TransferRL: Efficient Knowledge Transfer

• TransferRL is a framework that formalizes the transfer of knowledge between source and target reinforcement learning domains, aligning state and action spaces for efficient learning.
• It employs methods such as sample transfer, distribution matching, and trajectory-level optimization to boost sample efficiency and mitigate policy and dynamics discrepancies.
• TransferRL integrates various algorithmic techniques like transferred Q-learning, RPTO, and TvD, offering theoretical guarantees and empirical improvements in continuous control and multi-agent systems.

Transfer Reinforcement Learning (TransferRL) formalizes the problem of efficiently transferring knowledge from one or multiple source reinforcement learning (RL) domains to a target domain, leveraging similarities in environment dynamics, reward structure, or agent experience to accelerate learning, improve sample efficiency, and enhance generalization. Various approaches to TransferRL have been proposed, including sample transfer, distribution matching, trajectory-level optimization, and explicit modeling of relativity gaps across domains. The methodologies span classical offline regression, fitted Q-iteration, imitation learning, optimal transport, and fully integrated policy-dynamics updates.

1. Source and Target MDPs: Formal Setups

TransferRL typically assumes access to one or several source MDPs $$\mathcal{M}_m = \langle \mathcal{X}, \mathcal{A}, R_m, P_m, \gamma \rangle$$ sharing state-action spaces (sometimes differing distributions), and a target MDP $$\mathcal{M}_1 = \langle \mathcal{X}, \mathcal{A}, R_1, P_1, \gamma \rangle$$ (Lazaric et al., 2011, Chen et al., 2022). The agent's objective is to efficiently solve the target task, often with limited target samples, by leveraging source-domain data:

• State space alignment: Most frameworks assume shared or mappable state spaces, though some (e.g., "Transfer RL via the Undo Maps Formalism" (Gupta et al., 2022)) deal with intrinsic state-space drift via domain transformations.
• Policy and transition discrepancies: Key distinctions arise when either reward $R_m$, transition kernel $P_m$, or even the state representation itself differs between source and target, requiring correction mechanisms.

A common abstraction is the mixture model, where training sets are augmented by source samples, importance-weighted according to similarity metrics or auxiliary estimates.

2. Transfer Objectives and Distribution Matching

Early sample-transfer methods minimize the Bellman error across mixed datasets without explicit similarity weighting (Lazaric et al., 2011). Adaptive algorithms (BAT, BTT) optimize source mixing proportions $\lambda$ by minimizing empirical Bellman transfer error:

$$\hat{E}_\lambda(Q) = \frac{1}{S} \sum_{s=1}^S \left[ R_{s,1} - \sum_{m=2}^M \lambda_m R_{s,m} + \frac{\gamma}{T} \sum_{t=1}^T ( \max_{a'} Q(Y_{s,1}^t, a') - \sum_{m=2}^M \lambda_m \max_{a'} Q(Y_{s,m}^t, a') ) \right]^2$$

More recent approaches reframe the problem as trajectory-level distribution matching (Gupta et al., 2022). Here, discrepancies between environments are characterized by complex transformations ("undo maps"), and the goal is to learn policies whose induced trajectory distributions in the target domain optimally align with those from the source. This is typically formalized via optimal transport metrics over trajectory spaces.

Relative Policy-Transition Optimization (RPTO) (Xu et al., 2022) further refines this objective by decomposing the "relativity gap," quantifying return discrepancies between source and target induced by both policy and dynamics mismatch:

$\Delta_{\text{rel}} = J(P', \pi) - J(P, \pi')$

Where $J(P, \pi)$ is the expected cumulative return under transition kernel $P$ and policy $\pi$.

3. Algorithmic Frameworks and Update Schemes

The following table summarizes principal TransferRL algorithmic variants:

Method	Core Update	Discrepancy Handling
AST/BAT/BTT (Lazaric et al., 2011)	FQI + mixture regression	Bellman error estimator $\hat{E}_\lambda$, adaptive source weighting
Transferred Q-learning (Chen et al., 2022)	Transfer-Lasso, backward recursion	Re-targeting source future-values for "vertical" transfer, sparse reward difference
RPTO (Xu et al., 2022)	RPO (policy), RTO (dynamics), closed-loop	Explicit relativity gap minimization, joint policy-dynamics optimization
TvD (Gupta et al., 2022)	Policy updates via trajectory OT	Data-centric undo-map learning, trajectory alignment

In transferred Q-learning, each source sample is "re-targeted" so future values are estimated in the target model, focusing horizontal transfer at each stage and allowing vertical cascading (information flow, value backpropagation) via backward recursion (Chen et al., 2022).

RPTO alternates between adjusting the policy (RPO) to maximize target returns and tuning a source-model (RTO) to minimize transition kernel mismatch. Critically, these updates occur within a closed feedback loop against both environments, yielding fast transfer in continuous-control tasks (Xu et al., 2022).

