Proximal Point Newton Methods for Nonconvex Optimization

Tensor methods offer improved complexity guarantees for nonconvex optimization, but evaluating high-order derivatives and solving the corresponding subproblems can be computationally expensive. In this paper, we develop a proximal point Newton framework that exploits third-derivative smoothness using only gradients and Hessians. Under Hessian Lipschitz continuity, a single-loop proximal point Newton method that alternates between negative-curvature directions and Newton steps finds an $(\varepsilon_g,\varepsilon_H)$-approximate second-order stationary point within $\mathcal{O}(\max\{\varepsilon_g^{-3/2},\varepsilon_H^{-3}\})$ second-order oracle evaluations. When the third derivative is also Lipschitz continuous, we introduce an accelerated proximal point Newton method and obtain a dimension-free complexity of $\widetilde{\mathcal{O}}(\max\{\varepsilon_g^{-4/3},\varepsilon_H^{-2}\})$, matching that of the third-order tensor methods up to logarithmic factors without constructing tensors or solving tensor subproblems. We further develop an adaptive inexact Newton--CG variant that requires no prior knowledge of Lipschitz constants and preserves the improved oracle complexity with high probability. Under uniformly bounded queried Hessians, this variant achieves a Hessian-vector product complexity of $\widetilde{\mathcal{O}}(\max\{\varepsilon_g^{-5/3},\varepsilon_H^{-5/2}\})$, improving upon the previous $\widetilde{\mathcal{O}}(\max\{\varepsilon_g^{-3}\varepsilon_H^{5/2},\varepsilon_H^{-7/2}\})$ result attained by existing second-order methods such as cubic regularized Newton method and Newton--CG method. Preliminary numerical experiments are encouraging.