Fast Online Learning with Gaussian Prior-Driven Hierarchical Unimodal Thompson Sampling

In stochastic multi-armed bandits, rewards are generated by an unknown distribution; the world's reward uncertainty induces the exploration-exploitation trade-off. Thompson sampling (TS) addresses this trade-off elegantly, but requires knowing the reward model's parametric form to sample from its posterior and compute per-arm expected rewards ---which limits its applicability as it demands per-scenario model specification and hyperparameter tuning. We here adopt Bayesian nonparametric (BNP) mixture models to remove this requirement, extending TS to arbitrary reward distributions. We model each arm's unknown reward distribution with an independent BNP Gaussian mixture, which can approximate any continuous reward density, and adapts its complexity to sequentially observed data. Embedding the BNP mixture model into TS yields Nonparametric TS, an algorithm that sequentially learns the reward model that best approximates the true, yet unknown, per-arm reward distribution, avoiding case-by-case model specification and hyperparameter tuning. We empirically demonstrate that ($i$) Nonparametric TS matches the regret performance of Oracle TS policies that know the true reward model; ($ii$) it favorably compares (in cumulative regret and volatility) to a suite of state-of-the-art alternatives across multimodal, outlier-contaminated, and non-exponential-family reward distributions; and ($iii$) its Bayesian nonparametric inference remains computationally practical and accurate for regret minimization.

We prove that $ρ\text{-}\mathrm{NPTS}_{\mathrm{SG}}$, an anchor-free nonparametric Thompson Sampling algorithm for risk-averse bandits, achieves regret matching the instance-dependent lower bound to leading order in $\log n$, establishing it as asymptotically optimal for any continuous risk functional $ρ$ (CVaR, mean-variance, Sharpe ratio, distortion risk measures, and more) on the class of distributions with bounded density and sub-Gaussian tails, including Gaussian arms. Both this result and its bounded-support counterpart require only continuity of $ρ$: strictly weaker than the dominance condition of prior parametric Thompson Sampling results, and strictly weaker than the Lipschitz condition of UCB-type algorithms, yielding the first instance-optimal guarantees for non-Lipschitz functionals such as the Sharpe ratio without parametric reward assumptions. The bounded-support case is developed first as a stepping stone sharing the same proof structure. The key technical contributions are a discretisation lemma (bounded support) and a truncated discretisation lemma (sub-Gaussian tails), each projecting the growing-alphabet Dirichlet posterior onto a fixed grid via the Dirichlet aggregation property, holding all polynomial prefactors at fixed degree independent of sample size and breaking the super-exponential barrier that blocked prior proofs.

Thompson sampling (TS) is one of the most popular and earliest algorithms to solve stochastic multi-armed bandit problems. We consider a variant of TS, named $α$-TS, where we use a fractional or $α$-posterior ($α\in(0,1)$) instead of the standard posterior distribution. To compute an $α$-posterior, the likelihood in the definition of the standard posterior is tempered with a factor $α$. For $α$-TS we obtain both instance-dependent $\mathcal{O}\left(\sum_{k \neq i^*} Δ_k\left(\frac{\log(T)}{C(α)Δ_k^2} + \frac{1}{2} \right)\right)$ and instance-independent $\mathcal{O}(\sqrt{KT\log K})$ frequentist regret bounds under very mild conditions on the prior and reward distributions, where $Δ_k$ is the gap between the true mean rewards of the $k^{th}$ and the best arms, and $C(α)$ is a known constant. Both the sub-Gaussian and exponential family models satisfy our general conditions on the reward distribution. Our conditions on the prior distribution can be easily satisfied by a density that is positive, continuous, and bounded. We also establish another instance-dependent regret upper bound that matches (up to constants) to that of improved UCB [Auer and Ortner, 2010]. Our regret analysis carefully adapts and combines recent theoretical developments in the non-asymptotic concentration analysis and Bernstein-von Mises type results for the $α$-posterior distribution. Moreover, our analysis does not require additional structural properties such as closed-form posteriors or conjugate priors.