Matrix AdaGrad: Row-wise and Column-wise Adaptive Subgradient Methods

Wenpeng Zhang, Runsheng Yu, Peilin Zhao

Adaptive optimization methods such as AdaGrad and Adam are widely used in modern deep neural network training, but their adaptive scaling is primarily designed for vector-valued parameters and does not explicitly exploit matrix structure. Recent matrix-aware optimizers demonstrate the benefits of structured optimization, yet a general theoretical framework for deriving matrix-aware adaptivity comparable to that of AdaGrad remains lacking. In this work, we develop an Online Mirror Descent framework with adaptive proximal functions for matrix-valued parameters, providing a principled methodology for deriving matrix-aware adaptive optimization through online regret minimization. By introducing row-wise and column-wise matrix proximal functions, our framework explicitly reveals the trade-off governing adaptive scaling: increasing the scaling factors reduces the gradient-dependent dual norm term while increasing the cost of evolving the proximal geometry. In the row-wise setting, this trade-off becomes separable under diagonal parameterization, allowing the adaptive scaling for each row to be derived independently by minimizing its corresponding row-wise regret bound. The column-wise…