# Minimization of Stochastic First-order Oracle Complexity of Adaptive Methods for Nonconvex Optimization

arXiv: https://arxiv.org/abs/2112.07163 · HTML: https://mlsanity.com/paper/2112.07163/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 31 references that are in the map.
- First posted on arXiv 14 Dec 2021.

## Builds on (most cited first, 18 of 18)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](https://mlsanity.com/paper/1502.03044/) (2015)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradients](https://mlsanity.com/paper/2010.07468/) (2020)
- [Measuring the Effects of Data Parallelism on Neural Network Training](https://mlsanity.com/paper/1811.03600/) (2018)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence](https://mlsanity.com/paper/2002.10542/) (2020)
- [Descending through a Crowded Valley - Benchmarking Deep Learning Optimizers](https://mlsanity.com/paper/2007.01547/) (2020)
- [Which Algorithmic Choices Matter at Which Batch Sizes? Insights From a Noisy Quadratic Model](https://mlsanity.com/paper/1907.04164/) (2019)
- [Stochastic Optimization for Performative Prediction](https://mlsanity.com/paper/2006.06887/) (2020)
- [Convergence rates for the stochastic gradient descent method for non-convex objective functions](https://mlsanity.com/paper/1904.01517/) (2019)
- [Appropriate Learning Rates of Adaptive Learning Rate Optimization Algorithms for Training Deep Neural Networks](https://mlsanity.com/paper/2002.09647/) (2020)

## Abstract (arXiv)

Numerical evaluations have definitively shown that, for deep learning optimizers such as stochastic gradient descent, momentum, and adaptive methods, the number of steps needed to train a deep neural network halves for each doubling of the batch size and that there is a region of diminishing returns beyond the critical batch size. In this paper, we determine the actual critical batch size by using the global minimizer of the stochastic first-order oracle (SFO) complexity of the optimizer. To prove the existence of the actual critical batch size, we set the lower and upper bounds of the SFO complexity and prove that there exist critical batch sizes in the sense of minimizing the lower and upper bounds. This proof implies that, if the SFO complexity fits the lower and upper bounds, then the existence of these critical batch sizes demonstrates the existence of the actual critical batch size. We also discuss the conditions needed for the SFO complexity to fit the lower and upper bounds and provide numerical results that support our theoretical results.