# Learning Iterative Reasoning through Energy Minimization

arXiv: https://arxiv.org/abs/2206.15448 · HTML: https://mlsanity.com/paper/2206.15448/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 45 references that are in the map.
- First posted on arXiv 30 Jun 2022.

## Builds on (most cited first, 28 of 28)

- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Neural Turing Machines](https://mlsanity.com/paper/1410.5401/) (2014)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Melding the Data-Decisions Pipeline: Decision-Focused Learning for Combinatorial Optimization](https://mlsanity.com/paper/1809.05504/) (2018)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [PonderNet: Learning to Ponder](https://mlsanity.com/paper/2107.05407/) (2021)
- [Adaptive Neural Networks for Efficient Inference](https://mlsanity.com/paper/1702.07811/) (2017)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Neural GPUs Learn Algorithms](https://mlsanity.com/paper/1511.08228/) (2015)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Hierarchical Multiscale Recurrent Neural Networks](https://mlsanity.com/paper/1609.01704/) (2016)
- [VAEBM: A Symbiosis between Variational Autoencoders and Energy-based Models](https://mlsanity.com/paper/2010.00654/) (2020)
- [Neural Programmer-Interpreters](https://mlsanity.com/paper/1511.06279/) (2015)
- [Generalized Energy Based Models](https://mlsanity.com/paper/2003.05033/) (2020)
- [Unsupervised Learning of Compositional Energy Concepts](https://mlsanity.com/paper/2111.03042/) (2021)
- [Making Neural Programming Architectures Generalize via Recursion](https://mlsanity.com/paper/1704.06611/) (2017)
- [Energy-Based Models for Continual Learning](https://mlsanity.com/paper/2011.12216/) (2020)
- [Learning to Stop While Learning to Predict](https://mlsanity.com/paper/2006.05082/) (2020)
- [Constraint-based graph network simulator](https://mlsanity.com/paper/2112.09161/) (2021)
- [Implicit Generation and Generalization in Energy-Based Models](https://mlsanity.com/paper/1903.08689/) (2019)
- [Learning the Stein Discrepancy for Training and Evaluating Energy-Based Models without Sampling](https://mlsanity.com/paper/2002.05616/) (2020)

## Built on by (most cited first, 30 of 32)

- [Path Independent Equilibrium Models Can Better Exploit Test-Time Computation](https://mlsanity.com/paper/2211.09961/) (2022)
- [Artificial Kuramoto Oscillatory Neurons](https://mlsanity.com/paper/2410.13821/) (2024)
- [Learning Iterative Reasoning through Energy Diffusion](https://mlsanity.com/paper/2406.11179/) (2024)
- [Energy-Based Transformers are Scalable Learners and Thinkers](https://mlsanity.com/paper/2507.02092/) (2025)
- [Fixed-Point Reasoners: Stable and Adaptive Deep Looped Transformers](https://mlsanity.com/paper/2606.18206/) (2026)
- [Implicit Language Models are RNNs: Balancing Parallelization and Expressivity](https://mlsanity.com/paper/2502.07827/) (2025)
- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Thinking with Looped Flows](https://mlsanity.com/paper/2609.11801/) (2026)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [Energy-Inspired Self-Supervised Pretraining for Vision Models](https://mlsanity.com/paper/2302.01384/) (2023)
- [Improving Adversarial Energy-Based Model via Diffusion Process](https://mlsanity.com/paper/2403.01666/) (2024)
- [VFScale: Intrinsic Reasoning through Verifier-Free Test-time Scalable Diffusion Model](https://mlsanity.com/paper/2502.01989/) (2025)
- [GeoWorld: Geometric World Models](https://mlsanity.com/paper/2602.23058/) (2026)
- [nabla-Reasoner: LLM Reasoning via Test-Time Gradient Descent in Latent Space](https://mlsanity.com/paper/2603.04948/) (2026)
- [GRIG: Few-Shot Generative Residual Image Inpainting](https://mlsanity.com/paper/2304.12035/) (2023)
- [Explorative Modeling: Unlocking a Third Pretraining Axis and End-to-End Generation](https://mlsanity.com/paper/2607.27372/) (2026)
- [Diffusion Probabilistic Models for Structured Node Classification](https://mlsanity.com/paper/2302.10506/) (2023)
- [BECAUSE: Bilinear Causal Representation for Generalizable Offline Model-based Reinforcement Learning](https://mlsanity.com/paper/2407.10967/) (2024)
- [Self-Correction as Feedback Control: Error Dynamics, Stability Thresholds, and Prompt Interventions in LLMs](https://mlsanity.com/paper/2604.22273/) (2026)
- [Hyper-SET: Designing Transformers via Hyperspherical Energy Minimization](https://mlsanity.com/paper/2502.11646/) (2025)
- [Generalizable Reasoning through Compositional Energy Minimization](https://mlsanity.com/paper/2510.20607/) (2025)
- [Energy-guided Recursive Model](https://mlsanity.com/paper/2607.10128/) (2026)
- [Learning 3D Reconstruction with Priors in Test Time](https://mlsanity.com/paper/2604.03878/) (2026)
- [When in Doubt, Think Slow: Iterative Reasoning with Latent Imagination](https://mlsanity.com/paper/2402.15283/) (2024)
- [LIME-Eval: Rethinking Low-light Image Enhancement Evaluation via Object Detection](https://mlsanity.com/paper/2410.08810/) (2024)
- [You Only Look One Step: Accelerating Backpropagation in Diffusion Sampling with Gradient Shortcuts](https://mlsanity.com/paper/2505.07477/) (2025)
- [ReNF: Rethinking the Design of Neural Long-Term Time Series Forecasters](https://mlsanity.com/paper/2509.25914/) (2025)
- [PID-controlled Langevin Dynamics for Faster Sampling of Generative Models](https://mlsanity.com/paper/2511.12603/) (2025)
- [Controlled Dynamics Attractor Transformer](https://mlsanity.com/paper/2606.15207/) (2026)
- [Retrieval-Warmed Energy-Based Reasoning: A Five-Arm Ablation Methodology for Diffusion-as-Inference on Structured Reasoning Tasks](https://mlsanity.com/paper/2606.26476/) (2026)

## Abstract (arXiv)

Deep learning has excelled on complex pattern recognition tasks such as image classification and object recognition. However, it struggles with tasks requiring nontrivial reasoning, such as algorithmic computation. Humans are able to solve such tasks through iterative reasoning -- spending more time thinking about harder tasks. Most existing neural networks, however, exhibit a fixed computational budget controlled by the neural network architecture, preventing additional computational processing on harder tasks. In this work, we present a new framework for iterative reasoning with neural networks. We train a neural network to parameterize an energy landscape over all outputs, and implement each step of the iterative reasoning as an energy minimization step to find a minimal energy solution. By formulating reasoning as an energy minimization problem, for harder problems that lead to more complex energy landscapes, we may then adjust our underlying computational budget by running a more complex optimization procedure. We empirically illustrate that our iterative reasoning approach can solve more accurate and generalizable algorithmic reasoning tasks in both graph and continuous domains. Finally, we illustrate that our approach can recursively solve algorithmic problems requiring nested reasoning