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Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.

Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.

Stochastic approximation techniques have been used in various contexts in data science. We propose a stochastic version of the forward-backward algorithm for minimizing the sum of two convex functions, one of which is not necessarily smooth. Our framework can handle stochastic approximations of the gradient of the smooth function and allows for stochastic errors in the evaluation of the proximity operator of the nonsmooth function. The almost sure convergence of the iterates generated by the algorithm to a minimizer is established under relatively mild assumptions. We also propose a stochastic version of a popular primal-dual proximal splitting algorithm, establish its convergence, and apply it to an online image restoration problem.

We present algorithms for a class of problems called resource allocation problems, both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem for search queries, display-ads problem for webpage banner-advertisement, online network routing, Bayesian combinatorial auctions, etc. In the online setting we introduce a new distributional model called the adversarial stochastic input model (asi), which is a generalization of the model where the input elements to the algorithm are drawn i.i.d from a distribution unknown to the algorithm designer. In asi model, the distributions can change over time too. In this model, we give a near optimal approximation algorithm for the resource allocation problem under mild assumptions about the input. Our proof technique, which is based on a broader interpretation of pessimistic estimators, also gives a very simple proof that the natural greedy algorithm for the adwords problem has a competitive ratio of 1-1/e in the i.i.d model with unknown distributions, and more generally in the asi model, with no assumptions at all about the input. In the offline setting we give a fast near-linear time algorithm to approximately solve very large LPs with both packing and covering constraints. Joint work with Nikhil Devanur, Kamal Jain and Chris Wilkens ©2011 Microsoft Corporation. All rights reserved.

We present algorithms for a class of problems called resource allocation problems, both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem for search queries, display-ads problem for webpage banner-advertisement, online network routing, Bayesian combinatorial auctions, etc. In the online setting we introduce a new distributional model called the adversarial stochastic input model (asi), which is a generalization of the model where the input elements to the algorithm are drawn i.i.d from a distribution unknown to the algorithm designer. In asi model, the distributions can change over time too. In this model, we give a near optimal approximation algorithm for the resource allocation problem under mild assumptions about the input. Our proof technique, which is based on a broader interpretation of pessimistic estimators, also gives a very simple proof that the natural greedy algorithm for the adwords problem has a competitive ratio of 1-1/e in the i.i.d model with unknown distributions, and more generally in the asi model, with no assumptions at all about the input. In the offline setting we give a fast near-linear time algorithm to approximately solve very large LPs with both packing and covering constraints. Joint work with Nikhil Devanur, Kamal Jain and Chris Wilkens ©2011 Microsoft Corporation. All rights reserved.