# Optimal Learning Rate Schedule for Balancing Effort and Performance

arXiv: https://arxiv.org/abs/2601.07830 · HTML: https://mlsanity.com/paper/2601.07830/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 117 references that are in the map.
- First posted on arXiv 12 Jan 2026.

## Builds on (most cited first, 23 of 23)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Continual Lifelong Learning with Neural Networks: A Review](https://mlsanity.com/paper/1802.07569/) (2018)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [A Comprehensive Survey of Continual Learning: Theory, Method and Application](https://mlsanity.com/paper/2302.00487/) (2023)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [Descending through a Crowded Valley - Benchmarking Deep Learning Optimizers](https://mlsanity.com/paper/2007.01547/) (2020)
- [Dynamics of stochastic gradient descent for two-layer neural networks in the teacher-student setup](https://mlsanity.com/paper/1906.08632/) (2019)
- [On Empirical Comparisons of Optimizers for Deep Learning](https://mlsanity.com/paper/1910.05446/) (2019)
- [Meta-Learning with Adaptive Hyperparameters](https://mlsanity.com/paper/2011.00209/) (2020)
- [Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs](https://mlsanity.com/paper/2206.00939/) (2022)
- [Maslow's Hammer for Catastrophic Forgetting: Node Re-Use vs Node Activation](https://mlsanity.com/paper/2205.09029/) (2022)
- [Optimal Protocols for Continual Learning via Statistical Physics and Control Theory](https://mlsanity.com/paper/2409.18061/) (2024)
- [Toward Equation of Motion for Deep Neural Networks: Continuous-time Gradient Descent and Discretization Error Analysis](https://mlsanity.com/paper/2210.15898/) (2022)
- [Modelling continual learning in humans with Hebbian context gating and exponentially decaying task signals](https://mlsanity.com/paper/2203.11560/) (2022)
- [What does it mean to understand a neural network?](https://mlsanity.com/paper/1907.06374/) (2019)
- [Meta-Learning Strategies through Value Maximization in Neural Networks](https://mlsanity.com/paper/2310.19919/) (2023)
- [A model-based approach to meta-Reinforcement Learning: Transformers and tree search](https://mlsanity.com/paper/2208.11535/) (2022)
- [AdaLRS: Loss-Guided Adaptive Learning Rate Search for Efficient Foundation Model Pretraining](https://mlsanity.com/paper/2506.13274/) (2025)
- [A statistical physics framework for optimal learning](https://mlsanity.com/paper/2507.07907/) (2025)
- [Curriculum Meta-Learning for Few-shot Classification](https://mlsanity.com/paper/2112.02913/) (2021)
- [Lifelong Reinforcement Learning via Neuromodulation](https://mlsanity.com/paper/2408.08446/) (2024)
- [A Note on Linear Quadratic Regulator and Kalman Filter](https://mlsanity.com/paper/2308.15798/) (2023)

## Built on by (most cited first, 1 of 1)

- [Theory of Optimal Learning Rate Schedules and Scaling Laws for a Random Feature Model](https://mlsanity.com/paper/2602.04774/) (2026)

## Abstract (arXiv)

Learning how to learn efficiently is a fundamental challenge for biological agents and a growing concern for artificial ones. To learn effectively, an agent must regulate its learning speed, balancing the benefits of rapid improvement against the costs of effort, instability, or resource use. We introduce a normative framework that formalizes this problem as an optimal control process in which the agent maximizes cumulative performance while incurring a cost of learning. From this objective, we derive a closed-form solution for the optimal learning rate, which has the form of a closed-loop controller that depends only on the agent's current and expected future performance. Under mild assumptions, this solution generalizes across tasks and architectures and reproduces numerically optimized schedules in simulations. In simple learning models, we can mathematically analyze how agent and task parameters shape learning-rate scheduling as an open-loop control solution. Because the optimal policy depends on expectations of future performance, the framework predicts how overconfidence or underconfidence influence engagement and persistence, linking the control of learning speed to theories of self-regulated learning. We further show how a simple episodic memory mechanism can approximate the required performance expectations by recalling similar past learning experiences, providing a biologically plausible route to near-optimal behaviour. Together, these results provide a normative and biologically plausible account of learning speed control, linking self-regulated learning, effort allocation, and episodic memory estimation within a unified and tractable mathematical framework.