Dynamic pricing algorithms typically assume continuous price variables, which may not reflect real-world scenarios where prices are often discrete. This paper demonstrates that leveraging discrete price information within a semi-parametric model can substantially improve performance, depending on the size of the support set of the price variable relative to the time horizon. Specifically, we propose a novel semi-parametric contextual dynamic pricing algorithm, namely BayesCoxCP, based on a Bayesian approach to the Cox proportional hazards model. Our theoretical analysis establishes high-probability regret bounds that adapt to the sparsity level γ, proving that our algorithm achieves a regret upper bound of eO(T(1+γ)/2 + dT) for γ < 1/3 and eO(T2/3 + dT) for γ 1/3, where γ represents the sparsity of the price grid relative to the time horizon T. Through numerical experiments, we demonstrate that our proposed algorithm significantly outperforms an existing method, particularly in scenarios with sparse discrete price points.

As one of the most mainstream paradigms for sequential decision-making, RL has extensive applications in many real-world problems (Kober et al., 2013; Mnih et al.,

In this section, we first summarize frequently used notations in the following table. Table 4: Frequently used notations.Notation Description C Lemma 7. Let d = 1 . Combining the cases above completes the proof. Subsection B.2 proves several convergence rate lemmas. Subsection B.3 gives some technical We are now ready to prove Theorem 1. Proof of Theorem 1.

Variance-dependent regret bounds have received increasing attention in recent studies on contextual bandits. However, most of these studies are focused on upper confidence bound (UCB)-based bandit algorithms, while sampling based bandit algorithms such as Thompson sampling are still understudied. The only exception is the LinVDTS algorithm (Xu et al., 2023), which is limited to linear reward function and its regret bound is not optimal with respect to the model dimension. In this paper, we present FGTSVA, a variance-aware Thompson Sampling algorithm for contextual bandits with general reward function with optimal regret bound. At the core of our analysis is an extension of the decoupling coefficient, a technique commonly used in the analysis of Feel-good Thompson sampling (FGTS) that reflects the complexity of the model space. With the new decoupling coefficient denoted by $\mathrm{dc}$, FGTS-VA achieves the regret of $\tilde{O}(\sqrt{\mathrm{dc}\cdot\log|\mathcal{F}|\sum_{t=1}^Tσ_t^2}+\mathrm{dc})$, where $|\mathcal{F}|$ is the size of the model space, $T$ is the total number of rounds, and $σ_t^2$ is the subgaussian norm of the noise (e.g., variance when the noise is Gaussian) at round $t$. In the setting of contextual linear bandits, the regret bound of FGTSVA matches that of UCB-based algorithms using weighted linear regression (Zhou and Gu, 2022).