Adaptive Reward Function Design

• Adaptive reward function design is a framework that automatically tunes and refines reward functions in reinforcement learning by utilizing feedback and optimization techniques.
• It leverages a bi-level optimization strategy where an inner loop optimizes the policy under current rewards while an outer loop adjusts reward parameters using implicit gradients and meta-learning.
• The approach integrates behavioral metrics, human-in-the-loop signals, and Bayesian methods to enhance learning efficiency and mitigate risks of static or misleading reward schemes.

Adaptive reward function design is a principled framework for automatically constructing, tuning, or refining reward functions in reinforcement learning (RL), control, and related sequential decision-making domains. Unlike static, hand-crafted reward engineering, adaptive reward methods interleave reward optimization with policy learning or environmental feedback, leveraging both behavioral, empirical, and possibly human-in-the-loop signals to ensure that the shaped reward accelerates learning, aligns with intended high-level goals, and avoids unintended behaviors. This concept underpins a wide range of recent approaches, including bi-level optimization schemes, meta-gradient and teacher-driven criteria, Bayesian posterior inference, and automated reward synthesis via LLMs.

1. Formal Frameworks for Adaptive Reward Function Optimization

Adaptive reward function design is often cast as a (stochastic) bi-level optimization problem, where an outer loop optimizes the reward parameters to maximize true environment performance, while an inner loop trains the agent policy using the current reward parameterization. In the general schema (Gupta et al., 2023, Hu et al., 2020, Devidze et al., 10 Feb 2024):

• Let $S$, $A$ denote the state and action spaces; $p(s'|s,a)$ the dynamics; $r_p(s,a)$ the environment (primary) reward; $r_\mathrm{aux}(s,a)$ a designer-provided auxiliary (shaping) reward; $r_\phi(s,a)$ a learned or parameterized reward function; $\pi_\theta(a|s)$ a policy.
• The bi-level optimization is

$$\max_{\phi,\gamma} \quad J\bigl(\theta^*(\phi,\gamma)\bigr) - \lambda_\gamma \gamma$$

subject to

$$\theta^*(\phi,\gamma) = \operatorname{Alg}(r_\phi, \gamma) \approx \arg\max_\theta \mathbb{E}_{\pi_\theta}\Bigl[ \sum_{t=0}^T \gamma^t r_\phi(S_t, A_t) \Bigr]$$

with outer objective $$J(\theta) = \mathbb{E}_{\pi_\theta}\bigl[ \sum_{t=0}^T \gamma^t r_p(S_t, A_t) \bigr]$$ and $\lambda_\gamma \geq 0$ a regularization parameter encouraging informative (less delayed) rewards.

• The reward parameterization $r_\phi(s,a)$ can interpolate among base rewards, heuristics, and learned corrections, and may include a learned discount factor $\gamma_\phi$.

This formalism generalizes to structurally constrained or interpretable reward families (Devidze et al., 10 Feb 2024, Devidze, 27 Mar 2025), Bayesian reward posteriors (He et al., 2021, Liampas, 2023), LTL- or DFA-aligned shaping (Kwon et al., 14 Dec 2024), and model-driven or algorithmic aggregation strategies (Tang et al., 11 Jul 2025).

2. Algorithmic Schemes and Optimization Procedures

Adaptive reward design algorithms are characterized by alternating updates to both policy and reward parameters, exploiting implicit gradients, regularization, and task-specific inductive biases.

Inner-Outer Alternating Optimization

• The inner loop runs a policy optimization algorithm (e.g., Actor-Critic, PPO, DQN) under the current $r_\phi$, with possibly a learned discount factor, to (approximately) reach a fixed point $\theta^*(\phi, \gamma)$ (Gupta et al., 2023, Hu et al., 2020).
• The outer loop updates $\phi, \gamma$ via implicit differentiation: implicit gradients are approximated using Hessian-vector products $H^{-1}v$ (Neumann series or conjugate gradients). Outer gradients involve computing $\partial_\phi$ (and $\partial_\gamma$) of the fixed-point condition, as well as the policy-gradient with respect to the environment reward (Gupta et al., 2023).

Meta-Gradient and Bi-level Learning

• In meta-gradient approaches, the reward parameters (or shaping coefficients) are learned by differentiating through one or more policy update steps, allowing the reward adaptation to be tailored to the current policy and learning algorithm (Hu et al., 2020).
• Incremental variants carry forward Jacobian or Hessian estimates for efficiency in deep RL settings.

