AISTATS 2019 论文列表

The parameter space of a deep neural network is a Riemannian manifold, where the metric is defined by the Fisher information matrix. The natural gradient method uses the steepest descent direction in a Riemannian manifold, but it requires inversion of the Fisher matrix, however, which is practically difficult. The present paper uses statistical neurodynamical method to reveal the properties of the Fisher information matrix in a net of random connections. We prove that the Fisher information matrix is unit-wise block diagonal supplemented by small order terms of off-block-diagonal elements. We further prove that the Fisher information matrix of a single unit has a simple reduced form, a sum of a diagonal matrix and a rank 2 matrix of weight-bias correlations. We obtain the inverse of Fisher information explicitly. We then have an explicit form of the approximate natural gradient, without relying on the matrix inversion.

We consider the problem where an agent wants to find a hidden object that is randomly located in some vertex of a directed acyclic graph (DAG) according to a fixed but possibly unknown distribution. The agent can only examine vertices whose in-neighbors have already been examined. In this paper, we address a learning setting where we allow the agent to stop before having found the object and restart searching on a new independent instance of the same problem. Our goal is to maximize the total number of hidden objects found given a time budget. The agent can thus skip an instance after realizing that it would spend too much time on it. Our contributions are both to the search theory and multi-armed bandits. If the distribution is known, we provide a quasi-optimal and efficient stationary strategy. If the distribution is unknown, we additionally show how to sequentially approximate it and, at the same time, act near-optimally in order to collect as many hidden objects as possible.

This paper is concerned with a feature subset selection problem for the multinomial logit (MNL) model. There are several convex approximation algorithms for this problem, but to date the only exact algorithms are those for the binomial logit model. In this paper, we propose an exact algorithm to solve the problem for the MNL model. Our algorithm is based on a mixed-integer optimization approach with an outer approximation method. We prove the convergence properties of the algorithm for more general models including generalized linear models for multiclass classification. We also propose approximation of loss functions to accelerate the algorithm computationally. Numerical experiments demonstrate that our exact and approximation algorithms achieve better generalization performance than does an L1-regularization method.

SPIDER (Stochastic Path Integrated Differential EstimatoR) is an efficient gradient estimation technique developed for non-convex stochastic optimization. Although having been shown to attain nearly optimal computational complexity bounds, the SPIDER-type methods are limited to linear metric spaces. In this paper, we introduce the Riemannian SPIDER (R-SPIDER) method as a novel nonlinear-metric extension of SPIDER for efficient non-convex optimization on Riemannian manifolds. We prove that for finite-sum problems with $n$ components, R-SPIDER converges to an $\epsilon$-accuracy stationary point within $\mathcal{O}\big(\min\big(n+\frac{\sqrt{n}}{\epsilon^2},\frac{1}{\epsilon^3}\big)\big)$ stochastic gradient evaluations, which is sharper in magnitude than the prior Riemannian first-order methods. For online optimization, R-SPIDER is shown to converge with $\mathcal{O}\big(\frac{1}{\epsilon^3}\big)$ complexity which is, to the best of our knowledge, the first non-asymptotic result for online Riemannian optimization. Especially, for gradient dominated functions, we further develop a variant of R-SPIDER and prove its linear convergence rate. Numerical results demonstrate the computational efficiency of the proposed methods.

We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.

Modern machine learning focuses on highly expressive models that are able to fit or interpolate the data completely, resulting in zero training loss. For such models, we show that the stochastic gradients of common loss functions satisfy a strong growth condition. Under this condition, we prove that constant step-size stochastic gradient descent (SGD) with Nesterov acceleration matches the convergence rate of the deterministic accelerated method for both convex and strongly-convex functions. We also show that this condition implies that SGD can find a first-order stationary point as efficiently as full gradient descent in non-convex settings. Under interpolation, we further show that all smooth loss functions with a finite-sum structure satisfy a weaker growth condition. Given this weaker condition, we prove that SGD with a constant step-size attains the deterministic convergence rate in both the strongly-convex and convex settings. Under additional assumptions, the above results enable us to prove an $O(1/k^2)$ mistake bound for $k$ iterations of a stochastic perceptron algorithm using the squared-hinge loss. Finally, we validate our theoretical findings with experiments on synthetic and real datasets.

Composite convex optimization problems which include both a nonsmooth term and a low-rank promoting term have important applications in machine learning and signal processing, such as when one wishes to recover an unknown matrix that is simultaneously low-rank and sparse. However, such problems are highly challenging to solve in large-scale: the low-rank promoting term prohibits efficient implementations of proximal methods for composite optimization and even simple subgradient methods. On the other hand, methods which are tailored for low-rank optimization, such as conditional gradient-type methods, which are often applied to a smooth approximation of the nonsmooth objective, are slow since their runtime scales with both the large Lipchitz parameter of the smoothed gradient vector and with $1/\epsilon$, where $\epsilon$ is the target accuracy. In this paper we develop efficient algorithms for \textit{stochastic} optimization of a strongly-convex objective which includes both a nonsmooth term and a low-rank promoting term. In particular, to the best of our knowledge, we present the first algorithm that enjoys all following critical properties for large-scale problems: i) (nearly) optimal sample complexity, ii) each iteration requires only a single \textit{low-rank} SVD computation, and iii) overall number of thin-SVD computations scales only with $\log{1/\epsilon}$ (as opposed to $\textrm{poly}(1/\epsilon)$ in previous methods). We also give an algorithm for the closely-related finite-sum setting. We empirically demonstrate our results on the problem of recovering a simultaneously low-rank and sparse matrix.

