SOREL: A Stochastic Algorithm for Spectral Risks Minimization

The spectral risk has wide applications in machine learning, especially in real-world decision-making, where people are concerned with more than just average model performance. By assigning different weights to the losses of different sample points, rather than the same weights as in the empirical risk, it allows the model's performance to lie between the average performance and the worst-case performance. In this paper, we propose SOREL, the first stochastic gradient-based algorithm with convergence guarantees for spectral risks minimization. Previous approaches often rely on smoothing the spectral risk by adding a strongly concave function, thereby lacking convergence guarantees for the original spectral risk. We theoretically prove that our algorithm achieves a near-optimal rate of $\widetilde{O}(1/\sqrt{\epsilon})$ to obtain an $\epsilon$-optimal solution in terms $\epsilon$. Experiments on real datasets show that our algorithm outperforms existing ones in most cases, both in terms of runtime and sample complexity.