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" 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.

采用Hit-and-Run方法通过log-concave分布的采样来优化凸可行解，在log-concave分布的基础上扩展近似log-concave分布的分析， 并使1D采样器的实现需要新的方法和分析，应用于不同的激励问题，并讨论了这种方法的其他应用。

通过模拟退火逃离局部极小值：近似凸函数的优化