Domain-Adaptive Reward Functions

• Domain-adaptive reward functions are mechanisms in reinforcement and imitation learning that decouple reward specification from task execution, enabling robust transfer.
• They employ modular architectures that separate task representation, dynamics, and reward learning, allowing selective adaptation to domain shifts with minimal retraining.
• Practical applications in robotics, imitation learning, and RLHF demonstrate improved sample efficiency, policy alignment, and versatility across varying environments.

Domain-adaptive reward functions are mechanisms in reinforcement learning (RL) and imitation learning that enable the learned reward signals, or their surrogates, to generalize, transfer, or adapt across environments with different dynamics, morphologies, observation modalities, or task specifications. These methods address the brittleness of classical reward functions and engineered shaping in the face of domain shift, thereby facilitating transfer learning, sample-efficient adaptation, and general-purpose policy alignment.

1. Foundational Concepts and Theoretical Motivation

Domain-adaptive reward functions are predicated on decoupling the specification of "what to achieve" from "how to achieve it," so that changes in environment dynamics, observation modalities, or objectives primarily require modifying only targeted modules, not the entire learned system. The foundational idea, formalized in the decoupling paradigm (Zhang et al., 2018), is to separate task representation, dynamics (forward and inverse models), and reward learning:

• Task Representation: An encoder–decoder architecture maps raw observations $s_t \in \mathcal{S}$ into a latent space $z_t \in \mathcal{Z}$:

$z_t = f_{\mathrm{enc}}(s_t;\theta_\mathrm{enc})$

• Dynamics Modeling: Forward and inverse models parameterize state transitions and action-generation in $\mathcal{Z}$:

$$\hat{z}_{t+1} = f_{\mathrm{for}}(z_t, a_t; \theta_{\mathrm{for}})$$

$$\hat{a}_t = f_{\mathrm{inv}}(z_t, z_{t+1}; \theta_{\mathrm{inv}})$$

• Reward Module: Policy and value learning operate in $\mathcal{Z}$:

$\pi(a_t | z_t; \theta_\mathrm{actor})$

$V(z_t; \theta_\mathrm{critic})$

Such modular separation implies that if the reward or the environment shifts, only the corresponding module needs to be adapted (policy/value or dynamics), reusing the rest.

Formally, domain-adaptive reward functions leverage properties such as smoothness or bounded transformation between reward and value functions across reward/reward-weight spaces (Kusari et al., 2019), key for multi-objective or preference-driven adaptations.

2. Architectural and Algorithmic Implementations

Latent Representation Decoupling:

The encoder, forward, and inverse dynamics modules are trained independently of the reward policy. The joint dynamics loss is:

$$\mathcal{L}_{\mathrm{dynamics}}(\theta) = \sum_t [\lambda_{\mathrm{dec}} \mathcal{L}_{t, \mathrm{dec}} + \lambda_{\mathrm{for}} \mathcal{L}_{t, \mathrm{for}} + \lambda_{\mathrm{inv}} \mathcal{L}_{t, \mathrm{inv}}]$$

Ensuring a robust $z$ allows the reward function to be transferred by retraining only the policy/value heads for new tasks or objectives.

Reward Correction via Domain Classifiers:

In off-dynamics settings (Eysenbach et al., 2020), domain classifiers estimate the dynamics shift:

$$\Delta r(s_t, a_t, s_{t+1}) = \log p_\mathrm{target}(s_{t+1}|s_t,a_t) - \log p_\mathrm{source}(s_{t+1}|s_t,a_t)$$

Estimated using two classifiers over $(s_t, a_t, s_{t+1})$ and $(s_t, a_t)$, this term is added to the reward during training in the source domain, penalizing policies that exploit artifacts in the source but are not valid in the target.

