Collaborating Authors

Xiao, Lin

Understanding the Role of Momentum in Stochastic Gradient Methods

Neural Information Processing Systems

The use of momentum in stochastic gradient methods has become a widespread practice in machine learning. Different variants of momentum, including heavy-ball momentum, Nesterov's accelerated gradient (NAG), and quasi-hyperbolic momentum (QHM), have demonstrated success on various tasks. Despite these empirical successes, there is a lack of clear understanding of how the momentum parameters affect convergence and various performance measures of different algorithms. In this paper, we use the general formulation of QHM to give a unified analysis of several popular algorithms, covering their asymptotic convergence conditions, stability regions, and properties of their stationary distributions. In addition, by combining the results on convergence rates and stationary distributions, we obtain sometimes counter-intuitive practical guidelines for setting the learning rate and momentum parameters.

Using Statistics to Automate Stochastic Optimization

Neural Information Processing Systems

Despite the development of numerous adaptive optimizers, tuning the learning rate of stochastic gradient methods remains a major roadblock to obtaining good practical performance in machine learning. Rather than changing the learning rate at each iteration, we propose an approach that automates the most common hand-tuning heuristic: use a constant learning rate until "progress stops," then drop. We design an explicit statistical test that determines when the dynamics of stochastic gradient descent reach a stationary distribution. This test can be performed easily during training, and when it fires, we decrease the learning rate by a constant multiplicative factor. Our experiments on several deep learning tasks demonstrate that this statistical adaptive stochastic approximation (SASA) method can automatically find good learning rate schedules and match the performance of hand-tuned methods using default settings of its parameters.

A Stochastic Composite Gradient Method with Incremental Variance Reduction

Neural Information Processing Systems

We consider the problem of minimizing the composition of a smooth (nonconvex) function and a smooth vector mapping, where the inner mapping is in the form of an expectation over some random variable or a finite sum. We propose a stochastic composite gradient method that employs incremental variance-reduced estimators for both the inner vector mapping and its Jacobian. We show that this method achieves the same orders of complexity as the best known first-order methods for minimizing expected-value and finite-sum nonconvex functions, despite the additional outer composition which renders the composite gradient estimator biased. This finding enables a much broader range of applications in machine learning to benefit from the low complexity of incremental variance-reduction methods. Papers published at the Neural Information Processing Systems Conference.

Statistical Adaptive Stochastic Gradient Methods Machine Learning

We propose a statistical adaptive procedure called SALSA for automatically scheduling the learning rate (step size) in stochastic gradient methods. SALSA first uses a smoothed stochastic line-search procedure to gradually increase the learning rate, then automatically switches to a statistical method to decrease the learning rate. The line search procedure ``warms up'' the optimization process, reducing the need for expensive trial and error in setting an initial learning rate. The method for decreasing the learning rate is based on a new statistical test for detecting stationarity when using a constant step size. Unlike in prior work, our test applies to a broad class of stochastic gradient algorithms without modification. The combined method is highly robust and autonomous, and it matches the performance of the best hand-tuned learning rate schedules in our experiments on several deep learning tasks.

Dual Averaging Method for Regularized Stochastic Learning and Online Optimization

Neural Information Processing Systems

We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as L1-norm for sparsity. We develop a new online algorithm, the regularized dual averaging method, that can explicitly exploit the regularization structure in an online setting. In particular, at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient. This method achieves the optimal convergence rate and often enjoys a low complexity per iteration similar as the standard stochastic gradient method. Computational experiments are presented for the special case of sparse online learning using L1-regularization.

Coupled Variational Bayes via Optimization Embedding

Neural Information Processing Systems

Variational inference plays a vital role in learning graphical models, especially on large-scale datasets. Much of its success depends on a proper choice of auxiliary distribution class for posterior approximation. However, how to pursue an auxiliary distribution class that achieves both good approximation ability and computation efficiency remains a core challenge. In this paper, we proposed coupled variational Bayes which exploits the primal-dual view of the ELBO with the variational distribution class generated by an optimization procedure, which is termed optimization embedding. Theoretically, we establish an interesting connection to gradient flow and demonstrate the extreme flexibility of this implicit distribution family in the limit sense.

Learning SMaLL Predictors

Neural Information Processing Systems

We introduce a new framework for learning in severely resource-constrained settings. Our technique delicately amalgamates the representational richness of multiple linear predictors with the sparsity of Boolean relaxations, and thereby yields classifiers that are compact, interpretable, and accurate. We provide a rigorous formalism of the learning problem, and establish fast convergence of the ensuing algorithm via relaxation to a minimax saddle point objective. Papers published at the Neural Information Processing Systems Conference.

Q-LDA: Uncovering Latent Patterns in Text-based Sequential Decision Processes

Neural Information Processing Systems

In sequential decision making, it is often important and useful for end users to understand the underlying patterns or causes that lead to the corresponding decisions. However, typical deep reinforcement learning algorithms seldom provide such information due to their black-box nature. In this paper, we present a probabilistic model, Q-LDA, to uncover latent patterns in text-based sequential decision processes. The model can be understood as a variant of latent topic models that are tailored to maximize total rewards; we further draw an interesting connection between an approximate maximum-likelihood estimation of Q-LDA and the celebrated Q-learning algorithm. We demonstrate in the text-game domain that our proposed method not only provides a viable mechanism to uncover latent patterns in decision processes, but also obtains state-of-the-art rewards in these games.

An Accelerated Proximal Coordinate Gradient Method

Neural Information Processing Systems

We develop an accelerated randomized proximal coordinate gradient (APCG) method, for solving a broad class of composite convex optimization problems. In particular, our method achieves faster linear convergence rates for minimizing strongly convex functions than existing randomized proximal coordinate gradient methods. We show how to apply the APCG method to solve the dual of the regularized empirical risk minimization (ERM) problem, and devise efficient implementations that can avoid full-dimensional vector operations. For ill-conditioned ERM problems, our method obtains improved convergence rates than the state-of-the-art stochastic dual coordinate ascent (SDCA) method. Papers published at the Neural Information Processing Systems Conference.

End-to-end Learning of LDA by Mirror-Descent Back Propagation over a Deep Architecture

Neural Information Processing Systems

We develop a fully discriminative learning approach for supervised Latent Dirichlet Allocation (LDA) model using Back Propagation (i.e., BP-sLDA), which maximizes the posterior probability of the prediction variable given the input document. Different from traditional variational learning or Gibbs sampling approaches, the proposed learning method applies (i) the mirror descent algorithm for maximum a posterior inference and (ii) back propagation over a deep architecture together with stochastic gradient/mirror descent for model parameter estimation, leading to scalable and end-to-end discriminative learning of the model. As a byproduct, we also apply this technique to develop a new learning method for the traditional unsupervised LDA model (i.e., BP-LDA). Experimental results on three real-world regression and classification tasks show that the proposed methods significantly outperform the previous supervised topic models, neural networks, and is on par with deep neural networks. Papers published at the Neural Information Processing Systems Conference.