Reward-Guided Symbolic Calibration (RGSC)

• RGSC is a framework that integrates domain-specific symbolic constraints with reward signals to calibrate AI behavior, enhancing sample efficiency and interpretability.
• It leverages formal methods such as QBF, SAT solvers, genetic programming, and Bayesian calibration to guide decision-making in reinforcement learning, language models, and robotics.
• Experimental results demonstrate significant gains, including win rates improving from 17% to 85% in symbolic-advised MCTS and better accuracy in scientific discovery applications.

Reward-Guided Symbolic Calibration (RGSC) is a methodology for integrating domain-relevant symbolic knowledge and reward signals into the training or inference processes of decision-making agents, generative models, and robot controllers. Its core motivation is to enhance sample efficiency, interpretability, and performance by calibrating system behavior with rewards defined or shaped by logical, human-understandable constraints, formulas, or features. RGSC spans domains including reinforcement learning, symbolic regression, LLM alignment, multimodal generation, and scientific discovery, with implementations ranging from Monte Carlo Tree Search guided by formal advice to reward-influenced diffusion model training and controlled multimodal decoding.

1. Symbolic Advice and Structural Calibration

RGSC initially emerged in the context of Monte Carlo Tree Search (MCTS) for Markov Decision Processes, where symbolic advice is used to constrain selection and simulation actions (Busatto-Gaston et al., 2020). Formal domain properties—expressed as logical formulas (e.g., safety or reachability)—are encoded as selection advice (φ) and simulation advice (ψ), which respectively prune the action and trajectory sets in MCTS according to:

• Selection phase: restrict actions to those in σ₍φ₎ᴴ(p)
• Simulation phase: sample only descendant paths p′ that satisfy p * p′ ⊨ ψ

These constraints are operationalized using Quantified Boolean Formula (QBF) and SAT solvers to efficiently determine which actions and trajectories satisfy the advice. For example:

$$a^* \in \arg\max_{a \in \sigma_{(\varphi)}^H(p)} \left[ \text{value}(p, a) + C \sqrt{ \frac{\ln(\text{count}(p)) }{ \text{count}(p, a) } } \right]$$

Embedding such symbolic calibration not only improves empirical performance (e.g., reaching 85% win rate in symbolic-advised Pac-Man vs. 17% for vanilla MCTS) but also retains asymptotic theoretical guarantees equivalent to classical MCTS—convergence error bounded by $O((\ln n)/n)$ and probability of suboptimal actions vanishing with increasing simulations.

2. Symbolic Rewards and Feature Calibration

In deep RL and regression settings, RGSC involves discovering or constructing symbolic reward functions—often as low-dimensional, interpretable trees composed of arithmetic and logical operators, as opposed to neural network-based estimators (Sheikh et al., 2020). These symbolic trees are evolved using genetic programming, providing dense intrinsic rewards:

$\hat{r}(s) = f(s; \theta_{\text{ST}})$

Compared to neural methods, such symbolic rewards enable clearer analysis, more precise calibration, and often superior policy guidance, especially in sparse or complex tasks (e.g., outperforming neural ICM-based reward discovery on Mujoco, Atari, and Pygame domains).

In context-dependent RL problems, calibrated features serve as intermediate representations that adapt the saliency of base features according to environmental context (Forsey-Smerek et al., 17 Jun 2025). The modularity of RGSC here lies in learning functions $\varphi'_{\psi_i}(\varphi_i(s), s)$, capturing how, for example, proximity to a hazard may become more or less critical depending on the state (e.g., stove hot vs. cold). Feature calibration is isolated and tuned efficiently with paired comparison queries based on the Bradley-Terry model:

$$P_{\psi_i}(s_1 \succ s_2) = \frac{ \exp(\varphi'_{\psi_i}(s_1)) }{ \exp(\varphi'_{\psi_i}(s_1)) + \exp(\varphi'_{\psi_i}(s_2)) }$$

3. Reward Machines, Bayesian Calibration, and Uncertainty

RGSC has particular relevance for tasks where reward functions are encoded in automaton-based symbolic structures such as Reward Machines (RMs) (Li et al., 2022). Traditional RM approaches assume perfect symbol grounding, but real deployments face noisy or uncertain labelling. RGSC addresses this via:

• Belief modeling: learning a recurrent belief over RM states conditioned on observed transitions, as in Reward Machine State Modelling (RMSM).
• Explicit separation of context-invariant preference parameters from context-dependent saliency functions.
• Bayesian calibration: hierarchical inference to fill symbolic “holes” (free parameters) in automaton transitions, leveraging expert demonstrations to maximize discrimination between expert and non-expert trajectories (Zhou et al., 2022).
• POMDP formulation: policies are trained to maximize expected reward under uncertainty in RM state observation, yielding robust, adaptable behaviors in partially observable domains.

