Differentially Private Coordinate Descent for Composite Empirical Risk Minimization
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Machine learning models can leak information about the data used to train them. Differentially Private (DP) variants of optimization algorithms like Stochastic Gradient Descent (DP-SGD) have been designed to mitigate this, inducing a trade-off between privacy and utility. In this paper, we propose a new method for composite Differentially Private Empirical Risk Minimization (DP-ERM): Differentially Private proximal Coordinate Descent (DP-CD). We analyze its utility through a novel theoretical analysis of inexact coordinate descent, and highlight some regimes where DP-CD outperforms DP-SGD, thanks to the possibility of using larger step sizes. We also prove new lower bounds for composite DP-ERM under coordinate-wise regularity assumptions, that are, in some settings, nearly matched by our algorithm. In practical implementations, the coordinate-wise nature of DP-CD updates demands special care in choosing the clipping thresholds used to bound individual contributions to the gradients. A natural parameterization of these thresholds emerges from our theory, limiting the addition of unnecessarily large noise without requiring coordinate-wise hyperparameter tuning or extra computational cost.
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