Posterior Tempering Explains Variance Inflation in Linear and Generalized Linear Thompson Sampling
Prateek Jaiswal, Debdeep Pati, Anirban Bhattacharya, Bani K. Mallick
stat.ML · cs.IT · cs.LG · math.ST
Quick verdict
Formalizes TS variance inflation as α-posterior tempering, proving O(d^{3/2}√T log T) GLM-bandit regret; strong theory, but several proof gaps need fixing.
Summary
The paper studies Thompson Sampling (TS) for stochastic generalized linear bandits and formalizes the common variance-inflation heuristic through fractional posteriors. The proposed α-TS samples from an α-posterior in which the likelihood is tempered by α∈(0,1), inflating the posterior covariance; the authors show this coincides with the variance-inflated LinTS variants previously introduced ad hoc by Abeille–Lazaric and Agrawal–Goyal. The main result (Theorem 1) derives a general frequentist regret bound under structural conditions on prior and reward (Assumptions 1–7), combining the saturated/unsaturated action decomposition with first-order α-posterior contraction (Lemma 1) and a finite-sample Bernstein–von Mises lower bound (Lemma 2); no tractable posterior approximation is required. With α=d^{-1}, the bound becomes O(d^{3/2}√T log T) for both exponential-family and sub-Gaussian rewards (Corollaries 1–2), matching the best known LinTS rates and, the authors claim, extending them to GLM rewards for the first time. A hypercube construction (Propositions 1–2) yields an α-dependent lower bound: when αd=O(1), regret is Ω(d^{3/2}√T), suggesting the d^{3/2} rate is intrinsic to tempered-posterior sampling. Limitations: Proposition 2 is stated without a formal proof; the Rényi-divergence constant in Lemma 4 appears inconsistent with its own derivation; the analysis requires an Ω(d) warm-up; and the paper offers no simulations or code to corroborate the claimed dimension scaling.
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Assessment at a glance
4
risk flags identified
● high 1
● medium 1
● low 2
4 strengths · 4 weaknesses · 5 author questions
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Key claims under review
- α-TS, sampling from an α-tempered fractional posterior with α=d^{-1}, attains O(d^{3/2}√T log T) frequentist regret for generalized linear bandits under general regularity conditions (Theorem 1, Corollaries 1–2), matching the best known LinTS bounds and claimed as the first such result for GLM rewards.
- The α-posterior mechanism provably coincides with the variance-inflated sampling distributions used ad hoc in prior LinTS analyses, providing a principled Bayesian interpretation of the heuristic.
- An α-dependent lower bound shows the regret constant depends on the product αd, and for α∝d^{-1} the regret is Ω(d^{3/2}√T) on a hypercube instance, explaining the d^{3/2} factor of the upper bound (Propositions 1–2).
- The regret bounds are in expectation and free of a confidence hyperparameter δ, addressing a conjecture the paper attributes to Agrawal–Goyal.
Strengths
Principled unification of variance inflation
α-TS (Algorithm 1, eq. 2) exactly recovers the ad hoc covariance-inflated sampling distributions of Abeille–Lazaric [2] and Agrawal–Goyal [3]; the paper is explicit that no new algorithm is introduced, giving existing heuristics a coherent fractional-posterior interpretation.
General framework without closed-form posterior
Lemma 1 (α-posterior contraction, following Bhattacharya–Pati–Yang [7]) plus Lemma 2 (finite-sample BvM lower bound from Panov–Spokoiny [23]/Spokoiny [26]) replace Gaussian Mill's-ratio arguments; the modified discrepancy λα/2‖θ−θ0‖²+D^{(t)}_α aligning contraction with the V_t regret geometry is a real technical contribution.
Bounds in expectation, δ-free
Rare algorithm-matched lower bound
Weaknesses
Headline lower bound is unproven
Constant-level inconsistency at the core of the rate
No empirical validation whatsoever
Narrower scope than headline and loose early-time bookkeeping
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Scope & Claims
The paper (arXiv 2609.01999) studies Thompson Sampling (TS) for stochastic generalized linear bandits on compact action spaces. Its central object is α-TS (Algorithm 1), which samples from the fractional or α-posterior (eq. 2): the likelihood is tempered by α∈(0,1), inflating the sampling covariance. Three claims structure the contribution. First, this tempering is a principled re-reading of the variance-inflated sampling distributions introduced ad hoc in the LinTS analyses of Abeille–Lazaric [2] and Agrawal–Goyal [3] — the authors are explicit that no new algorithm is proposed. Second, Theorem 1 proves a general frequentist regret bound under regularity conditions on prior and reward (Assumptions 1–7) without any tractable posterior approximation; specialized to α=d^{-1}, it yields O(d^{3/2}√T log T) for exponential-family (Corollary 1) and smooth sub-Gaussian (Corollary 2) rewards, claimed as the first such result for GLM bandits. Third, Propositions 1–2 give an α-dependent lower bound on a hypercube instance: the regret constant depends on the product αd, and for α∝d^{-1} the regret is Ω(d^{3/2}√T), 'explaining' the d^{3/2} factor of the upper bound.
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Frequently asked questions
What is α-Thompson sampling (α-TS)?
α-TS is Thompson Sampling where sampling uses the α-posterior (fractional posterior): the likelihood is raised to a power α∈(0,1) before combining with the prior. With α<1 the posterior covariance is inflated, making exploration deliberately more aggressive. The paper shows this exactly reproduces the 'variance-inflated' Linear TS variants that earlier analyses introduced ad hoc.
Why is α chosen as 1/d in α-TS?
The general regret bound contains terms that grow exponentially in the dimension d unless tempering offsets posterior over-concentration. Choosing α=d^{-1} keeps the product αd constant, maintains a constant probability of sampling optimistic parameters, and yields the near-optimal O(d^{3/2}√T log T) regret; smaller α (e.g., d^{-(1+ε)}) degrades the rate.
What regret does α-TS achieve for generalized linear bandits?
Under the paper's regularity conditions, the upper bound is O(d^{3/2}√T log T) for both exponential-family and sub-Gaussian rewards with α=d^{-1}, matching the best known bounds for Gaussian-conjugate Linear TS (Agrawal–Goyal; Abeille–Lazaric). A lower bound Ω(d^{3/2}√T) for α∝d^{-1} is given on a hypercube instance, though its final statement is not formally proved in the paper.
Does the paper include experiments or code?
No. The paper is purely theoretical: there are no simulations, numerical experiments, or released code; all results are theorems with proofs in the appendices.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
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