Iterative methods lie at the heart of numerous optimization and reasoning algorithms, ranging from classical mirror descent and dynamic programming to modern deep learning architectures that exhibit chain-of-thought reasoning. Traditional acceleration techniques, such as Nesterov's momentum, have shown that carefully designed iterative schemes can significantly improve convergence rates in convex settings. However, many practical applications operate in non-Euclidean spaces and are subject to state-dependent perturbations or even adversarial disturbances, motivating the need for a more general analysis. In this work, we develop a comprehensive framework that unifies a wide class of iterative reasoning processes using the language of Bregman divergences.

We introduce iterative reasoning through energy diffusion (IRED), a novel framework for learning to reason for a variety of tasks by formulating reasoning and decision-making problems with energy-based optimization. IRED learns energy functions to represent the constraints between input conditions and desired outputs. After training, IRED adapts the number of optimization steps during inference based on problem difficulty, enabling it to solve problems outside its training distribution -- such as more complex Sudoku puzzles, matrix completion with large value magnitudes, and pathfinding in larger graphs. Key to our method's success is two novel techniques: learning a sequence of annealed energy landscapes for easier inference and a combination of score function and energy landscape supervision for faster and more stable training. Our experiments show that IRED outperforms existing methods in continuous-space reasoning, discrete-space reasoning, and planning tasks, particularly in more challenging scenarios. Code and visualizations at https://energy-based-model.github.io/ired/

In an unfamiliar setting, a model-based reinforcement learning agent can be limited by the accuracy of its world model. In this work, we present a novel, training-free approach to improving the performance of such agents separately from planning and learning. We do so by applying iterative inference at decision-time, to fine-tune the inferred agent states based on the coherence of future state representations. Our approach achieves a consistent improvement in both reconstruction accuracy and task performance when applied to visual 3D navigation tasks. We go on to show that considering more future states further improves the performance of the agent in partially-observable environments, but not in a fully-observable one. Finally, we demonstrate that agents with less training pre-evaluation benefit most from our approach.