Learning Goals From Motion: A New Approach to Reward Machine Inference

Author: Denis Avetisyan

Researchers have developed a novel framework for automatically discovering reward functions from observed behavior, bypassing the need for manual labeling.

The study investigates reinforcement learning within grid world environments-specifically, a standard room layout and a more complex Tetris-inspired arrangement-utilizing a patrol-based reward mechanism to guide agent behavior.

This work introduces a SAT-solving and active learning strategy for inferring reward machines and labeling functions directly from raw state trajectories.

Specifying complex, multi-stage tasks for robots often requires hand-crafting intricate reward structures, a process that can be both time-consuming and prone to error. This paper, ‘Active Reward Machine Inference From Raw State Trajectories’, addresses this challenge by presenting a novel framework for learning reward machines – automaton-like representations of task memory – directly from observed state trajectories, without requiring access to reward signals or labeled data. By leveraging a SAT-based approach combined with an active learning strategy, the authors demonstrate efficient inference of both the reward machine structure and the underlying labeling function. Could this approach unlock more scalable and adaptable robotic systems capable of learning complex behaviors from minimal supervision?

The Fragility of Reward: A Challenge for Robotics

Traditionally, instructing a robot involves crafting a reward function – a mathematical formula assigning values to different states and actions, effectively telling the robot what constitutes ‘good’ behavior. However, these reward functions are surprisingly fragile; even slight deviations from the intended environment or task can lead to unexpected and undesirable outcomes. A reward function might perfectly guide a robot through a training simulation, yet fail spectacularly when deployed in the real world due to unmodeled physics or unforeseen obstacles. This brittleness arises from the difficulty of anticipating every possible scenario and precisely quantifying the desired behavior with a single, all-encompassing equation, often necessitating extensive manual tuning and refinement – a process that can be both time-consuming and prone to error.

Rather than painstakingly crafting reward functions that dictate every nuance of a robot’s desired actions, researchers are exploring the use of reward machines – essentially, finite state machines that formally capture task specifications. These machines function as blueprints for behavior, defining states representing different stages of a task and transitions triggered by specific conditions. For example, a robot tasked with setting a table might have states like ‘locate plate’, ‘grasp plate’, and ‘place plate’, with transitions occurring when the robot successfully completes each action. This approach offers increased robustness and clarity, allowing for more complex behaviors to be defined in a structured and easily verifiable manner, ultimately reducing the brittleness often associated with traditional reward function design.

Despite the promise of reward machines as a robust alternative to hand-crafted reward functions, a key obstacle in deploying them lies in the difficulty of automated learning. Current reinforcement learning techniques often struggle to infer the underlying finite state machine representing the desired task directly from expert demonstrations. These demonstrations, while illustrative of correct behavior, rarely provide sufficient information to uniquely determine the reward machine’s structure and transition criteria. The ambiguity inherent in translating observed actions into a concise, generalizable state representation necessitates sophisticated algorithms capable of discerning the essential task logic from potentially noisy or incomplete data. Overcoming this challenge requires advancements in areas such as inverse reinforcement learning and state abstraction, enabling robots to not simply mimic expert behavior, but to internalize the intent behind it and generalize to novel situations.

Formalizing Behavior: The Logic of Reward Machines

The process of learning a reward machine is formalized as a Boolean satisfiability (SAT) problem by representing the reward machine’s states, transitions, and the desired behavior as Boolean variables and clauses. Specifically, each state in the reward machine is assigned a unique Boolean variable. Transitions between states, as dictated by the robot’s actions and the environment, are encoded as logical implications. The desired behavior, defined by the reward function, is translated into clauses that must be satisfied for the generated reward machine to be considered valid. This transformation allows the use of established SAT solvers to efficiently search for a reward machine that satisfies the specified constraints and reward structure; a solution to the SAT problem directly yields the structure and labeling function of the learned reward machine.

