Summary of Directional Smoothness and Gradient Methods: Convergence and Adaptivity, by Aaron Mishkin et al.
Directional Smoothness and Gradient Methods: Convergence and Adaptivity
by Aaron Mishkin, Ahmed Khaled, Yuanhao Wang, Aaron Defazio, Robert M. Gower
First submitted to arxiv on: 6 Mar 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Optimization and Control (math.OC)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper develops new sub-optimality bounds for gradient descent (GD) that rely on the conditioning of the objective along the optimization path rather than global constants. The authors leverage directional smoothness, a measure of gradient variation, to derive upper-bounds on the objective and design strongly adapted step-sizes that minimize these bounds. They demonstrate that solving implicit equations yields new guarantees for two classical step-sizes in convex quadratic cases and reveal that Polyak’s and normalized GD achieve fast rates despite lacking knowledge of directional smoothness. The paper provides tighter convergence guarantees than the classical theory based on L-smoothness, as demonstrated by experiments on logistic regression. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary The paper explores a new way to understand how gradient descent works for optimizing functions. Instead of relying on general rules that work everywhere, it focuses on the specific path that the optimization takes. This approach leads to more accurate predictions about when and why gradient descent will succeed or fail. The researchers show that by using this new understanding, they can design better step-sizes for the algorithm and get closer to the optimal solution. |
Keywords
* Artificial intelligence * Gradient descent * Logistic regression * Optimization
