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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.19277 |
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| _version_ | 1866915953422368768 |
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| author | Zuo, Jierui Qin, Hanzhang |
| author_facet | Zuo, Jierui Qin, Hanzhang |
| contents | We study online assortment optimization under stochastic choice when a decision maker simultaneously values cumulative revenue performance and the quality of post-hoc inference on revenue contrasts. We analyze a forced-exploration optimism-in-the-face-of-uncertainty (OFU) scheme that combines two regularized maximum-likelihood estimators: one based on all observations for sequential decision making, and one based only on exploration rounds for inference. Our general theory is developed under predictable score proxies and per-round action-dependent curvature domination. Under these conditions we establish a self-normalized concentration inequality, a likelihood-based ellipsoidal confidence-set theorem, and a regret bound for approximate optimistic actions that explicitly accounts for optimization error. For the multinomial logit (MNL) model we derive explicit score and curvature proxies and show that a balanced spaced singleton-exploration schedule yields realized coordinate coverage, implying regret $\Otilde(n_T + T/\sqrt{n_T})$ and revenue-contrast error $\Otilde(1/\sqrt{n_T})$ up to fixed problem-dependent factors. A hard two-assortment subclass yields a matching lower bound at the product level. Consequently, within the polynomial exploration family $n_T \asymp T^α$, the regret and inference rates become $\Otilde(T^{\max\{α,1-α/2\}})$ and $\Otilde(T^{-α/2})$, respectively; hence $α\in[2/3,1)$ is the rate-wise Pareto-undominated interval and $α=2/3$ is the unique balancing point that minimizes the regret exponent. Finally, for the Exponomial Choice and Nested Logit models we state verifiable sufficient conditions that would instantiate the general framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_19277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Pareto Optimality for Parametric Choice Bandits Zuo, Jierui Qin, Hanzhang Machine Learning We study online assortment optimization under stochastic choice when a decision maker simultaneously values cumulative revenue performance and the quality of post-hoc inference on revenue contrasts. We analyze a forced-exploration optimism-in-the-face-of-uncertainty (OFU) scheme that combines two regularized maximum-likelihood estimators: one based on all observations for sequential decision making, and one based only on exploration rounds for inference. Our general theory is developed under predictable score proxies and per-round action-dependent curvature domination. Under these conditions we establish a self-normalized concentration inequality, a likelihood-based ellipsoidal confidence-set theorem, and a regret bound for approximate optimistic actions that explicitly accounts for optimization error. For the multinomial logit (MNL) model we derive explicit score and curvature proxies and show that a balanced spaced singleton-exploration schedule yields realized coordinate coverage, implying regret $\Otilde(n_T + T/\sqrt{n_T})$ and revenue-contrast error $\Otilde(1/\sqrt{n_T})$ up to fixed problem-dependent factors. A hard two-assortment subclass yields a matching lower bound at the product level. Consequently, within the polynomial exploration family $n_T \asymp T^α$, the regret and inference rates become $\Otilde(T^{\max\{α,1-α/2\}})$ and $\Otilde(T^{-α/2})$, respectively; hence $α\in[2/3,1)$ is the rate-wise Pareto-undominated interval and $α=2/3$ is the unique balancing point that minimizes the regret exponent. Finally, for the Exponomial Choice and Nested Logit models we state verifiable sufficient conditions that would instantiate the general framework. |
| title | On Pareto Optimality for Parametric Choice Bandits |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2501.19277 |