This paper investigates a subgradient-based algorithm to solve the system identification problem for linear time-invariant systems with non-smooth objectives. This is essential for robust system identification in safety-critical applications. While existing work provides theoretical exact recovery guarantees using optimization solvers, the design of fast learning algorithms with convergence guarantees for practical use remains unexplored. We analyze the subgradient method in this setting where the optimization problems to be solved change over time as new measurements are taken, and we establish linear convergence results for both the best and Polyak step sizes after a burn-in period. Additionally, we characterize the asymptotic convergence of the best average sub-optimality gap under diminishing and constant step sizes. Finally, we compare the time complexity of standard solvers with the subgradient algorithm and support our findings with experimental results. This is the first work to analyze subgradient algorithms for system identification with non-smooth objectives.

翻译：本文研究了一种基于次梯度的算法，用于解决具有非光滑目标的线性时不变系统的系统辨识问题。这对于安全关键应用中的鲁棒系统辨识至关重要。虽然现有工作利用优化求解器提供了理论上的精确恢复保证，但针对实际应用、具有收敛保证的快速学习算法设计仍未得到探索。我们分析了在此设置下的次梯度方法，其中待求解的优化问题随着新测量数据的获取而随时间变化，并证明了在预热期后，采用最优步长和Polyak步长均能实现线性收敛。此外，我们刻画了在递减步长和恒定步长下，最优平均次优性间隙的渐近收敛性。最后，我们比较了标准求解器与次梯度算法的时间复杂度，并通过实验结果支持了我们的发现。这是首个分析针对非光滑目标系统辨识的次梯度算法的工作。