研究背景与问题
Modern machine learning models are often overparameterized, i.e., have more parameters than training data. This typically implies that the loss function admits multiple minimizers so that training methods (and their hyperparameter choices) have significant influence on the computed solution. We study this so-called implicit bias phenomenon in the context of training deep linear diagonal networks.
Previous Work
Previous work has shown that the gradient flow for learning such networks converges to (almost) sparse solutions when initialized small.
Our Contribution
For an arbitrary number of layers, we extend this analysis to the practically more realistic (stochastic) gradient descent iterations.
Approach
Our approach identifies (stochastic) gradient descent as a perturbed (stochastic) mirror gradient descent.
Results
This allows us to establish convergence and characterize the limiting solution, relating it to weighted $\ell_1$-minimization. Moreover, we provide sufficient conditions on the step sizes and quantify the influence of the step sizes on the implicit bias.