We examine the continuous-time counterpart of mirror descent, namely mirror flow, on classification problems which are linearly separable. Such problems are minimised `at infinity' and have many possible solutions; we study which solution is preferred by the algorithm depending on the mirror potential. For exponential tailed losses and under mild assumptions on the potential, we show that the iterates converge in direction towards a $\phi_\infty$-maximum margin classifier. The function $\phi_\infty$ is the \textit{horizon function} of the mirror potential and characterises its shape `at infinity'. When the potential is separable, a simple formula allows to compute this function. We analyse several examples of potentials and provide numerical experiments highlighting our results.

翻译：本文研究镜像下降的连续时间对应方法——镜像流在线性可分分类问题上的表现。此类问题在无穷远处取得最小值，且存在多个可能的解；我们探讨算法根据镜像势函数倾向于选择何种解。对于指数尾部损失函数，在势函数满足温和假设的条件下，我们证明迭代方向会收敛于一个$\phi_\infty$-最大间隔分类器。函数$\phi_\infty$是镜像势函数的\textit{水平函数}，刻画了其在无穷远处的形态。当势函数可分离时，可通过简单公式计算该函数。我们分析了多种势函数示例，并通过数值实验验证了研究结果。