Optimization-Based Sampling

• Optimization-based sampling is a methodology that employs Lyapunov potentials and functional inequalities to analyze and accelerate the convergence of sampling algorithms for complex target distributions.
• It leverages key isoperimetric inequalities, such as Poincaré and log-Sobolev, to deliver explicit non-asymptotic convergence rates and rapid mixing guarantees.
• The framework extends to non-log-concave scenarios by adapting local Lyapunov functions and providing actionable discrete-time algorithms with validated error bounds.

Optimization-based sampling is a unifying field at the interface of probability, functional inequalities, and the theory of gradient flows, which enables both the design and analysis of sampling algorithms for complex target distributions through the lens of optimization dynamics and Lyapunov potentials. The approach leverages functionals (typically Lyapunov potentials) to characterize convergence of both optimization and sampling procedures and connects such convergence to key isoperimetric inequalities like the Poincaré and log-Sobolev inequalities. This framework enables efficient sampling from (potentially non-log-concave) Gibbs measures and establishes precise conditions under which rapid mixing and non-asymptotic guarantees hold.

1. Optimization and Sampling via Lyapunov Potentials

A principal technique advanced in this framework is the use of Lyapunov functions—originally applied to certify convergence of optimization algorithms—for analyzing the convergence of sampling dynamics. Specifically, a Lyapunov potential $V$ is constructed so that, along the trajectories $\mu_t$ of the sampling dynamics (such as those governed by Langevin or Fokker–Planck equations), one has

$\frac{d}{dt} V(\mu_t) \le -\lambda V(\mu_t),$

for some rate $\lambda > 0$. For continuous-time processes, the evolution of probability densities $\rho_t$ associated with $\mu_t$ is described by the Fokker–Planck PDE: $$\frac{\partial \rho_t}{\partial t} = \nabla \cdot (\rho_t \nabla V) + \Delta \rho_t.$$ This approach yields exponential decay of the Lyapunov potential and, consequently, establishes rapid convergence of the distribution towards a target Gibbs measure, providing a direct bridge from optimization theory to non-asymptotic sampling guarantees.

2. Poincaré and Log-Sobolev Inequalities

The analysis of optimization-based sampling hinges on two key functional inequalities:

• Poincaré Inequality: For a measure $\mu$ on $\mathbb{R}^d$ and suitable test function $f$, the inequality

$$\lambda \, \mathrm{Var}_\mu(f) \le \int \|\nabla f\|^2 \, d\mu$$

establishes a quantitative measure of variance contraction (spectral gap $\lambda$).

• Log-Sobolev Inequality (LSI): For entropy $\mathrm{Ent}_\mu(f^2)$,

$$\rho\, \mathrm{Ent}_\mu(f^2) \le 2\int \|\nabla f\|^2\, d\mu$$

and constant $\rho$ controls exponential decay of entropy under the sampling dynamics.

These inequalities are central in certifying rapid convergence to equilibrium for both optimization and sampling algorithms. The paper demonstrates that, under mild regularity assumptions, optimizability of a function $F$ (i.e., convergence of gradient flow from all initializations) implies that low-temperature Gibbs measures $\mu_\beta = e^{-\beta F}/Z$ satisfy a Poincaré inequality with explicit constant $O(C'+1/\beta)$ for $\beta \ge \Omega(d)$, where $C'$ is the Poincaré constant in a neighborhood of global minimizers.

Under additional mild conditions on $F$, the work establishes that $\mu_\beta$ also satisfies a log-Sobolev inequality with constant $O(\beta\max(S,1)\max(C',1))$ (where $S$ is the second moment), thus ensuring strong ergodicity and entropy contraction for the sampling process.

3. Gradient Flow Interpretation and Gibbs Measures

A unifying aspect of the approach is the interpretation of Langevin (and related) sampling dynamics as gradient flows on the space of probability measures equipped with the Wasserstein metric. For a target Gibbs measure (with density proportional to $e^{-V(x)}$), the measured-valued process $\mu_t$ evolves as the gradient flow of the relative entropy functional: $\mathcal{F}(\mu) := \mathrm{Ent}(\mu \| \pi)$ where $\pi$ is the equilibrium Gibbs measure. This formalism connects the convergence of the sampling process to the minimization of a free energy/objective, recasting sampling as infinite-dimensional optimization. For the theory to rigorously apply, the work assumes sufficient regularity (e.g., smoothness and (local) convexity of the potential $V$) to guarantee existence, uniqueness, and stability of the gradient flow.

4. Sampling Beyond Log-Concavity

While many classical results focus on log-concave densities, the presented framework significantly advances to cover non-log-concave settings. The authors construct modified Lyapunov potentials that accommodate regions of local convexity or multimodal, heavy-tailed structure, and prove convergence by patching together local analysis with regularity conditions. Techniques include

• adapting Lyapunov functions to local geometry,
• employing regularization terms to counteract non-convexity, and
• controlling metastability via local mixing estimates.

This enables efficient sampling from a broader class of target distributions, including several new non-log-concave examples for which efficient sampling was previously unknown.

5. Sampling from Most Initializations and Weak Poincaré Inequalities

The results distinguish between optimizing from all initializations and from most initializations in terms of functional inequalities satisfied. If gradient flow converges from all starting points, the associated Gibbs measure satisfies a full Poincaré and log-Sobolev inequality (implying global fast mixing and sampling). When $F$ is only optimizable from almost every point (i.e., fails on a small set $S$), the low-temperature measure $\mu_\beta$ satisfies a Weak Poincaré Inequality: $$\mathrm{Var}_\mu(f) \leq O(C'+1/\beta)\int \|\nabla f\|^2 d\mu + O(\mu_\beta(S))\|f\|_{\infty}^2$$ for $\beta = \Omega(d)$. This result implies efficient sampling from suitable “warm starts” (i.e., initializations outside a small bad set), and formalizes a sharp delineation between global and local convergence for optimization-based sampling.

6. Discrete-Time Sampling Algorithms

An important corollary of the Lyapunov-potential analysis is the derivation of concrete discrete-time sampling algorithms. The work gives explicit step-size conditions and error bounds for sampling log-concave measures under weaker regularity assumptions than classical smoothness, analogous to the results of Lehec (2023). This advances algorithmic implementation, ensuring that favorable continuous-time convergence extends to practical, computationally realizable MCMC schemes.

7. Implications and Applications

The synthesis of optimization and sampling via Lyapunov potentials and functional inequalities has several ramifications:

• For optimization, these results clarify the connection between landscape regularity and the spectral properties of associated Gibbs measures, yielding new perspectives on complexity and convergence.
• For sampling, the techniques enable efficient sampling in high dimensions for measures outside classical log-concave regimes, including practical models in Bayesian inference, statistical physics, and machine learning.
• The discrete-time results guide the design of fast-mixing MCMC and Langevin-based algorithms for complex distributions with quantifiable non-asymptotic mixing rates.

Overall, these developments establish optimization-based sampling as a central methodology for efficient, theoretically grounded exploration of complex, high-dimensional probability distributions, unifying concepts from analysis, probability, and optimization into a coherent algorithmic framework.