The translation of desired robot behaviors and operational constraints into a logical formula facilitates a formal representation amenable to automated reasoning. This encoding process involves defining Boolean variables representing states, actions, and conditions, then constructing a propositional logic expression – typically in Conjunctive Normal Form (CNF) – that evaluates to true only when the robot’s actions satisfy the specified criteria. Constraints, such as obstacle avoidance or task completion requirements, are expressed as clauses within this formula. The structure of the formula directly mirrors the desired behavior, allowing a SAT solver to determine if a consistent set of variable assignments – and thus a valid robot behavior – exists.

Resolution of the Boolean satisfiability (SAT) problem directly produces the structural components of the reward machine, specifically defining the states and transitions. The resulting truth assignment determines the labeling function, which maps states and actions to reward values. In Task 1, this process successfully reconstructed the known, correct reward machine structure, differing only in the arbitrary naming of states – a functionally equivalent result. This recovery confirms the viability of formulating reward machine learning as a SAT problem and demonstrates the approach’s capacity to accurately represent desired robot behaviors within the logical framework.

The solution count for the warehouse grid world pick-and-drop task, as shown in (a) and (b), decreases with increasing search depth, exhibiting a [latex]\pm 1[/latex] standard deviation as indicated by the shaded area, and is cut off in the negative region.

Refining Hypotheses: The Efficiency of Active Learning

The Active Extension Algorithm is implemented to mitigate the computational demands of exhaustive search within the hypothesis space. This algorithm operates by iteratively selecting and evaluating trajectory pairs – demonstrations of desired robot behavior – rather than assessing all possible combinations. The selection process prioritizes trajectories that offer the greatest potential to refine the current hypothesis, effectively focusing computational resources on the most informative examples. This targeted approach allows for efficient reduction of the hypothesis space, accelerating the learning process and enabling convergence to an optimal solution with fewer iterations compared to methods requiring complete exploration.

The Active Extension Algorithm prioritizes data efficiency by not uniformly sampling trajectory pairs for analysis. Instead, it selectively queries pairs based on their potential to maximize information gain regarding the robot’s control policy. This selection process focuses on demonstrations that exhibit the greatest disagreement between the current hypothesis and the observed behavior, effectively identifying areas where the hypothesis is most uncertain. The algorithm utilizes a scoring function to quantify this disagreement, allowing it to efficiently pinpoint and query the trajectory pairs most relevant for refining the hypothesis and reducing the search space.

Selective querying of informative trajectory pairs directly impacts learning efficiency by minimizing the search space for optimal hypotheses. Rather than exhaustively evaluating all possible demonstrations, the Active Extension Algorithm prioritizes examples that yield the greatest reduction in uncertainty regarding the underlying solution. This focused approach effectively constrains the hypothesis space, allowing the algorithm to converge more rapidly on accurate solutions. The resultant acceleration in learning is demonstrated by a 96.6% convergence rate observed in Task 2 with [latex]N_{active} = 200[/latex], achieving convergence to the ground truth solution set by depth 13.

The presented active learning methodology demonstrates high performance when guided by a History Policy, which incorporates demonstrations of optimal behavior. Specifically, in Task 2, utilizing [latex]N_{active} = 200[/latex] actively queried trajectory pairs, the method achieved a 96.6% convergence rate to the established ground truth solution set. This convergence was realized by depth 13, indicating the number of iterative refinements required to reach the solution within the specified performance threshold. The History Policy, therefore, serves as a critical component in focusing the search and accelerating the learning process by providing informative examples for hypothesis refinement.

Scaling Complexity: The Power of History Restriction

The complexity of reinforcement learning often stems from the need to consider extensive past experiences when making decisions. To address this, a history restriction-specifically, limiting the depth, denoted as ‘l’, of considered past states-was implemented within the policy. This restriction effectively reduces the computational burden by focusing only on the most recent ‘l’ states, thereby simplifying the learning process. By discarding irrelevant historical data, the algorithm can more efficiently identify patterns and correlations crucial for optimal behavior, leading to faster training and improved scalability, particularly in dynamic and complex environments where an exhaustive consideration of the entire past is computationally prohibitive.

