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 for constant privacy parameters, demonstrating that privacy incurs only a small cost in this setting. Our result assumes that the input sequence is oblivious, i.e., it is fixed in advance and independent of the algorithm's outputs. We also prove that an $\tildeΩ(\sqrt{T})$ regret is inevitable for adaptively generated inputs, even when $L^\star = 0$.