Zeroth-order Optimization with Weak Dimension Dependency

Abstract

Zeroth-order optimization is a fundamental research topic that has been a focus of various learning tasks, such as black-box adversarial attacks, bandits, and reinforcement learning. However, in theory, most complexity results assert a linear dependency on the dimension of optimization variable, which implies paralyzations of zeroth-order algorithms for high-dimensional problems and cannot explain their effectiveness in practice. In this paper, we present a novel zeroth-order optimization theory characterized by complexities that exhibit weak dependencies on dimensionality. The key contribution lies in the introduction of a new factor, denoted as $\mathrm{ED}_α=\sup_{x\in \mathbb{R}^d}\sum_{i=1}^dσ_i^α(\nabla^2 f(x))$ ($α>0$, $σ_i(\cdot)$ is the $i$-th singular value in non-increasing order), which effectively functions as a measure of dimensionality. The algorithms we propose demonstrate significantly reduced complexities when measured in terms of the factor $\mathrm{ED}_α$. Specifically, we first study a well-known zeroth-order algorithm from Nesterov and Spokoiny (2017) on quadratic objectives and show a complexity of $\mathcal{O}\left(\frac{\mathrm{ED}_1}{σ_d}\log(1/ε)\right)$ for the strongly convex setting. Furthermore, we introduce novel algorithms that leverages the Heavy-ball mechanism. Our proposed algorithm exhibits a complexity of $\mathcal{O}\left(\frac{\mathrm{ED}_{1/2}}{\sqrt{σ_d}}\cdot\log{\frac{L}μ}\cdot\log(1/ε)\right)$. We further expand the scope of the method to encompass generic smooth optimization problems under an additional Hessian-smooth condition. The resultant algorithms demonstrate remarkable complexities which improve by an order in $d$ under appropriate conditions. Our analysis lays the foundation for zeroth-order optimization methods for smooth functions within high-dimensional settings.