Private Online Prediction from Experts with Small Losses

We consider online prediction from experts, a fundamental problem in machine learning, under differential privacy constraints. Existing private algorithms achieve near-optimal regret $\tilde{O}(\sqrt{T})$, where $T$ is the time horizon. However, this bound becomes suboptimal when $L^\star$, the cumulative loss of the best expert, is significantly less than $T$. In this work, we present the first differentially private algorithm with regret $\tilde{O}(\sqrt{L^\star})$, offering a substantial improvement in the low-loss regime. Our regret bound matches the non-private lower bound up to poly-logarithmic factors, demonstrating that privacy incurs only a small cost in this setting.