Online prediction from experts is a fundamental problem in machine learning and several works have studied this problem under privacy constraints. We propose and analyze new algorithms for this problem that improve over the regret bounds of the best existing algorithms for non-adaptive adversaries. For approximate differential privacy, our algorithms achieve regret bounds of $\tilde{O}(\sqrt{T \log d} + \log d/\varepsilon)$ for the stochastic setting and $\tilde O(\sqrt{T \log d} + T^{1/3} \log d/\varepsilon)$ for oblivious adversaries (where $d$ is the number of experts). For pure DP, our algorithms are the first to obtain sub-linear regret for oblivious adversaries in the high-dimensional regime $d \ge T$. Moreover, we prove new lower bounds for adaptive adversaries. Our results imply that unlike the non-private setting, there is a strong separation between the optimal regret for adaptive and non-adaptive adversaries for this problem. Our lower bounds also show a separation between pure and approximate differential privacy for adaptive adversaries where the latter is necessary to achieve the non-private $O(\sqrt{T})$ regret.

翻译：专家的在线预测是机器学习的根本问题,一些作品在隐私限制下研究了这个问题。我们建议和分析解决这个问题的新算法,这些算法改善了非适应性对手现有最佳算法的遗憾范围。对于近似差异隐私,我们的算法实现了$\tilde{O}(sqrt{T\log d}+\log d/\varepsilon)的遗憾范围。此外,我们证明适应性对手的新下限。我们的结果表明,与非私人环境不同的是,适应性对手和非适应性对手(即专家人数为美元)之间有强烈的遗憾。对于纯粹的 DP,我们的算法是第一个为在高空间制度中被忽略的对手获得亚线性遗憾的范围 $d\ge t$。 此外,我们证明,适应性对手的新下限。我们的结果是,与非私人环境不同的是,适应性和非适应性对手之间有强烈的遗憾。 对于这一问题,适应性和非适应性对手(即专家人数为美元 ) 。 对于纯度,我们低限的排序的排序也显示, 实现必要的稳定之间的最差。