用概率空间变形提升非凸优化效率，解决传统方法易陷局部最优的问题。
Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization
- 通过变形玻尔兹曼分布，加权聚合扰动梯度，引导搜索向低能区
- 在高维非凸问题和稀疏恢复中表现优异，优于经典梯度法
- 融合贝叶斯去噪与扩散平滑，适合复杂优化场景研究者
我们提出概率高斯同伦（PGH），一种基于概率空间的非凸优化延续框架。与经典高斯同伦不同，PGH不均匀平均梯度，而是变形关联的玻尔兹曼分布，实现扰动梯度的玻尔兹曼加权聚合，使下降方向指数偏向低能量区域。我们证明PGH对应一个log-sum-exp（软最小）同伦，在尺度λ>0时平滑非凸目标函数，当λ→0时恢复原始目标，形成莫雷映射的后验均值推广。我们推导了沿退火同伦路径极小值演化所遵循的动力学系统，建立了高斯延续、贝叶斯去噪与扩散式平滑之间的原理性联系。我们进一步提出基于蒙特卡洛梯度估计的概率高斯同伦优化（PGHO）算法，并在高维非凸基准测试和稀疏恢复问题上表现出色，而经典梯度方法和目标空间平滑常在此类问题上失效。
原文摘要 · Abstract (English)
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 $λ>0$ and recovers the original objective as $λ\to 0$, 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.
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