Manifold Identification in Dual Averaging Methods for Regularized Stochastic Online Learning
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Date
2012-03-07
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Abstract
Iterative methods that calculate their steps from approximate subgradient directions have
proved to be useful for stochastic learning problems over large and streaming data sets.
When the objective consists of a loss function plus a nonsmooth regularization term whose
purpose is to induce structure in the solution, the solution often lies on a low-dimensional
manifold of parameter space along which the regularizer is smooth. (When an l1 regularizer
is used to induce sparsity in the solution, for example, this manifold is defined by the set
of nonzero components of the parameter vector.) This paper shows that a regularized dual
averaging algorithm can identify this manifold, with high probability, before reaching the
solution. This observation motivates an algorithmic strategy in which, once an iterate is
suspected of lying on an optimal or near-optimal manifold, we switch to an “local phase”
algorithm that searches in this manifold, thus converging rapidly to a near-optimal point.
Computational results are presented to verify the identification property and to illustrate
the effectiveness of this approach.
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Keywords
dual averaging, manifold identification, partly smooth manifold, regularization