sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In prior research, the sample complexity of revenue maximization has been studied primarily in the single-item or the more general single-dimensional settings [Elkind, 2007, Cole and Roughgarden, 2014, Huang et al., 2015, Medina and Mohri, 2014, Morgenstern and Roughgarden, 2015, Roughgarden and Schrijvers, 2016, Devanur et al., 2016, as well as some multi-dimensional settings which are reducible to the single-bidder setting [Morgenstern and Roughgarden, 2016].
Sentence 2: the combinatorial settings that we study are much more complex since the revenue functions consist of multi-stage optimization procedures that cannot be reduced to a single-bidder setting.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Comparison to [1] Our bound (2.4) improves a quadratic dependence on µ 2 to a linear dependence and replaces the dependence on the average squared smoothness EL 2 i with a linear dependence on the smoothness bound sup L. When all Lipschitz constants L i are of similar magnitude, this is a quadratic improvement in the number of required iterations.
Sentence 2: when different components f i have widely different scaling, i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: ( 4), with a translation-invariance factor that decays with σ n . The dependence on a notion of depth (the number of layers n here) also appears in [17], with a factor equal to the maximal length of scattering paths, and with the same condition ∇τ ∞ ≤ 1/2.
Sentence 2: while the norm of the scattering representation is preserved as the length of these paths goes to infinity, the norm of Φ(x) can decrease with depth due to pooling layers, though this concern may be alleviated by using an additional non-linear prediction layer, as discussed previously (see also [3]).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, λ is usually learnt from data through cross-validation or through Lepski's method [24] for which guarantees similar to the one presented in Theorem 5 can be provided.
Sentence 2: irrespective of the data-dependent/independent choice for λ, checking for the admissibility of Spectral-KMSE (similar to the one in Theorem 1) is very difficult and we intend to consider it in future work.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: CTC [5] is a popular framework for end-to-end sequence learning tasks, such as speech recognition [6,8,1,2,22], scene text recognition [29,19,7], sign language recognition [4] and video segmentation [10,18].
Sentence 2: cTC tends to output highly peaky distribution [5,22], which is a sign for model overconfidence.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, v is increased by each presynaptic spike, and in the absence of spikes it decays to its resting value, v 0 , with membrane time constant τ .
Sentence 2: the effect of each spike on v is scaled by x which itself is decreased after each spike and increases between spikes back towards one with time constant τ D . Thus, the postsynaptic potential, v, behaves much like the posterior mean of the optimal estimator, µ, while the dynamics of the synaptic resource variable, x, closely resemble that of the posterior variance of the optimal estimator, σ 2 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This means as f is allowed to deviate further from F , we must smooth over a larger radius-choose a larger σ-to guarantee the same smoothness as before.
Sentence 2: part (2) implies that choosing a large σ increases the upper bound on the gradient and Hessian difference between fσ and F . Smoothing is a form of local averaging, so choosing a too-large radius will erase information about local geometry.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For instance, HPN tracks do not move towards macro-goals in unrealistic straight lines, but often take a curved route, indicating that the policy balances moving towards macro-goals with short-term responses to the current state (see also Figure 6b).
Sentence 2: the baseline model often generates more constrained behavior, such as moving in straight lines or remaining stationary for long periods of time.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the experiments we discuss in the following, we set the hyperparameters as u The Bayesian method calculates p(θ, Z|X, M ), which is a posterior of unknown parameters and hidden variable set Z = (Y, S) given the data and model structure, M (in our case, this indicates the number of hidden states, and correlation structure).
Sentence 2: the calculation of the posterior involves a difficult integral.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 7 to contribute equally during updating.
Sentence 2: since the dimension of the data is typically much greater than the number of classes, the dynamics of the conditional herding system will be largely driven by W. To negate this effect, we rescaled the standard deviation of the Gaussian by a factor 1/M with M the total number of elements of the parameter involved (e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For other matrices such that B¨c £ B § c is a linear function of B¨c (such as the one-against-all matrix), we can prove that loss-based decoding (with L y £ ay) predicts the same class as the non-iterative estimates.
Sentence 2: in this case, the non-iterative estimates do not necessarily predict the same class as the iterative ones.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: With the added layers for the classifier, the conjugate-exponential character of the model is retained, sustaining the ability to perform VB or MCMC inference with analytic update equations.
Sentence 2: note that the model in (2) may be employed for unlabeled data, and the extension above may be employed for the available labeled data; consequently, all data (labeled and unlabeled) may be processed jointly to infer D. In the above discussion, we implicitly assumed all data D = {x i , y i } i=1,N are used together to infer the dictionary D. in some applications N may be large, and therefore such a "batch" approach is undesirable.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: An ideal observer who knows the inertia perfectly will greatly benefit from displays with high inertia in which uncertainty will be low, and will be able to track with the same level of accuracy at greater speeds given a particular spacing.
