A Mirror Descent Perspective of Smoothed Sign Descent
This work provides a theoretical analysis for non-gradient algorithms in optimization, which is incremental as it builds on existing mirror descent perspectives to address a specific limitation.
The paper tackles the problem of analyzing optimization algorithms that use non-gradient update directions, such as smoothed sign descent, by extending the mirror descent framework to characterize their dynamics for regression problems. The result shows that the convergent solution approximates a KKT point, with tuning of the stability constant ε reducing KKT error.
Recent work by Woodworth et al. (2020) shows that the optimization dynamics of gradient descent for overparameterized problems can be viewed as low-dimensional dual dynamics induced by a mirror map, explaining the implicit regularization phenomenon from the mirror descent perspective. However, the methodology does not apply to algorithms where update directions deviate from true gradients, such as ADAM. We use the mirror descent framework to study the dynamics of smoothed sign descent with a stability constant $\varepsilon$ for regression problems. We propose a mirror map that establishes equivalence to dual dynamics under some assumptions. By studying dual dynamics, we characterize the convergent solution as an approximate KKT point of minimizing a Bregman divergence style function, and show the benefit of tuning the stability constant $\varepsilon$ to reduce the KKT error.