Multi-Task Reward Function Decomposition

• Multi-Task Reward Function Decomposition is a method that splits a global reward into simpler sub-rewards, enabling improved credit assignment and modular learning in reinforcement learning settings.
• Various algorithmic approaches—such as parallel decomposition, independently obtainable subrewards, and symbolic reward machines—enhance sample efficiency and transferability across tasks.
• This decomposition framework improves interpretability and multi-task coordination while addressing challenges like non-Markovian rewards, sensitivity in subreward selection, and interdependent credit assignment.

Multi-Task Reward Function Decomposition refers to the set of algorithmic, representational, and theoretical techniques used to partition a complex, often high-dimensional or non-Markovian, reward function into multiple, simpler sub-reward functions—each corresponding to different subtasks, agents, or structural components of a problem. In multi-task and cooperative multi-agent reinforcement learning (RL), this decomposition enables improved credit assignment, sample efficiency, transferability, and modularity, and is critical both for flat RL settings as well as hierarchical and symbolic task representations.

1. Fundamental Concepts and Mathematical Formulations

The central principle in multi-task reward function decomposition is to represent the global reward $R_\mathrm{env}(s,a,s')$ as a sum or structured aggregate of component reward functions:

$$R_\mathrm{env}(s, a, s') = \sum_{k=1}^n R_k(s, a, s')$$

where each $R_k$ is constructed to focus on a particular feature, subgoal, event, or agent. This decomposition is useful when the original value function or return cannot be efficiently approximated by a single low-dimensional function.

Different frameworks instantiate these components as:

• Low-dimensional reward heads in deep Q-networks (HRA, (Seijen et al., 2017)).
• Independently obtainable reward functions enforcing policy disentanglement (Grimm et al., 2019).
• Sub-reward distributions reflecting latent channels (Lin et al., 2019).
• Symbolic representations using reward machines to encode temporal and logical subtasks (Neary et al., 2020, Icarte et al., 2020, Furelos-Blanco et al., 2022, Shah et al., 19 Feb 2025).

Modules may sum ($Q^*_\mathrm{global}(s,a) = \sum_k Q_k(s,a)$), be composed by convex combination, or, in hierarchical cases, follow temporal or logical orderings specified as automata transitions.

2. Algorithmic Methodologies

Approaches to multi-task reward function decomposition fall into several classes:

• Parallel Decomposition and Hybrid Reward Architectures: Each reward head operates on a restricted subspace, with separate value functions learned per reward. Aggregation is often via summation (Seijen et al., 2017), allowing tractable function approximation within high-dimensional domains.
• Independently Obtainable Subrewards: Methods explicitly force policies for one subreward to avoid collecting other subrewards, optimizing objectives of the form

$$J_\mathrm{disentangled} = J_\mathrm{nontriv} - J_\mathrm{indep}$$

where $J_\mathrm{nontriv}$ encourages each optimal policy for $R_i$ to collect non-trivial reward only for $i$ and $J_\mathrm{indep}$ penalizes policies for $j \neq i$ from collecting $R_i$ (Grimm et al., 2019).

• Distributional and Latent Channel Techniques: The total return is modeled as a convolution over distributional estimates from each channel; a disentanglement regularizer (e.g., pairwise KL divergence of subreturn distributions under their maximizing actions) improves the separation of learned channels (Lin et al., 2019).
• Hierarchical and Symbolic Decomposition: Tasks are encoded as reward machines (RMs)—finite state automata with logical or event-based transitions—then formally decomposed into machine projections or submodules (Neary et al., 2020, Icarte et al., 2020, Furelos-Blanco et al., 2022). Hierarchies (HRMs, MAHRM) further enable each RM to call other RMs as subtasks (Furelos-Blanco et al., 2022, Zheng et al., 8 Mar 2024).
• Meta-Learning and Inverse RL: Multi-task IRL methods infer a family of reward functions or a reward decomposition by regularizing for proximity to a task-mean (Gleave et al., 2018), clustering via Dirichlet processes (Arora et al., 2020), or learning parameter initialization for rapid adaptation (Gleave et al., 2018) or transferable constraints (Jang et al., 2023, Glazer et al., 17 Feb 2024).

3. Application Domains and Empirical Results

Empirical validation covers a range of domains:

