# Gradient Descent as Loss Landscape Navigation: a Normative Framework for Deriving Learning Rules

arXiv: https://arxiv.org/abs/2510.26997 · HTML: https://mlsanity.com/paper/2510.26997/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 30 Oct 2025.

## Builds on (most cited first, 26 of 26)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Measuring Catastrophic Forgetting in Neural Networks](https://mlsanity.com/paper/1708.02072/) (2017)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)
- [Fast and Scalable Bayesian Deep Learning by Weight-Perturbation in Adam](https://mlsanity.com/paper/1806.04854/) (2018)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)
- [On the Generalization Benefit of Noise in Stochastic Gradient Descent](https://mlsanity.com/paper/2006.15081/) (2020)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [Noether's Learning Dynamics: Role of Symmetry Breaking in Neural Networks](https://mlsanity.com/paper/2105.02716/) (2021)
- [Addressing Loss of Plasticity and Catastrophic Forgetting in Continual Learning](https://mlsanity.com/paper/2404.00781/) (2024)
- [In Search of Adam's Secret Sauce](https://mlsanity.com/paper/2505.21829/) (2025)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)
- [Can We Remove the Square-Root in Adaptive Gradient Methods? A Second-Order Perspective](https://mlsanity.com/paper/2402.03496/) (2024)
- [A Comparison of Optimization Algorithms for Deep Learning](https://mlsanity.com/paper/2007.14166/) (2020)
- [Generalization through variance: how noise shapes inductive biases in diffusion models](https://mlsanity.com/paper/2504.12532/) (2025)
- [I-Con: A Unifying Framework for Representation Learning](https://mlsanity.com/paper/2504.16929/) (2025)
- [Deep RL Needs Deep Behavior Analysis: Exploring Implicit Planning by Model-Free Agents in Open-Ended Environments](https://mlsanity.com/paper/2506.06981/) (2025)
- [Nature's Cost Function: Simulating Physics by Minimizing the Action](https://mlsanity.com/paper/2303.02115/) (2023)
- [Is All Learning (Natural) Gradient Descent?](https://mlsanity.com/paper/2409.16422/) (2024)
- [Dynamical symmetries in the fluctuation-driven regime: an application of Noether's theorem to noisy dynamical systems](https://mlsanity.com/paper/2504.09761/) (2025)

## Built on by (most cited first, 4 of 4)

- [A Variational Manifold Embedding Framework for Nonlinear Dimensionality Reduction](https://mlsanity.com/paper/2511.22128/) (2025)
- [A Thermodynamic Theory of Learning Part II: History-Dependent Reachability and Continual Learning](https://mlsanity.com/paper/2602.07950/) (2026)
- [Metaplasticity as adaptive gradient preconditioning for incremental learning](https://mlsanity.com/paper/2608.14634/) (2026)
- [Normative Loss Landscape Navigation: A Trajectory-Based Approach to Mitigating Forgetting in Incremental Learning](https://mlsanity.com/paper/2609.35926/) (2026)

## Abstract (arXiv)

Learning rules -- prescriptions for updating model parameters to improve performance -- are typically assumed rather than derived. Why do some learning rules work better than others, and under what assumptions can a given rule be considered optimal? We propose a theoretical framework that casts learning rules as policies for navigating (partially observable) loss landscapes, and identifies optimal rules as solutions to an associated optimal control problem. A range of well-known rules emerge naturally within this framework under different assumptions: gradient descent from short-horizon optimization, momentum from longer-horizon planning, natural gradients from accounting for parameter space geometry, non-gradient rules from partial controllability, and adaptive optimizers like Adam from online Bayesian inference of loss landscape shape. We further show that continual learning strategies like weight resetting can be understood as optimal responses to task uncertainty. By unifying these phenomena under a single objective, our framework clarifies the computational structure of learning and offers a principled foundation for designing adaptive algorithms.