Framework Introduces Sample-Complexity Guarantees for Zeroth-Order Optimization with Generative Priors

Global: Framework Introduces Sample-Complexity Guarantees for Zeroth-Order Optimization with Generative Priors

A team of computer scientists has presented a new framework that tackles zeroth-order optimization problems while respecting qualitative constraints by leveraging deep generative models such as large language models (LLMs). The approach, detailed in an arXiv preprint released in March 2025, aims to find solutions that both minimize a black‑box objective function and remain probable under a learned prior, offering theoretical guarantees on the number of samples required.

Background on Zeroth-Order Optimization

Zeroth-order optimization focuses on minimizing a function using only function evaluations, a scenario common when gradients are unavailable or costly to compute. In many real‑world applications, feasible solutions must also satisfy complex constraints or adhere to a prior distribution that captures domain knowledge.

Generative Priors and Target Distribution

The proposed framework models these constraints with an initial generative prior L(·), which can be instantiated by an LLM. The goal is to sample solutions s from a target distribution proportional to L(s)·e^{-T·d(s)}, where d(s) is the objective value and T is a temperature parameter that balances objective minimization against prior likelihood.

Coarse Learnability Assumption

To enable rigorous analysis, the authors introduce a “coarse learnability” assumption. Under this premise, an agent accessing a polynomial number of samples from L can learn a model whose point‑wise density approximates the true prior within a polynomial factor. This assumption underlies the derivation of sample‑complexity bounds for the optimization process.

Algorithmic Approach

Building on the assumption, the paper describes an iterative algorithm that incorporates a Metropolis‑Hastings correction step. The algorithm provably approximates the target distribution using only a polynomial number of queries to the objective function, marking one of the first formal sample‑complexity results for model‑based optimization with deep generative priors.

Theoretical Foundations

The authors argue that maximum likelihood estimation naturally satisfies the coarse learnability condition. They demonstrate this for standard exponential families and extend the argument to misspecified models, providing a theoretical bridge between statistical learning and black‑box optimization.

Empirical Validation

Experimental results show that contemporary LLMs can adjust their output distributions in response to zeroth‑order feedback, successfully solving combinatorial optimization tasks that were previously intractable for pure black‑box methods.

Implications and Future Directions

If the coarse learnability framework holds broadly, it could enable scalable optimization across domains where constraints are best expressed through generative models, including natural‑language design, protein engineering, and automated theorem proving. The authors suggest further investigation into tighter bounds and alternative generative architectures.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.