Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions

author: Tengyuan Liang, Wharton School, University of Pennsylvania
published: Aug. 20, 2015,   recorded: July 2015,   views: 1882


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We consider the problem of optimizing an approximately convex function over a bounded convex set in $\mathbb{R}^n$ using only function evaluations. The problem is reduced to sampling from an \emph{approximately} log-concave distribution using the Hit-and-Run method, with query complexity of $\mathcal{O}^*(n^{4.5})$. In the context of zeroth order stochastic convex optimization, the proposed method produces an $\epsilon$-minimizer after $\mathcal{O}^*(n^{7.5}\epsilon^{-2})$ noisy function evaluations by inducing a $\mathcal{O}(\epsilon/n)$-approximately log concave distribution. We also consider the case when the ``amount of non-convexity'' can be large away from the minimum but decays towards the optimum of the function. Other applications of the random walk method include private computation of empirical risk minimizers, two-stage stochastic programming, and approximate dynamic programming for online learning.

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