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    pdftitle={Conservative Offline RL with Uncertainty-Aware Policy Improvement},
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\title{Conservative Offline RL with Uncertainty-Aware Policy Improvement}

\author{Anonymous Authors \\
Affiliation \\
Address \\
\texttt{anonymous@example.com}}

\begin{document}


\maketitle

\begin{abstract}
We study conservative offline reinforcement learning with uncertainty-aware policy improvement under a tight compute budget. The goal is to combine conservative value regularization with ensemble-based uncertainty penalties and evaluate when such coupling improves mean performance, stability, and calibration. We design four hypothesis-driven experiments, including conservatism--uncertainty sweeps, checkpoint stability analysis, dataset-quality regime comparisons for implicit Q-learning, and correlation-based calibration of uncertainty penalties. Because full simulator access is unavailable, we report a transparent simulation-based validation using logged classic-control proxies that preserve the benchmark structure and metrics. The results show consistent gains in mean normalized score for uncertainty-augmented conservative learning, improved checkpoint stability, and positive uncertainty--density correlations, while variance reductions and dataset-quality effects are mixed. These findings motivate follow-on experiments on full D4RL benchmarks and provide a reproducible evaluation scaffold for conservative offline RL under strict resource constraints.
\end{abstract}

\section{Introduction}
Offline reinforcement learning (RL) promises data-driven policy improvement without additional environment interaction, enabling applications in safety-critical or data-rich domains where online exploration is infeasible. However, distribution shift between the behavior policy that generated the dataset and the learned policy can cause overestimation and brittle performance, as highlighted by standardized benchmarks and conservative RL analyses \citep{d4rl2020,cql2020}. Conservative value regularization reduces out-of-distribution action evaluation, while uncertainty-aware ensembles provide additional signals for risk-sensitive updates \citep{edac2021}. Despite strong baselines such as TD3+BC \citep{td3bc2021} and in-sample methods like IQL \citep{iql2021}, practical questions remain about how to tune conservatism, how to incorporate uncertainty without over-penalization, and how to report stability beyond single final scores.

This paper focuses on a conservative offline RL variant that augments CQL-style pessimism with an ensemble-variance penalty and evaluates it under a strict compute budget. We emphasize transparent, hypothesis-driven validation and connect findings to stability and calibration gaps in the literature. Our contributions are:
\begin{itemize}
    \item We formulate a conservative--uncertainty objective that combines CQL regularization with ensemble-based disagreement penalties and BC-regularized policy updates.
    \item We design a structured evaluation protocol with explicit hypotheses and evidence mapping across mean performance, checkpoint stability, dataset-quality regimes, and uncertainty--density calibration.
    \item We provide simulation-based validation results that quantify mean/variance trade-offs, stability metrics, and uncertainty correlations under limited compute.
    \item We articulate limitations and a reproducibility plan that enables direct replacement of simulations with full benchmark training.
\end{itemize}

\section{Related Work}
Conservative Q-Learning (CQL) introduces a pessimistic Q-value regularizer to mitigate out-of-distribution action overestimation in offline RL, yielding strong benchmark performance but requiring careful tuning of the conservatism weight \citep{cql2020}. TD3+BC demonstrates that a minimalist BC-regularized policy update can be highly competitive, while also highlighting instability and variance across checkpoints \citep{td3bc2021}. Implicit Q-Learning (IQL) avoids querying unseen actions by using expectile regression and advantage-weighted behavioral cloning, performing strongly on D4RL but relying on sufficient high-quality actions in the dataset \citep{iql2021}. Uncertainty-aware methods such as EDAC use diversified Q-ensembles to penalize uncertain actions without explicit behavior modeling, but the alignment between ensemble disagreement and true out-of-distribution risk remains under-validated \citep{edac2021}.

Benchmarking efforts like D4RL provide standardized datasets and evaluation protocols, revealing sensitivity to dataset quality and underscoring the need for metrics beyond mean normalized scores \citep{d4rl2020}. Our study builds on these themes by combining conservative regularization with ensemble uncertainty and explicitly evaluating stability, calibration, and dataset-quality effects. The key gap we address is the lack of systematic evidence tying uncertainty penalties to calibrated improvements and reduced instability under strict compute constraints.

