# Rethinking Deep Thinking: Stable Learning of Algorithms using Lipschitz Constraints

arXiv: https://arxiv.org/abs/2410.23451 · HTML: https://mlsanity.com/paper/2410.23451/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 25 references that are in the map.
- First posted on arXiv 30 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Neural Turing Machines](https://mlsanity.com/paper/1410.5401/) (2014)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Spectral Norm Regularization for Improving the Generalizability of Deep Learning](https://mlsanity.com/paper/1705.10941/) (2017)
- [Learning Latent Permutations with Gumbel-Sinkhorn Networks](https://mlsanity.com/paper/1802.08665/) (2018)
- [Can You Learn an Algorithm? Generalizing from Easy to Hard Problems with Recurrent Networks](https://mlsanity.com/paper/2106.04537/) (2021)
- [Residual Connections Encourage Iterative Inference](https://mlsanity.com/paper/1710.04773/) (2017)
- [Deep Set Prediction Networks](https://mlsanity.com/paper/1906.06565/) (2019)
- [Parametric Exponential Linear Unit for Deep Convolutional Neural Networks](https://mlsanity.com/paper/1605.09332/) (2016)
- [Datasets for Studying Generalization from Easy to Hard Examples](https://mlsanity.com/paper/2108.06011/) (2021)

## Built on by (most cited first, 9 of 9)

- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Encode, Think, Decode: Scaling test-time reasoning with recursive latent thoughts](https://mlsanity.com/paper/2510.07358/) (2025)
- [Probabilistic Tiny Recursive Model](https://mlsanity.com/paper/2605.19943/) (2026)
- [Chain-of-Experts: Unlocking the Communication Power of Mixture-of-Experts Models](https://mlsanity.com/paper/2506.18945/) (2025)
- [Efficient Parallel Samplers for Recurrent-Depth Models and Their Connection to Diffusion Language Models](https://mlsanity.com/paper/2510.14961/) (2025)
- [L-Lipschitz Gershgorin ResNet Network](https://mlsanity.com/paper/2502.21279/) (2025)
- [LDLT L-Lipschitz Network: Generalized Deep End-To-End Lipschitz Network Construction](https://mlsanity.com/paper/2512.05915/) (2025)
- [DART: Distributional Adversarial Recurrent Training for Algorithm Learning](https://mlsanity.com/paper/2609.05988/) (2026)
- [Resample the Answer: Where to Spend Test-Time Compute in Recursive Reasoning](https://mlsanity.com/paper/2610.16427/) (2026)

## Abstract (arXiv)

Iterative algorithms solve problems by taking steps until a solution is reached. Models in the form of Deep Thinking (DT) networks have been demonstrated to learn iterative algorithms in a way that can scale to different sized problems at inference time using recurrent computation and convolutions. However, they are often unstable during training, and have no guarantees of convergence/termination at the solution. This paper addresses the problem of instability by analyzing the growth in intermediate representations, allowing us to build models (referred to as Deep Thinking with Lipschitz Constraints (DT-L)) with many fewer parameters and providing more reliable solutions. Additionally our DT-L formulation provides guarantees of convergence of the learned iterative procedure to a unique solution at inference time. We demonstrate DT-L is capable of robustly learning algorithms which extrapolate to harder problems than in the training set. We benchmark on the traveling salesperson problem to evaluate the capabilities of the modified system in an NP-hard problem where DT fails to learn.