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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.24669 |
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| _version_ | 1866908741207588864 |
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| author | Ma, Wanteng Cai, T. Tony |
| author_facet | Ma, Wanteng Cai, T. Tony |
| contents | Contextual bandits are a central framework for sequential decision-making, with applications ranging from recommendation systems to clinical trials. While nonparametric methods can flexibly model complex reward structures, they suffer from the curse of dimensionality. We address this challenge using a single-index model, which projects high-dimensional covariates onto a one-dimensional subspace while preserving nonparametric flexibility.
We first develop a nonasymptotic theory for offline single-index regression for each arm, combining maximum rank correlation for index estimation with local polynomial regression. Building on this foundation, we propose a single-index bandit algorithm and establish its convergence rate. We further derive a matching lower bound, showing that the algorithm achieves minimax-optimal regret independent of the ambient dimension $d$, thereby overcoming the curse of dimensionality.
We also establish an impossibility result for adaptation: without additional assumptions, no policy can adapt to unknown smoothness levels. Under a standard self-similarity condition, however, we construct a policy that remains minimax-optimal while automatically adapting to the unknown smoothness. Finally, as the dimension $d$ increases, our algorithm continues to achieve minimax-optimal regret, revealing a phase transition that characterizes the fundamental limits of single-index bandit learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_24669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonparametric Bandits with Single-Index Rewards: Optimality and Adaptivity Ma, Wanteng Cai, T. Tony Statistics Theory Contextual bandits are a central framework for sequential decision-making, with applications ranging from recommendation systems to clinical trials. While nonparametric methods can flexibly model complex reward structures, they suffer from the curse of dimensionality. We address this challenge using a single-index model, which projects high-dimensional covariates onto a one-dimensional subspace while preserving nonparametric flexibility. We first develop a nonasymptotic theory for offline single-index regression for each arm, combining maximum rank correlation for index estimation with local polynomial regression. Building on this foundation, we propose a single-index bandit algorithm and establish its convergence rate. We further derive a matching lower bound, showing that the algorithm achieves minimax-optimal regret independent of the ambient dimension $d$, thereby overcoming the curse of dimensionality. We also establish an impossibility result for adaptation: without additional assumptions, no policy can adapt to unknown smoothness levels. Under a standard self-similarity condition, however, we construct a policy that remains minimax-optimal while automatically adapting to the unknown smoothness. Finally, as the dimension $d$ increases, our algorithm continues to achieve minimax-optimal regret, revealing a phase transition that characterizes the fundamental limits of single-index bandit learning. |
| title | Nonparametric Bandits with Single-Index Rewards: Optimality and Adaptivity |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2512.24669 |