NeurIPS 2018 论文列表

This paper studies the problem of sparse regression where the goal is to learn a sparse vector that best optimizes a given objective function. Under the assumption that the objective function satisfies restricted strong convexity (RSC), we analyze orthogonal matching pursuit (OMP), a greedy algorithm that is used heavily in applications, and obtain support recovery result as well as a tight generalization error bound for OMP. Furthermore, we obtain lower bounds for OMP, showing that both our results on support recovery and generalization error are tight up to logarithmic factors. To the best of our knowledge, these support recovery and generalization bounds are the first such matching upper and lower bounds (up to logarithmic factors) for {\em any} sparse regression algorithm under the RSC assumption.

We introduce a framework to transfer knowledge acquired from a repository of (heterogeneous) supervised datasets to new unsupervised datasets. Our perspective avoids the subjectivity inherent in unsupervised learning by reducing it to supervised learning, and provides a principled way to evaluate unsupervised algorithms. We demonstrate the versatility of our framework via rigorous agnostic bounds on a variety of unsupervised problems. In the context of clustering, our approach helps choose the number of clusters and the clustering algorithm, remove the outliers, and provably circumvent Kleinberg's impossibility result. Experiments across hundreds of problems demonstrate improvements in performance on unsupervised data with simple algorithms despite the fact our problems come from heterogeneous domains. Additionally, our framework lets us leverage deep networks to learn common features across many small datasets, and perform zero shot learning.

Generalization performance is a central goal in machine learning, particularly when learning representations with large neural networks. A common strategy to improve generalization has been through the use of regularizers, typically as a norm constraining the parameters. Regularizing hidden layers in a neural network architecture, however, is not straightforward. There have been a few effective layer-wise suggestions, but without theoretical guarantees for improved performance. In this work, we theoretically and empirically analyze one such model, called a supervised auto-encoder: a neural network that predicts both inputs (reconstruction error) and targets jointly. We provide a novel generalization result for linear auto-encoders, proving uniform stability based on the inclusion of the reconstruction error---particularly as an improvement on simplistic regularization such as norms or even on more advanced regularizations such as the use of auxiliary tasks. Empirically, we then demonstrate that, across an array of architectures with a different number of hidden units and activation functions, the supervised auto-encoder compared to the corresponding standard neural network never harms performance and can significantly improve generalization.

We study the problem of maximizing deep submodular functions (DSFs) subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use a strategy similar to the continuous greedy approach, but we show that the multilinear extension of any DSF has a natural and computationally attainable concave relaxation that we can optimize using gradient ascent. Our results show a guarantee of $\max_{0<\delta<1}(1-\epsilon-\delta-e^{-\delta^2\Omega(k)})$ with a running time of $O(\nicefrac{n^2}{\epsilon^2})$ plus time for pipage rounding to recover a discrete solution, where $k$ is the rank of the matroid constraint. This bound is often better than the standard $1-1/e$ guarantee of the continuous greedy algorithm, but runs much faster. Our bound also holds even for fully curved ($c=1$) functions where the guarantee of $1-c/e$ degenerates to $1-1/e$ where $c$ is the curvature of $f$. We perform computational experiments that support our theoretical results.

Natural scenes contain many layers of part-subpart structure, and distributions over them are thus naturally represented by stochastic image grammars, with one production per decomposition of a part. Unfortunately, in contrast to language grammars, where the number of possible split points for a production $A \rightarrow BC$ is linear in the length of $A$, in an image there are an exponential number of ways to split a region into subregions. This makes parsing intractable and requires image grammars to be severely restricted in practice, for example by allowing only rectangular regions. In this paper, we address this problem by associating with each production a submodular Markov random field whose labels are the subparts and whose labeling segments the current object into these subparts. We call the result a submodular field grammar (SFG). Finding the MAP split of a region into subregions is now tractable, and by exploiting this we develop an efficient approximate algorithm for MAP parsing of images with SFGs. Empirically, we present promising improvements in accuracy when using SFGs for scene understanding, and show exponential improvements in inference time compared to traditional methods, while returning comparable minima.

Let $\PP=\{ p_1, p_2, \ldots p_n \}$ and $\QQ = \{ q_1, q_2 \ldots q_m \}$ be two point sets in an arbitrary metric space. Let $\AA$ represent the $m\times n$ pairwise distance matrix with $\AA_{i,j} = d(p_i, q_j)$. Such distance matrices are commonly computed in software packages and have applications to learning image manifolds, handwriting recognition, and multi-dimensional unfolding, among other things. In an attempt to reduce their description size, we study low rank approximation of such matrices. Our main result is to show that for any underlying distance metric $d$, it is possible to achieve an additive error low rank approximation in sublinear time. We note that it is provably impossible to achieve such a guarantee in sublinear time for arbitrary matrices $\AA$, and our proof exploits special properties of distance matrices. We develop a recursive algorithm based on additive projection-cost preserving sampling. We then show that in general, relative error approximation in sublinear time is impossible for distance matrices, even if one allows for bicriteria solutions. Additionally, we show that if $\PP = \QQ$ and $d$ is the squared Euclidean distance, which is not a metric but rather the square of a metric, then a relative error bicriteria solution can be found in sublinear time. Finally, we empirically compare our algorithm with the SVD and input sparsity time algorithms. Our algorithm is several hundred times faster than the SVD, and about $8$-$20$ times faster than input sparsity methods on real-world and and synthetic datasets of size $10^8$. Accuracy-wise, our algorithm is only slightly worse than that of the SVD (optimal) and input-sparsity time algorithms.

