Beyond Ordinary Lipschitz Constraints: Differentially Private Optimization with TNC

Abstract

We study Stochastic Convex Optimization in Differential Privacy model (DP-SCO). Unlike previous studies, here we assume the population risk function satisfies the Tsybakov Noise Condition (TNC) with some parameter $\theta>1$, where the Lipschitz constant of the loss could be extremely large or even unbounded, but the $\ell_2$-norm gradient of the loss has bounded $k$-th moment with $k\geq 2$. For the Lipschitz case with $\theta\geq 2$, we first propose an $(\epsilon, \delta)$-DP algorithms whose utility bound is $\tilde{O}\left(\left(\tilde{r}_{2k}(\frac{1}{\sqrt{n}}+(\frac{\sqrt{d}}{n\epsilon}))^\frac{k-1}{k}\right)^\frac{\theta}{\theta-1}\right)$ in high probability, where $n$ is the sample size, $d$ is the model dimension, and $\tilde{r}_{2k}$ is a term that only depends on the $2k$-th moment of the gradient. It is notable that such an upper bound is independent of the Lipschitz constant. We then extend to the case where $\theta\geq \bar{\theta}> 1$ for some known constant $\bar{\theta}$. Moreover, when the privacy budget $\epsilon$ is small enough, we show an upper bound of $\tilde{O}\left(\left(\tilde{r}_{k}(\frac{1}{\sqrt{n}}+(\frac{\sqrt{d}}{n\epsilon}))^\frac{k-1}{k}\right)^\frac{\theta}{\theta-1}\right)$ even if the loss function is not Lipschitz. For the lower bound, we show that for any $\theta\geq 2$, the private minimax rate for $\rho$-zero Concentrated Differential Privacy is lower bounded by $\Omega\left(\left(\tilde{r}_{k}(\frac{1}{\sqrt{n}}+(\frac{\sqrt{d}}{n\sqrt{\rho}}))^\frac{k-1}{k}\right)^\frac{\theta}{\theta-1}\right)$.