These approaches bridge formal verification and data-driven RL, leading to interpretable, generalizable reward models and improved sample efficiency, especially in multi-task or meta-IRL settings.

4. Decoding-Time Reward Guidance in Language and Multimodal Models

RGSC extends to LLMs and multimodal LLMs by incorporating reward signals directly into the inference (decoding) stage. For textual models, reward-guided decoding adjusts the probability of next-token selection by interpolating LLM scores with reward model outputs aligned to human preferences (Khanov et al., 23 Jan 2024, Mao et al., 25 Feb 2024):

$s(v, x_{<t}) = LM(v | x_{<t}) + w r([x_{<t}, v])$

Where $r(\cdot)$ is provided by a reward model trained on preference data, and $w$ tunes alignment strength. In value-based calibration (VCB), RGSC explicitly ties probability ratios to normalized reward gaps:

$$\Delta^\pi_{y_1} - \Delta^\pi_{y_2} = \frac{1}{\gamma} \frac{ r(x, y_1) - r(x, y_2) }{ \sigma^r_{\text{sft}}(x) }$$

For multimodal models (image captioning, grounded generation), RGSC constructs separate reward models for object precision (hallucination mitigation) and recall, enabling controllable trade-offs via a weighted sum during beam search (Mañas et al., 15 Aug 2025):

$$s(x_v, x_q, y) = w \cdot r_{\text{hal}}(x_v, x_q, y) + (1-w) \cdot r_{\text{rec}}(x_v, x_q, y)$$

The controllability features include dynamic adjustment of precision vs. recall, breadth of search, and compute budget, with measurable improvements over conventional hallucination mitigation and reward-guided approaches.

5. Scientific Discovery, Symbolic Regression, and Model Refinement

In scientific domains, RGSC fuses visual induction, symbolic reasoning, and RL-based calibration to generate interpretable formulas explaining empirical data (Liu et al., 24 Aug 2025). For example, in the VIPER-R1 system, MSI forms an initial symbolic hypothesis using causal chain-of-thought and visual data. RGSC then refines this topology via RL, employing a composite reward function:

$$R(S_i) = w_f R_{\text{format}}(S_i) + w_s R_{\text{structural}}(S_i, S_{GT}) + w_a R_{\text{accuracy}}(S_i, S_{GT})$$

Relative advantage normalization supports stable policy updates:

$$A_i = \frac{ r_i - \text{mean}(r_1, \ldots, r_G) }{ \text{std}(r_1, \ldots, r_G) + \epsilon }$$

Experiments confirm improved structural scores (0.812), accuracy, and MSE relative to SOTA baselines—demonstrating both data-fit and interpretability.

Related approaches in symbolic regression employ MCTS and double Q-learning to efficiently explore expression trees anchored in predefined operators, further refined by modulated subtree discovery blocks (Xu et al., 2023). The reward function blends fit with parsimony, guiding the search toward equations that not only match data but are easily understood and analyzed.

6. Training Pipelines, Practical Considerations, and Limitations

RGSC implementations leverage both offline calibration (reward model training, Bayesian inference, symbolic regression) and online calibration (decoding-time reward guidance, guided simulation). General-purpose solvers (QBF, SAT) allow real-time enforcement of symbolic advice and constraints. Sample efficiency is a recurring benefit: approaches typically require an order of magnitude fewer queries or demonstrations than non-calibrated baselines (Forsey-Smerek et al., 17 Jun 2025, Guan et al., 2022).

Effective RGSC relies on the availability and interpretability of base features or symbolic constraints, as well as robust reward models. Limitations can arise from coverage of continuous state spaces, quality of feature construction, and calibration errors propagating from unreliable reward signals (especially in RLHF). Active learning, adaptive calibration strategies, and external integration (e.g., symbolic regression tools such as SR² in VIPER-R1) are ongoing research directions.

7. Applications and Future Directions

RGSC underpins solutions across robot planning, safety-critical RL, multi-agent systems, program synthesis, multimodal reasoning, and scientific law discovery. Its modular separation of preference and saliency, integration of formal verification, and support for controllable inference position it for widespread application in adaptable, interpretable, and personalized AI systems.

Future research will extend RGSC to:

• Multi-objective reward calibration in generative modeling (Zhang et al., 2023)
• Integration with active learning and uncertainty estimation (Li et al., 2022, Leng et al., 13 Oct 2024)
• Fine-grained symbolic feedback for logical reasoning tasks (Jha et al., 26 May 2024)
• Calibration in noisy or partially observable environments
• Scaling to real-world scientific data and complex dynamical systems (Liu et al., 24 Aug 2025)

In summary, Reward-Guided Symbolic Calibration is a unifying framework for embedding formal, interpretable, and reward-driven structure into AI systems, providing principled control and guidance for learning and generation across a range of domains and model classes.