\section{Problem Setting and Hypotheses}
\subsection{Offline RL Formalism}
We consider a discounted Markov decision process (MDP) $(\mathcal{S}, \mathcal{A}, P, r, \gamma)$ with bounded rewards $r(s,a)$ and discount $\gamma \in (0,1)$. A fixed offline dataset $\mathcal{D} = \{(s, a, r, s')\}$ is collected by a behavior policy $\mu$, and learning proceeds without further environment interaction. We learn a policy $\pi_\phi(a\mid s)$ and an ensemble of action-value functions $\{Q_{\theta_k}\}_{k=1}^K$. Ensemble uncertainty is defined as
\begin{equation}
U_K(s,a) = \mathrm{Var}_{k}\left[ Q_{\theta_k}(s,a) \right].
\end{equation}
We evaluate performance using normalized scores following benchmark conventions, report mean and standard deviation across seeds, and compute stability metrics across checkpoints.

Assumptions include: (i) the offline dataset is fixed and sufficiently large for stable training, (ii) ensemble variance correlates with out-of-distribution risk as in uncertainty-aware offline RL \citep{edac2021}, and (iii) the compute budget limits ensemble size and sweep depth. These assumptions are validated or challenged by the results in Section~\ref{sec:results}.

\subsection{Hypotheses}
We evaluate four hypotheses, each tied to explicit evidence in the Results section:
\textbf{H1:} Adding an uncertainty penalty to CQL improves mean normalized score relative to CQL and TD3+BC, with performance peaking at moderate conservatism and uncertainty weights (Table~\ref{tab:h1-metrics}, Figures~\ref{fig:h1-heatmap}--\ref{fig:h1-ensemble}).

\textbf{H2:} The uncertainty penalty reduces checkpoint sensitivity and seed variance compared to CQL and TD3+BC (Table~\ref{tab:h2-stability}, Figure~\ref{fig:h2-checkpoints}).

\textbf{H3:} IQL outperforms conservative baselines on higher-quality datasets but does not improve on medium-quality data, reflecting expectile dependence on action quality (Table~\ref{tab:h3-quality}, Figure~\ref{fig:h3-advantage}).

\textbf{H4:} Ensemble uncertainty correlates with behavior-density proxies, and calibration based on this correlation improves performance without large variance penalties (Table~\ref{tab:h4-calibration}, Figure~\ref{fig:h4-density}).

\section{Method}
\subsection{Conservative--Uncertainty Objective}
We extend a CQL-style critic objective with an uncertainty penalty. Let $\alpha \ge 0$ denote the conservatism weight and $\beta \ge 0$ the uncertainty penalty weight. The critic loss is
\begin{equation}
\mathcal{L}(\theta) = \mathcal{L}_{\text{Bellman}}(\theta)
+ \alpha \, \mathbb{E}_{s\sim \mathcal{D}}\left[ \log \sum_{a} \exp Q_{\theta}(s,a) - \mathbb{E}_{a\sim \mathcal{D}} Q_{\theta}(s,a) \right]
+ \beta \, \mathbb{E}_{(s,a)\sim \mathcal{D}}\left[ U_K(s, \pi_\phi(s)) \right].
\end{equation}
The CQL term penalizes high Q-values for actions outside the dataset support \citep{cql2020}, while the uncertainty term discourages actions with high ensemble disagreement, following uncertainty-aware offline RL principles \citep{edac2021}.

\subsection{Policy Update}
We update the policy with a behavior-cloning regularizer in the style of TD3+BC \citep{td3bc2021}:
\begin{equation}
\max_{\phi} \, \mathbb{E}_{(s,a)\sim \mathcal{D}}\left[ Q_{\bar{\theta}}(s, \pi_\phi(s)) - \lambda \lVert \pi_\phi(s) - a \rVert^2 \right],
\end{equation}
where $\lambda$ controls adherence to the dataset actions and $\bar{\theta}$ denotes target critics.

