Progressive Power Homotopy for Non-convex Optimization

Abstract

We propose a novel first-order method for non-convex optimization of the form $\max_{\bm{w}\in\mathbb{R}^d}\mathbb{E}_{\bm{x}\sim\mathcal{D}}[f_{\bm{w}}(\bm{x})]$, termed Progressive Power Homotopy (Prog-PowerHP). The method applies stochastic gradient ascent to a surrogate objective obtained by first performing a power transformation and then Gaussian smoothing, $F_{N,\sigma}(\bm{\mu}):=\mathbb{E}_{\bm{w}\sim\mathcal{N}(\bm{\mu},\sigma^2I_d),\bm{x}\sim\mathcal{D}}[e^{Nf_w(\bm{x})}]$, while progressively increasing the power parameter $N$ and decreasing the smoothing scale $\sigma$ along the optimization trajectory. We prove that, under mild regularity conditions, Prog-PowerHP converges to a small neighborhood of the global optimum with an iteration complexity scaling nearly as $O(d^2\varepsilon^{-2})$. Empirically, Prog-PowerHP demonstrates clear advantages in phase retrieval when the samples-to-dimension ratio approaches the information-theoretic limit, and in training two-layer neural networks in under-parameterized regimes. These results suggest that Prog-PowerHP is particularly effective for navigating cluttered non-convex landscapes where standard first-order methods struggle.