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 evolve over time as new measurements are collected, and we establish linear convergence to the ground-truth system for both the best and Polyak step sizes after a burn-in period. We further characterize sublinear convergence of the iterates under constant and diminishing step sizes, which require only minimal information and thus offer broad applicability. 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。尽管现有工作利用优化求解器提供了理论上的精确恢复保证，但面向实际应用、具有收敛保证的快速学习算法设计仍未被探索。我们分析了该场景下的次梯度方法，其中待求解的优化问题随着新测量数据的收集而随时间演化，并证明了在预热期后，无论是采用最优步长还是Polyak步长，算法均能线性收敛至真实系统。我们进一步刻画了在恒定步长与衰减步长下迭代序列的次线性收敛性，这些步长仅需极少信息，因而具有广泛的适用性。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。这是首个针对非光滑目标系统辨识的次梯度算法分析工作。