Sentence 2: if inertia is incorrectly assumed to be zero, high-and low-inertia iso-difficulty contours will be quite similar (Figure 3).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Many systems-such as Factorie [14], OpenBugs [12], PGibbs [5], DimmWitted [28], and others [15,22,25]-use Gibbs sampling for inference because it is fast to run, simple to implement, and often produces high quality empirical results.
Sentence 2: theoretical guarantees about Gibbs are lacking.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The core idea is to align distributions of points (in the feature space) belonging to same concept class across the two domains (i.e., aligning g#P s (•|y) and g#P t (•|y) where g is a measurable feature generator mapping and g#P denotes the push-forward of a distribution P ), and the prediction performance in target domain directly depends on the correctness of this alignment.
Sentence 2: the right alignment of class conditional distributions can be challenging to achieve without access to any labels in the target domain.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Another closely related problem is learning to hash, which includes BoostSSC [20], Spectral Hashing [26], Restricted Boltzmann Machines [19], Semi-Supervised Hashing [24], Hashing with Graphs [14], etc.
Sentence 2: the motivation of hashing problem is fundamentally different from that of the search problem considered in this paper.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Lastly, there are some recent works that learn hierarchical graph representations by combining GNNs with deterministic graph clustering algorithms [8,35,13], following a two-stage approach.
Sentence 2: unlike these previous approaches, we seek to learn the hierarchical structure in an end-to-end fashion, rather than relying on a deterministic graph clustering subroutine.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Their complexity bound is of the order of (1/ε) 2+log κ/(log(1/γ))+o (1) , where κ ∈ [0, KN ] is a problem-dependent quantity.
Sentence 2: their κ defined as lim ε→0 max(κ 1 , κ 2 ) (in their Theorem 2) is somehow difficult to interpret as a measure of the quantity of near-optimal nodes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Rather than making updates on each step to improve the estimate, these methods maintain compact summaries of all observed state transitions and rewards and solve for the value function which has zero expected TD error over the observed data.
Sentence 2: although LSTD and LSTD(λ) make more efficient use of the data, they require O(n 2 ) computation per time step, which is often impractical for the large feature sets needed in many applications.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Many machine learning problems have been shown to benefit from the negative dependence properties of Strongly Rayleigh measures: measures based on elementary symmetric polynomials -including (dual) volume sampling -have been applied to experimental design; DPPs have been applied successfully to fields ranging from automatic summarization to minibatch selection and neural network pruning.
Sentence 2: tuning the strength of the quality/diversity tradeoff of SR measures requires significant effort.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In particular, the non-zero elements of the optimal sparse factor x are exactly equal to the elements of u * k , the principal eigenvector of A * k . This underscores the inherent combinatorial nature of sparse PCA and the equivalent class of cardinality-constrained optimization problems.
Sentence 2: despite providing an exact formulation and revealing necessary conditions for optimality (and in such simple matrix terms), this proposition does not suggest an efficient method for actually finding the principal submatrix A * k -short of an enumerative exhaustive search, which is impractical for n > 30 due to the exponential growth of possible submatrices.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The main challenge in using the curriculum learning strategy is that it requires the identification of easy and hard samples in a given training dataset.
Sentence 2: in many real-world applications, such a ranking of training samples may be onerous or conceptually difficult for a human to provide -even if this additional human supervision can be provided, what is intuitively "easy" for a human may not match what is easy for the algorithm in the feature and hypothesis space employed for the given application.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These two quantities are independent for a centered multivariate normal, and therefore if P L µ = 0 then P L Y 2 follows a χ k distribution even if we have conditioned on dir L (Y ).
Sentence 2: in the general case where P L µ = 0, we do not have independence between the length and the direction of P L Y , and so while P L Y 2 is marginally distributed as a noncentral χ k , this is no longer true after conditioning on dir L (Y ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We plot the empirical success probability as a function of k/ √ n for several values of p, with p = n. The empirical success probability was computed by using 100 independent instances of the problem.
