Variance-sensitive Thompson sampling for generalised linear bandits, revisited

We study a variant of the Thompson Sampling (TS) algorithm, called $α$-TS, for solving stochastic generalized linear bandit problems. Existing analyses of TS require inflating the posterior variance to derive near-optimal regret guarantees. We formalize the idea of variance inflation by introducing $α$-TS that uses a fractional or $α$-posterior instead of the standard posterior. Our main contribution is to identify general regularity conditions on the prior and reward distributions that enable a regret analysis of $α$-TS without assuming any tractable approximation of the posterior distribution, unlike previous works. For a specific choice of $α\propto d^{-1}$, our general regret bound yields the best known regret bound of $O(d^{3/2}\sqrt{T}\log T)$ for both the exponential and sub-Gaussian families of reward distributions. We further provide an $α$-dependent lower bound showing that the regret constant depends on the product $αd$, and that when $α\propto d^{-1}$ the regret scales as $Ω(d^{3/2}\sqrt{T})$, explaining the origin of the $d^{3/2}$ factor in the upper bound. Our proof technique adapts and combines recent advancements in the analysis of linear bandit problems with first- and second-order posterior concentration theory from the Bayesian statistics literature.

We prove that Thompson sampling exhibits $\tilde{O}(σd \sqrt{T} + d r \sqrt{\mathrm{Tr}(Σ_0)})$ Bayesian regret in the linear-Gaussian bandit with a $\mathcal{N}(μ_0, Σ_0)$ prior distribution on the coefficients, where $d$ is the dimension, $T$ is the time horizon, $r$ is the maximum $\ell_2$ norm of the actions, and $σ^2$ is the noise variance. In contrast to existing regret bounds, this shows that to within logarithmic factors, the prior-dependent burn-in'' term $d r \sqrt{\mathrm{Tr}(Σ_0)}$ decouples additively from the minimax (long run) regret $σd \sqrt{T}$. Previous regret bounds exhibit a multiplicative dependence on these terms. We establish these results via a new elliptical potential'' lemma, and also provide a lower bound indicating that the burn-in term is unavoidable.

In stochastic linear bandits, the canonical Upper Confidence Bound (UCB) algorithm admits a simple frequentist regret analysis but can be computationally demanding, while Thompson Sampling (TS) is computationally attractive yet typically harder to analyze due to its non-optimistic nature. We propose Absolute Thompson Sampling (ATS), a simple modification of TS that ensures optimism in expectation by replacing the signed exploration noise with its absolute value. This preserves the computational efficiency of TS while avoiding the technically involved anti-concentration arguments common in TS analyses, enabling a simple UCB-style regret analysis. We show that ATS achieves $\tilde{O}(d^{3/2}\sqrt{K})$ regret, matching existing bounds for TS in linear bandits. We further introduce Ensemble Absolute Thompson Sampling (EATS), which takes the maximum over multiple absolute perturbations with normalization by the ensemble size. As the ensemble size grows, EATS converges to the UCB objective, recovering UCB behavior in the limit. Experiments show that moderate ensemble sizes already yield strong performance. Our results point to a bridge between randomized exploration and deterministic optimism both in theory and practice.