Faster Rates of Private Stochastic Convex Optimization
This work improves privacy-preserving optimization rates for specific function classes, offering incremental advances in machine learning with differential privacy.
The paper tackles the problem of differentially private stochastic convex optimization by providing faster excess population risk bounds for functions satisfying the Tysbakov Noise Condition and strongly convex functions with additional assumptions, achieving upper bounds like $ ilde{O}(( rac{1}{\sqrt{n}}+ rac{\sqrt{d\log rac{1}δ}}{nε})^ racθ{θ-1})$ and $O( rac{d\log rac{1}δ}{n^2ε^2}+ rac{1}{n^τ})$, and establishing matching lower bounds for TNC functions.
In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) and provide excess population risks for some special classes of functions that are faster than the previous results of general convex and strongly convex functions. In the first part of the paper, we study the case where the population risk function satisfies the Tysbakov Noise Condition (TNC) with some parameter $θ>1$. Specifically, we first show that under some mild assumptions on the loss functions, there is an algorithm whose output could achieve an upper bound of $\tilde{O}((\frac{1}{\sqrt{n}}+\frac{\sqrt{d\log \frac{1}δ}}{nε})^\fracθ{θ-1})$ for $(ε, δ)$-DP when $θ\geq 2$, here $n$ is the sample size and $d$ is the dimension of the space. Then we address the inefficiency issue, improve the upper bounds by $\text{Poly}(\log n)$ factors and extend to the case where $θ\geq \barθ>1$ for some known $\barθ$. Next we show that the excess population risk of population functions satisfying TNC with parameter $θ\geq 2$ is always lower bounded by $Ω((\frac{d}{nε})^\fracθ{θ-1}) $ and $Ω((\frac{\sqrt{d\log \frac{1}δ}}{nε})^\fracθ{θ-1})$ for $ε$-DP and $(ε, δ)$-DP, respectively. In the second part, we focus on a special case where the population risk function is strongly convex. Unlike the previous studies, here we assume the loss function is {\em non-negative} and {\em the optimal value of population risk is sufficiently small}. With these additional assumptions, we propose a new method whose output could achieve an upper bound of $O(\frac{d\log\frac{1}δ}{n^2ε^2}+\frac{1}{n^τ})$ for any $τ\geq 1$ in $(ε,δ)$-DP model if the sample size $n$ is sufficiently large.