Convex Optimization with Local Label Differential Privacy: Tight Bounds in All Privacy Regimes
For researchers working on private convex optimization with label privacy, this work provides tight bounds and a practical algorithm that significantly reduces the dependence on label space size.
This paper resolves the open question of whether the linear cost in label space size is fundamental for stochastic convex optimization under local label differential privacy. The authors propose a new algorithm that achieves an excess risk of O(√(K/(εn))) in the high-privacy regime and O(√(K/(e^ε n))) in the medium-privacy regime, quadratically improving over prior O(K) dependence, and prove matching lower bounds.
We study the problem of Stochastic Convex Optimization (SCO) under the constraint of local Label Differential Privacy (L-LDP). In this setting, the features are considered public, but the corresponding labels are sensitive and must be randomized by each user locally before being sent to an untrusted analyzer. Prior work for SCO under L-LDP (Ghazi et al., 2021) established an excess population risk bound with a \emph{linear} dependence on the size of the label space, $K$: $O\left({\frac{K}{ε\sqrt{n}}}\right)$ in the high-privacy regime ($ε\leq 1$) and $O\left({\frac{K}{e^ε \sqrt{n}}}\right)$ in the medium-privacy regime ($1 \leq ε\leq \ln K$). This left open whether this linear cost is fundamental to the L-LDP model. In this note, we resolve this question. First, we present a novel and efficient non-interactive L-LDP algorithm that achieves an excess risk of $O\left({\sqrt{\frac{K}{εn}}}\right)$ in the high-privacy regime ($ε\leq 1$) and $O\left({\sqrt{\frac{K}{e^ε n}}}\right)$ in the medium-privacy regime ($1 \leq ε\leq \ln K$). This quadratically improves the dependency on the label space size from $O(K)$ to $O(\sqrt{K})$. Second, we prove a matching information-theoretic lower bound across all privacy regimes for any sufficiently large $n$.