Blind deconvolution is a ubiquitous problem of recovering two unknown signals from their convolution. Unfortunately, this is an ill-posed problem in general. This paper focuses on the {\em short and sparse} blind deconvolution problem, where the one unknown signal is short and the other one is sparsely and randomly supported. This variant captures the structure of the unknown signals in several important applications. We assume the short signal to have unit $\ell^2$ norm and cast the blind deconvolution problem as a nonconvex optimization problem over the sphere. We demonstrate that (i) in a certain region of the sphere, every local optimum is close to some shift truncation of the ground truth, and (ii) for a generic short signal of length $k$, when the sparsity of activation signal $\theta\lesssim k^{-2/3}$ and number of measurements $m\gtrsim\poly\paren{k}$, a simple initialization method together with a descent algorithm which escapes strict saddle points recovers a near shift truncation of the ground truth kernel.

Convolutional neural networks (CNNs) are inherently subject to invariable filters that can only aggregate local inputs with the same topological structures. It causes that CNNs are allowed to manage data with Euclidean or grid-like structures (e.g., images), not ones with non-Euclidean or graph structures (e.g., traffic networks). To broaden the reach of CNNs, we develop structure-aware convolution to eliminate the invariance, yielding a unified mechanism of dealing with both Euclidean and non-Euclidean structured data. Technically, filters in the structure-aware convolution are generalized to univariate functions, which are capable of aggregating local inputs with diverse topological structures. Since infinite parameters are required to determine a univariate function, we parameterize these filters with numbered learnable parameters in the context of the function approximation theory. By replacing the classical convolution in CNNs with the structure-aware convolution, Structure-Aware Convolutional Neural Networks (SACNNs) are readily established. Extensive experiments on eleven datasets strongly evidence that SACNNs outperform current models on various machine learning tasks, including image classification and clustering, text categorization, skeleton-based action recognition, molecular activity detection, and taxi flow prediction.

We study the problem of identifying the best action in a sequential decision-making setting when the reward distributions of the arms exhibit a non-trivial dependence structure, which is governed by the underlying causal model of the domain where the agent is deployed. In this setting, playing an arm corresponds to intervening on a set of variables and setting them to specific values. In this paper, we show that whenever the underlying causal model is not taken into account during the decision-making process, the standard strategies of simultaneously intervening on all variables or on all the subsets of the variables may, in general, lead to suboptimal policies, regardless of the number of interventions performed by the agent in the environment. We formally acknowledge this phenomenon and investigate structural properties implied by the underlying causal model, which lead to a complete characterization of the relationships between the arms' distributions. We leverage this characterization to build a new algorithm that takes as input a causal structure and finds a minimal, sound, and complete set of qualified arms that an agent should play to maximize its expected reward. We empirically demonstrate that the new strategy learns an optimal policy and leads to orders of magnitude faster convergence rates when compared with its causal-insensitive counterparts.

Several algorithms for solving constraint satisfaction problems are based on survey propagation, a variational inference scheme used to obtain approximate marginal probability estimates for variable assignments. These marginals correspond to how frequently each variable is set to true among satisfying assignments, and are used to inform branching decisions during search; however, marginal estimates obtained via survey propagation are approximate and can be self-contradictory. We introduce a more general branching strategy based on streamlining constraints, which sidestep hard assignments to variables. We show that streamlined solvers consistently outperform decimation-based solvers on random k-SAT instances for several problem sizes, shrinking the gap between empirical performance and theoretical limits of satisfiability by 16.3% on average for k = 3, 4, 5, 6.

We study the statistical and computational aspects of kernel principal component analysis using random Fourier features and show that under mild assumptions, $O(\sqrt{n} \log n)$ features suffices to achieve $O(1/\epsilon^2)$ sample complexity. Furthermore, we give a memory efficient streaming algorithm based on classical Oja's algorithm that achieves this rate

The state-of-the-art methods for solving optimization problems in big dimensions are variants of randomized coordinate descent (RCD). In this paper we introduce a fundamentally new type of acceleration strategy for RCD based on the augmentation of the set of coordinate directions by a few spectral or conjugate directions. As we increase the number of extra directions to be sampled from, the rate of the method improves, and interpolates between the linear rate of RCD and a linear rate independent of the condition number. We develop and analyze also inexact variants of these methods where the spectral and conjugate directions are allowed to be approximate only. We motivate the above development by proving several negative results which highlight the limitations of RCD with importance sampling.

Regularized empirical risk minimization problem with linear predictor appears frequently in machine learning. In this paper, we propose a new stochastic primal-dual method to solve this class of problems. Different from existing methods, our proposed methods only require O(1) operations in each iteration. We also develop a variance-reduction variant of the algorithm that converges linearly. Numerical experiments suggest that our methods are faster than existing ones such as proximal SGD, SVRG and SAGA on high-dimensional problems.

