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To the best of our knowledge, there are no methods today for trainingdifferentially private regression models on sparse input data. To remedy this,we adapt the Frank-Wolfe algorithm for $L_1$ penalized linear regression to beaware of sparse inputs and to use them effectively. In doing so, we reduce thetraining time of the algorithm from $\mathcal{O}( T D S + T N S)$ to$\mathcal{O}(N S + T \sqrt{D} \log{D} + T S^2)$, where $T$ is the number ofiterations and a sparsity rate $S$ of a dataset with $N$ rows and $D$ features.Our results demonstrate that this procedure can reduce runtime by a factor ofup to $2,200\times$, depending on the value of the privacy parameter $\epsilon$and the sparsity of the dataset.

Scaling Up Differentially Private LASSO Regularized Logistic Regression via Faster Frank-Wolfe Iterations