Adversarial Domain-Invariant Feature Extractors:

Adversarial learning schemes, e.g., DARL (Chen et al., 2019), employ Q-learning agents to select source data (instances) scored by domain-relevance, measured by

$\phi(x) = \mu_i \cdot D(F(x))_{(d)}$

with $\mu_i$ as class relevance and $D$ as a domain classifier. The reward is binary and episode-terminating: $R_t = +1$ if $\phi(x) > \tau$, $-1$ otherwise, driving careful source selection for adaptation. Domain-adversarial minimax games between feature extractors and discriminators promote domain-invariant representations.

Imitation Learning with Dynamics Embedding and Adversarial Losses:

Methods such as ADAIL (Lu et al., 2020) condition the policy on learned dynamics embeddings and employ discriminators with gradient reversal layers to enforce invariance to environment physics, so the reward function focuses solely on imitation-relevant behavioral features.

Hybrid Reward Shaping:

Reward functions adaptively composed of environment and auxiliary (heuristics, constraints) terms are optimized with architecture-aware (bi-level) objectives (Gupta et al., 2023) to counter misspecification and domain-specific guidance:

$$r_\phi(s,a) = f_{\phi_1}(s,a) + \phi_2 r_p(s,a) + \phi_3 r_{\mathrm{aux}}(s,a)$$

Inner policies are optimized on $r_\phi$; outer-loop optimizes $\phi$ for best original-task performance.

Constraint-based Reward Replacement:

The CaR framework (Ishihara et al., 8 Jan 2025) expresses complex tasks as sets of constraints $c_i(s,a) \leq 0$ and solves for policies with the Lagrangian formulation:

$$L(\pi, \lambda) = J(\pi) + \sum_i \lambda_i [c_i(s,a)]$$

Adaptive adjustment of Lagrange multipliers automatically balances objectives without manual tuning.

3. Transferability Mechanisms and Empirical Validation

Transfer Across Reward and Dynamics:

The decoupled modules exhibited in continuous (MuJoCo Swimmer, Ant, HalfCheetah) and discrete (Maze) tasks allow efficient re-optimizing only the relevant policy/reward modules (when rewards change) or only the dynamics modules (when environment changes), always reusing the shared latent backbone (Zhang et al., 2018). For example, DDR Prior achieved transfer rewards of 508 vs. 24.3 (A3C) on Ant after reward negation; planning in $z$ further improved performance under noise and transfer.

Reward Interpolation and Preference Adaptation:

Given scalarized multi-objective settings, Gaussian process regression can interpolate optimal value functions $V^*(s;w)$ across the reward weight space $w$, avoiding the need for retraining for every preference shift (Kusari et al., 2019). In autonomous driving, this translates into instant adaptation to driver preferences or environment randomization.

Adaptation Under Dynamics Shift:

Approaches based on learned correction terms, such as DARC and ODIRL (Eysenbach et al., 2020, Kang et al., 2021), shown in both gridworlds and high-dimensional control, match the trajectory distributions relevant for the target domain, penalizing spurious source-domain behaviors. Ablations confirm both classifier-based corrections and adversarial mechanisms as essential for alignment.

Imitation and Reward Transfer with Abstractions:

Abstract state-space reward inference (T-IRL) (Gui et al., 3 Jan 2025) learns environment-agnostic reward representations, yielding robust transfer across different dynamic instantiations of the same task, with evaluation on OpenAI Gym/AssistiveGym confirming improved transfer and correlation with expert returns. This suggests intrinsic preferences distilled via abstraction are key for transferability.

Router and Model Merging for Reward Models in LLMs:

In RLHF for LLMs, domain-adaptive reward models are obtained by merging pre-trained general models with domain-specific SFT models (DogeRM (Lin et al., 1 Jul 2024)), using weighted parameter averaging. Additionally, routing mechanisms (MoRE, RODOS, ARLISS) (Namgoong et al., 24 Jul 2024) modularize or select reward experts, improving domain specificity while reducing parameter size and training cost.

