poly-logarithmically on 1/ε.
Sentence 2: the worst-case 1-SVD complexity (1.3) of FW and Garber depends on ε −1/2 because the gap σ 1 − σ 2 can be as small as zero.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In Algorithm 1 we denote P r = |{p : r 2 p}|, we define (µ) as In Appendix D we show that the convergence rate of Algorithm 1 is O(1/ ✏), similar to subgradient in the dual.
Sentence 2: algorithm 1 has several advantages over subgradient.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: ( 15) implies that span(L mAP ) = span(F mAP )).
Sentence 2: as noted in Section 6 of [18], although the matrix F mAP is convenient from the consistency point of view (in the setup of [18]), it leads to the prediction problem max σ∈Sr (F mAP θ) σ , which is a quadratic assignment problem (QAP), and most QAPs are NP-hard.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More precisely, when the data are well clustered around a few patterns, the accumulated updates (difference between the learned weights and the initialization) should be approximately low rank, which can be seen from checking the SGD updates.
Sentence 2: when the difference is small compared to the initialization, the spectrum of the final weight matrix is dominated by that of the initialization and thus will tend to closer to that of a random matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: k , then a trace is given, even in this exploit case, because the teacher does not know V A and cannot distinguish this case from the "explain" case above.
Sentence 2: this trace will still be helpful, because U T ≤ U A , so at least U T < V T − k (satisfying iii), and again by the simulation lemma, the trace will help us learn a parameter and there are a limited number of such mistakes, so (ii) holds.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [2] exposed rats to a tone T immediately followed by a click train C. In a second phase, the tone was paired with a footshock US that either immediately followed tone offset (variant A), or occurred 5 sec after tone offset (variant B).
Sentence 2: they found that when C and US both immediately follow T , little conditioned response is elicited by the presentation of C. when the US occurs 5 sec after tone offset, so that it occurs later than C (measured relative to T), then C does come to elicit a CR.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We can show that the correction to e' is typically by a factor of O(l/m) smaller than the leading term.
Sentence 2: our approximation becomes worse with decreasing noise variance a 2 . a = 0 is a singular case for which (at least for some GPs with slowly decreasing eigenvalues) it can be shown that our approximation for eg decays to zero at the wrong rate.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Manifold regression can be interpreted as a vector-valued learning setting where outputs are constrained to be in C ⊆ R T and prediction errors are measured according to the geodesic distance.
Sentence 2: note that the two problems are also significantly different since, 1) in MTL noise could make output examples y i lie close but not exactly on the constraint set C and moreover, 2) the loss functions used in MTL typically measure errors independently for each task (as in Eq.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In Figures 1c and 1d we fix m and vary n. Observe in Figure 1c how KBRL and KBSF have similar performances, and both improve as n → ∞.
Sentence 2: since KBSF is using a model of fixed size, its computational cost depends only linearly on n, whereas KBRL's cost grows with n 3 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This behavior is captured by our robustness statistics-the baseline neural net has lower adversarial frequency (so it has fewer adversarial examples with ρ(f, x * ) ≤ ) but also has worse adversarial severity (since its adversarial examples are on average closer to the original points x * ).
Sentence 2: according to our metrics shown in Figure 3 (b), our neural net is substantially more robust than the baseline neural net.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Even adding the positive semi-definite constraint M 1 (µ) 0, for which TRW must be optimized using conditional gradient and semidefinite programming for the projection step, does not improve the accuracy by much.
Sentence 2: both entropy approximations give significantly better pseudomarginals when used by our algorithm together with the cycle inequalities (see "TRW + Cycle" and "Logdet + Cycle" in the figures).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Now we expect that after c 1 d log d more levels, the size should go down further by a factor of 2 thereby giving us our desired result.
Sentence 2: given the large number of nodes at this level and the fact that the success probability in Fact 2 is just greater than a constant bounded away from 1, it is not possible to argue that after c 1 d log d more levels the descendants of all these 2 c1d log d nodes will be of radius ∆/4 or less.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Hence, they are suited to analog VLSI implementation, as the main barrier to the implementation of on-chip learning, the long term storage of precise analog weight values, can be rendered unimportant.
Sentence 2: weight-independent STDP creates a highly unstable learning process that may hinder learning when only low levels of spike-timing correlations exist and neurons have few synapses.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The recursive grouping procedure is similar to the short quartet method from phylogenetics [15], which also guarantees efficient reconstruction in the context of evolutionary trees.
Sentence 2: our method and analysis applies to considerably more general high-dimensional settings; for instance, our sample complexity bound is given in terms of natural correlation con-ditions that generalize the more restrictive effective depth conditions of previous works [15,21].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The other extreme choice for the proposal, that has been considered in [10], is to apply Gibbs sampling where we iteratively draw samples from each posterior conditional density p(f i |f −i , y) with f −i = f \f i .
Sentence 2: gibbs sampling can be extremely slow for densely discretized functions, as in the regression problem of Figure 1, where the posterior GP process is highly correlated.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 6 means that if we route a query point x through the tree to its leaf cell A(x), then its nearest neighbor is likely to be in the same cell, which is quite appealing when trees are used for nearest-neighbor search.