• Single-Agent Multi-Reward Domains: Atari games like Ms. Pac-Man, Seaquest, and gridworld fruit collection tasks demonstrate that decomposing per-object or per-region reward yields higher scores and faster convergence than monolithic or manually pruned architectures (Seijen et al., 2017, Lin et al., 2019, Grimm et al., 2019).
• Hierarchical and Non-Markovian Tasks: Office worlds, Minecraft-like environments, and long-horizon sparse reward domains benefit substantially from RM- or LTL-based decomposition, as flat approaches suffer exponentially with added complexity (Icarte et al., 2020, Liu et al., 2 Nov 2024, Furelos-Blanco et al., 2022).
• Cooperative Multi-Agent Settings: Explicit decomposition into agent-specific or group-specific RMs, sometimes learned automatically (Shah et al., 19 Feb 2025), enables efficient credit assignment in tasks requiring coordination, overlapping subtasks, or concurrent events (Neary et al., 2020, Ardon et al., 2023, Zheng et al., 8 Mar 2024, Liu et al., 2 Nov 2024).
• Constraint and Common-Sense Decomposition: By separating transferable constraint signals from task-centric rewards, greater robustness and safety is achieved, particularly in robotics (tray-carrying, wall-following, and manipulation tasks) (Jang et al., 2023, Glazer et al., 17 Feb 2024).
• Language-Grounded and Semantically Aligned Decompositions: Pretrained LLMs propose decompositions aligned with high-level semantics, boosting sample efficiency in collaborative games like Overcooked and MiniRTS (Li et al., 2023).

4. Theoretical Guarantees and Properties

• Consistency and Optimality: When decomposed Q-functions sum exactly to the global optimal Q-function, greedy policies are guaranteed to be optimal. If not, policies may be “semi-consistent,” usually retaining robust performance as long as decompositions respect the environment’s causal structure (Seijen et al., 2017).
• Disentanglement and Saturation: With suitable objectives (e.g., maximizing the difference $U_i^{\pi_i^*} - U_i^{\pi_j^*}$), the resultant policies have non-overlapping state visitation frequencies, supporting theoretical guarantees about the independence of subtask contributions (Grimm et al., 2019).
• Value Bounds in Multi-Agent Combinations: In decentralized settings, the global value is bounded by the sum and minimum of agent-local value functions, certifying joint completion only when all subtasks are solved (Neary et al., 2020).
• Hierarchical Compactness: HRMs and MAHRM reduce sample and computational complexity relative to flat RMs, averting exponential state space blowup in temporally extended or compositional tasks (Furelos-Blanco et al., 2022, Zheng et al., 8 Mar 2024).

5. Challenges, Open Problems, and Limitations

• Decomposition Choice Sensitivity: The effectiveness of decomposition depends heavily on choosing subrewards that minimize cross-head interference while preserving the overall semantics of the original reward. Suboptimal decomposition can lead to redundancy, trivial heads, or poor alignment between subtasks (Seijen et al., 2017, Grimm et al., 2019).
• Credit Assignment with Interdependencies: In domains with codependent agent dynamics or overlapping subtasks, static decompositions can reduce performance; conditioning policies on both individual and overall reward machine states is crucial (Shah et al., 19 Feb 2025).
• Learning Decompositions Automatically: Deriving optimal reward decompositions without domain knowledge is nontrivial. Recent methods employ candidate enumeration and selection (UCB-based or curriculum-driven), symbolic projection, or meta-learning (Shah et al., 19 Feb 2025, Ardon et al., 2023, Furelos-Blanco et al., 2022).
• Handling Non-Markovianity and Logic Constraints: For complex tasks specified via temporal logic or automata (e.g., LTL), decomposing non-Markovian rewards into Markovian surrogates or subgoal-based requirements requires careful progression and reward shaping strategies (Icarte et al., 2020, Liu et al., 2 Nov 2024).

6. Implications for Transfer, Safety, and Multi-Task Generalization

• Transferability: Additive or constraint-based decompositions enable the reuse of constraint components or common-sense rewards across tasks, supporting rapid adaptation, safety, and generalization to novel scenarios (Jang et al., 2023, Glazer et al., 17 Feb 2024).
• Sample Efficiency and Modularization: Decomposition isolates more easily learnable parts, reduces the effective problem size, and permits modular insertion or replacement of subtask components—critical for scaling to multi-task or multi-agent environments.
• Interpretability and Debugging: Symbolic and logical decompositions yield interpretable subgoal structures, supporting debugging, trust, and transparent policy verification, especially in safety-critical domains (Neary et al., 2020, Icarte et al., 2020, Furelos-Blanco et al., 2022).

7. Comparative Analysis and Future Directions

• Contrast with Traditional RL: Flat reward structures obscure the role of subtasks, impede transfer, and challenge scalability. Decomposition—either explicit (engineered or projected) or implicit (latent channels, IRL-derived)—offers systematic credit assignment and modularity.
• Integration with Language, Meta-Learning, and Hierarchy: Recent trends seek to leverage LLMs for semantically meaningful decompositions (Li et al., 2023), meta-learn reward initializations (Gleave et al., 2018), or autonomously build symbolic hierarchies (Furelos-Blanco et al., 2022, Zheng et al., 8 Mar 2024, Shah et al., 19 Feb 2025).
• Open Problems: Construction of principled decompositions in continuous, high-dimensional, or highly stochastic environments remains open. The interplay between symbolic and learned decompositions, handling highly interdependent subtasks, and robust automated curriculum generation are active areas of research.

In sum, multi-task reward function decomposition has evolved into a multi-faceted paradigm encompassing architectural, optimization, symbolic, and meta-learning advances. It underpins many of the recent breakthroughs in learning effectively in environments with complex, compositional, and collaborative structure.