Distribution-matching approaches (TvD) avoid direct policy adaptation, instead matching trajectory-level statistics, and are particularly robust to environment drift that induces intrinsic state-space transformations (Gupta et al., 2022).

4. Theoretical Guarantees and Analytical Results

Sample-transfer algorithms establish explicit bounds on target Q-function estimation error and policy regret in both offline and online transfer settings:

• AST/BAT/BTT error bounds: One-step FQI error is bounded by inherent approximation error, transfer error $E_\lambda(Q)$, and estimation variance:

$$\| T(Q^k) - T_1 Q^{k-1}\|_\mu \leq 4\|f^* - T_1 Q^{k-1}\|_\mu + 5\sqrt{E_\lambda(Q^{k-1})} + O((d/L)^{1/2})$$

Over $K$ iterations, final policy error scales with the best achievable Bellman discrepancy (Lazaric et al., 2011).

• Transferred Q-learning: Guarantees minimax-optimal rates for Q-function convergence and reduced cumulative regret when the reward discrepancy $\delta_t^{(k)}$ is sparse (Chen et al., 2022):

$$\|\hat{\theta}_t - \theta_t\|^2 \lesssim \|\hat{\theta}_{t+1} - \theta_{t+1}\|^2 + \frac{s \log p}{N_{\text{src}}} + h \sqrt{ \frac{ \log p }{ n_0 } }$$

With sufficient source data and reward similarity, transferred Q-learning uniformly improves estimation.

• RPTO bounds: Every RPO step provably increases target environment return up to a total-variation error constant, while RTO reduces dynamics-induced return gap, guaranteeing convergence when model class matches the target (Xu et al., 2022).

Distribution-matching objectives (TvD) are supported by guarantees on trajectory-level optimal transport minimization, though detailed guarantees depend on environmental assumptions (Gupta et al., 2022).

5. Empirical Evaluation and Practical Performance

Empirical validation spans domains including continuous-chain navigation (Lazaric et al., 2011), gridworlds (Gupta et al., 2022), MuJoCo continuous control (Xu et al., 2022), multi-agent ride-sharing (Castagna et al., 2021), and real clinical data (MIMIC-III sepsis) (Chen et al., 2022):

• Continuous Chain/MDP mixtures: BAT achieves robust performance when target sample is scarce; BTT automatically balances estimation variance and transfer bias, down-weighting poor sources as target data grows.
• Multi-agent systems: Filtering source experiences by epistemic confidence yields superior early jumpstart and improved operational efficiency (higher %served requests, reduced detour ratios) in large fleets (Castagna et al., 2021).
• RL with environment drift: TvD achieves successful transfer where state-space transformations render direct policy transfer ineffective (Gupta et al., 2022).
• Continuous control (RPTO): RPTO is empirically optimal, requiring 2–5× fewer samples than strong baselines (SAC/TRPO warmstart, MBPO, SLBO, etc.) and maintaining higher final returns.
• Medical RL: Transferred Q-learning provides consistent improvements in cumulative reward and policy quality even in high-dimensional, sparse-reward healthcare settings (Chen et al., 2022).

6. Challenges, Limitations, and Open Directions

While TransferRL offers theoretical and empirical advantages, practical deployment faces several challenges:

• Negative transfer: When source MDPs cannot approximate the target, excessive transfer increases bias; adaptive algorithms mitigate this but require reliable similarity estimation (Lazaric et al., 2011, Chen et al., 2022).
• State-space misalignment: Undo-map and trajectory-matching formalisms are required where intrinsic drift alters the state distribution (Gupta et al., 2022).
• Model capacity and curriculum: RTO and similar approaches depend critically on expressive models and careful tuning of update pace; insufficient capacity leads to underfitting, instability (Xu et al., 2022).
• Transfer selection: Dynamic role assignment, online estimation of decision-relevant transfer content, and batch size/quality tuning remain active areas of research (Castagna et al., 2021).
• Feature space restriction: Most methods require aligned feature spaces; extending to heterogeneous or nonstationary settings is an open problem.

Potential extensions include meta-learning update schedules, domain randomization for broader transfer, and representation learning for cross-domain adaptation beyond trajectory statistics (Xu et al., 2022, Castagna et al., 2021).

7. Connections and Comparative Perspectives

TransferRL acts as an overarching principle linking supervised transfer learning, model-based RL, imitation learning, and offline RL in diverse settings. Whether through adaptive sample weighting, explicit distribution alignment, or joint optimization of policy and environment models, successful transfer hinges on quantifying and minimizing discrepancies between domains, often formalized through Bellman error, relativity gaps, or optimal transport distances. These frameworks collectively advance RL toward sample-efficient, robust deployment in complex, shifting environments and multi-agent systems.