{"ID":6023450,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-10T08:15:11.905439937Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05991","arxiv_id":"2607.05991","title":"A Semismooth Newton Augmented Lagrangian Method for Sparse Spectral Risk Optimization","abstract":"Empirical risk minimization is a standard and effective paradigm for learning predictive models by minimizing average loss. In high-stakes decision-making, however, an average-loss criterion may underrepresent rare but severe losses. Spectral risk measures (SRMs) provide a principled framework by incorporating weighted order statistics of losses, but the induced nonsmoothness and nonseparability from sorting make the resulting optimization problems challenging. We propose a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems. Exploiting a dual reformulation and properties of the Moreau envelope, we reduce the subproblems to structured dual-variable formulations, significantly simplifying computation. We provide explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation. Numerical results on synthetic and real-data instances show that the proposed method attains lower running times than the tested ADMM baseline while producing comparable stationarity residuals and sparse solutions.","short_abstract":"Empirical risk minimization is a standard and effective paradigm for learning predictive models by minimizing average loss. In high-stakes decision-making, however, an average-loss criterion may underrepresent rare but severe losses. Spectral risk measures (SRMs) provide a principled framework by incorporating weighted...","url_abs":"https://arxiv.org/abs/2607.05991","url_pdf":"https://arxiv.org/pdf/2607.05991v1","authors":"[\"Rufeng Xiao\",\"Rujun Jiang\",\"Xudong Li\",\"Defeng Sun\"]","published":"2026-07-07T08:23:24Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}