Reinforcement Learning with Mixed Rewards

• Reinforcement Learning with Mixed Rewards is a paradigm where multiple heterogeneous reward signals are combined to form a unified objective for optimal policy learning.
• It employs diverse methodologies such as occupancy measures, PAC policy search, and risk-sensitive techniques to manage latent, delayed, or exogenous rewards.
• Applications span multi-agent systems, robotics, creative writing, and resource allocation, addressing challenges like unidentifiability and high sample complexity.

Reinforcement Learning with Mixed Rewards (RLMR) denotes a class of reinforcement learning (RL) problems and algorithmic frameworks where multiple sources or types of reward signals—potentially heterogeneous in provenance, granularity, or semantics—jointly drive policy optimization. The core challenge in RLMR is to design methods that can robustly combine or integrate these signals to yield policies that are near-optimal with respect to some joint or compositionally defined objective, even when the mixture structure is latent, dynamically varying, or injects statistical and structural complexities. This paradigm appears prominently in multi-agent partially observable domains, multi-objective and constrained RL, risk-sensitive and trajectory-level optimization settings, and modern applications such as dialogue, creative writing, recommendation, and alignment for large models.

1. Formal Problem Definition and Motivation

RLMR fundamentally departs from canonical RL settings where the agent seeks to maximize a scalar, temporally additive reward. Instead, reward signals may be subject to mixing at multiple levels:

• Objective composition: The agent’s true target is a joint function $f(\lambda_1(\pi), \ldots, \lambda_K(\pi))$ of per-objective long-term averages $\lambda_k(\pi)$ (Agarwal et al., 2019). $f(\cdot)$ may be nonlinear, non-additive (e.g., fairness, risk, coverage) and may preclude standard BeLLMan recursion.
• Statistical mixing: At each episode, the reward function may be drawn from a latent distribution over possible reward models, so the observed signal entwines the influence of several models without explicit annotation (Kwon et al., 2021, Kwon et al., 2022).
• Granularity/structure: Rewards could arrive in delayed or aggregated “bagged” form (over windows/trajectories) (Tang et al., 6 Feb 2024), or may be composed of endogenous and exogenous (non-controllable) parts (Trimponias et al., 2023).

Formally, the RLMR agent must select a policy $\pi^*$ such that: $$\pi^* = \arg\max_{\pi} f(\lambda_1(\pi), \ldots, \lambda_K(\pi))$$ where each $\lambda_k(\pi)$ is a marginal or expected return (over possibly latent or delayed reward sources), and $f: \mathbb{R}^K \to \mathbb{R}$ encodes the mixing structure, which may be designed, discovered, or adapted during training.

2. Algorithmic Methodologies for Mixed Rewards

The diversity of reward-mixing structures motivates a range of algorithmic solutions:

a. Model-Free and Model-Based Occupancy Methods: For scenarios where the objective is a nonlinear function $f$ as above, classical dynamic programming collapses. One approach is to optimize over steady-state occupancy measures (Agarwal et al., 2019):

• Model-based: Estimate transition distributions, formulate the objective as a convex program over $d(s,a)$ with constraints from Markovianity. Regret bounds are established (e.g., $\tilde{O}(L K D S \sqrt{A/T})$).
• Model-free: Employ direct policy-gradient methods, computing $\nabla_\theta f$ via chain rule and policy gradient estimators (e.g., REINFORCE).

b. Probabilistic (PAC) Policy Search for Multiagent or POMDPs: In multiagent or partially observed settings where mixed rewards are induced by unobserved agent behaviors, the Monte Carlo Exploring Starts for POMDPs (MCES-P) and its extensions offer policy-based hill-climbing approaches with sample complexity guarantees that ensure $\epsilon$-optimality under the Probably Approximately Correct (PAC) framework (Ceren, 2019). These involve:

