Automated Reward Function Design

• Automated reward function design is a framework that systematically synthesizes and refines rewards for reinforcement learning by decomposing tasks into independent, manageable subproblems.
• It leverages techniques such as divide-and-conquer specification, Bayesian statistical inference, and Monte Carlo methods to fuse environment-specific proxy rewards into a unified function.
• Empirical studies demonstrate significant improvements, including reduced tuning time and lower regret, thereby ensuring better generalization and scalability in complex real-world applications.

Automated reward function design refers to the set of methodologies and algorithmic frameworks developed to systematically synthesize, optimize, and refine reward functions for reinforcement learning (RL) and robot planning. Rather than relying solely on manual, trial-and-error engineering, these approaches automate or partially automate the generation of reward specifications, the integration of task objectives and domain features, or the inference of shared rewards across different scenarios. Automation addresses key challenges such as cognitive overload, suboptimal generalization, and engineering inefficiency in conventional reward design. Techniques in this area include divide-and-conquer specification, statistical inference, Bayesian posterior estimation, and procedures that leverage human data, inverse design principles, and task decomposition.

1. Divide-and-Conquer Specification of Rewards

Automated reward function design can be significantly improved via divide-and-conquer methodologies, in which the overall reward engineering task is decomposed into independent subproblems by environment. In the canonical approach, a reward designer separately specifies a proxy reward for each training environment, focusing only on the subset of features and trade-offs relevant to that scenario. This independence reduces the cognitive load required to balance competing objectives across disparate environments and allows for faster, more user-friendly reward tuning.

The core of this approach is the use of each environment-specific proxy reward as an independent, noisy observation of the underlying true reward. By framing reward design as a form of statistical inference, one can leverage the following factorization for the posterior over reward parameters $\theta$:

$$P(\theta \mid r_{1:N}, M_{1:N}) \propto \left[ \prod_{i=1}^N P(r_i \mid \theta, M_i) \right] P(\theta)$$

Here, $r_i$ denotes the proxy reward for environment $M_i$, and the observation model $P(r_i \mid \theta, M_i)$ is formulated based on how well the proxy’s induced trajectory maximizes the true reward. Intractable normalization constants are handled with Monte Carlo integration and Metropolis sampling.

This divide-and-conquer strategy both accelerates the design process and reduces the subjective frustration of having to satisfy global, cross-environmental requirements simultaneously.

2. Statistical Inference and Reward Fusion

After specifying proxy rewards for each environment, the next step is to infer a single, shared reward function valid across all environments. This is systematically handled via a Bayesian inference process where each proxy reward is viewed as a noisy measurement of the true reward function, parameterized by $\theta$.

Assuming an observation model of the form:

$$P(r_i \mid \theta, M_i) \propto \exp(\beta R(\xi^*_ \theta; \theta))$$

where $\xi^*_\theta$ is the optimal trajectory induced by $\theta$ in environment $M_i$ and $\beta$ parameterizes assumed designer optimality, the joint posterior is:

$$P(\theta \mid r_{1:N}, M_{1:N}) \propto \prod_{i=1}^N P(r_i \mid \theta, M_i) P(\theta)$$

Planning in novel environments can then proceed using the mean of this posterior or via expected value maximization. Monte Carlo methods are used to handle normalization, making this procedure scalable in moderately complex spaces. This inference-based fusion robustly aggregates environment-specific insights and produces reward functions that generalize across scenarios.

3. Empirical Evaluation and Sensitivity Analysis

User studies in both abstract (grid world) and applied (7-DOF manipulator) domains provide quantitative and qualitative evidence for the efficiency and effectiveness of automated, environment-wise reward specification and inference. The measurable criteria include:

• Design Time: Environment-wise independent reward design achieves a mean reduction in total reward-tuning time by approximately 51.4% compared to standard joint (global) approaches.
• Solution Quality (Regret): Policies induced from fused rewards over independently specified proxies exhibit up to 69.8% lower regret on held-out environments relative to joint design. Regret was measured as the discrepancy between the reward-maximizing behavior under the designed/inferred reward and that under the ground-truth reward.

Sensitivity analyses reveal that the advantages of independent reward design (and subsequent automated fusion) are most pronounced when:

• Environments each express only a subset of the overall features (increasing decomposability),
• The feasible reward sets per environment do not trivially intersect,
• The number of environments is sufficiently high to enable robust inference.

If reward subproblems cannot be separated due to highly entangled features or trivial feasible sets across environments, some of the observed gains are diminished.

4. Comparison with Traditional Joint Reward Design

Joint reward design, in which a single reward function is globally specified for all training environments, is fraught with iterative cycles in which improving performance in one scenario often degrades another. This can increase the time to convergence as well as user frustration. In contrast, the divide-and-conquer paradigm decouples complex trade-offs, allowing simpler, more focused design decisions. Subjective user studies confirm higher ease-of-use ratings and less frustration in the divide-and-conquer approach.

However, the benefit of the divide-and-conquer method is greatest when the reward design problem is divisible by feature structure or environment. When subproblems are not naturally decomposable, the approximation to the true posterior may lose efficacy.

5. Principles for Automation and Practical Implications

Automating reward function design under this framework involves two essential steps:

• Human designers interact with simpler, environment-specific reward tuning interfaces, substantially reducing the manual search space and associated frustration.
• A statistical fusion mechanism automatically computes a reward function that is consistent with the set of observed proxies, generalizing better to held-out environments.

In robotics and autonomous systems operating in varied or safety-critical settings (such as motion planning for household manipulators or autonomous vehicles), this paradigm provides a data-efficient, low-regret path to robust reward function synthesis. Automation reduces the manual tuning cycle and the engineering burden of repeatedly re-specifying rewards, particularly in tasks with expensive simulation, costly real-world testing, or complex trade-offs.

The characteristics that make this method particularly attractive are:

• Fast, efficient iterations for reward tuning,
• Stronger generalization of the resultant reward to previously unseen environments,
• Ease of use for designers or operators,
• Scalability as task/environmental complexity increases.

These attributes position divide-and-conquer reward design, with automated statistical inference, as an effective, scalable strategy for deploying RL in complex domains where robust reward design is a central bottleneck.