Sentence 2: a few observations are of interest: (i) Covariance Thresholding appears to have a significantly larger success probability in the 'difficult' regime where Diagonal Thresholding starts to fail; (ii) The curves for Diagonal Thresholding appear to decrease monotonically with p indicating that k proportional to √ n is not the right scaling for this algorithm (as is known from theory); (iii) the curves for Covariance Thresholding become steeper for larger p, and, in particular, the success probability increases with p for k ≤ 1.1 √ n. This indicates a sharp threshold for k = const • √ n, as suggested by our theory.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: (2016) later improved this model by simplifying the binarization by simply taking T B = sign(T ) and adding a gradient for this operation, namely the straight-through estimator: The authors showed that the straight-through estimator allowed the binarization of activations as well as weights without collapse of model performance.
Sentence 2: they did not attempt to train a model on ImageNet in this work.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: With this setting of RMSPROP and batch size of 320, we obtained a top-1 and top-5 error of 21.63% and 5.79% respectively, under single crop evaluation on the Imagenet validation set comprising of 50K images (slightly lower than the published results).
Sentence 2: using YOGI with essentially the default parameters (i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We encode the manifold smoothness into the empirical data distribution used in boosting algorithms for semi-supervised learning, while their implementation adaptively adjusts the edge offset in the marginal Adaboost algorithm for a weight decay used in the linear combination of based learners [19].
Sentence 2: our implementation is simpler yet applicable to any boosting algorithms for semi-supervised learning, while theirs needs to be fulfilled via the marginal Adaboost algorithm even though their regularized marginal Adaboost is applicable to both supervised and semi-supervised learning indeed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: More recentely, frameworks such as conditional Generative Adversarial Networks (cGAN) [12], Style and Structure Generative Adversarial Network (S 2 -GAN) [30], and VAE-GAN [17] are proposed to address the problem of supervised imageto-image translation.
Sentence 2: in many real-world applications, collecting paired training data is laborious and expensive [37].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Benchmark for video tasks There exist benchmarks for several video tasks like online object tracking [29] and video object segmentation [21].
Sentence 2: there is no benchmark for the precipitation nowcasting problem, which is also a video task but has its unique properties since radar echo map is a completely different type of data and the data is highly imbalanced (as mentioned in Section 1).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The integral in ( 12) is of the form ( 8), but is intractable since solving it corresponds to a GP prediction at uncertain inputs [14], resulting in non-Gaussian predictive distributions.
Sentence 2: the mean and covariance of a Gaussian approximation Z to Z can be computed analytically: either using exact moment matching [14,3], or approximately by expected linearization of the posterior GP [8]; details are given in [4].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: While these results are not directly comparable, we have also first results where we ran the original implementation with value iteration and our implementation with Minimax LPA* on a very similar environment and the original implementation expanded one to two orders of magnitude more states than ours even though its number of searches and its final number of states was smaller.
Sentence 2: these results are very preliminary since the time per state expansion is different for the different implementations and it is future work to compare the various implementations of the parti-game algorithm in a common testbed.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Hence, the scalar product p (γ(t)), γ (t) coincides with the directional derivative of p in direction T , so the right hand side of Equation ( 1) coincides with the left hand side of the equation in the proposition.
Sentence 2: it is well known that the line integral over a gradient field only depends on the starting and end point of γ and is given by This coincides with the right hand side of the equation in the proposition.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For convex problems, an alternative to proximal splitting is the subgradient method; similarly, for nonconvex problems one may use a generalized subgradient method [7,12].
Sentence 2: as in the convex case, the use of subgradients has drawbacks: it fails to exploit the composite structure, and even when using sparsity promoting regularizers it does not generate intermediate sparse iterates [11].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, one has to resort to "nonparametric" methods and search for good estimators in a suitably defined class whose complexity grows with n. Some nonparametric methods proposed in the literature, such as kernels [4] and orthogonal expansions [5,6], either have very slow rates of MSE convergence or are computationally prohibitive for large d. For example, the kernel method [4] requires O(n 2 d) operations to compute the estimate at any x ∈ B d , yet its MSE decays as O(n −4/(4+d) ) [7], which is extremely slow when d is large.
Sentence 2: orthogonal function methods generally have much better MSE decay rates, but rely on estimating 2 d coefficients in a fixed basis, which requires enormous computational resources for large d. For instance, using the Fast Hadamard Transform to estimate the coefficients in the so-called Walsh basis using n samples requires O(nd2 d ) operations [5].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Therefore, not all the pursuit methods, such as basis pursuit and LARS/Homotopy methods, that are applicable to the standard sparse representation method can be directly applied to optimize J 2 (X, λ) and J 3 (X, λ 1 , λ 2 ).
Sentence 2: the iterative optimization methods employed in the matching pursuit and the orthogonal matching pursuit provide a direct reference to the optimization of J 2 (X, λ) and J 3 (X, λ 1 , λ 2 ).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact, one can assume a generalized noise model X T j1 X i1 with probability p P ij with probability 1 − p where X i1 , 1 ≤ i ≤ n are pre-defined underlying permutations from object S i to the first object S 1 .