The ability of a reinforcement learning algorithm to generalize beyond simplified scenarios is often limited by computational demands as task complexity increases. This research demonstrates that restricting the historical depth considered during policy learning provides a critical pathway to scalability. By focusing on a finite, relevant history, the algorithm significantly reduces the state space it must explore, thereby curbing exponential growth in computational cost. This streamlined approach allows the algorithm to tackle increasingly intricate tasks and adapt to more dynamic and unpredictable environments that would otherwise overwhelm exhaustive methods. Consequently, the system maintains robust performance even with limited resources, opening doors to real-world robotic applications demanding both adaptability and efficiency.

Robotic systems can now reliably execute complex tasks, such as pick-and-drop manipulation and dynamic patrolling, through an efficient reward machine learning process. This advancement significantly reduces computational demands; the methodology requires just 0.147 GB of memory – a dramatic decrease from the 24.76 GB needed by exhaustive approaches at a depth of 9. Furthermore, runtime is nearly halved, achieving 3544.76 seconds compared to the 7100 seconds required by conventional methods. This substantial improvement in both memory efficiency and processing speed enables the deployment of sophisticated robotic behaviors in real-world, dynamic environments where resource constraints are often a limiting factor.

The robustness of the learned robotic behaviors stems from a crucial design element: the labeling function’s reliance on atomic propositions. Rather than abstracting task requirements into complex, potentially brittle symbolic representations, this approach directly connects actions to observable environmental features. By grounding the learning process in these fundamental, concrete observations – such as the presence of an object, a robot’s proximity to a landmark, or the status of a gripper – the system avoids the pitfalls of generalization based on incomplete or misleading abstractions. This ensures that the learned policies are intrinsically linked to the physical reality of the environment, promoting reliable performance even when faced with unexpected variations or disturbances. Consequently, the robot’s behavior remains interpretable and adaptable, as it is fundamentally driven by verifiable, real-world conditions rather than high-level, potentially ambiguous commands.

The pursuit of efficient policy inference, as detailed in this work, often leads to unnecessarily complex systems. One might observe a tendency to overengineer solutions, building elaborate frameworks to obscure the fundamental challenge of distilling intent from raw data. As Donald Davies observed, “Simplicity is a prerequisite for reliability.” This sentiment rings true; the paper’s approach, leveraging SAT solving and active learning to minimize search space, demonstrates a mature understanding that elegance isn’t merely aesthetic-it’s integral to practical application. The framework isn’t built to impress, but to work, effectively extracting reward machines from state trajectories with focused precision.

Where To Now?

The demonstrated confluence of trajectory analysis, Boolean satisfiability, and active learning offers a reduction in complexity – a welcome gesture in a field often burdened by its own expanding parameter space. Yet, the framework’s efficacy remains tethered to the quality of the initial trajectory data. Noise, sparsity, or ambiguity within these observations inevitably propagate through the SAT solver, demanding robust filtering or pre-processing – a necessary addition, but one that reintroduces complexity. Future work must address the question of inherent data resilience.

Moreover, the current approach implicitly assumes a degree of stationarity in the underlying reward function. Real-world systems rarely adhere to such constraints. Adaptation to non-stationary rewards – the capacity to refine the learned reward machine during operation – represents a significant, and likely arduous, extension. The elegance of the current formulation may necessitate sacrifice for genuine generality.

Ultimately, the pursuit of lossless compression in policy inference is not merely an engineering challenge, but a philosophical one. To truly distill intent from observation requires not just efficient algorithms, but a principled understanding of what constitutes ‘essential’ information. The framework provides a valuable tool, but the question of what it should learn remains open.

Original article: https://arxiv.org/pdf/2604.07480.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Fragility of Reward: A Challenge for Robotics

Formalizing Behavior: The Logic of Reward Machines

Refining Hypotheses: The Efficiency of Active Learning

Scaling Complexity: The Power of History Restriction

Where To Now?

See also: