Zeroth-order Stochastic Cubic Newton Method Revisited

Abstract: This paper studies stochastic minimization of a finite-sum loss $ F (\mathbf{x}) = \frac{1}{N} \sum_{\xi=1}^N f(\mathbf{x};\xi) $. In many real-world scenarios, the Hessian matrix of such objectives exhibits a low-rank structure on a batch of data. At the same time, zeroth-order optimization has gained prominence in important applications such as fine-tuning LLMs. Drawing on these observations, we propose a novel stochastic zeroth-order cubic Newton method that leverages the low-rank Hessian structure via a matrix recovery-based estimation technique. Our method circumvents restrictive incoherence assumptions, enabling accurate Hessian approximation through finite-difference queries. Theoretically, we establish that for most real-world problems in $\mathbb{R}^n$, $\mathcal{O}\left(\frac{n}{\eta^{\frac{7}{2}}}\right)+\widetilde{\mathcal{O}}\left(\frac{n^2 }{\eta^{\frac{5}{2}}}\right)$ function evaluations suffice to attain a second-order $\eta$-stationary point with high probability. This represents a significant improvement in dimensional dependence over existing methods. This improvement is mostly due to a new Hessian estimator that achieves superior sample complexity; This new Hessian estimation method might be of separate interest. Numerical experiments on matrix recovery and machine learning tasks validate the efficacy and scalability of our approach.