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Revisiting the Approximation Bound for Stochastic Submodular Cover

Journal of Artificial Intelligence Research

Deshpande et al. presented a k(ln R 1) approximation bound for Stochastic Submodular Cover, where k is the state set size, R is the maximum utility of a single item, and the utility function is integer-valued. This bound is similar to the ln Q/(eta 1) bound given by Golovin and Krause, whose analysis was recently found to have an error. Here Q R is the goal utility and eta is the minimum gap between Q and any attainable utility Q' Q.


Stochastic Approximation for Canonical Correlation Analysis

arXiv.org Machine Learning

We propose novel first-order stochastic approximation algorithms for canonical correlation analysis (CCA). Algorithms presented are instances of inexact matrix stochastic gradient (MSG) and inexact matrix exponentiated gradient (MEG), and achieve $\epsilon$-suboptimality in the population objective in $\operatorname{poly}(\frac{1}{\epsilon})$ iterations. We also consider practical variants of the proposed algorithms and compare them with other methods for CCA both theoretically and empirically.


Asynchronous Stochastic Approximation with Differential Inclusions

arXiv.org Machine Learning

The asymptotic pseudo-trajectory approach to stochastic approximation of Benaim, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of an equivalent synchronous process. In addition, this allows many of the restrictive assumptions previously associated with asynchronous stochastic approximation to be removed. The framework is extended for a coupled asynchronous stochastic approximation process with set-valued mean fields. Two-timescales arguments are used here in a similar manner to the original work in this area by Borkar. The applicability of this approach is demonstrated through learning in a Markov decision process.


Non-strongly-convex smooth stochastic approximation with convergence rate O(1/n)

Neural Information Processing Systems

We consider the stochastic approximation problem where a convex function has to be minimized, given only the knowledge of unbiased estimates of its gradients at certain points, a framework which includes machine learning methods based on the minimization of the empirical risk. We focus on problems without strong convexity, for which all previously known algorithms achieve a convergence rate for function values of $O(1/\sqrt{n})$. We consider and analyze two algorithms that achieve a rate of $O(1/n)$ for classical supervised learning problems. For least-squares regression, we show that averaged stochastic gradient descent with constant step-size achieves the desired rate. For logistic regression, this is achieved by a simple novel stochastic gradient algorithm that (a) constructs successive local quadratic approximations of the loss functions, while (b) preserving the same running time complexity as stochastic gradient descent. For these algorithms, we provide a non-asymptotic analysis of the generalization error (in expectation, and also in high probability for least-squares), and run extensive experiments showing that they often outperform existing approaches.