We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret ${O} \big( \varepsilon^{-1} \log^{1.5}{d} \big)$ where $d$ is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are ${O} \big( \varepsilon^{-1} \min\big\{d, T^{1/3}\log d\big\} \big)$. We also develop an adaptive algorithm for the small-loss setting with regret $O(L^\star\log d + \varepsilon^{-1} \log^{1.5}{d})$ where $L^\star$ is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret $O \big(\varepsilon^{-1} d^{1.5} \big)$, as well as an algorithm for the smooth case with regret $O \big( \varepsilon^{-2/3} (dT)^{1/3} \big)$, both significantly improving over existing bounds in the non-realizable regime.

翻译：我们考虑可实现场景下的在线学习问题，其中存在零损失解，并提出了新的差分隐私（DP）算法，获得近最优的遗憾界。对于专家预测在线问题，我们设计了新算法，获得近最优遗憾界 ${O} \big( \varepsilon^{-1} \log^{1.5}{d} \big)$，其中 $d$ 为专家数量。这显著优于现有DP非可实现场景下最优的遗憾界 ${O} \big( \varepsilon^{-1} \min\big\{d, T^{1/3}\log d\big\} \big)$。我们还为小损失场景开发了自适应算法，遗憾界为 $O(L^\star\log d + \varepsilon^{-1} \log^{1.5}{d})$，其中 $L^\star$ 为最佳专家的总损失。此外，我们考虑可实现场景下的DP在线凸优化，提出了一个近最优遗憾界为 $O \big(\varepsilon^{-1} d^{1.5} \big)$ 的算法，以及针对平滑情况的遗憾界为 $O \big( \varepsilon^{-2/3} (dT)^{1/3} \big)$ 的算法，两者均显著优于非可实现场景下的现有界。