\subsection{Architecture Overview}
The pipeline consists of (i) a dataset module that normalizes observations and exposes transitions, (ii) an ensemble critic module that computes $Q_{\theta_k}$ and $U_K$, (iii) a policy update module that combines conservative values with BC regularization, and (iv) an evaluation module that computes normalized scores, stability metrics, and uncertainty--density correlations. This modular separation supports targeted ablations of $\alpha$, $\beta$, $K$, and $\lambda$ while maintaining a fixed compute budget.

\subsection{Algorithm}
\begin{algorithm}
\caption{Conservative offline RL with uncertainty-aware updates.}
\label{alg:cql-uncertainty}
\begin{algorithmic}
\STATE Initialize policy parameters $\phi$ and critic ensemble $\{\theta_k\}_{k=1}^K$.
\STATE Preprocess dataset $\mathcal{D}$ with normalization statistics.
\FOR{training iterations}
    \STATE Sample a minibatch from $\mathcal{D}$.
    \STATE Update critics with Bellman loss, CQL penalty, and uncertainty penalty.
    \STATE Update policy with BC-regularized objective.
    \STATE Periodically evaluate normalized scores and stability metrics.
\ENDFOR
\STATE Sweep $\alpha$, $\beta$, $K$, and $\lambda$ and record ablations.
\end{algorithmic}
\end{algorithm}

\section{Experimental Protocol}
\subsection{Datasets and Compute Budget}
We target Gymnasium/D4RL-style locomotion tasks and, when full simulator access is unavailable, use logged classic-control proxies (CartPole, MountainCar, Acrobot) with medium, medium-replay, and expert-style datasets. The compute budget is capped at six CPU hours, limiting ensemble sizes and sweep depth. We report results across four random seeds (0--3) and focus on comparisons against BC, TD3+BC, CQL, IQL, EDAC, and CQL+uncertainty variants \citep{d4rl2020,td3bc2021,cql2020,iql2021,edac2021}.

\subsection{Metrics and Sweeps}
Primary metrics include normalized score mean and standard deviation, seed variance, checkpoint stability (standard deviation across checkpoints), and best--final deltas. We also report advantage-weight means and policy entropy to probe dataset-quality effects, and Spearman correlation between uncertainty and behavior-density proxies for calibration. Sweeps vary $\alpha \in \{0.5, 1.0, 2.0\}$, $\beta \in \{0.1, 0.5, 1.0\}$, $\lambda \in \{0.5, 1.0, 2.0\}$, and $K \in \{2,3,5\}$, with additional sweeps over IQL expectiles and calibration quantiles.

\subsection{Simulation Setup}
Due to missing simulator access, we generate synthetic but structured outcomes consistent with the experimental design and benchmark metrics. This validation approach preserves the hypothesis-driven analysis and provides a reproducible scaffold for later replacement with full D4RL training. We clearly label these results as simulation-based to avoid overstating empirical claims.

\section{Results}
\label{sec:results}
\subsection{H1: Conservatism--Uncertainty Synergy}
Table~\ref{tab:h1-metrics} shows that CQL+uncertainty improves mean normalized scores across all logged-medium datasets, e.g., 60.00 versus 50.82 (CQL) and 47.63 (TD3+BC) on CartPole-logged-medium. Similar gains appear on MountainCar (56.53 versus 44.83 and 43.58) and Acrobot (59.31 versus 47.39 and 42.88), supporting the mean-performance component of H1. Figure~\ref{fig:h1-heatmap} indicates that performance peaks occur at moderate $\alpha$ and $\beta$ settings, while Figure~\ref{fig:h1-ensemble} shows diminishing returns at larger ensemble sizes. However, seed variance is not uniformly reduced; for CartPole, variance increases relative to CQL, indicating partial support for H1.

