Sample-Efficient Optimization over Generative Priors via Coarse Learnability
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| Format: | Preprint |
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2025
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| _version_ | 1866917461592375296 |
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| author | Awasthi, Pranjal Gollapudi, Sreenivas Kumar, Ravi Munagala, Kamesh |
| author_facet | Awasthi, Pranjal Gollapudi, Sreenivas Kumar, Ravi Munagala, Kamesh |
| contents | We study zeroth-order optimization where solutions must minimize a cost $d(s)$ while maintaining high probability under a complex generative prior $L(s)$ (e.g., a parameterized model). This reduces to sampling from a target distribution proportional to $L(s) e^{-T \cdot d(s)}$. Since classical model-based optimization (MBO) lacks finite-sample guarantees for expressive approximate learners, we introduce "coarse learnability", a flexible statistical assumption requiring only that a learned model covers the target's probability mass within a polynomial factor. Leveraging this assumption, we design an iterative MBO algorithm called \alift with a sample correction step that provably approximates the target using only a polynomial number of samples. We apply this framework to globally optimizing non-convex objectives bounded by a quadratic envelope in $R^n$, where we show this assumption is naturally satisfied for a family of "optimistic" posterior distributions. To reach global $\varepsilon$-optimality, this implies a sample complexity of $\widetilde{O}(\log 1/\varepsilon)$, a rate characteristic of optimistic space-partitioning methods. We further justify coarse learnability as an assumption for generative priors theoretically, proving that in simple settings, parametric maximum likelihood estimation and over-smoothed kernel density estimators naturally satisfy it. Finally, one motivation for our framework comes from inference-time alignment. Though our primary contribution pertains to the theoretical foundations of MBO, we provide qualitative evidence that, in simple settings, even primitive LLMs can shift their distributions toward lower-cost regions when fine-tuned with zeroth-order feedback. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_06917 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sample-Efficient Optimization over Generative Priors via Coarse Learnability Awasthi, Pranjal Gollapudi, Sreenivas Kumar, Ravi Munagala, Kamesh Machine Learning Data Structures and Algorithms We study zeroth-order optimization where solutions must minimize a cost $d(s)$ while maintaining high probability under a complex generative prior $L(s)$ (e.g., a parameterized model). This reduces to sampling from a target distribution proportional to $L(s) e^{-T \cdot d(s)}$. Since classical model-based optimization (MBO) lacks finite-sample guarantees for expressive approximate learners, we introduce "coarse learnability", a flexible statistical assumption requiring only that a learned model covers the target's probability mass within a polynomial factor. Leveraging this assumption, we design an iterative MBO algorithm called \alift with a sample correction step that provably approximates the target using only a polynomial number of samples. We apply this framework to globally optimizing non-convex objectives bounded by a quadratic envelope in $R^n$, where we show this assumption is naturally satisfied for a family of "optimistic" posterior distributions. To reach global $\varepsilon$-optimality, this implies a sample complexity of $\widetilde{O}(\log 1/\varepsilon)$, a rate characteristic of optimistic space-partitioning methods. We further justify coarse learnability as an assumption for generative priors theoretically, proving that in simple settings, parametric maximum likelihood estimation and over-smoothed kernel density estimators naturally satisfy it. Finally, one motivation for our framework comes from inference-time alignment. Though our primary contribution pertains to the theoretical foundations of MBO, we provide qualitative evidence that, in simple settings, even primitive LLMs can shift their distributions toward lower-cost regions when fine-tuned with zeroth-order feedback. |
| title | Sample-Efficient Optimization over Generative Priors via Coarse Learnability |
| topic | Machine Learning Data Structures and Algorithms |
| url | https://arxiv.org/abs/2503.06917 |