arXiv 13 Dec 2021 · Econometrics · publishedEconometrica (2025) · 2 citations (OpenAlex)
arXiv:2112.06363 · PDF · DOI · OpenAlex · Extracted main text
We provide a decision theoretic analysis of bandit experiments under local asymptotics. Working within the framework of diffusion processes, we define suitable notions of asymptotic Bayes and minimax risk for these experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distributions of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and thereby suggests a practical strategy for dimension reduction. The PDEs characterizing minimal Bayes risk can be solved efficiently using sparse matrix routines or Monte-Carlo methods. We derive the optimal Bayes and minimax policies from their numerical solutions. These optimal policies substantially dominate existing methods such as Thompson sampling; the risk of the latter is often twice as high.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | T. Lattimore and C. Szepesvári, Bandit algorithms. 1em plus 0.5em mi… (2020) | 1.000 | 5 | 3 | 100% |
| 2 | X. Kuang and S. Wager, “Weak signal asymptotics for sequentially ran… (2024) Weak signal asymptotics for sequentially randomized experiments | 0.874 | 6 | 2 | 100% |
| 3 | M. Kasy and A. Sautmann, “Adaptive treatment assignment in experimen… (2021) Adaptive treatment assignment in experiments for policy choice | 0.843 | 4 | 3 | 75% |
| 4 | L. Fan and P. W. Glynn, “Diffusion approximations for thompson sampl… (2021) Diffusion approximations for thompson sampling | 0.811 | 4 | 2 | 100% |
| 5 | L. Le Cam and G. L. Yang, Asymptotics in Statistics: Some basic conc… (2000) | 0.737 | 5 | 3 | 40% |
| 6 | M. G. Crandall, H. Ishii, and P.-L. Lions, “User's guide to viscosit… (1992) User's guide to viscosity solutions of second order partial differential equations | 0.737 | 4 | 3 | 50% |
| 7 | T. L. Lai, “Adaptive treatment allocation and the multi-armed bandit… (1987) Adaptive treatment allocation and the multi-armed bandit problem | 0.737 | 3 | 2 | 100% |
| 8 | A. W. Van der Vaart, Asymptotic statistics. 1em plus 0.5em minus 0.4… (2000) | 0.669 | 10 | 3 | 30% |
| 9 | K. Hirano and J. R. Porter, “Asymptotics for statistical treatment r… (2009) Asymptotics for statistical treatment rules | 0.644 | 2 | 2 | 100% |
| 10 | G. Barles and P. E. Souganidis, “Convergence of approximation scheme… (1991) Convergence of approximation schemes for fully nonlinear second order equations | 0.585 | 4 | 3 | 25% |
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