Papers › Escaping saddle points in zeroth-order optimization: the power of two-point estimators
Escaping saddle points in zeroth-order optimization: the power of two-point estimators
Zhaolin Ren, Yujie Tang, Na Li
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Two-point zeroth order methods are important in many applications of zeroth-order optimization, such as robotics, wind farms, power systems, online optimization, and adversarial robustness to black-box attacks in deep neural networks, where the problem may be high-dimensional and/or time-varying. Most problems in these applications are nonconvex and contain saddle points. While existing works have shown that zeroth-order methods utilizing Ω(d) function valuations per iteration (with d denoting the problem dimension) can escape saddle points efficiently, it remains an open question if zeroth-order methods based on two-point estimators can escape saddle points. In this paper, we show that by adding an appropriate isotropic perturbation at each iteration, a zeroth-order algorithm based on $2m$ (for any 1 ≤m ≤d) function evaluations per iteration can not only find ϵ-second order stationary points polynomially fast, but do so using only Õ(d/(mϵ²ψ̅)) function evaluations, where ψ̅ ≥Ω̃(√(ϵ)) is a parameter capturing the extent to which the function of interest exhibits the strict saddle property.
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