Re$^3$MCN: Cubic Newton + Variance Reduction + Momentum + Quadratic Regularization for Finite-sum Non-convex Problems

Abstract: We analyze a stochastic cubic regularized Newton method for finite sum optimization $\textstyle\min_{x\in\mathbb{R}^d} F(x) \;=\; \frac{1}{n}\sum_{i=1}^n f_i(x)$, that uses SARAH-type recursive variance reduction with mini-batches of size $b\sim n^{1/2}$ and exponential moving averages (EMA) for gradient and Hessian estimators. We show that the method achieves a $(\varepsilon,\sqrt{L_2\varepsilon})$-second-order stationary point (SOSP) with total stochastic oracle calls $n + \widetilde{\mathcal{O}}(n^{1/2}\varepsilon^{-3/2})$ in the nonconvex case (Theorem 8.3) and convergence rate $\widetilde{\mathcal{O}}(\frac{L R^3}{T^2} + \frac{\sigma_2 R^2}{T^2} + \frac{\sigma_1 R}{\sqrt{T}})$ in the convex case (Theorem 6.1). We also treat the matrix-free variant based on Hutchinson's estimator for Hessian and present a fast inner solver for the cubic subproblem with provable attainment of the required inexactness level.