4. Adaptive and Policy-Aware Reward Design

Reward Informativeness and Adaptive Reward Design:

Expert-driven, policy-aware adaptation (Devidze et al., 10 Feb 2024) formulates reward informativeness $I_h(R)$ as the expected improvement in the learner’s current policy, measured as:

$$I_h(R_\phi\,|\,\bar{R},\pi^T,\pi^L) := \mathbb{E}_{s, a \sim \mu^L} \Big[ \mu^s_{(\pi^T)} \cdot \pi^L(a|s) \cdot \left( A_{(\bar{R})}^{(\pi^T)}(s,a) - A_{(\bar{R})}^{(\pi^T)}(s,\pi^L(s)) \right) A_{(R_\phi), h}^{(\pi^L)}(s,a) \Big]$$

Maximizing $I_h$ adaptively constructs rewards to maximize expected alignment per learning round, thus tuning to both domain and learner progress. Experiments show superior convergence and interpretability.

Adaptive Progression Shaping via Formal Languages:

Adaptive progression and hybrid reward functions for LTL-specified (or DFA-based) tasks (Kwon et al., 14 Dec 2024) use scheduled updates based on agent progress, dynamically adjusting rewards to reflect empirical difficulty of subgoals and transitions. This allows for robust learning even in partially infeasible scenarios or when DFA-state transitions do not align with observed task difficulty.

5. Limitations, Scalability, and Open Directions

While domain-adaptive reward functions offer pronounced gains in transfer, sample efficiency, and robustness, some recurring limitations exist:

• Dynamics Approximation Sensitivity:

Accuracy of classifiers or models estimating the domain shift determines penalization efficacy. Highly stochastic or mismatched domains may degrade correction quality (Kang et al., 2021, Eysenbach et al., 2020).

• Expressiveness and Alignment:

Success depends on the underlying representation capturing true task invariants (e.g., in T-IRL, abstraction quality governs transferability (Gui et al., 3 Jan 2025)). Poor alignment between learned latent spaces and intrinsic behavior can hinder generalization.

• Trade-offs in Multi-Domain Merging:

Router and model merging frameworks can introduce performance degradation in non-adapted domains (e.g., safety, general chat) if the weighting is not carefully managed (Lin et al., 1 Jul 2024, Namgoong et al., 24 Jul 2024).

• Complexity of Reward or Constraint Spaces:

As the number of constraints or objectives grows (as in CaR (Ishihara et al., 8 Jan 2025)), scalability and convergence become more challenging, requiring advances in Lagrange multiplier adaptation and non-convex optimization.

6. Applications and Broader Implications

Domain-adaptive reward function research is directly applicable to:

• Robotics: Efficient sim-to-real transfer when direct reward engineering is infeasible or unsafe, as in standing-up tasks or manipulation involving morphology/dynamics variation (Ishihara et al., 8 Jan 2025, Gui et al., 3 Jan 2025, Kang et al., 2021).
• Imitation Learning: Visual Imitation under morphology or observation shift, using domain-invariant behavioral features (Choi et al., 2023).
• RLHF for LLMs: Fast adaptation of reward models to new domains by router mechanisms or parameter merging, avoiding retraining from scratch and reducing memory footprint (Lin et al., 1 Jul 2024, Namgoong et al., 24 Jul 2024).
• Multi-objective and Preference-based RL: Preference adaptation and online policy update via reward interpolation or informative reward optimization (Kusari et al., 2019, Devidze et al., 10 Feb 2024).
• Resource-constrained Evaluation: Adaptive exploration allocates sampling budget efficiently across policies/rewards, ensuring coverage of uncertain regions for robust evaluation and transfer (Russo et al., 4 Feb 2025).
• Policy Warmstarting and Action Pruning: Q-Manipulation efficiently combines multiple pre-trained behaviors for a target domain, enabling zero-shot or few-shot adaptation (Vora et al., 17 Mar 2025).

A plausible implication is that future RL systems deployed in open, safety-critical, or human-interactive settings will increasingly employ domain-adaptive reward frameworks as fundamental infrastructure, leveraging representational abstraction, adversarial regularization, and explicit constraint specification to ensure safe, efficient, and robust behavior across diverse and dynamically shifting environments.