Parameter identification and comparison of dynamical systems is a challenging task in many fields. Bayesian approaches based on Gaussian process regression over time-series data have been successfully applied to infer the parameters of a dynamical system without explicitly solving it. While the benefits in computational cost are well established, the theoretical foundation has been criticized in the past. We offer a novel interpretation which leads to a better understanding, improvements in state-of-the-art performance in terms of accuracy and robustness and a decrease in run time due to a more efficient setup for general nonlinear dynamical systems.

We consider a reformulation of Reduced-Rank Regression (RRR) and Sparse Reduced-Rank Regression (SRRR) as a non-convex non-differentiable function of a single of the two matrices usually introduced to parametrize low-rank matrix learning problems. We study the behavior of proximal gradient algorithms for the minimization of the objective. In particular, based on an analysis of the geometry of the problem, we establish that a proximal Polyak-{Ł}ojasiewicz inequality is satisfied in a neighborhood of the set of optima under a condition on the regularization parameter. We can consequently derive linear convergence rates for the proximal gradient descent with line search and for related algorithms in a neighborhood of the optima. Our experiments show that our formulation leads to much faster learning algorithms for RRR and especially for SRRR.

Mixture of exponential family models are among the most fundamental and widely used statistical models. Stochastic variational inference (SVI), the state-of-the-art algorithm for parameter estimation in such models is inherently serial. Moreover, it requires the parameters to fit in the memory of a single processor; this poses serious limitations on scalability when the number of parameters is in billions. In this paper, we present extreme stochastic variational inference (ESVI), a distributed, asynchronous and lock-free algorithm to perform variational inference for mixture models on massive real world datasets. ESVI overcomes the limitations of SVI by requiring that each processor only access a subset of the data and a subset of the parameters, thus providing data and model parallelism simultaneously. Our empirical study demonstrates that ESVI not only outperforms VI and SVI in wallclock-time, but also achieves a better quality solution. To further speed up computation and save memory when fitting large number of topics, we propose a variant ESVI-TOPK which maintains only the top-k important topics. Empirically, we found that using top 25% topics suffices to achieve the same accuracy as storing all the topics.

Normalization techniques such as Batch Normalization have been applied very successfully for training deep neural networks. Yet, despite its apparent empirical benefits, the reasons behind the success of Batch Normalization are mostly hypothetical. We here aim to provide a more thorough theoretical understanding from a classical optimization perspective. Our main contribution towards this goal is the identification of various problem instances in the realm of machine learning where Batch Normalization can provably accelerate optimization. We argue that this acceleration is due to the fact that Batch Normalization splits the optimization task into optimizing length and direction of the parameters separately. This allows gradient-based methods to leverage a favourable global structure in the loss landscape that we prove to exist in Learning Halfspace problems and neural network training with Gaussian inputs. We thereby turn Batch Normalization from an effective practical heuristic into a provably converging algorithm for these settings. Furthermore, we substantiate our analysis with empirical evidence that suggests the validity of our theoretical results in a broader context.

We study a general online linear optimization problem(OLO). At each round, a subset of objects from a fixed universe of $n$ objects is chosen, and a linear cost associated with the chosen subset is incurred. To measure the performance of our algorithms, we use the notion of regret which is the difference between the total cost incurred over all iterations and the cost of the best fixed subset in hindsight. We consider Full Information and Bandit feedback for this problem. This problem is equivalent to OLO on the $\{0,1\}^n$ hypercube. The Exp2 algorithm and its bandit variant are commonly used strategies for this problem. It was previously unknown if it is possible to run Exp2 on the hypercube in polynomial time. In this paper, we present a polynomial time algorithm called PolyExp for OLO on the hypercube. We show that our algorithm is equivalent Exp2 on $\{0,1\}^n$, Online Mirror Descent(OMD), Follow The Regularized Leader(FTRL) and Follow The Perturbed Leader(FTPL) algorithms. We show PolyExp achieves expected regret bound that is a factor of $\sqrt{n}$ better than Exp2 in the full information setting under $L_\infty$ adversarial losses. Because of the equivalence of these algorithms, this implies an improvement on Exp2’s regret bound in full information. We also show matching regret lower bounds. Finally, we show how to use PolyExp on the $\{-1,+1\}^n$ hypercube, solving an open problem in Bubeck et al (COLT 2012).

