This work proposes a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems and provides explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation.
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.
Empirical results reveal that VR-DR remains highly effective for nonsmooth loss functions, significantly broadening its practical utility beyond its theoretical constraints.
Zehui Jia, Denghui Li, Zhiyu Liu et al.· Journal of Scientific Comput...· 0 citations
In this paper, we propose a balanced augmented Lagrangian method based on accelerated stochastic ADMM (b-ASADMM) to efficiently solve structured separable nonconvex optimization problems subject to linear constraints. The objective function in this problem comprises potentially nonsmooth and smooth functions, where the smooth function is an average of multiple nonconvex smooth functions. The involved smooth subproblem is tackled by an accelerated stochastic gradient method based on weighting of stochastic item and pre-variable. The involved nonsmooth subproblem is solved under incorporation of Bregman distance to avoid the case that subproblem does not have a closed-form solution due to the complicated quadratic term or other hindering. The involved balanced augmented Lagrangian method advances the original ALM by balancing its subproblems and improving its implementation. In contrast to most deterministic and stochastic ADMMs, our dual variable allows a more flexible and larger step-size region. By standard smoothness assumption, we establish the global convergence and iteration complexity of the generated sequence. Furthermore, we provide a linear convergence rate of b-ASADMM under a local error bound condition and the weakly convex property of the nonsmooth component. Numerical experiments on the graph-guided fused Lasso problem and the smooth clipped absolute deviation penalty problem are conducted to verify the effectiveness of b-ASADMM.
This work derives a finite-dimensional dual formulation of PrO inference that separates sampling fluctuation, approximation under a divergence budget, regularization, and numerical optimization error and uses an exactly solvable categorical example to show that predictive-risk convergence can imply convergence to a unique predictive distribution even though the parameter distributions have no weak limit on the original parameter space.
Aurya Javeed, D. Kouri, Teresa Portone et al.· 0 citations
This paper proposes a tractable stochastic approach based on an entropic regularization of the distributionally robust value function, which makes it possible to compute stochastic gradient estimators, and the combination of these estimators with a stochastic Frank-Wolfe algorithm, allowing us to optimize the regularized robust objective while naturally handling constraints.
Wasserstein distributionally robust optimization (DRO) is a versatile and widely adopted framework for decision-making under uncertainty, yet its standard deterministic reformulations generally contain non-convex inner subproblems that are challenging to solve. To address this issue, we propose a shrinkage path heuristic that reduces the solution of a DRO problem to a one-dimensional search over the line segment connecting the (typically benign) sample average approximation (SAA) and the (more demanding but practically solvable) classical robust optimization solution. We derive a priori suboptimality bounds in stylized settings and, for the general case, a posteriori bounds obtained by applying a similar heuristic to a dual formulation. Numerical experiments on a multi-item newsvendor and an appointment scheduling problem show that the shrinkage path heuristic attains 85-110% (resp. 45-70%) of the out-of-sample performance improvements of Wasserstein DRO over SAA, at a fraction of the computational cost.
Ling-Jun Meng, Ryan Cory-Wright, W. Wiesemann· 0 citations
This paper studies trajectory-wise estimation of generalization error for primal--dual algorithms in non-smooth regression. Motivating examples include \(\ell_1\)-penalized least absolute deviations regression and square-root Lasso regression, where the data-fitting loss is non-differentiable and existing risk estimators for gradient-type optimization paths do not apply directly. We develop a general recursive framework that includes the Chambolle--Pock algorithm and related primal--dual splitting methods. We estimate risk by correcting each in-sample fitted value with a weighted combination of past dual iterates. The ideal weights are Stein derivative contractions and depend on the design covariance. We construct replacement weights from observable derivative contractions of the fitted-signal trajectory, yielding a covariance-free, data-driven correction. For high-dimensional Gaussian designs and fixed finite iteration horizon, we prove finite-sample guarantees for both estimators. For square-root ridge, we further establish a matched-Gaussian universality result beyond Gaussian designs. Numerical experiments show that the proposed estimators accurately track the out-of-sample risk along finite optimization paths.
