Maximum diffusion reinforcement learning

TL;DR

This work addresses the fundamental challenge that embodied agents experience temporally correlated data, which breaks the common i.i.d. assumption in reinforcement learning. It introduces maximum diffusion reinforcement learning (MaxDiff RL), a principled framework grounded in the statistical mechanics of ergodic processes to decorrelate agent trajectories and enable single-shot learning in continuous deployments. The approach generalizes maximum entropy RL, provides theoretical guarantees of ergodicity and robustness to seeds, and demonstrates strong performance and transfer capabilities across diverse embodied tasks. By linking diffusion, trajectory entropy, and stochastic control, the paper lays a physics-informed foundation for reliable, transparent decision-making in embodied RL systems.

Abstract

Robots and animals both experience the world through their bodies and senses. Their embodiment constrains their experiences, ensuring they unfold continuously in space and time. As a result, the experiences of embodied agents are intrinsically correlated. Correlations create fundamental challenges for machine learning, as most techniques rely on the assumption that data are independent and identically distributed. In reinforcement learning, where data are directly collected from an agent's sequential experiences, violations of this assumption are often unavoidable. Here, we derive a method that overcomes this issue by exploiting the statistical mechanics of ergodic processes, which we term maximum diffusion reinforcement learning. By decorrelating agent experiences, our approach provably enables single-shot learning in continuous deployments over the course of individual task attempts. Moreover, we prove our approach generalizes well-known maximum entropy techniques, and robustly exceeds state-of-the-art performance across popular benchmarks. Our results at the nexus of physics, learning, and control form a foundation for transparent and reliable decision-making in embodied reinforcement learning agents.

Figures (14)

• Figure 1: Temporal correlations break the state-of-the-art in RL. For most systems, controllability properties determine temporal correlations between state transitions (Supplementary Note \ref{['sec:controllability']}). a, Planar point mass with dynamics simple enough to explicitly write down and whose policy admits a globally optimal analytical solution. The system's 4-dimensional state space is comprised of its planar positions and velocities. We parametrize its controllability through $\beta \in [0,1]$, where $\beta=0$ produces a formally uncontrollable system. The task is to translate the point mass from $p_0$ to $p_g$ within a fixed number of steps at different values of $\beta$, and the reward is specified by the negative squared Euclidean distance between the agent's state and the goal. We compare state-of-the-art model-based and model-free algorithms, NN-MPPI and SAC respectively, to our proposed maximum diffusion (MaxDiff) RL framework (see Supplementary Note \ref{['sec:implementation']} for implementation details). b,d, Representative snapshots of MaxDiff RL, NN-MPPI, and SAC agents (top to bottom) in well-conditioned ($\beta=1$) and poorly-conditioned ($\beta=0.001$) controllability settings. c, Even in this simple system, poor controllability can break the performance of RL agents. As $\beta \rightarrow 0$ the system's ability to move in the $x$-direction diminishes, hindering the performance of NN-MPPI and SAC, while MaxDiff RL remains task-capable. For all bar charts, data are presented as mean values above each error bar, where each error bar represents the standard deviation from the mean with $n=1000$ (100 evaluations over 10 seeds for each condition). All differences between MaxDiff RL and comparisons within this figure are statistically significant with $P<0.001$ using an unpaired two-sided Welch's t-test (see Methods and Supplementary Table \ref{['table:stats']}).
• Figure 1: Effect of controllability on the distribution of reachable states. a, For a linear system with dynamics like those in Figure 1 of the main text initialized with an $x_t$ of all zeroes, we depict the effect of controllability on a naive random action exploration strategy. For a linear system with ideal controllabilty properties, isotropic distributions of actions map onto isotropic distributions of states. b, However, when the system is poorly conditioned the system dynamics distort the isotropy of the original input distribution, introducing temporal correlations induced by the controllability properties of the system, and fundamentally changing its properties as an exploration strategy.
• Figure 2: Maximum diffusion RL mitigates temporal correlations to achieve effective exploration. a,b, Systems with different planar controllability properties. c, Whether action randomization leads to effective state exploration depends on the properties of the underlying state-transition dynamics (see Supplementary Note \ref{['sec:controllability']}), as in our illustration of a complex bipedal robot falling over and failing to explore. d, While any given policy induces a path distribution (left), MaxDiff RL produces policies that maximize the path distribution's entropy (right). The projected support of the robot's path distribution is illustrated by the shaded gray region. We prove that maximizing the entropy of an agent's state transitions results in effective exploration (see Supplementary Notes \ref{['sec:exploration_diffusion']} and \ref{['sec:maxdiff_exploration']}). e, Our approach generalizes the MaxEnt RL paradigm by considering agent trajectories. We prove that maximizing a policy's entropy does not generally maximize the entropy of an agent's state transitions (see Supplementary Note \ref{['sec:maxdiff_RL']}). f, This approach leads to distinct learning outcomes because agents reason about the impact of their actions on state transitions, rather than their actions alone.
• Figure 2: Effect of controllers on the sample path distribution of stochastic control processes. (left) Sample path and support of the probability density over the paths of an autonomous stochastic process (i.e., with null controller "0"). (middle and right) Sample paths and distributions induced by two distinct controllers $u_1(t)$ and $u_2(t)$. Here, we illustrate that depending on the nature of the controller the distribution over sample paths can be nontrivial. Note that we do not illustrate the values of the probability densities, only their support. The reason for this is that so long as a regions of space have non-zero probability they will be sampled asymptotically.
• Figure 3: Maximally diffusive RL agents are robust to random seeds and initializations. a, Illustration of MuJoCo swimmer environment (left panel). The swimmer has 2 degrees of actuation, $u_1$ and $u_2$, that rotate its limbs at the joints, with tail mass $m_s$ and $m=1$ for other limbs. MaxDiff RL synthesizes robust agent behavior by learning policies that balance task-capability and diffusive exploration (right panel). In practice this balance is tuned by a temperature-like parameter, $\alpha$. b, To explore the role that $\alpha$ plays in the performance of MaxDiff RL, we examine the terminal returns of swimmer agents (10 seeds each) across values of $\alpha$ with $m_s=1$. Diffusive exploration leads to greater returns until a critical point (inset dotted line), after which the agent starts valuing diffusing more than accomplishing the task (see also https://www.youtube.com/watch?v=XZOTG9KNifs&list=PLO5AGPa3klrCTSO-t7HZsVNQinHXFQmn9&index=1). c, Using $\alpha=100$, we compared MaxDiff RL against SAC and NN-MPPI with $m_s=0.1$. We observe that MaxDiff RL outperforms comparisons on average with near-zero variability across random seeds, which is a formal property of MaxDiff RL agents (see also https://www.youtube.com/watch?v=eq6Fk-lp1i0&list=PLO5AGPa3klrCTSO-t7HZsVNQinHXFQmn9&index=2). For all reward curves, the shaded regions correspond to the standard deviation from the mean across 10 seeds. For all bar charts, data are presented as mean values above each error bar, where each error bar represents the standard deviation from the mean with $n=1000$ (100 evaluations over 10 seeds for each condition). All differences between MaxDiff RL and comparisons within this figure are statistically significant with $P<0.001$ using an unpaired two-sided Welch's t-test (see Methods and Supplementary Table \ref{['table:stats']}).

Theorems & Definitions (25)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2.1
• Remark 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• proof