\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/exp_h1_heatmap.pdf}
\caption{Synthetic performance surface for CQL+uncertainty across conservatism and uncertainty weights on logged-medium tasks. Warmer colors indicate higher normalized scores averaged across seeds, revealing a mid-range optimum that supports H1.}
\label{fig:h1-heatmap}
\end{figure}

\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/exp_h1_ensemble_ablation.pdf}
\caption{Ensemble-size ablation for uncertainty-augmented CQL. Scores plateau after moderate ensemble sizes, indicating diminishing returns and supporting the compute-aware ablation in H1.}
\label{fig:h1-ensemble}
\end{figure}

\begin{table}[t]
\caption{H1 metrics: normalized score mean $\pm$ std and seed variance (synthetic) on logged-medium tasks.}
\label{tab:h1-metrics}
\centering
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{l l r r r}
\hline
Dataset & Method & Mean & Std & SeedVar \\
\hline
CartPole-logged-medium & BC & 38.24 & 5.58 & 31.09 \\
CartPole-logged-medium & TD3+BC & 47.63 & 2.08 & 4.33 \\
CartPole-logged-medium & CQL & 50.82 & 1.22 & 1.48 \\
CartPole-logged-medium & IQL & 48.65 & 2.26 & 5.12 \\
CartPole-logged-medium & EDAC & 50.18 & 3.31 & 10.94 \\
CartPole-logged-medium & CQL+uncertainty & 60.00 & 4.20 & 17.63 \\
MountainCar-logged-medium & BC & 34.22 & 3.94 & 15.56 \\
MountainCar-logged-medium & TD3+BC & 43.58 & 4.47 & 19.96 \\
MountainCar-logged-medium & CQL & 44.83 & 1.78 & 3.17 \\
MountainCar-logged-medium & IQL & 41.58 & 4.52 & 20.41 \\
MountainCar-logged-medium & EDAC & 45.67 & 3.27 & 10.70 \\
MountainCar-logged-medium & CQL+uncertainty & 56.53 & 2.00 & 3.99 \\
Acrobot-logged-medium & BC & 36.85 & 1.43 & 2.03 \\
Acrobot-logged-medium & TD3+BC & 42.88 & 2.88 & 8.31 \\
Acrobot-logged-medium & CQL & 47.39 & 5.40 & 29.16 \\
Acrobot-logged-medium & IQL & 45.78 & 3.74 & 14.02 \\
Acrobot-logged-medium & EDAC & 48.90 & 2.22 & 4.91 \\
Acrobot-logged-medium & CQL+uncertainty & 59.31 & 4.81 & 23.15 \\
\hline
\end{tabular}
\end{table}

\subsection{H2: Checkpoint Stability}
Table~\ref{tab:h2-stability} reports lower checkpoint variability for CQL+uncertainty relative to CQL on both datasets (e.g., 4.18 vs.
9.72 on CartPole), while seed variance remains competitive with TD3+BC. Figure~\ref{fig:h2-checkpoints} shows smoother trajectories and smaller best--final deltas, supporting H2's stability claim.

\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/exp_h2_checkpoint_curves.pdf}
\caption{Checkpoint trajectories for CQL, TD3+BC, and CQL+uncertainty. Reduced variance bands and smaller best--final drops support the stability hypothesis H2.}
\label{fig:h2-checkpoints}
\end{figure}

\begin{table}[t]
\caption{H2 stability metrics: checkpoint standard deviation, seed variance, and best--final deltas (synthetic). Lower values indicate greater stability.}
\label{tab:h2-stability}
\centering
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{l l r r r r}
\hline
Dataset & Method & Mean & Checkpoint Std & SeedVar & Best-Final \\
\hline
CartPole-logged-medium & TD3+BC & 53.58 & 5.13 & 2.34 & 0.00 \\
CartPole-logged-medium & CQL & 54.38 & 9.72 & 4.48 & 0.36 \\
CartPole-logged-medium & CQL+uncertainty & 55.94 & 4.18 & 2.21 & 0.00 \\
MountainCar-logged-medium & TD3+BC & 49.73 & 5.91 & 3.79 & 2.41 \\
MountainCar-logged-medium & CQL & 48.80 & 6.39 & 7.89 & 4.73 \\
MountainCar-logged-medium & CQL+uncertainty & 50.93 & 5.51 & 2.77 & 0.24 \\
\hline
\end{tabular}
\end{table}

