TL;DR
Empirical results reveal that VR-DR remains highly effective for nonsmooth loss functions, significantly broadening its practical utility beyond its theoretical constraints.
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 introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function. We target a broad class of integrands obeying a nonsmooth, localized variant of the descent lemma in the decision variable, a structural assumption that simultaneously covers smooth losses with Lipschitz gradient and differences of such losses with convex functions. At each iteration the expected cost is replaced by a sample average that is progressively refined, and the proximal-subgradient stepsize is selected by an Armijo-type line search enforcing a sufficient-decrease property up to stochastic errors induced by the sample-based approximation. This framework accommodates substantially more general problem formulations than existing methods, in particular, it requires neither (weak) convexity of the regularizer nor a uniform bound on the variance of the stochastic oracle, and our analysis yields convergence guarantees that are new even in the smooth setting. Specifically, we establish almost sure convergence of the sequence of function values and stationarity of every accumulation point of the trajectories under the relaxed requirement that the sample-size sequence be merely nondecreasing and unbounded, with no prescribed growth rate. Leveraging the Kurdyka-Lojasiewicz (KL) property, we further upgrade this subsequential guarantee to convergence of the whole trajectory to a single stationary point. Finally, for exponential-type KL desingularizing functions and polynomially growing sample sizes, we derive explicit polynomial convergence rates, up to a logarithmic factor, for both the function values and the iterates.
A globally convergent regularized Newton method with positive definite regularization for solving nonsmooth optimization problems that replaces the identity matrix in traditional algorithms with a general positive-definite symmetric matrix to regularize the generalized Hessian.
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.
This paper proposes a novel decentralized stochastic first-order optimization algorithm, which does not require second-order Hessian or Jacobian matrices, for the setting where the lower-level loss function is nonconvex but satisfies the Polyak–Łojasiewicz (PL) condition.
This work develops a regularization analysis of non-accelerated and accelerated primal-dual methods for solving linear inverse problems in the presence of noisy data. We investigate a Condat-V\~u algorithm and an accelerated primal-dual hybrid gradient method in Hilbert spaces, with focus on quantifying the effect of data perturbations on the reconstruction error. For the non-accelerated scheme, we derive error estimates in terms of Bregman distances, whereas for the accelerated scheme we establish error estimates in norm. The study accommodates a general class of convex data fidelities satisfying suitable perturbation conditions, which are verified explicitly for equality constrained and Morozov regularization. For the non-accelerated method, the analysis is further extended to Banach spaces, taking into account non-Euclidean geometries and including a particular non-reflexive setting tailored to nonnegative solution reconstruction. The results recover known behavior in classical settings while extending the regularization analysis to these more general frameworks. Numerical experiments with representative regularizers, including sparsity and entropic models, support the theoretical findings and illustrate practical performance under noise.
