This work proposes a novel single-loop algorithm based on a constrained reformulation in which lower-level stationarity is imposed as a constraint, and constructs a regularized Lagrangian by introducing a quadratic regularizer and restricting the dual variable to a bounded domain.
Abstract
In this work, we study the oracle complexity of finding an $\epsilon$-stationary point for nonconvex-strongly-convex (NC-SC) bilevel optimization using only first-order oracles. Existing methods achieving the best-known complexity guarantees typically rely on double-loop, penalty-based procedures. We propose a novel single-loop algorithm based on a constrained reformulation in which lower-level stationarity is imposed as a constraint. Specifically, we construct a regularized Lagrangian by introducing a quadratic regularizer and restricting the dual variable to a bounded domain, and then apply Smoothed Gradient Descent Ascent [Zhang et al., 2020], with Hessian-vector products approximated via finite differences of gradients. We refer to the resulting deterministic and stochastic algorithms as SGHA and Stoc-SGHA, respectively. In the deterministic setting, SGHA achieves an oracle complexity of $O(\bar{\kappa}_y^{5}\epsilon^{-2})$, where $\bar{\kappa}_y$ denotes the relevant condition number. In the stochastic setting, Stoc-SGHA achieves an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\rho^{-3}\right)$ with probability at least $1-\rho$ for any $\rho\in(0,1)$, and an oracle complexity of $O\left(\bar{\kappa}_y^{17}\epsilon^{-6}\right)$ in expectation under an additional bounded-iterate assumption. Moreover, under an additional stochastic smoothness assumption imposed only on the lower-level objective, the stochastic oracle complexity of Stoc-SGHA improves to $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\rho^{-2}\right)$ with high probability and $O\left(\bar{\kappa}_y^{11}\epsilon^{-4}\right)$ in expectation, matching the $\epsilon$-dependence of the lower bounds.
In this paper, we study last-iterate convergence of stochastic first-order methods for constrained smooth convex--concave minimax optimization under the standard bounded-variance stochastic oracle. A fundamental challenge is that the last iterates of vanilla stochastic extragradient (S-EG) and stochastic optimistic gradient descent--ascent (S-OGDA) may fail to converge in the presence of stochastic gradient noise, even for simple bilinear problems. To overcome this difficulty, we introduce a simple perturbation framework that regularizes the original convex--concave problem into a strongly convex--strongly concave one. Applying S-EG and S-OGDA to the perturbed problem yields two simple single-loop methods, referred to as perturbed S-EG (PS-EG) and perturbed S-OGDA (PS-OGDA). We establish last-iterate convergence by first deriving convergence in terms of the squared distance to the saddle point of the perturbed problem and then translating this estimate into guarantees for the restricted primal--dual gap. Based on this framework, we establish two types of convergence guarantees. When the optimization horizon is known \emph{a priori}, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/4})$ last-iterate convergence rate for the restricted primal--dual gap, which coincides with the standard primal--dual gap on compact feasible domains. When the optimization horizon is unknown, we develop an anytime variant based on diminishing perturbations and diminishing stepsizes. For general closed convex feasible sets, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/5})$ last-iterate convergence rate for the restricted primal--dual gap. Furthermore, in the unconstrained setting, PS-EG admits a sharper $\mathcal{O}(T^{-1/4})$ anytime convergence rate in terms of the gradient norm.
Taoli Zheng, Jiajin Li, Anthony Man-Cho So· 0 citations
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.
Yihan Zhang, Xinwen Zhang, My T. Thai et al.· 0 citations
This work considers the design of first-order convex optimization algorithms and convergence proofs. In particular, we consider nonsmooth Lipschitz and smooth problems accessed through a subgradient or gradient oracle, respectively. For the general class of fixed-step first-order methods, prior work on Performance Estimation Problems (PEPs) has shown that structured, tight convergence proofs typically exist. Under mild conditions, we further show that any first-order method guaranteeing a bound on the primal objective gap $f(x_N)-f(x_\star)$ assuming only a bound on $\|x_0-x_\star\|$ actually has a stronger guarantee on an explicit, computable primal-dual gap at the same rate. These implicit optimal dual certificates, which take the form of affine lower bounds, also provide insight into the role of auxiliary sequences in momentum methods.