• Neighbor-wise statistical switching on empirical Q-value differences with bounds from Hoeffding’s inequality.
• Opponent-modeling (MCESIP+PAC): Bayesian update of belief over opponent models, tagging Q-values with most likely opponent action patterns and tightening sample complexity.
• Multiagent decomposition (MCESMP+PAC): Enforcing joint improvements with distributed Q-value estimation and composite PAC bounds.

c. Distributional and Risk-Sensitive Methods: Risk-sensitive RL—especially in cooperative MARL—leverages return distribution learning and CVaR-based policies to accommodate rewards that mix rare adverse and frequent moderate returns (Qiu et al., 2021). Here, the policy is dynamically tuned by risk level predictors and a BeLLMan operator over return distributions, with decentralized execution guided by learned risk measures.

d. Reward Decomposition and Exogenous Filtering: In environments where reward includes both controllable (endogenous) and uncontrollable (exogenous) components, the optimal policy is found by projecting onto the endogenous subspace and focusing on that reward (Trimponias et al., 2023). Algorithms (GRDS/SRAS) discover exogenous state variables via conditional independence criteria and reduce variance by removing exogenous reward.

e. Bagged and Delayed Reward Redistribtion: For bagged rewards, where only aggregate feedback is available for a window or trajectory, transformer-based models equipped with bidirectional attention accurately redistribute the bagged reward back to individual steps, scaling to long bags and complex dependencies (Tang et al., 6 Feb 2024).

f. Symbolic and Preference-Based Approaches: In RL for language, alignment, recommendation, or human-in-the-loop domains, reward signals may consist of explicit ratings, comparative preferences, or compositional mixture models. Algorithms such as integrated reward and policy learning (Wu et al., 13 Jan 2025), multi-phase human-in-the-loop reward mixing (Wang et al., 3 Mar 2025), and dual-feedback actor architectures (Khorasani et al., 15 Aug 2025) fuse numeric, preference, and high-level evaluative signals via unified update rules or KL-penalized policy shaping.

3. Theoretical Guarantees and Analytical Insights

RLMR frameworks offer rigorous performance guarantees contingent on reward mixing structure:

• PAC Bounds: Sample complexity for policy improvement in MCES-P-based methods is controlled by the reward range $\Lambda$, the number of neighbors $N$, and the confidence level $\delta$, as shown in: $$k = \left\lceil 2\left(\frac{\Lambda(\pi)}{\epsilon}\right)^2 \ln \frac{2N}{\delta} \right\rceil$$ Tighter bounds result from opponent-modeling, reducing $\Lambda$ by conditioning on likely opponent behaviors.
• Nonlinear Joint-Objective Regret: In multi-objective settings, regret vanishes at rate $\tilde{O}(L K D S \sqrt{A/T})$ when $f$ is $L$-Lipschitz and concave (Agarwal et al., 2019).
• Method-of-Moments for Latent Mixing: For episodic RMMDPs with $M$ latent contexts, matching moments up to order $d = \min\{2M - 1, H\}$ suffices for near-optimality. Sample complexity is $$\tilde{O}(\epsilon^{-2}(SA)^d \cdot \mathrm{poly}(H, Z)^d)$$, and exponential dependence on $M$ is information-theoretically unavoidable (Kwon et al., 2022).
• Hardness for Global Rewards: In global RL, the degree of non-additivity (curvature of submodular/supermodular reward) governs the achievable approximation ratio and computational hardness (Santi et al., 13 Jul 2024).