Teacher-Driven, Policy-Aware Criteria

• When an expert or target policy $\pi^*$ is available, adaptive reward design uses teacher-driven criteria to construct $R_t$ at each epoch that maximally improves the learner, using an informativeness objective that measures expected gain in true-task advantage across the learner's current state occupancy (Devidze et al., 10 Feb 2024, Devidze, 27 Mar 2025).
• For a one-step greedy learner, the criterion (Devidze et al., 10 Feb 2024) is

$$I_1(R) = \mathbb{E}_{s\sim d^{\pi_t}, a\sim\pi_t}\bigl[ (A^{\pi^*}(s,a) - A^{\pi^*}(s,\pi_t(s)))\cdot (R(s,a) - \mathbb{E}_{b\sim\pi_t} R(s,b)) \bigr]$$

maximizing this over $R$ under structure/policy-invariance ($\mathcal{R}$) yields interpretable adaptive rewards.

Data-Driven and Bayesian Posterior Methods

• In settings with human-in-the-loop feedback or noisy human-designed proxies, Bayesian learning over reward weights is integrated with batch or active environment querying, with risk-averse planning for deployment safety (Liampas, 2023, He et al., 2021, Ratner et al., 2018).
• Batches of environments are used to amortize human labeling; Bayesian updates propagate information to new, previously unseen reward-feature dimensions for continual adaptation (Liampas, 2023).

Automated and LLM-Driven Adaptive Pipelines

• LLMs synthesize candidate reward code, perform reward-logic reasoning, and iteratively critique/refine reward schemes based on metric evaluations (LEARN-Opt (Cardenoso et al., 24 Nov 2025), URDP (Yang et al., 3 Jul 2025), Auto MC-Reward (Li et al., 2023)).
• LLM agents may self-consistently filter redundant candidates, integrate Bayesian optimization for code-component intensity tuning, and interleave reward execution/evaluation with empirical feedback.

3. Adaptation to Model Misspecification and Heuristic Reward Quality

Adaptive reward function frameworks provide resilience against the dangers of auxiliary/heuristic reward misspecification:

• If the auxiliary reward $r_\mathrm{aux}$ is well-aligned, the optimal reward $r_\phi(s,a)$ learns to integrate it (positive weights), enhancing density and temporal credit assignment (Gupta et al., 2023).
• If $r_\mathrm{aux}$ conflicts with long-term task goals (misleading), the learned weight is driven to zero, and $r_\phi$ approximates the primary reward. Learned discounting $\gamma$ is automatically tuned to discourage excess horizon, preventing reward shaping from fostering suboptimal loops or distractors (Gupta et al., 2023, Hu et al., 2020).
• Regularization (e.g., penalty on $\gamma$, L2 or entropy on $\theta$) improves stability, maintains mapping smoothness from $\phi\to\theta$, and ensures well-behaved gradients (Gupta et al., 2023).

Empirically, these mechanisms have been shown to recover high performance even under adversarial shaping, partially aligned heuristics, or dynamically changing features in real-world environments (Gupta et al., 2023, Liampas, 2023).

4. Representative Domains, Empirical Results, and Quantitative Metrics

Adaptive reward design is validated in both low-dimensional discrete control and high-dimensional continuous domains:

Environment	Misalignment Regime	Standard Baseline	Adaptive Method (BARFI)
CartPole	adversarial shaping	return ≈ 10 (failure)	≈ 475/500 (near-optimal)
MountainCar	partially aligned shaping	fails	≈ 0.9 success rate
HalfCheetah (MuJoCo)	wrong action-cost weight	fails	high velocity, full recovery

Experimental metrics include final (true) return, sample efficiency, robustness (stability under misspecification), and learning curve profiles. Adaptive approaches consistently avoided performance collapse seen in naïve or statically shaped schemes, and maintained robustness when confronted with misleading auxiliary signals (Gupta et al., 2023, Hu et al., 2020).

High-dimensional and non-tabular tasks (e.g., reward-driven generative molecular design (Urbonas et al., 2023), portfolio management via recursive aggregation (Tang et al., 11 Jul 2025), complex composed tasks with LTL/DFA (Kwon et al., 14 Dec 2024)) further exemplify the practical reach of adaptive reward design, with iterative retraining or retraining cycles yielding monotonic improvements and robust evaluation performance.