Direct policy optimization is widely used in reinforcement learning and control, but generally leads to nonconvex optimization problems. For state-feedback $H_\infty$ control, the policy objective is also nonsmooth, despite possessing a benign landscape whose hidden convexity can be revealed by the recently developed extended convex lifting framework. Motivated by recent advances in hidden convex optimization, we study zeroth-order optimization of nonsmooth, nonconvex problems admitting a convex lifting. We propose a zeroth-order proximal point algorithm: An inexact proximal-point outer loop constructs strongly convex subproblems, while an inner loop approximately solves each subproblem using only function evaluations. With probability at least $1-\delta$, our proposed algorithm returns an $\epsilon$-optimal solution using $\widetilde{O}\left(d\epsilon^{-3}\right)$ function evaluations, while all iterates remain feasible without explicit projection. Finally, we verify that the assumptions underlying our analysis hold for discrete-time state-feedback $H_\infty$ policy optimization, yielding an oracle complexity of $\widetilde{O}\left(n_u n_x\epsilon^{-3}\right)$ for attaining a prescribed objective value gap, where $n_u\times n_x$ is the dimension of the feedback gain to be optimized over.
We consider a quadratic minmax problem with coupled inner constraints and propose a method to compute a class of stationary points. To motivate the need to compute such stationary points, we first show that they are meaningful, in the sense that they can be locally optimal for our problem under suitable{non-degeneracy} conditions. Then based on a suitable log barrier function, we build an infeasible interior point-type {single loop method} (which does not explicitly distinguish between the outer and inner problem) and prove that a non-degenerate stationary point is an attraction point as the algorithm moves along the designed central path. We show in particular that our method is polynomial in the special case where the inner feasible set of our constrained minmax problem is independent from outer variables. Our numerical experiments, on both synthetic data and a class of min-cost flow problems, showcase the behavior of our method and how it outperforms existing algorithms from the literature in terms of the quality of the computed stationary points.
S. Cipolla, O. Stein, Alain B. Zemkoho· 0 citations
Augmented Lagrangian (AL) methods are a classical framework for constrained optimization, but for directly verifiable approximate KKT points, known first-order complexity bounds for standard inexact AL methods are suboptimal, while the best known proximal augmented Lagrangian (PAL) bounds retain an additional logarithmic factor. We consider linearly constrained convex composite problems with a smooth convex term and a possibly nonsmooth closed proper convex term with compact domain. We develop three inexact AL schemes that preserve the standard AL subproblem structure and attain the optimal primal-dual complexity $\mathcal O(\epsilon^{-1})$ in the convex setting, improving prior AL bounds of $\mathcal O(\epsilon^{-4/3})$, $\mathcal O(\epsilon^{-7/4})$, and $\mathcal O(\epsilon^{-2})$, and removing the logarithmic factor from PAL guarantees. Two variants are parameter-free, and all three admit nonergodic guarantees, including a stronger last-iterate guarantee for one variant. These results show that proximal regularization, ergodic averaging, and prior knowledge of problem-dependent constants are not intrinsic requirements for attaining optimal verifiable primal-dual complexity within the standard AL framework. A key ingredient is a parameter-free accelerated method that computes verifiable stationarity certificates for the standard, unregularized AL subproblems with optimal complexity. In the strongly convex setting, our methods attain near-optimal complexity $\mathcal O(\epsilon^{-1/2}\log(\epsilon^{-1}))$, with two parameter-free variants. Numerical experiments on six problem classes, including elastic-net least-squares regression, group-sparse Huberized support vector machines, and a quantum semidefinite program (SDP), demonstrate substantial computational advantages over a representative PAL method, with speedups frequently ranging from $5$ to $50$ times.
Arnesh Sujanani, Saeed Ghadimi, Henry Wolkowicz· 0 citations