Gokularam Muthukrishnan

Lakshmi Jayalal, Gokularam Muthukrishnan, Sheetal Kalyani

The performance of the standard Online Robust Principal Component Analysis (OR-PCA) technique depends on the optimum tuning of the explicit regularizers and this tuning is dataset sensitive. We aim to remove the dependency on these tuning parameters by using implicit regularization. We propose to use the implicit regularization effect of various modified gradient descents to make OR-PCA tuning free. Our method incorporates three different versions of modified gradient descent that separately but naturally encourage sparsity and low-rank structures in the data. The proposed method performs comparable or better than the tuned OR-PCA for both simulated and real-world datasets. Tuning-free ORPCA makes it more scalable for large datasets since we do not require dataset-dependent parameter tuning.

Gokularam Muthukrishnan, Anshoo Tandon

Differentially private $K$-means clustering enables releasing cluster centers derived from a dataset while protecting the privacy of the individuals. Non-interactive clustering techniques based on privatized histograms are attractive because the released data synopsis can be reused for other downstream tasks without additional privacy loss. The choice of the number of grids for discretizing the data points is crucial, as it directly controls the quantization bias and the amount of noise injected to preserve privacy. The widely adopted strategy selects a grid size that is independent of the number of clusters and also relies on empirical tuning. In this work, we revisit this choice and propose a refined grid-size selection rule derived by minimizing an upper bound on the expected deviation in the K-means objective function, leading to a more principled discretization strategy for non-interactive private clustering. Compared to prior work, our grid resolution differs both in its dependence on the number of clusters and in the scaling with dataset size and privacy budget. Extensive numerical results elucidate that the proposed strategy results in accurate clustering compared to the state-of-the-art techniques, even under tight privacy budgets.