Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs
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| Format: | Preprint |
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2025
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| _version_ | 1866910993190223872 |
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| author | Chen, Shulun Zhou, Runlong Zhang, Zihan Fazel, Maryam Du, Simon S. |
| author_facet | Chen, Shulun Zhou, Runlong Zhang, Zihan Fazel, Maryam Du, Simon S. |
| contents | We consider the gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\tilde{O}\left(\left(\sum_{Δ_h(s,a)>0} \frac{H^2 \log K \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_h(s,a)} +\sum_{Δ_h(s,a)=0}\frac{ H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_{\mathrm{min}}} + SAH^4 (S \lor H) \right) \log K\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $Δ_h(s,a) =V_h^* (a) - Q_h^* (s, a)$ represents the suboptimality gap and $Δ_{\mathrm{min}} := \min_{Δ_h (s,a) > 0} Δ_h(s,a)$. The term $\mathtt{Var}_{\max}^{\text{c}}$ denotes the maximum conditional total variance, calculated as the maximum over all $(π, h, s)$ tuples of the expected total variance under policy $π$ conditioned on trajectories visiting state $s$ at step $h$. $\mathtt{Var}_{\max}^{\text{c}}$ characterizes the maximum randomness encountered when learning any $(h, s)$ pair. Our result stems from a novel analysis of the weighted sum of the suboptimality gap and can be potentially adapted for other algorithms. To complement the study, we establish a lower bound of $$Ω\left( \sum_{Δ_h(s,a)>0} \frac{H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_h(s,a)}\cdot \log K\right),$$ demonstrating the necessity of dependence on $\mathtt{Var}_{\max}^{\text{c}}$ even when the maximum unconditional total variance (without conditioning on $(h, s)$) approaches zero. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_06521 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs Chen, Shulun Zhou, Runlong Zhang, Zihan Fazel, Maryam Du, Simon S. Machine Learning We consider the gap-dependent regret bounds for episodic MDPs. We show that the Monotonic Value Propagation (MVP) algorithm achieves a variance-aware gap-dependent regret bound of $$\tilde{O}\left(\left(\sum_{Δ_h(s,a)>0} \frac{H^2 \log K \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_h(s,a)} +\sum_{Δ_h(s,a)=0}\frac{ H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_{\mathrm{min}}} + SAH^4 (S \lor H) \right) \log K\right),$$ where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $K$ is the number of episodes. Here, $Δ_h(s,a) =V_h^* (a) - Q_h^* (s, a)$ represents the suboptimality gap and $Δ_{\mathrm{min}} := \min_{Δ_h (s,a) > 0} Δ_h(s,a)$. The term $\mathtt{Var}_{\max}^{\text{c}}$ denotes the maximum conditional total variance, calculated as the maximum over all $(π, h, s)$ tuples of the expected total variance under policy $π$ conditioned on trajectories visiting state $s$ at step $h$. $\mathtt{Var}_{\max}^{\text{c}}$ characterizes the maximum randomness encountered when learning any $(h, s)$ pair. Our result stems from a novel analysis of the weighted sum of the suboptimality gap and can be potentially adapted for other algorithms. To complement the study, we establish a lower bound of $$Ω\left( \sum_{Δ_h(s,a)>0} \frac{H^2 \land \mathtt{Var}_{\max}^{\text{c}}}{Δ_h(s,a)}\cdot \log K\right),$$ demonstrating the necessity of dependence on $\mathtt{Var}_{\max}^{\text{c}}$ even when the maximum unconditional total variance (without conditioning on $(h, s)$) approaches zero. |
| title | Sharp Gap-Dependent Variance-Aware Regret Bounds for Tabular MDPs |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2506.06521 |