Year: 2023
Type: preprint
Abstract: We study private empirical risk minimization (ERM) problem for losses satisfying the $(γ,κ)$-Kurdyka-Łojasiewicz (KL) condition. The Polyak-Łojasiewicz (PL) condition is a special case of this condition when $κ=2$. Specifically, we study this problem under the constraint of $ρ$ zero-concentrated differential privacy (zCDP). When $κ\in[1,2]$ and the loss function is Lipschitz and smooth over a sufficiently large region, we provide a new algorithm based on variance reduced gradient descent that achieves the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^κ\big)$ on the excess empirical risk, where $n$ is the dataset size and $d$ is the dimension. We further show that this rate is nearly optimal. When $κ\geq 2$ and the loss is instead Lipschitz and weakly convex, we show it is possible to achieve the rate $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^κ\big)$ with a private implementation of the proximal point method. When the KL parameters are unknown, we provide a novel modification and analysis of the noisy gradient descent algorithm and show that this algorithm achieves a rate of $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^{\frac{2κ}{4-κ}}\big)$ adaptively, which is nearly optimal when $κ= 2$. We further show that, without assuming the KL condition, the same gradient descent algorithm can achieve fast convergence to a stationary point when the gradient stays sufficiently large during the run of the algorithm. Specifically, we show that this algorithm can approximate stationary points of Lipschitz, smooth (and possibly nonconvex) objectives with rate as fast as $\tilde{O}\big(\frac{\sqrt{d}}{n\sqrtρ}\big)$ and never worse than $\tilde{O}\big(\big(\frac{\sqrt{d}}{n\sqrtρ}\big)^{1/2}\big)$. The latter rate matches the best known rate for methods that do not rely on variance reduction.
Source: arXiv (Cornell University)
Institutions: The Ohio State University, Johns Hopkins University
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: AI Institute for Future Edge Networks and Distributed Intelligence (AI-EDGE), ALGORITHMIC STABILITY AND DIFFERENTIAL PRIVACY IN OPTIMIZATION MACHINE LEARNING AND MARKET EQUILIBRIA, INFORMATION AND COMPUTATION IN MARKET DESIGN, CAREER: Extending the Foundations of Privacy-Preserving Machine Learning, G6321998864, G6671297155, CAREER: Understanding the Inductive Biases in Modern Machine Learning, AF: Small: Collaborative Research: Rigorous Approaches for Scalable Privacy-preserving Deep Learning, NATIONAL CENTER FOR ARTIFICIAL INTELLIGENCE RESEARCH (NAIR), BIGDATA: F: Privacy in Unsupervised Learning
OpenAlex ID: W4388963771