The communication overhead has become a significant bottleneck in data-parallel network with the increasing of model size and data samples. In this work, we propose a new algorithm LPC-SVRG with quantized gradients and its acceleration ALPC-SVRG to effectively reduce the communication complexity while maintaining the same convergence as the unquantized algorithms. Specifically, we formulate the heuristic gradient clipping technique within the quantization scheme and show that unbiased quantization methods in related works [3, 33, 38] are special cases of ours. We introduce double sampling in the accelerated algorithm ALPC-SVRG to fully combine the gradients of full-precision and low-precision, and then achieve acceleration with fewer communication overhead. Our analysis focuses on the nonsmooth composite problem, which makes our algorithms more general. The experiments on linear models and deep neural networks validate the effectiveness of our algorithms.

This paper studies the problem of identifying any $k$ distinct arms among the top $\rho$ fraction (e.g., top 5%) of arms from a finite or infinite set with a probably approximately correct (PAC) tolerance $\epsilon$. We consider two cases: (i) when the threshold of the top arms’ expected rewards is known and (ii) when it is unknown. We prove lower bounds for the four variants (finite or infinite arms, and known or unknown threshold), and propose algorithms for each. Two of these algorithms are shown to be sample complexity optimal (up to constant factors) and the other two are optimal up to a log factor. Results in this paper provide up to $\rho n/k$ reductions compared with the “$k$-exploration” algorithms that focus on finding the (PAC) best $k$ arms out of $n$ arms. We also numerically show improvements over the state-of-the-art.

Probabilistic classifiers output a probability distribution on target classes rather than just a class prediction. Besides providing a clear separation of prediction and decision making, the main advantage of probabilistic models is their ability to represent uncertainty about predictions. In safety-critical applications, it is pivotal for a model to possess an adequate sense of uncertainty, which for probabilistic classifiers translates into outputting probability distributions that are consistent with the empirical frequencies observed from realized outcomes. A classifier with such a property is called calibrated. In this work, we develop a general theoretical calibration evaluation framework grounded in probability theory, and point out subtleties present in model calibration evaluation that lead to refined interpretations of existing evaluation techniques. Lastly, we propose new ways to quantify and visualize miscalibration in probabilistic classification, including novel multidimensional reliability diagrams.

We develop a general method for estimating a finite mixture of non-normalized models. A non-normalized model is defined to be a parametric distribution with an intractable normalization constant. Existing methods for estimating non-normalized models without computing the normalization constant are not applicable to mixture models because they contain more than one intractable normalization constant. The proposed method is derived by extending noise contrastive estimation (NCE), which estimates non-normalized models by discriminating between the observed data and some artificially generated noise. In particular, the proposed method provides a probabilistically principled clustering method that is able to utilize a deep representation. Applications to clustering of natural images and neuroimaging data give promising results.

Multivariate Bernoulli autoregressive (BAR) processes model time series of events in which the likelihood of current events is determined by the times and locations of past events. These processes can be used to model nonlinear dynamical systems corresponding to criminal activity, responses of patients to different medical treatment plans, opinion dynamics across social networks, epidemic spread, and more. Past work examines this problem under the assumption that the event data is complete, but in many cases only a fraction of events are observed. Incomplete observations pose a significant challenge in this setting because the unobserved events still govern the underlying dynamical system. In this work, we develop a novel approach to estimating the parameters of a BAR process in the presence of unobserved events via an unbiased estimator of the complete data log-likelihood function. We propose a computationally efficient estimation algorithm which approximates this estimator via Taylor series truncation and establish theoretical results for both the statistical error and optimization error of our algorithm. We further justify our approach by testing our method on both simulated data and a real data set consisting of crimes recorded by the city of Chicago.

We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions – which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation scaling as $(k\ast/n)\log(p/k\ast)$: n,p is the size of the design matrix and k* the dimension of the theoretical loss minimizer. This is done under standard assumptions, for which we derive stronger versions of a cone condition and a restricted strong convexity. Our bound holds with high probability and in expectation and applies to an L1-regularized estimator and to a recently introduced Slope estimator, which we generalize for classification problems. Slope presents the advantage of adapting to unknown sparsity. Thus, we propose a tractable proximal algorithm to compute it and assess its empirical performance. Our results match the best existing bounds for classification and regression problems.

Empirical risk minimization is the main tool for prediction problems, but its extension to relational data remains unsolved. We solve this problem using recent ideas from graph sampling theory to (i) define an empirical risk for relational data and (ii) obtain stochastic gradients for this empirical risk that are automatically unbiased. This is achieved by considering the method by which data is sampled from a graph as an explicit component of model design. By integrating fast implementations of graph sampling schemes with standard automatic differentiation tools, we provide an efficient turnkey solver for the risk minimization problem. We establish basic theoretical properties of the procedure. Finally, we demonstrate relational ERM with application to two non-standard problems: one-stage training for semi-supervised node classification, and learning embedding vectors for vertex attributes. Experiments confirm that the turnkey inference procedure is effective in practice, and that the sampling scheme used for model specification has a strong effect on model performance.