arXiv:2503.06917v3 Announce Type: replace-cross 
Abstract: In zeroth-order optimization, we seek to minimize a function $d(\cdot)$, which may encode combinatorial feasibility, using only function evaluations. We focus on the setting where solutions must also satisfy qualitative constraints or conform to a complex prior distribution. To address this, we introduce a new framework in which such constraints are represented by an initial generative prior $\L(\cdot)$, for example, a Large Language Model (LLM). The objective is to find solutions $s$ that minimize $d(s)$ while having high probability under $\L(s)$, effectively sampling from a target distribution proportional to $\L(s) \cdot e^{-T \cdot d(s)}$ for a temperature parameter $T$.
  While this framework aligns with classical Model-Based Optimization (e.g., the Cross-Entropy method), existing theory is ill-suited for deriving sample complexity bounds in black-box deep generative models. We therefore propose a novel learning assumption, which we term \emph{coarse learnability}, where an agent with access to a polynomial number of samples can learn a model whose point-wise density approximates the target within a polynomial factor. Leveraging this assumption, we design an iterative algorithm that employs a Metropolis-Hastings correction to provably approximate the target distribution using a polynomial number of samples. To the best of our knowledge, this is one of the first works to establish such sample-complexity guarantees for model-based optimization with deep generative priors.
  We provide two lines of evidence supporting the coarse learnability assumption. Theoretically, we show that maximum likelihood estimation naturally induces the required coverage properties, holding for both standard exponential families and for misspecified models. Empirically, we demonstrate that LLMs can adapt their learned distributions to zeroth-order feedback to solve combinatorial optimization problems.

تم تقديم إطار جديد للتحسين من الدرجة صفر، يركز على تقليل دالة مع الالتزام بالقيود النوعية من خلال استخدام أولويات توليدية، مثل نماذج اللغة الكبيرة (LLMs). يهدف هذا النهج إلى أخذ عينات من الحلول التي لا تقلل فقط من الدالة المستهدفة، ولكنها تتماشى أيضًا مع توزيع محدد، مما يعزز كفاءة عمليات التحسين.

Se ha introducido un nuevo marco para la optimización de orden cero, centrado en minimizar una función mientras se cumplen restricciones cualitativas mediante el uso de priors generativos, como los Modelos de Lenguaje Grande (LLMs). Este enfoque busca muestrear soluciones que no solo minimicen la función objetivo, sino que también se alineen con una distribución especificada, mejorando así la eficiencia de los procesos de optimización.

Un nouveau cadre pour l'optimisation de zéro ordre a été introduit, se concentrant sur la minimisation d'une fonction tout en respectant des contraintes qualitatives grâce à l'utilisation de prioris génératifs, tels que les grands modèles de langage (LLMs). Cette approche vise à échantillonner des solutions qui minimisent non seulement la fonction cible, mais qui s'alignent également sur une distribution spécifiée, améliorant ainsi l'efficacité des processus d'optimisation.

A new framework for zeroth-order optimization has been introduced, focusing on minimizing a function while adhering to qualitative constraints through the use of generative priors, such as Large Language Models (LLMs). This approach aims to sample solutions that not only minimize the target function but also align with a specified distribution, enhancing the efficiency of optimization processes.

Sample-Efficient Optimization over Generative Priors via Coarse Learnability

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