Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime

A team of researchers from Caltech, led by Reza Ghane, Danil Akhtiamov, and Babak Hassibi, has published a significant theoretical paper titled "Dual Space Preconditioning for Gradient Descent in the Overparameterized Regime." The work provides a rigorous mathematical proof that a broad class of optimization algorithms, which they term Dual Space Preconditioned Gradient Descent (DSPGD), is guaranteed to converge to a perfect solution when training overparameterized linear models. This class encompasses practically ubiquitous optimizers like Adam, Normalized Gradient Descent, and Gradient Clipping. Their proof introduces a novel version of the Bregman Divergence, a tool for analyzing optimization, offering new techniques of independent interest to the machine learning theory community.

The paper also investigates the 'implicit bias' of these algorithms—what specific solution they converge to among the infinite possibilities that fit the training data perfectly. They found that for a subclass called 'isotropic preconditioners,' the final solution is identical to the one found by standard Gradient Descent: the solution closest to the initialization point in terms of Frobenius norm. For more general preconditioners, they prove the solution is at most a constant multiplicative factor away from the Gradient Descent solution. This work provides a crucial theoretical backbone, explaining why these adaptive optimizers work so reliably in practice for modern, overparameterized neural networks, even though their dynamics are more complex than vanilla gradient descent.