5. Extensions, Limitations, and Practical Considerations

Key limitations and open directions for adaptive reward function design include:

• Computational efficiency: Implicit-gradient and Hessian-vector product calculation can be a bottleneck. Neumann series or conjugate gradient approximations mitigate, but analysis under approximate inner-loop convergence remains open (Gupta et al., 2023).
• Expressivity: The learned reward $r_\phi$ (or reward surrogate) must be sufficiently expressive to capture necessary corrections or alignments; over-constrained parameterizations can fail to compensate for algorithmic or heuristic biases (Gupta et al., 2023, Hu et al., 2020).
• Generalization: Extensions to POMDPs, structurally constrained designs (e.g., temporal logic, reward machines), and complex, noisy, or multi-task settings are ongoing (Kwon et al., 14 Dec 2024, Tang et al., 11 Jul 2025).
• Human-in-the-loop and automated methods: Progressive integration with human feedback (e.g., RLHF), and LLM-powered logic/metric construction, represent emerging axes of progress (Yang et al., 3 Jul 2025, Cardenoso et al., 24 Nov 2025, Li et al., 2023).
• Uncertainty and Safety: Risk-averse planning under reward uncertainty (distributional approaches) and adaptive incorporation of novel features (active batch Bayesian frameworks) are crucial for real-world deployments (Liampas, 2023, He et al., 2021).

Practical usage suggests alternating updates at frequencies tailored to the task and learner, regular hyperparameter sweeps (discount, regularization), use of interpretable or human-auditable reward components, and rigorous rollout validation. Integration with automated code synthesis or Bayesian meta-level strategies is increasingly common in complex or high-dimensional settings.

6. Theoretical Guarantees and Analysis

Adaptive reward frameworks admit both theoretical and empirical analyses:

• Policy invariance and correction: Policy-optimality constraints can ensure that no spurious policies are induced and that intended optima remain unchanged (Devidze et al., 10 Feb 2024, Devidze, 27 Mar 2025).
• Variance and bias correction: Adaptive schemes (e.g., BARFI) can match unbiased discounted policy gradients or correct for off-policy bias without explicit importance sampling, ensuring both efficacy and sample efficiency (Gupta et al., 2023).
• Convergence rates: In teacher-driven or one-step greedy tabular cases, optimality is provably reached in $O(|A|)$ iterations, compared to exponentially many for static sparse or random schemes (Devidze et al., 10 Feb 2024, Devidze, 27 Mar 2025).
• Robustness: Bi-level and meta-gradient formulations guarantee that, in the limit, the reward weights adapt to nullify the effects of harmful or misleading shaping, while fully utilizing helpful signals (Hu et al., 2020).

A plausible implication is that, even in nonconvex or high-noise settings, the outer loop can reliably optimize agent performance, provided sufficient expressivity, sample coverage, and regularization are enforced.

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References:

• "Behavior Alignment via Reward Function Optimization" (Gupta et al., 2023)
• "Learning to Utilize Shaping Rewards: A New Approach of Reward Shaping" (Hu et al., 2020)
• "Reward Design for Reinforcement Learning Agents" (Devidze, 27 Mar 2025)
• "Informativeness of Reward Functions in Reinforcement Learning" (Devidze et al., 10 Feb 2024)
• "Automating reward function configuration for drug design" (Urbonas et al., 2023)
• "Adaptive Reward Design for Reinforcement Learning" (Kwon et al., 14 Dec 2024)
• "Recursive Reward Aggregation" (Tang et al., 11 Jul 2025)
• "Risk-averse Batch Active Inverse Reward Design" (Liampas, 2023)
• "Assisted Robust Reward Design" (He et al., 2021)
• "Uncertainty-aware Reward Design Process" (Yang et al., 3 Jul 2025)
• "Leveraging LLMs for reward function design in reinforcement learning control tasks" (Cardenoso et al., 24 Nov 2025)
• "Auto MC-Reward: Automated Dense Reward Design with LLMs for Minecraft" (Li et al., 2023)
• "Simplifying Reward Design through Divide-and-Conquer" (Ratner et al., 2018)
• "Adaptive Incentive Design" (Ratliff et al., 2018)