Towards Quantifying the Preconditioning Effect of Adam
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910328424497152 |
|---|---|
| author | Das, Rudrajit Agarwal, Naman Sanghavi, Sujay Dhillon, Inderjit S. |
| author_facet | Das, Rudrajit Agarwal, Naman Sanghavi, Sujay Dhillon, Inderjit S. |
| contents | There is a notable dearth of results characterizing the preconditioning effect of Adam and showing how it may alleviate the curse of ill-conditioning -- an issue plaguing gradient descent (GD). In this work, we perform a detailed analysis of Adam's preconditioning effect for quadratic functions and quantify to what extent Adam can mitigate the dependence on the condition number of the Hessian. Our key finding is that Adam can suffer less from the condition number but at the expense of suffering a dimension-dependent quantity. Specifically, for a $d$-dimensional quadratic with a diagonal Hessian having condition number $κ$, we show that the effective condition number-like quantity controlling the iteration complexity of Adam without momentum is $\mathcal{O}(\min(d, κ))$. For a diagonally dominant Hessian, we obtain a bound of $\mathcal{O}(\min(d \sqrt{d κ}, κ))$ for the corresponding quantity. Thus, when $d < \mathcal{O}(κ^p)$ where $p = 1$ for a diagonal Hessian and $p = 1/3$ for a diagonally dominant Hessian, Adam can outperform GD (which has an $\mathcal{O}(κ)$ dependence). On the negative side, our results suggest that Adam can be worse than GD for a sufficiently non-diagonal Hessian even if $d \ll \mathcal{O}(κ^{1/3})$; we corroborate this with empirical evidence. Finally, we extend our analysis to functions satisfying per-coordinate Lipschitz smoothness and a modified version of the Polyak-Łojasiewicz condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07114 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Towards Quantifying the Preconditioning Effect of Adam Das, Rudrajit Agarwal, Naman Sanghavi, Sujay Dhillon, Inderjit S. Machine Learning Numerical Analysis Optimization and Control There is a notable dearth of results characterizing the preconditioning effect of Adam and showing how it may alleviate the curse of ill-conditioning -- an issue plaguing gradient descent (GD). In this work, we perform a detailed analysis of Adam's preconditioning effect for quadratic functions and quantify to what extent Adam can mitigate the dependence on the condition number of the Hessian. Our key finding is that Adam can suffer less from the condition number but at the expense of suffering a dimension-dependent quantity. Specifically, for a $d$-dimensional quadratic with a diagonal Hessian having condition number $κ$, we show that the effective condition number-like quantity controlling the iteration complexity of Adam without momentum is $\mathcal{O}(\min(d, κ))$. For a diagonally dominant Hessian, we obtain a bound of $\mathcal{O}(\min(d \sqrt{d κ}, κ))$ for the corresponding quantity. Thus, when $d < \mathcal{O}(κ^p)$ where $p = 1$ for a diagonal Hessian and $p = 1/3$ for a diagonally dominant Hessian, Adam can outperform GD (which has an $\mathcal{O}(κ)$ dependence). On the negative side, our results suggest that Adam can be worse than GD for a sufficiently non-diagonal Hessian even if $d \ll \mathcal{O}(κ^{1/3})$; we corroborate this with empirical evidence. Finally, we extend our analysis to functions satisfying per-coordinate Lipschitz smoothness and a modified version of the Polyak-Łojasiewicz condition. |
| title | Towards Quantifying the Preconditioning Effect of Adam |
| topic | Machine Learning Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2402.07114 |