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Multi-Objective Bi-Level Optimization (MOBLO) addresses nested multi-objective optimization problems common in a range of applications. However, its multi-objective and hierarchical bilevel nature makes it notably complex. Gradient-based MOBLO algorithms have recently grown in popularity, as they effectively solve crucial machine learning problems like meta-learning, neural architecture search, and reinforcement learning. Unfortunately, these algorithms depend on solving a sequence of approximation subproblems with high accuracy, resulting in adverse time and memory complexity that lowers their numerical efficiency. To address this issue, we propose a gradient-based algorithm for MOBLO, called gMOBA, which has fewer hyperparameters to tune, making it both simple and efficient. Additionally, we demonstrate the theoretical validity by accomplishing the desirable Pareto stationarity. Numerical experiments confirm the practical efficiency of the proposed method and verify the theoretical results. To accelerate the convergence of gMOBA, we introduce a beneficial L2O neural network (called L2O-gMOBA) implemented as the initialization phase of our gMOBA algorithm. Comparative results of numerical experiments are presented to illustrate the performance of L2O-gMOBA., Comment: accepted by SCIENCE CHINA Mathematics

Constrained bilevel optimization tackles nested structures present in constrained learning tasks like constrained meta-learning, adversarial learning, and distributed bilevel optimization. However, existing bilevel optimization methods mostly are typically restricted to specific constraint settings, such as linear lower-level constraints. In this work, we overcome this limitation and develop a new single-loop, Hessian-free constrained bilevel algorithm capable of handling more general lower-level constraints. We achieve this by employing a doubly regularized gap function tailored to the constrained lower-level problem, transforming constrained bilevel optimization into an equivalent single-level optimization problem with a single smooth constraint. We rigorously establish the non-asymptotic convergence analysis of the proposed algorithm under the convexity of lower-level problem, avoiding the need for strong convexity assumptions on the lower-level objective or coupling convexity assumptions on lower-level constraints found in existing literature. Additionally, the generality of our method allows for its extension to bilevel optimization with minimax lower-level problem. We evaluate the effectiveness and efficiency of our algorithm on various synthetic problems, typical hyperparameter learning tasks, and generative adversarial network.

Bilevel programming has emerged as a valuable tool for hyperparameter selection, a central concern in machine learning. In a recent study by Ye et al. (2023), a value function-based difference of convex algorithm was introduced to address bilevel programs. This approach proves particularly powerful when dealing with scenarios where the lower-level problem exhibits convexity in both the upper-level and lower-level variables. Examples of such scenarios include support vector machines and $\ell_1$ and $\ell_2$ regularized regression. In this paper, we significantly expand the range of applications, now requiring convexity only in the lower-level variables of the lower-level program. We present an innovative single-level difference of weakly convex reformulation based on the Moreau envelope of the lower-level problem. We further develop a sequentially convergent Inexact Proximal Difference of Weakly Convex Algorithm (iP-DwCA). To evaluate the effectiveness of the proposed iP-DwCA, we conduct numerical experiments focused on tuning hyperparameters for kernel support vector machines on simulated data.

Gradient-based optimization methods for hyperparameter tuning guarantee theoretical convergence to stationary solutions when for fixed upper-level variable values, the lower level of the bilevel program is strongly convex (LLSC) and smooth (LLS). This condition is not satisfied for bilevel programs arising from tuning hyperparameters in many machine learning algorithms. In this work, we develop a sequentially convergent Value Function based Difference-of-Convex Algorithm with inexactness (VF-iDCA). We show that this algorithm achieves stationary solutions without LLSC and LLS assumptions for bilevel programs from a broad class of hyperparameter tuning applications. Our extensive experiments confirm our theoretical findings and show that the proposed VF-iDCA yields superior performance when applied to tune hyperparameters., Comment: 19 pages