4. Practical Applications and Empirical Validation

RLMR methods have been deployed in a variety of settings:

• Precision Agriculture: Adaptive team-based decision frameworks using MCES-P-based approaches coordinate heterogeneous sensors for early detection of crop stress, integrating mixed imaging and sensor data with statistical PAC guarantees for sector-level decisions (Ceren, 2019).
• Communications and Resource Allocation: Cellular scheduling and queueing, where objectives include fairness or delay satisfaction, benefit from joint optimization frameworks that achieve higher fairness and system throughput than linear RL baselines (Agarwal et al., 2019).
• Robotics and Exploration: Curiosity-driven RL augmented with auxiliary tasks and intrinsic symbolic rewards provides robust exploration mechanisms in sparse-reward robotic and gaming environments (Hare, 2019, Sheikh et al., 2020).
• Creative Writing and MLLM Training: Dynamic weighed mixing of subjective and hard objective rewards, tight constraint verification models, and groupwise advantage normalizations enable reinforcement learning to optimize both literary quality and compliance in LLMs (Liao et al., 26 Aug 2025, Xu et al., 30 May 2025).
• Human-in-the-Loop and Alignment: Frameworks for integrating rating-based, comparative, and structured human feedback in both single-agent and multi-agent settings support improved alignment with human intent, robustness to feedback quality, and interpretability (Wu et al., 13 Jan 2025, Wang et al., 3 Mar 2025, Khorasani et al., 15 Aug 2025).

5. Challenges and Limitations

Despite significant progress, RLMR introduces unique challenges:

• Unidentifiability: In reward-mixing MDPs, latent reward components may be fundamentally unidentifiable from trajectory data; only certain mixture properties (moments) are estimable (Kwon et al., 2021, Kwon et al., 2022).
• Sample Complexity: For a general $M$, learning optimal policies with mixed latent rewards often incurs super-polynomial sample complexity in $M$.
• Non-Markovian and Delayed Reward: Aggregated and delayed feedback complicates credit assignment, necessitating advanced reward redistribution or temporal attention mechanisms (Tang et al., 6 Feb 2024).
• Non-additive and Global Objectives: Extension to non-additive objectives (submodular, supermodular, trajectory-level) requires iterative approximation and may be computationally hard to optimize even approximately (Santi et al., 13 Jul 2024).
• Hyperparameter Sensitivity: Algorithms with multiple reward signal weights (including dynamically adjusted or groupwise normalizations) can exhibit sensitivity to penalty magnitudes or group sizes, particularly in multi-objective or human-in-the-loop setups (Liao et al., 26 Aug 2025, Wu et al., 13 Jan 2025).

6. Directions for Future Research

Key open areas include:

• Scaling to Many Latent Contexts: Designing sample-efficient algorithms for RMMDPs as $M$ grows, potentially leveraging structure or smoothness in the set of reward models (Kwon et al., 2022).
• Online and Non-stationary Adaptation: RLMR algorithms capable of real-time adaptation to shifting reward distributions or human intent, especially under resource or feedback limitations (Wang et al., 3 Mar 2025).
• Generalization to Global and History-Dependent Rewards: Developing efficient surrogate or semi-gradient schemes for high-dimensional trajectory-level (global) objectives and formalizing their convergence and quality guarantees (Santi et al., 13 Jul 2024).
• Interpretable and Transparent Reward Integration: Methods for discovering interpretable decompositions (e.g., symbolic trees, human-aligned reward machines) that permit inspection and intervention (Sheikh et al., 2020, Icarte et al., 2021).
• Unified Architectures: Frameworks integrating numerical, symbolic, preference-based, and structural reward feedback, applicable both in continuous control and over complex, abstract domain spaces.

7. Significance and Synthesis

Reinforcement Learning with Mixed Rewards has matured into a core paradigm for multi-objective, multiagent, and human-centric RL. Recent theoretical results demonstrate that with appropriately designed algorithms—often blending model-based, direct policy, and belief-update strategies—joint near-optimality is achievable with provable sample efficiency, even in the presence of noise, hidden mixing, or conflicting feedback. Empirically, RLMR frameworks have shown superiority to conventional single-objective RL in high-stakes multi-agent settings, fair resource allocation, creative and language tasks, robotics, and where reward feedback is distributed, delayed, or contaminated. The field continues to evolve toward settings involving deeper integration of structured, human, and latent sources in the reward mix, promising advances in RL’s robustness, alignment, and applicability to complex, real-world domains.