An official website of the United States government
In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an $$\mathcal{O}(\epsilon^{-1.5})$$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires $$\mathcal{O}(\epsilon^{-1.5})$$ iterations (each using $$\mathcal{O}(1)$$ samples and only first-order gradient information) to find an $$\epsilon$$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an $$\mathcal{O}(\epsilon^{-1.5})$$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization.
more »
« less
This content will become publicly available on June 15, 2026
Improved Complexity for Smooth Nonconvex Optimization: A Two-Level Online Learning Approach with Quasi-Newton Methods
We study the problem of finding an π-first-order stationary point (FOSP) of a smooth function, given access only to gradient information. The best-known gradient query complexity for this task, assuming both the gradient and Hessian of the objective function are Lipschitz continuous, is O(πβ7/4). In this work, we propose a method with a gradient complexity of O(π1/4πβ13/8), where π is the problem dimension, leading to an improved complexity when π = O(πβ1/2). To achieve this result, we design an optimization algorithm that, underneath, involves solving two online learning problems. Specifically, we first reformulate the task of finding a stationary point for a nonconvex problem as minimizing the regret in an online convex optimization problem, where the loss is determined by the gradient of the objective function. Then, we introduce a novel optimistic quasi-Newton method to solve this online learning problem, with the Hessian approximation update itself framed as an online learning problem in the space of matrices. Beyond improving the complexity bound for achieving an π-FOSP using a gradient oracle, our result provides the first guarantee suggesting that quasi-Newton methods can potentially outperform gradient descent-type methods in nonconvex settings.
more »
« less
- PAR ID:
- 10631738
- Publisher / Repository:
- ACM
- Date Published:
- Page Range / eLocation ID:
- 2225 to 2236
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In this paper, we consider a nonconvex unconstrained optimization problem minimizing a twice differentiable objective function with HΓΆlder continuous Hessian. Specifically, we first propose a Newton-conjugate gradient (Newton-CG) method for finding an approximate first- and second-order stationary point of this problem, assuming the associated HΓΆlder parameters are explicitly known. Then, we develop a parameter-free Newton-CG method without requiring any prior knowledge of these parameters. To the best of our knowledge, this method is the first parameter-free second-order method achieving the best-known iteration and operation complexity for finding an approximate first- and second-order stationary point of this problem. Finally, we present preliminary numerical results to demonstrate the superior practical performance of our parameter-free Newton-CG method over a well-known regularized Newton method. Funding: C. He was partially financially supported by the Wallenberg AI, Autonomous Systems and Software Program funded by the Knut and Alice Wallenberg Foundation. H. Huang was partially financially supported by the National Science Foundation [Award IIS-2347592]. Z. Lu was partially financially supported by the National Science Foundation [Award IIS-2211491], the Office of Naval Research [Award N00014-24-1-2702], and the Air Force Office of Scientific Research [Award FA9550-24-1-0343].more » « less
-
Variance reduction techniques like SVRG provide simple and fast algorithms for optimizing a convex finite-sum objective. For nonconvex objectives, these techniques can also find a first-order stationary point (with small gradient). However, in nonconvex optimization it is often crucial to find a second-order stationary point (with small gradient and almost PSD hessian). In this paper, we show that Stabilized SVRG (a simple variant of SVRG) can find an \eps-second-order stationary point using only O(n^{2/3}/\eps^2+n/\eps^{1.5}) stochastic gradients. To our best knowledge, this is the first second-order guarantee for a simple variant of SVRG. The running time almost matches the known guarantees for finding \eps-first-order stationary points.more » « less
-
In this paper, we present an accelerated quasi-Newton proximal extragradient method for solving unconstrained smooth convex optimization problems. With access only to the gradients of the objective function, we prove that our method can achieve a convergence rate of $${\bigO}\bigl(\min\{\frac{1}{k^2}, \frac{\sqrt{d\log k}}{k^{2.5}}\}\bigr)$$, where $$d$$ is the problem dimension and $$k$$ is the number of iterations. In particular, in the regime where $$k = \bigO(d)$$, our method matches the \emph{optimal rate} of $$\mathcal{O}(\frac{1}{k^2})$$ by Nesterov's accelerated gradient (NAG). Moreover, in the the regime where $$k = \Omega(d \log d)$$, it outperforms NAG and converges at a \emph{faster rate} of $$\mathcal{O}\bigl(\frac{\sqrt{d\log k}}{k^{2.5}}\bigr)$$. To the best of our knowledge, this result is the first to demonstrate a provable gain for a quasi-Newton-type method over NAG in the convex setting. To achieve such results, we build our method on a recent variant of the Monteiro-Svaiter acceleration framework and adopt an online learning perspective to update the Hessian approximation matrices, in which we relate the convergence rate of our method to the dynamic regret of a specific online convex optimization problem in the space of matrices.more » « less
-
Quasi-Newton algorithms are among the most popular iterative methods for solving unconstrained minimization problems, largely due to their favorable superlinear convergence property. However, existing results for these algorithms are limited as they provide either (i) a global convergence guarantee with an asymptotic superlinear convergence rate, or (ii) a local non-asymptotic superlinear rate for the case that the initial point and the initial Hessian approximation are chosen properly. In particular, no current analysis for quasi-Newton methods guarantees global convergence with an explicit superlinear convergence rate. In this paper, we close this gap and present the first globally convergent quasi-Newton method with an explicit non asymptotic superlinear convergence rate. Unlike classical quasi-Newton methods, we build our algorithm upon the hybrid proximal extragradient method and propose a novel online learning framework for updating the Hessian approximation matrices. Specifically, guided by the convergence analysis, we formulate the Hessian approximation update as an online convex optimization problem in the space of matrices, and we relate the bounded regret of the online problem to the superlinear convergence of our method.more » « less