Tensor decompositions are fundamental tools for multiway data analysis. Existing approaches, however, ignore the valuable temporal information along with data, or simply discretize them into time steps so that important temporal patterns are easily missed. Moreover, most methods are limited to multilinear decomposition forms, and hence are unable to capture intricate, nonlinear relationships in data. To address these issues, we formulate event-tensors, to preserve the complete temporal information for multiway data, and propose a novel Bayesian nonparametric decomposition model. Our model can (1) fully exploit the time stamps to capture the critical, causal/triggering effects between the interaction events, (2) flexibly estimate the complex relationships between the entities in tensor modes, and (3) uncover hidden structures from their temporal interactions. For scalable inference, we develop a doubly stochastic variational Expectation-Maximization algorithm to conduct an online decomposition. Evaluations on both synthetic and real-world datasets show that our model not only improves upon the predictive performance of existing methods, but also discovers interesting clusters underlying the data.

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration. For smooth nonconvex functions, the proposed algorithm converges to an $\epsilon$-approximate first-order stationary point (i.e., $\|\nabla F(\mathbf{x})\|_2\leq \epsilon$) within $\tilde O(n\land \epsilon^{-2}+\epsilon^{-3}\land n^{1/2}\epsilon^{-2})$\footnote{$\tilde O(\cdot)$ hides the logarithmic factors, and $a\land b$ means $\min(a,b)$.} number of stochastic gradient evaluations. This improves the best known gradient complexity of SVRG $O(n+n^{2/3}\epsilon^{-2})$ and that of SCSG $O(n\land \epsilon^{-2}+\epsilon^{-10/3}\land n^{2/3}\epsilon^{-2})$. For gradient dominated functions, our algorithm also achieves better gradient complexity than the state-of-the-art algorithms. Thorough experimental results on different nonconvex optimization problems back up our theory.

Expectation-Maximization (EM) is a popular tool for learning latent variable models, but the vanilla batch EM does not scale to large data sets because the whole data set is needed at every E-step. Stochastic Expectation Maximization (sEM) reduces the cost of E-step by stochastic approximation. However, sEM has a slower asymptotic convergence rate than batch EM, and requires a decreasing sequence of step sizes, which is difficult to tune. In this paper, we propose a variance reduced stochastic EM (sEM-vr) algorithm inspired by variance reduced stochastic gradient descent algorithms. We show that sEM-vr has the same exponential asymptotic convergence rate as batch EM. Moreover, sEM-vr only requires a constant step size to achieve this rate, which alleviates the burden of parameter tuning. We compare sEM-vr with batch EM, sEM and other algorithms on Gaussian mixture models and probabilistic latent semantic analysis, and sEM-vr converges significantly faster than these baselines.

This paper proposes a stochastic variant of a classic algorithm---the cubic-regularized Newton method [Nesterov and Polyak]. The proposed algorithm efficiently escapes saddle points and finds approximate local minima for general smooth, nonconvex functions in only $\mathcal{\tilde{O}}(\epsilon^{-3.5})$ stochastic gradient and stochastic Hessian-vector product evaluations. The latter can be computed as efficiently as stochastic gradients. This improves upon the $\mathcal{\tilde{O}}(\epsilon^{-4})$ rate of stochastic gradient descent. Our rate matches the best-known result for finding local minima without requiring any delicate acceleration or variance-reduction techniques.

We study stochastic composite mirror descent, a class of scalable algorithms able to exploit the geometry and composite structure of a problem. We consider both convex and strongly convex objectives with non-smooth loss functions, for each of which we establish high-probability convergence rates optimal up to a logarithmic factor. We apply the derived computational error bounds to study the generalization performance of multi-pass stochastic gradient descent (SGD) in a non-parametric setting. Our high-probability generalization bounds enjoy a logarithmical dependency on the number of passes provided that the step size sequence is square-summable, which improves the existing bounds in expectation with a polynomial dependency and therefore gives a strong justification on the ability of multi-pass SGD to overcome overfitting. Our analysis removes boundedness assumptions on subgradients often imposed in the literature. Numerical results are reported to support our theoretical findings.

A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral function is generally of cubic complexity, as such gradient descent methods are rather expensive for optimizing objectives involving the spectral function. Thus, one naturally turns to stochastic gradient methods in hope that they will provide a way to reduce or altogether avoid the computation of full gradients. However, here a new challenge appears: there is no straightforward way to compute unbiased stochastic gradients for spectral functions. In this paper, we develop unbiased stochastic gradients for spectral-sums, an important subclass of spectral functions. Our unbiased stochastic gradients are based on combining randomized trace estimators with stochastic truncation of the Chebyshev expansions. A careful design of the truncation distribution allows us to offer distributions that are variance-optimal, which is crucial for fast and stable convergence of stochastic gradient methods. We further leverage our proposed stochastic gradients to devise stochastic methods for objective functions involving spectral-sums, and rigorously analyze their convergence rate. The utility of our methods is demonstrated in numerical experiments.