Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization

Eshed Gal; Samy Wu Fung; Eldad Haber

cs.LGMar 13, 2026

Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization

Authors: Eshed Gal, Samy Wu Fung, Eldad Haber

Organizations: University of British Columbia, Vancouver, BC, Canada · Colorado School of Mines, Golden, CO, USA

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 $λ>0$ 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.

Figures & tables

Figure 1: Gaussian homotopy for the function $f(x_{1},x_{2})=\log(Rosen(x_{1},x_{2})+\alpha(1+\sin(3x_{1})\sin(3x_{2})))$ , where $Rosen$ denotes the banana Rosenbrock function. We estimate the homotopy-smoothed objective by Monte Carlo sampling. The surface begins nearly flat and gradually deforms into a highly multimodal landscape that is much harder to optimize.

F_{t_{k}}^{(K)}(x_{i}^{k})=-\lambda(t)\log\!\left(\frac{1}{K}\sum_{j=1}^{K}\exp\big(-\lambda(t_{k})^{-1}f(\alpha(t_{k})x_{i}^{k}+\beta(t_{k})z_{i,j})\big)\right)

(20)

Algorithm 1 Probabilistic Gaussian Homotopy Optimization (PGHO)

Panel of Figure 2 — Figure 2: Ackley and Griewank plotted for the 2D case with the trajectories obtained by the PGH algorithms starting from different points

Function	PGH-GD	PGH-Adam	GH	CMAES	Diff.Eval.	BasinHop.	Anneal.	Impl.Filt.	Part.Swarm	PRS
Ackley	205	601	3,894	804	6,683	N	97,646	240	3,663	N
Griewank	183	631	3,248	1,088	8,937	14,120	99,785	315	4,891	N
Alpine1	192	557	3,479	1,176	11,367	N	97,195	308	4,187	N
Levy	3,067	562	3,141	817	5,731	N	85,933	N	3,146	N

Table 1: Mean number of function evaluations required to reach success (“N” indicates no successful runs), where success is defined as

f(x)<5\times 10^{-2}

. Each run uses a maximum budget of

2\times 10^{5}

function evaluations. Results are averaged over 30 runs at dimension

n=10

Figure 3: Success rate vs. dimension on Ackley. PGH maintains high success rate across all dimensions, while competing methods degrade significantly on higher dimensions. Budget fixed at $10^{5}$ function evaluations, averaged over 30 runs.

Figure 4: Sparse recovery results along the regularization path. Left: tradeoff between reconstruction fidelity and sparsity. Right: attained smooth objective value as a function of the regularization parameter. In both plots, the PGH variants improve over the corresponding GD and Adam baselines.

Nonconvex Optimization

Nonconvex optimization develops methods for problems whose objectives or constraints can have multiple local optima, saddle points, or nonsmooth structure.

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Nov 17, 2025J/K move · Enter open · S save

Power Homotopy for Zeroth-Order Non-Convex Optimizations

Chen Xu

Department of Engineering, Shenzhen MSU-BIT University, China.

Zeroth-Order Optimization Nonconvex Optimization

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The existing method of GS-PowerOpt solves the non-convex optimization problem of the form

\max_{\boldsymbol{x} \in \mathbb{R}^d} f(\boldsymbol{x})

through maximizing a Gaussian-smoothed surrogate

F_{N,σ}(\boldsymbolμ) = \mathbb{E}_{\boldsymbol{x}\sim\mathcal{N}(\boldsymbolμ,σ^2 I_d)}[e^{N f(\boldsymbol{x})}]

. We analyze the role of the smoothing radius

σ>0

and identify a limitation of the fixed-

σ

design used in GS-PowerOpt. Specifically,

σ

induces an inherent exploration--refinement tradeoff: a larger

σ

improves global exploration and finite-time surrogate optimization, but may distort the location of the surrogate maximizer; in contrast, a smaller

σ

better preserves local structure but can weaken gradient signals away from high-value regions. To address this limitation, we propose GS-PowerHP, a power-smoothed homotopy method with an incrementally decaying

σ

schedule. The proposed mechanism uses larger smoothing radii in early iterations to maintain informative gradient signals when the iterate is far from high-value regions, and gradually decreases

σ

to improve local refinement near the maximizer. We provide theoretical results showing that this decaying schedule improves the exploration--refinement tradeoff of fixed-

σ

power smoothing. Empirically, GS-PowerHP consistently outperforms the fixed-

σ

baseline and exhibits robust performance across different optimization tasks, including adversarial attacks on ImageNet (

d=150{,}528

), where it substantially improves over other smoothing-based zeroth-order methods.

Abstract

Nonconvex Optimization

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