In this paper, we propose and analyze zeroth-order stochastic approximation
algorithms for nonconvex and convex optimization, with a focus on addressing
constrained optimization, high-dimensional setting and saddle-point avoiding.
To handle constrained optimization, we first propose generalizations of the
conditional gradient algorithm achieving rates similar to the standard
stochastic gradient algorithm using only zeroth-order information. To
facilitate zeroth-order optimization in high-dimensions, we explore the
advantages of structural sparsity assumptions. Specifically, (i) we highlight
an implicit regularization phenomenon where the standard stochastic gradient
algorithm with zeroth-order information adapts to the sparsity of the problem
at hand by just varying the step-size and (ii) propose a truncated stochastic
gradient algorithm with zeroth-order information, whose rate of convergence
depends only poly-logarithmically on the dimensionality. We next focus on
avoiding saddle-points in non-convex setting. Towards that, we interpret the
Gaussian smoothing technique for estimating gradient based on zeroth-order
information as an instantiation of first-order Stein's identity. Based on this,
we provide a novel linear-(in dimension) time estimator of the Hessian matrix
of a function using only zeroth-order information, which is based on
second-order Stein's identity. We then provide an algorithm for avoiding
saddle-points, which is based on a zeroth-order cubic regularization Newton's
method and discuss its convergence rates.

本文提出了针对非凸和凸优化的零阶随机逼近算法，并关注解决约束优化、高维设置和避免鞍点等问题。我们探索了结构稀疏假设的优点，并提出了一种使用零阶信息的被截断随机梯度算法和一种避免鞍点的算法，并讨论了它们的收敛率。