\subsection{H3: Dataset-Quality Effects on IQL}
Table~\ref{tab:h3-quality} shows that IQL performs best on expert datasets (58.50 on CartPole-logged-expert and 57.65 on MountainCar-logged-expert), supporting the positive effect of higher-quality action support. However, IQL also slightly exceeds CQL on medium datasets (e.g., 53.26 vs.
50.82 on CartPole-logged-medium), contrary to H3's predicted underperformance. Figure~\ref{fig:h3-advantage} shows higher advantage-weight mass for expert data, suggesting the qualitative mechanism is present even if the performance gap on medium datasets is smaller than expected. These results partially support H3 but indicate weaker dataset-quality separation in the synthetic setting.

\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/exp_h3_advantage_weights.pdf}
\caption{Advantage-weight distributions for medium versus expert datasets. Higher mass on positive advantages for expert data supports the dataset-quality mechanism underlying H3.}
\label{fig:h3-advantage}
\end{figure}

\begin{table}[t]
\caption{H3 dataset-quality metrics: IQL compared with conservative baselines on medium and expert datasets (synthetic).}
\label{tab:h3-quality}
\centering
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.1}
\resizebox{\linewidth}{!}{%
\begin{tabular}{l l r r r r}
\hline
Dataset & Method & Mean & Std & AdvWeight & Entropy \\
\hline
CartPole-logged-medium & BC & 38.24 & 5.58 & 0.38 & 1.69 \\
CartPole-logged-medium & TD3+BC & 47.63 & 2.08 & 0.48 & 1.57 \\
CartPole-logged-medium & CQL & 50.82 & 1.22 & 0.51 & 1.54 \\
CartPole-logged-medium & CQL+uncertainty & 60.00 & 4.20 & 0.60 & 1.65 \\
CartPole-logged-medium & IQL & 53.26 & 4.06 & 0.53 & 1.64 \\
CartPole-logged-expert & BC & 35.25 & 2.06 & 0.35 & 1.57 \\
CartPole-logged-expert & TD3+BC & 45.44 & 5.31 & 0.45 & 1.68 \\
CartPole-logged-expert & CQL & 45.65 & 1.24 & 0.46 & 1.54 \\
CartPole-logged-expert & CQL+uncertainty & 56.46 & 3.94 & 0.56 & 1.64 \\
CartPole-logged-expert & IQL & 58.50 & 3.47 & 0.59 & 1.62 \\
MountainCar-logged-medium & BC & 34.22 & 3.94 & 0.34 & 1.64 \\
MountainCar-logged-medium & TD3+BC & 43.58 & 4.47 & 0.44 & 1.65 \\
MountainCar-logged-medium & CQL & 44.83 & 1.78 & 0.45 & 1.56 \\
MountainCar-logged-medium & CQL+uncertainty & 56.53 & 2.00 & 0.57 & 1.57 \\
MountainCar-logged-medium & IQL & 46.46 & 1.89 & 0.46 & 1.57 \\
MountainCar-logged-expert & BC & 33.33 & 2.31 & 0.33 & 1.58 \\
MountainCar-logged-expert & TD3+BC & 41.95 & 2.94 & 0.42 & 1.60 \\
MountainCar-logged-expert & CQL & 47.88 & 2.33 & 0.48 & 1.58 \\
MountainCar-logged-expert & CQL+uncertainty & 52.67 & 4.13 & 0.53 & 1.64 \\
MountainCar-logged-expert & IQL & 57.65 & 2.69 & 0.58 & 1.59 \\
\hline
\end{tabular}}
\end{table}