Sentence 2: since P ij are independent of X i1 .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In order to compute an (1 + ε) approximation to the k-segment mean of the original signal P , it suffices to compute a (1 + ε) approximation to the k-segment mean of the coreset, where cost is replaced by cost .
Sentence 2: since D is not a simple signal, but a more involved data structure, it is not clear how to run existing algorithms on D. In the supplementary material we show how to apply such algorithms on our coresets.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, in [2], the Laplacian kernel k(x, y) = exp(−γ∥x − y∥ 1 ) can be decomposed through both the Fourier basis and the Random Binning basis, while in [7], the Laplacian kernel can be obtained from the integrating of an infinite number of decision trees.
Sentence 2: multiple kernels can be derived from the same set of basis functions via different distribution p(h).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This can be visualized by noting that the label image deforms in the same way as the input image, as show for example in figure 3.
Sentence 2: for learning, constraint (1) can be incorporated in a loss function as follows: minimizing this loss has the significant drawback that a global optimum is obtained by simply setting Φ(x, u) = const.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: One might be tempted to conclude that the causal relation is non-recoverable because P (w, z, y) is non-recoverable (by theorem 2) and P (y|z, w) is not recoverable (by corollary 2).
Sentence 2: p (y|do(z)) is recoverable as demonstrated below: P (y|do(z), w, R ′ y ) = P (y|z, w, R ′ y ) (by Rule-2 of do-calculus (Pearl [2009])) (3) Substituting ( 3) and ( 4) in (2) we get: The recoverability of P (y|do(z)) in the previous example follows from the notion of d*separability and dormant independence [Shpitser and Pearl, 2008].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In short, these kinds of distributed representations avoid the binding problem in the first place by ensuring that relevant conjunctions are encoded, instead of representing different features using entirely separate, localist units (which is what gives rise to binding problems in the first place).
Sentence 2: this form of binding cannot rapidly encode novel bindings required for specific tasksthe phonological loop mechanism can thus complement the basic cortical mechanism by providing flexible, transient bindings on an ad-hoc basis.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: It is useful to rewrite the matching probabilities as: Adding E(π 1:t−1 , θ) to each item's energy does not change the probabilities because this term cancels out during normalization (but it does change the partition function, denoted by Z * t here).
Sentence 2: in this form we see that p(v j |u σ(t) , π 1:t−1 ) is directly related to the change in the partial energy from π 1:t−1 to H(v j , u σ(t) , π 1:t−1 ) -the larger the change the bigger the resulting probability will be.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In fact Kalai and Vempala [2003], address the computational difficulty of handling a large number of experts under certain linear assumptions on the reward functions.
Sentence 2: their algorithm is not directly applicable to our setting, due to the fact that in our setting, decisions must be made with respect to the current state of the agent (and the reward could be changing frequently), while in their setting the decisions are only made with respect to a single state.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: However, optimizing these weights freely is not permissible, since this procedure allows us to choose d i = d j even if s (i) = s (j) , which violates the weights in the original LMI.
Sentence 2: if we additionally require that s (i) = s (j) ⇒ d i = d j (or more appropriately for continuous features and states, for example that d i − d j → 0 as φ(s (i) ) − φ(s (j) ) → 0 according to some norm) then we are free to optimize over these empirical distribution weights.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Many alternative methods (e.g., Topographic ICA) would require some form of distributed optimization algorithm to achieve a similar result, which requires many feed-forward and feed-back passes over the dataset.
Sentence 2: the above method is typically less expensive than a single feed-forward pass (to compute the feature values x (i) ) and is thus very fast compared to other conceivable solutions.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Considering d as the target dimension and d i × d i as the convolutional filter size of the i-th layer, the weight matrix would be inaccurately reshaped in the case of A more reasonable approach would be to resize the DCT coefficient matrices of convolutional filters in the frequency domain, because high-frequency coefficients are generally small and discarding them only has a small impact on the result (d i > d).
Sentence 2: the introduced zeros will be immediately compressed by CSR since we do not need to encode or store them (d i < d).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 4 This theorem, which can be viewed as the tail bound version of the expected bound in [11], implies that the risk of the average hypothesis is close to 3 $ ¡ 8£ for "most" samples 3 .
Sentence 2: note that it is unlikely that ¤ 3 fe F6 P @Q gi 6 $ ¡ 8£ concentrates around T vV h e ¡ ¤£ , at least without taking strong assumptions on the underlying on-line algorithm.
Relationship: contrasting