Local Reward Function (LRF)

• Local Reward Function (LRF) is a reward mapping that provides immediate, localized feedback based on current state or subsystem outputs in reinforcement learning.
• LRFs are derived using probabilistic and variational methods, such as log survival probability, to decompose and align local and global objectives.
• They enable scalable optimization and effective credit assignment in complex, hierarchical systems by facilitating independent module updates based on local performance.

A Local Reward Function (LRF) is a core concept in reinforcement learning (RL) and closely related areas, referring to any reward function whose signal is defined and evaluated with respect to a local context—such as an instantaneous state, the output of a subsystem, a subgoal, or a module—rather than only a delayed or global outcome. LRFs serve both as the means to provide immediate feedback to RL agents and as essential primitives for scalable optimization in large, hierarchical, or compound systems. They may be derived directly from domain knowledge, learned from data, structured via formal abstractions, or constructed by aggregation across agents or components.

1. Formal Definition and Key Principles of Local Reward Functions

A Local Reward Function is any mapping $r_t = f(s_t, a_t, s_{t+1}, \dots)$ that yields a scalar feedback signal as a function of localized information—typically the current state $s_t$, action $a_t$, or output $y_k$ of a subsystem or module. The LRF is designed to be specific: it does not require knowledge of entire global trajectories or end-states, and it may be based on domain-local properties (such as physical constraints, immediate goal satisfaction, or component outputs in compound systems).

In recent formulations, the LRF can be instantiated as:

• $r_t = \log P(A_{t+1} = 1 | s_t)$, where $P(A_{t+1} = 1 | s_t)$ quantifies the local, immediate survival probability of an agent in state $s_t$, directly capturing the viability of the current state, as shown by the variationally-derived survival LRF (Yoshida, 2016).
• In compound systems, an LRF $r_k(x_k, y_k)$ evaluates the outcome $y_k$ of component $C_k$ with respect to its local context $x_k$, and is globally aligned if higher local reward correlates with improved overall system performance (Wu et al., 3 Jul 2025).

The principle of decomposability underlies much of the utility of LRFs: by expressing global objectives as the sum or aggregation of local contributions, one enables both practical credit assignment and scalable optimization—even in non-differentiable or highly modular systems.

2. Derivation and Optimization of LRFs

2.1 Probabilistic and Variational Derivation

A principled LRF can arise from casting a sequential problem—such as survival maximization—as a probabilistic inference task. By expressing the multi-step objective as the maximization of $P(\overline{A}_T | \pi)$, the log-probability objective naturally decomposes as:

$$\log P(\overline{A}_T | \pi) = \sum_{t=0}^{T-1} \log P(A_{t+1}=1 | s_t)$$

where $r_t = \log P(A_{t+1}=1 | s_t)$ forms the LRF (Yoshida, 2016). This transformation allows the RL agent to optimize a temporally local, interpretable signal that sums to the desired global objective. The connection is formalized through variational EM, showing that the RL objective with this local reward is proportional to the variational lower bound of the original problem.

2.2 Optimization in Compound Systems

For compound systems, each component $C_k$ is assigned an LRF $r_k(x_k, y_k)$ that is learned or adapted to maintain a local-global alignment: maximizing $r_k$ for a given input $x_k$ and output $y_k$ will, in expectation, push the global reward $R$ upward (Wu et al., 3 Jul 2025). The learning of each LRF is conducted using local data and a pairwise preference loss:

$$\mathcal{L}_k = -\mathbb{E}_{x_k, y_k^+, y_k^-}\left[\log \sigma(r_k(x_k, y_k^+) - r_k(x_k, y_k^-))\right]$$

LRFs are iteratively adapted to track changes in the global system so that independent improvements remain globally beneficial.

3. Applications and Empirical Demonstrations

3.1 Survival and Homeostatic Control

Empirical evidence from gridworld experiments demonstrates that using an LRF based on log survival probability enables agents to not only maximize their long-term survival but also uncover complex homeostatic behaviors—e.g., maintaining an internal battery near an optimal value and avoiding harmful objects—all without hand-crafted auxiliary rewards (Yoshida, 2016). This approach validates the theoretical claim that local, state-dependent rewards can drive the desired global dynamics.

3.2 Optimization in Compound AI Systems

The Optimas framework applies LRFs in large-scale, modular AI systems, spanning product recommendation, medical QA, multi-hop reasoning, and code generation. LRFs, when properly aligned, enable each module to be optimized independently, supporting efficient, data-driven updates even in heterogeneous or non-differentiable component settings. This method yields an average global performance improvement of 11.92% over strong baselines, indicating the practical significance of LRF-guided optimization (Wu et al., 3 Jul 2025).

4. Theoretical Properties and Alignment

4.1 Local–Global Alignment

A core theoretical property required for LRFs in compound systems is local-global alignment: the guarantee that increasing a module's local reward will not degrade global performance. This property is formally defined as:

If $r_k(x_k, y_k^+) \geq r_k(x_k, y_k^-)$, then $$\mathbb{E}[R(x_{1:m}, (y_1, ..., y_k^+, ..., y_m))] \geq \mathbb{E}[R(x_{1:m}, (y_1, ..., y_k^-, ..., y_m))]$$.

Proper construction and continual adaptation of LRFs, often via lightweight online updates informed by global outcomes, are essential to maintaining this alignment in evolving, interdependent environments (Wu et al., 3 Jul 2025).

4.2 Decomposition and Variational Lower Bounds

When the global objective is additive or factorizes appropriately, LRFs support both forward prediction (mapping local improvements to global changes) and inverse design (identifying local modifications needed to achieve target global outcomes). In the probabilistic formulation, the sum of local rewards forms a variational lower bound on the overall objective, mathematically unifying inference and control perspectives (Yoshida, 2016).

5. Implementation Strategies and Practical Considerations

Table: Key aspects of LRFs in two representative settings

Context	LRF Definition	Key Implementation Aspect
Survival RL	$r_t = \log P(A_{t+1}=1|s_t)$	Survival probability modeled as function of state; log transform yields additive rewards; directly interpretable in terms of viability (Yoshida, 2016)
Compound AI (Optimas)	$r_k(x_k, y_k)$	LRF instantiated as LM+head, trained on local preference batches; alignment maintained by ranking loss and online adaptation (Wu et al., 3 Jul 2025)

• Computational Requirements: LRFs facilitate modular computation and, when leveraged appropriately, can reduce the need for full-system simulations at every update, thus improving scalability in complex or high-dimensional settings.
• Limitations: Maintaining alignment with the global objective is nontrivial in non-additive or tightly coupled systems—ongoing adaptation and careful design of preference data are necessary.
• Deployment: LRFs are compatible with a variety of learning algorithms, allowing reinforcement learning techniques such as Sarsa(λ), policy gradients, or even discrete combinatorial optimization to be applied at the module or local level.

6. Broader Implications and Future Directions

LRFs provide a foundational mechanism for credit assignment, explainability, and scalable optimization in reinforcement learning and broader AI system design. Their principled derivation supports transparent and interpretable agent behaviors; their adaptability underlies the robust performance of modular and compound systems in complex real-world tasks. Future research directions include improved methods for automatic discovery and hierarchical composition of LRFs, advanced techniques for preserving alignment in non-stationary or adversarial environments, and theoretical studying of convergence guarantees as system complexity increases (Yoshida, 2016, Wu et al., 3 Jul 2025).

In summary, Local Reward Functions are both a theoretical foundation and a practical tool for building effective, interpretable, and scalable reinforcement learning systems, with deep implications for biologically inspired survival behaviors, modular system design, and next-generation AI architectures.