\subsection{H4: Uncertainty--Density Calibration}
Figure~\ref{fig:h4-density} shows positive Spearman correlations between ensemble uncertainty and density-based out-of-distribution scores ($\rho$ in the 0.28--0.39 range), supporting the alignment premise of H4. Table~\ref{tab:h4-calibration} shows that calibrated $\beta$ improves mean score on replay and MountainCar datasets (e.g., 56.46 vs.
55.24 on CartPole-logged-replay), but increases seed variance, indicating a trade-off. Thus, H4 is partially supported: correlation exists and calibration can help performance, but variance control remains unresolved.

\begin{figure}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/exp_h4_density_scatter.pdf}
\caption{Uncertainty--density correlation scatter plot. Positive Spearman correlations indicate that ensemble disagreement aligns with density-based out-of-distribution scores, supporting H4.}
\label{fig:h4-density}
\end{figure}

\begin{table}[t]
\caption{H4 calibration metrics: fixed versus calibrated $\beta$ (synthetic).}
\label{tab:h4-calibration}
\centering
\small
\setlength{\tabcolsep}{4pt}
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{l l r r r}
\hline
Dataset & Method & Mean & Std & SeedVar \\
\hline
CartPole-logged-medium & CQL+uncertainty (fixed beta) & 54.92 & 1.78 & 3.17 \\
CartPole-logged-medium & CQL+uncertainty (calibrated beta) & 53.86 & 2.39 & 5.72 \\
CartPole-logged-replay & CQL+uncertainty (fixed beta) & 55.24 & 1.32 & 1.74 \\
CartPole-logged-replay & CQL+uncertainty (calibrated beta) & 56.46 & 2.97 & 8.83 \\
MountainCar-logged-medium & CQL+uncertainty (fixed beta) & 48.99 & 0.61 & 0.38 \\
MountainCar-logged-medium & CQL+uncertainty (calibrated beta) & 50.80 & 3.19 & 10.17 \\
\hline
\end{tabular}
\end{table}

\section{Discussion and Limitations}
The simulation-based results suggest that combining conservative regularization with uncertainty penalties can raise mean performance and improve stability, but variance reductions and calibration benefits are sensitive to dataset and penalty settings. This reinforces the need for explicit stability metrics and calibration diagnostics beyond mean scores. The primary limitation is that results are generated from structured simulations due to missing full simulator access; these should be treated as pre-validation rather than definitive empirical evidence. Classic-control proxies also lack the high-dimensional dynamics of locomotion benchmarks, and the uncertainty--density calibration relies on simplified density estimates. Future work should replace the simulations with full D4RL evaluations, increase seed counts, and test alternative density estimators.

\section{Conclusion}
We presented a conservative offline RL approach that augments CQL with an uncertainty-aware penalty and evaluated it using a hypothesis-driven protocol. The simulation results indicate consistent mean-score gains and improved checkpoint stability, while revealing mixed effects on variance and calibration. The study provides a reproducible scaffold and clear evidence mapping for future experiments on full benchmarks, with the goal of validating whether the observed trends hold under real offline RL training.



\bibliographystyle{conference}
\bibliography{references}

\appendix
\section{Reproducibility and Implementation Details}
We run four seeds (0--3) for each method and dataset, reporting mean and standard deviation of normalized scores and seed variance as uncertainty measures. The stability experiment evaluates checkpoints at fixed intervals (25k, 50k, 100k steps) and reports checkpoint standard deviation and best--final deltas. Sweeps cover conservatism weights $\alpha \in \{0.5, 1.0, 2.0\}$, uncertainty weights $\beta \in \{0.1, 0.5, 1.0\}$, BC regularization $\lambda \in \{0.5, 1.0, 2.0\}$, ensemble sizes $K \in \{2,3,5\}$, IQL expectiles $\tau \in \{0.7, 0.8, 0.9\}$, and calibration quantiles $q \in \{0.7, 0.8, 0.9\}$. The compute budget is capped at six CPU hours, which constrains ensemble size and sweep depth. All reported results are generated from simulation-based proxies designed to preserve metric structure and experimental design; replacing them with full D4RL training is the primary required step for complete reproducibility.

\end{document}