Temporal Difference Flows (2503.09817v1)

Abstract: Predictive models of the future are fundamental for an agent's ability to reason and plan. A common strategy learns a world model and unrolls it step-by-step at inference, where small errors can rapidly compound. Geometric Horizon Models (GHMs) offer a compelling alternative by directly making predictions of future states, avoiding cumulative inference errors. While GHMs can be conveniently learned by a generative analog to temporal difference (TD) learning, existing methods are negatively affected by bootstrapping predictions at train time and struggle to generate high-quality predictions at long horizons. This paper introduces Temporal Difference Flows (TD-Flow), which leverages the structure of a novel BeLLMan equation on probability paths alongside flow-matching techniques to learn accurate GHMs at over 5x the horizon length of prior methods. Theoretically, we establish a new convergence result and primarily attribute TD-Flow's efficacy to reduced gradient variance during training. We further show that similar arguments can be extended to diffusion-based methods. Empirically, we validate TD-Flow across a diverse set of domains on both generative metrics and downstream tasks including policy evaluation. Moreover, integrating TD-Flow with recent behavior foundation models for planning over pre-trained policies demonstrates substantial performance gains, underscoring its promise for long-horizon decision-making.

An Essay on "Temporal Difference Flows"

The paper "Temporal Difference Flows," authored by Jesse Farebrother et al., offers a significant advancement in predictive modeling within the field of Reinforcement Learning (RL). The central premise revolves around improving long-horizon generative models for future state prediction, an area previously plagued by the "curse of horizon." This setback primarily arises from the accumulation of errors over extended prediction periods due to the iterative nature of traditional predictive frameworks.

Core Contribution

The authors propose a novel approach known as Temporal Difference Flows (TD-Flows), which aims at reducing variance and promoting stability in long-horizon prediction tasks by leveraging the temporal difference structure inherent in the successor measure. The paper introduces three variants: TD-Conditional Flow Matching (TD-CFM), Coupled TD-Conditional Flow Matching (TD-CFM(c)), and TD${}^2$-Conditional Flow Matching (TD$^2$-CFM), each designed to better handle the intricacies of long-horizon predictions by incorporating forward-thinking strategies from flow matching frameworks.

Technical Insights

A notable aspect of the paper is the application of probabilistic generative models, specifically flow matching and denoising diffusion methods, adapted to learn the successor measure. Here, the authors delineate a robust framework that leverages geometric heuristics and bootstrapped learning, working through iterative embeddings and employing neural Ordinary Differential Equations (ODEs).

• TD-CFM and TD-CFM(c): These methods employ a flow matching strategy in which the conditional probability paths are constructed between the noise and data distributions, offering a potential reduction in computational variance.
• TD$^2$-CFM: This variant further extends the reduction of variance by embedding the bootstrapped sample into the optimization problem itself, thereby leveraging the structure of the BeLLMan operator akin to traditional RL approaches but with a novel generative twist.

The theoretical significance of these models lies in their convergence properties, with the paper demonstrating the contraction nature of the proposed methods within the 1-Wasserstein distance framework, ensuring sensitivity stability across transitions and iterative sample-based gradient estimations.

Empirical Validation

The empirical section substantiates the theoretical advancements through a series of extensive experiments spanning numerous domains (Maze, Walker, Cheetah, Quadruped). The results echo the robustness of TD$^2$-CFM methods, which consistently outperform baseline models like Generative Adversarial Networks (GANs) and Variational Auto-Encoders (VAEs) in terms of accuracy in long-term decision-making tasks.

Moreover, the investigation into effective horizons illustrates the impressive resilience of TD$^2$-based methodologies against increasing temporal prediction lengths—a pivotal requirement for real-world applications demanding reliable future state predictions.

Implications and Future Directions

The paper acknowledges the broader implications of stable long-horizon predictive modeling, particularly within planning, exploration, and representation learning in RL contexts. Future work could potentially explore consistent models and one-step distillation processes to further mitigate computational costs inherent in sampling.

In practical terms, TD-Flows are poised to revolutionize AI systems relying on robust long-term predictions, including autonomous navigation and strategic gaming applications, where precision over extended periods is crucial.

In conclusion, "Temporal Difference Flows" provides an innovative framework that not only addresses the inherent limitations of traditional deep RL models in handling long-horizon predictions but also establishes a notable theoretical and empirical foundation for further exploration and development in predictive modeling strategies.