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We consider the problem of optimizing an approximately convex function over abounded convex set in $\mathbb{R}^n$ using only function evaluations. Theproblem is reduced to sampling from an \emph{approximately} log-concavedistribution using the Hit-and-Run method, which is shown to have the same$\mathcal{O}^*$ complexity as sampling from log-concave distributions. Inaddition to extend the analysis for log-concave distributions to approximatelog-concave distributions, the implementation of the 1-dimensional sampler ofthe Hit-and-Run walk requires new methods and analysis. The algorithm then isbased on simulated annealing which does not relies on first order conditionswhich makes it essentially immune to local minima.  We then apply the method to different motivating problems. In the context ofzeroth order stochastic convex optimization, the proposed method produces an$\epsilon$-minimizer after $\mathcal{O}^*(n^{7.5}\epsilon^{-2})$ noisy functionevaluations by inducing a $\mathcal{O}(\epsilon/n)$-approximately log concavedistribution. We also consider in detail the case when the "amount ofnon-convexity" decays towards the optimum of the function. Other applicationsof the method discussed in this work include private computation of empiricalrisk minimizers, two-stage stochastic programming, and approximate dynamicprogramming for online learning.

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