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Probabilistic Gaussian Homotopy (PGH) introduces a novel optimization framework that biases descent directions toward low-energy regions by deforming Boltzmann distributions, offering a principled bridge between Gaussian continuation, Bayesian denoising, and diffusion-style smoothing. The proposed PGHO algorithm demonstrates robust performance on high-dimensional nonconvex benchmarks and sparse recovery tasks where classical gradient methods often struggle. This work advances the theoretical understanding of nonconvex optimization by establishing a dynamical system for minimizer evolution along annealed homotopy paths.
Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization
arXiv:2603.13546v2 Announce Type: replace Abstract: We introduce Probabilistic Gaussian Homotopy (PGH), a probability-space continuation framework for nonconvex optimization. Unlike classical Gaussian homotopy, which smooths the objective and uniformly averages gradients, PGH deforms the associated Boltzmann distribution and induces Boltzmann-weighted aggregation of perturbed gradients, which exponentially biases descent directions toward low-energy regions. We show that PGH corresponds to a log-sum-exp (soft-min) homotopy that smooths a nonconvex objective at scale $\lambda>0$ and recovers the original objective as $\lambda\to 0$, yielding a posterior-mean generalization of the Moreau envelope, and we derive a dynamical system governing minimizer evolution along an annealed homotopy path. This establishes a principled connection between Gaussian continuation, Bayesian denoising, and diffusion-style smoothing. We further propose Probabilistic Gaussian Homotopy Optimization (PGHO), a practical stochastic algorithm based on Monte Carlo gradient estimation, and demonstrate strong performance on high-dimensional nonconvex benchmarks and sparse recovery problems where classical gradient methods and objective-space smoothing frequently fail.
Abstract:We introduce Probabilistic Gaussian Homotopy (PGH), a probability-space continuation framework for nonconvex optimization. Unlike classical Gaussian homotopy, which smooths the objective and uniformly averages gradients, PGH deforms the associated Boltzmann distribution and induces Boltzmann-weighted aggregation of perturbed gradients, which exponentially biases descent directions toward low-energy regions. We show that PGH corresponds to a log-sum-exp (soft-min) homotopy that smooths a nonconvex objective at scale $\lambda>0$ and recovers the original objective as $\lambda\to 0$, yielding a posterior-mean generalization of the Moreau envelope, and we derive a dynamical system governing minimizer evolution along an annealed homotopy path. This establishes a principled connection between Gaussian continuation, Bayesian denoising, and diffusion-style smoothing. We further propose Probabilistic Gaussian Homotopy Optimization (PGHO), a practical stochastic algorithm based on Monte Carlo gradient estimation, and demonstrate strong performance on high-dimensional nonconvex benchmarks and sparse recovery problems where classical gradient methods and objective-space smoothing frequently fail.
Machine Learning (cs.LG)
arXiv:2603.13546 [cs.LG]
(or arXiv:2603.13546v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.13546 Focus to learn more arXiv-issued DOI via DataCite
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