On the Growth of Mistakes in Differentially Private Online Learning: A Lower Bound Perspective
This work addresses a foundational problem in privacy-preserving machine learning by providing theoretical lower bounds for differentially private online learning, which is incremental as it matches prior upper bounds and partially resolves an open question.
The paper tackles the problem of establishing lower bounds for differentially private online learning algorithms, showing that for a broad class of such algorithms, the expected number of mistakes grows as Ω(log(T/δ)), matching existing upper bounds and contrasting with non-private settings where mistakes are independent of T.
In this paper, we provide lower bounds for Differentially Private (DP) Online Learning algorithms. Our result shows that, for a broad class of $(\varepsilon,δ)$-DP online algorithms, for number of rounds $T$ such that $\log T\leq O(1 / δ)$, the expected number of mistakes incurred by the algorithm grows as $Ω(\log \frac{T}δ)$. This matches the upper bound obtained by Golowich and Livni (2021) and is in contrast to non-private online learning where the number of mistakes is independent of $T$. To the best of our knowledge, our work is the first result towards settling lower bounds for DP-Online learning and partially addresses the open question in Sanyal and Ramponi (2022).