Consider an online convex optimization problem where the loss functions are self-concordant barriers, smooth relative to a convex function $h$, and possibly non-Lipschitz. We analyze the regret of online mirror descent with $h$. Then, based on the result, we prove the following in a unified manner. Denote by $T$ the time horizon and $d$ the parameter dimension. 1. For online portfolio selection, the regret of $\widetilde{\text{EG}}$, a variant of exponentiated gradient due to Helmbold et al., is $\tilde{O} ( T^{2/3} d^{1/3} )$ when $T > 4 d / \log d$. This improves on the original $\tilde{O} ( T^{3/4} d^{1/2} )$ regret bound for $\widetilde{\text{EG}}$. 2. For online portfolio selection, the regret of online mirror descent with the logarithmic barrier is $\tilde{O}(\sqrt{T d})$. The regret bound is the same as that of Soft-Bayes due to Orseau et al. up to logarithmic terms. 3. For online learning quantum states with the logarithmic loss, the regret of online mirror descent with the log-determinant function is also $\tilde{O} ( \sqrt{T d} )$. Its per-iteration time is shorter than all existing algorithms we know.

翻译：考虑一个在线凸优化问题，其中损失函数为自洽壁垒函数，相对于凸函数 $h$ 光滑，且可能非利普希茨连续。我们分析了采用 $h$ 的在线镜像下降法的遗憾值。然后，基于该结果，我们以统一的方式证明了以下结论。记 $T$ 为时间跨度，$d$ 为参数维度。1. 对于在线投资组合选择，由 Helmbold 等人提出的指数梯度变体 $\widetilde{\text{EG}}$ 的遗憾值为 $\tilde{O} ( T^{2/3} d^{1/3} )$，当 $T > 4 d / \log d$ 时。这改进了 $\widetilde{\text{EG}}$ 原有的 $\tilde{O} ( T^{3/4} d^{1/2} )$ 遗憾界。2. 对于在线投资组合选择，采用对数壁垒函数的在线镜像下降法的遗憾值为 $\tilde{O}(\sqrt{T d})$。该遗憾界与 Orseau 等人提出的 Soft-Bayes 的遗憾界（忽略对数项）相同。3. 对于采用对数损失的量子态在线学习，使用对数行列式函数的在线镜像下降法的遗憾值同样为 $\tilde{O} ( \sqrt{T d} )$。其每次迭代时间比我们已知的所有现有算法都要短。