Dynamical Priors as a Training Objective in Reinforcement Learning

Abstract: Standard reinforcement learning (RL) optimizes policies for reward but imposes few constraints on how decisions evolve over time. As a result, policies may achieve high performance while exhibiting temporally incoherent behavior such as abrupt confidence shifts, oscillations, or degenerate inactivity. We introduce Dynamical Prior Reinforcement Learning (DP-RL), a training framework that augments policy gradient learning with an auxiliary loss derived from external state dynamics that implement evidence accumulation and hysteresis. Without modifying the reward, environment, or policy architecture, this prior shapes the temporal evolution of action probabilities during learning. Across three minimal environments, we show that dynamical priors systematically alter decision trajectories in task-dependent ways, promoting temporally structured behavior that cannot be explained by generic smoothing. These results demonstrate that training objectives alone can control the temporal geometry of decision-making in RL agents.

• The paper introduces a novel auxiliary loss using dynamical priors to impose biologically inspired temporal coherence in standard RL policies.
• It demonstrates that integrating explicit dynamical priors reduces oscillations and timing variance while enhancing evidence accumulation in decision-making.
• The approach decouples temporal structuring from architectural modifications, paving the way for designing robust, time-aware RL agents.

Dynamical Priors as a Training Objective in Reinforcement Learning

Introduction

The paper "Dynamical Priors as a Training Objective in Reinforcement Learning" (2604.21464) investigates the effects of integrating explicit dynamical priors, derived from external state dynamics (ESD), into the objective function during RL policy training. The motivation is rooted in addressing the lack of temporal constraints present in standard policy gradient algorithms, which often yields policies that maximize reward yet display abrupt, temporally incoherent decision patterns. By leveraging auxiliary losses grounded in evidence accumulation and hysteresis, the authors aim to systematically bias policy evolution toward biologically inspired, temporally structured behaviors without architectural modifications.

Methodology

Environments and Policy Structure

Experiments are conducted in three controlled, minimal Markov environments with scalar observations and a binary action set:

• Drift Environment: Tests response to sustained versus transient evidence.
• Threshold Hover Environment: Challenges the agent to differentiate genuine threshold crossings from noisy fluctuations.
• Decision Window Environment: Measures the ability to develop gradual confidence for temporally precise decisions.

All agents utilize identical feedforward MLP architectures (no recurrence or memory), ensuring policy differences are solely attributed to variations in training objectives.

Dynamical Prior Formulation

DP-RL introduces an auxiliary loss computed as the squared deviation between the action probability and a latent state $Z_t$, which evolves per a second-order hysteretic dynamical system:

• $Z_t$ integrates $S_t$ with asymmetrical rates (controlled by $a_{up}$ and $a_{down}$), fostering evidence accumulation and resistance to rapid reversals.
• An additional velocity variable introduces temporal smoothing and momentum, emulating properties of biological integrators.

The auxiliary loss $L_{ESD}$ is combined additively with the standard REINFORCE loss, using a fixed mixing coefficient.

Evaluation Metrics

To quantify temporal properties of policy output trajectories, the following metrics are introduced:

• Jerk: Maximal instantaneous change in decision probability—a proxy for abruptness.
• Oscillation Count: Number of threshold crossings, quantifying indecisiveness or flip-flopping.
• Timing Variance: Variability in when the policy first commits to a confident decision, reflecting temporal consistency.

These are computed from deterministic rollouts, isolating policy response from environmental or actuation stochasticity.

Results

Task-Dependent Temporal Restructuring

Across all environments, the introduction of the dynamical prior was found to exert systematic, environment-dependent alterations on policy temporal profiles:

• Drift Environment: DP-RL policies demonstrate substantially reduced oscillations and lower decision timing variance compared to standard REINFORCE, reflecting more stable and consistent evidence integration. The higher jerk indicates responsiveness aligned with emerging signals rather than erratic activity.
• Threshold Hover Environment: DP-RL policies overcome the degenerate inactivity of REINFORCE, showing dynamic adaptation to the evidence while avoiding rapid, noise-induced policy switches.
• Decision Window Environment: DP-RL produces a gradual, ramp-like confidence buildup, enabling consistent entry into the critical temporal action window. In contrast, plain REINFORCE remains inert, reflecting a conservative “wait and see” approach that avoids penalties at the cost of indecision.

Quantitative Comparison

Empirically, the DP-RL agent achieves:

• Lower oscillation counts and timing variance in settings where excessive flip-flopping or inconsistent timing would otherwise emerge.
• Increased (but structured) jerk and oscillations in environments where active, temporally extended decision engagement is desirable, whereas REINFORCE solutions collapse to inert or static behavior.
• Temporal variability metrics reveal that apparent stability in standard RL may often reflect trivial inactivity rather than genuine temporal coherence.

Non-Equivalence to Smoothing

A key assertion is that dynamical priors are not reducible to generic temporal smoothing; they shape the decision landscape in a task-appropriate manner. This is evidenced by instances where the jerk increases under DP-RL, but such increases correspond to heightened, context-sensitive responsiveness rather than increased noise.

Theoretical and Practical Implications

Inducing Temporal Inductive Bias via Loss Functions

The central insight is that explicit, dynamical priors in the training objective alone are sufficient to induce desired temporal structures in policy output—even in the absence of architectural temporal mechanisms (e.g., recurrence, memory). This finding advances the perspective that the policy's temporal geometry can be an explicit design target, controlled by the learning objective rather than only by network design or reward engineering.

Relationship to Biological Decision-Making

DP-RL's ESD mechanism draws direct inspiration from models of neural evidence integration and hysteresis, which underpin robust, temporally coherent behaviors in biological agents. The emergence of structured policy dynamics through optimization—rather than through architectural embedding—suggests that biological realism in RL agents is achievable through appropriate objective augmentation. This is orthogonal and potentially complementary to recent work on recurrent or memory-augmented agents.

Evaluation and Interpretation of Policy Variability

The findings challenge prevalent assumptions in RL evaluation: temporal smoothness and low variability may signify trivial inactivity rather than robust, well-calibrated policy dynamics. Oscillations, under DP-RL, often correspond to functional evidence gathering and timely engagement, indicating the necessity for more nuanced behavioral metrics in RL.

Future Directions

While the framework is validated in minimal, low-dimensional tasks, its extension to domains with richer observations, nonstationary environments, and partial observability is an immediate avenue for further study. Applying dynamical priors to more complex architectures (e.g., RNNs, multi-agent systems) may enable explicit control over high-level temporal strategies, such as strategic delay, anticipation, or coordinated action. Moreover, systematically varying the structure of the ESD prior could create families of RL agents tuned for distinct temporal robustness or responsiveness requirements.

Conclusion

The paper demonstrates that dynamical priors, operationalized as auxiliary losses derived from external, task-independent dynamical systems, can systematically sculpt the temporal geometry of RL policy decisions. Crucially, this is achieved without modification to reward designs, environments, or architectures. This paradigm decouples performance optimization from temporal behavior structuring, suggesting that RL research and applications can benefit from explicit design and analysis of policy temporal dynamics via training objectives. The results invite future exploration of temporally-aware RL, bridging biologically inspired modeling and practical policy design.