Sentence 2: via Lemma 1, we can now show that such trees lead to inconsistent forests whenever we grow the trees deep and disable subsampling.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the ∞ attack model where l = x− 1 and u = x+ 1, we have s j = W x− W j 1 and t j = W x+ W j 1 for j = 1,2,...m. From Figure 2a, we see that for a single ReLU unit taken in isolation, the LP is tighter than the SDP.
Sentence 2: when we have multiple units, the SDP is tighter than the LP.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For each VQA v1 question, VQA v2 was constructed to also contain an image which is similar to the VQA v1 image, but has a different answer to the same question -effectively reducing the sharpness of question-only priors.
Sentence 2: even with this additional 'balancing' there exist significant biases in the dataset.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Figure 2 shows the power of the tests for the above setting with ρ = 1 and growing n. We find that when r = 2, that is, true rank is known, both bootstrap tests perform well under alternative hypothesis, and outperform Asymp-TW, although Boot-ASE has a high type-I error rate.
Sentence 2: when an over-estimate of rank is used (r = 4), both bootstrap tests break down -Boot-EPA always rejects while Boot-ASE always accepts -but the performance of Asymp-TW is robust to this parameter change.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Given a specific decoy, the problem of estimating in a real-life problem may seem difficult.
Sentence 2: as decoys are not new-even though the use we make of them here is-a number of methods [Heath and Chatterjee, 1995] have been designed to estimate the quality of a decoy, which is directly related to , and, with limited work, this parameter may be estimated as well.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In this case, the optimal solution for the relaxation is also an optimal solution for the original model, i.e., r-map ⋆ = map ⋆ .
Sentence 2: if a relaxation's optimal assignment sets X i and X j to different states, then it is not a valid assignment for the original model M, as it violates the equivalence constraint and thus has log probability −∞.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Similarly, work on target/conditional shift focuses on matching marginal distributions on labels Y and adjusting conditional distributions P (X|Y ), such that P (X) can be matched across domains.
Sentence 2: covariate shift assumes that the support of test P (X) is contained in the support of training P (X), i.e., the training set is richer than the test set.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Specifically, Deep Voice 1's segmentation model computes the output of each layer as where h (l) is the output of the l-th layer, W (l) is the convolution filterbank, b (l) is the bias vector, and ⇤ is the convolution operator.
Sentence 2: deep Voice 2's segmentation model layers instead compute where BN is batch normalization (Ioffe and Szegedy, 2015).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The obvious probabilistic approach to clustering genes would be to simply apply a Dirichlet process mixture (DPM) of Gaussians, but considering the genes (our dimensions) as samples, and our samples as "features" so that the partitioning would be over the genes.
Sentence 2: this approach would not achieve the desired result of clustering correlated variables, and would rather cluster together variables close in terms of Euclidean distance.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: [11,12] have recently suggested that short-term synaptic plasticity arranges for each local dendritic postsynaptic potential, v d i , to (approximately) represent the posterior mean of the corresponding presynaptic membrane potential: Thus, it would be tempting to say that in order to achieve the computational goal of Eq.
Sentence 2: 2, the way the dendritic tree (together with the soma) should integrate these local potentials, as given by f , should be directly determined by the function that needs to be computed: f = f . it is easy to see that in general this is going to be incorrect: where the l.h.s.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Readers who are familiar with DPP might identify the conditional distribution is also a DPP, restricted to the ground set Y t . The conditional probability is defined in such a way that at time t, the subset selected should be diverse among Y t as well as be diverse from previously selected y t−1 .
Sentence 2: beyond those two priors, the subset is not constrained by subsets selected in the distant past.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Although the variants of ADMM with fast convergence rates have been developed under smoothness, strong convexity and other regularity conditions (e.g., the matrix A has full rank), the best iteration complexities of deterministic ADMM and stochastic ADMM for general convex optimization remain O(1/ ) and O(1/ 2 ), respectively.
Sentence 2: many studies have reported that the performance of ADMM is very sensitive to the penalty parameter β.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To obtain the prediction on one test sample, Nyström approximation only needs O(md) flops to compute c, and O(m) flops to compute the decision value cβ, so it becomes an effective ways to improve the prediction speed.
Sentence 2: nyström approximation usually needs more than 100 landmark points to achieve reasonable good accuracy, which is still expensive for large-scale applications.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The entities which have visual properties, i.e., a yellow tennis suit and white tennis shoes in Figure 4a, and a denim shirt in Figure 4b, are correct.
Sentence 2: a relatively small object (e.g., a cigarette in Figure 4b) and the entity that requires semantic inference (e.g., a male conductor in Figure 4c) are incorrect.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We find iMedLDA is the most efficient, which benefits from (1) generateing samples outside the iteration loop and uses them for all iterations; and (2) using the MAP estimates to collapse the sample space of (Θ, Φ) to a "single sample" for efficiency.
Sentence 2: both gMedLDA and fMedLDA have to iteratively update the variables or variational parameters.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, the narrowestover-threshold (NOT) algorithm [1] is more suitable when different segments between the change points have comparable lengths, and the stepwise marginal likelihood (SML) method [5] works better to identify frequent change points.
Sentence 2: in practice it is often the case that distances between consecutive change points may vary dramatically, while only those with certain distance gaps are of interest.
Relationship: contrasting