# A Trainable Optimizer

arXiv: https://arxiv.org/abs/2508.01764 · HTML: https://mlsanity.com/paper/2508.01764/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 23 references that are in the map.
- First posted on arXiv 3 Aug 2025.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives](https://mlsanity.com/paper/1407.0202/) (2014)
- [Momentum-Based Variance Reduction in Non-Convex SGD](https://mlsanity.com/paper/1905.10018/) (2019)
- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Learning to Optimize](https://mlsanity.com/paper/1606.01885/) (2016)
- [Neural Optimizer Search with Reinforcement Learning](https://mlsanity.com/paper/1709.07417/) (2017)
- [SUPER-ADAM: Faster and Universal Framework of Adaptive Gradients](https://mlsanity.com/paper/2106.08208/) (2021)
- [The Impact of the Mini-batch Size on the Variance of Gradients in Stochastic Gradient Descent](https://mlsanity.com/paper/2004.13146/) (2020)
- [Divergence Results and Convergence of a Variance Reduced Version of ADAM](https://mlsanity.com/paper/2210.05607/) (2022)

## Abstract (arXiv)

The concept of learning to optimize involves utilizing a trainable optimization strategy rather than relying on manually defined full gradient estimations such as ADAM. We present a framework that jointly trains the full gradient estimator and the trainable weights of the model. Specifically, we prove that pseudo-linear TO (Trainable Optimizer), a linear approximation of the full gradient, matches SGD's convergence rate while effectively reducing variance. Pseudo-linear TO incurs negligible computational overhead, requiring only minimal additional tensor multiplications. To further improve computational efficiency, we introduce two simplified variants of Pseudo-linear TO. Experiments demonstrate that TO methods converge faster than benchmark algorithms (e.g., ADAM) in both strongly convex and non-convex settings, and fine tuning of an LLM.