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Overcoming Lower-Level Constraints in Bilevel Optimization: A Novel Approach with Regularized Gap Functions

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arxiv 2406.01992 v1 pith:EWLA2XUK submitted 2024-06-04 math.OC

Overcoming Lower-Level Constraints in Bilevel Optimization: A Novel Approach with Regularized Gap Functions

classification math.OC
keywords lower-levelbileveloptimizationconstrainedconstraintsproblemalgorithmconvexity
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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    math.OC 2026-05 unverdicted novelty 7.0

    Optimistic bilevel optimization with manifold lower-level minimizers is differentiable if the optimistic selection is unique, yielding a pseudoinverse hyper-gradient and a convergent HG-MS algorithm whose rate depends...

  2. On the Stability and Generalization of First-order Bilevel Minimax Optimization

    cs.LG 2026-04 unverdicted novelty 7.0

    Provides the first systematic generalization analysis via algorithmic stability for single-timescale and two-timescale stochastic gradient descent-ascent in bilevel minimax problems.

  3. Optimization under Persistent State-Dependent Bias: Gradient-based Method and Complexity Analysis

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    Residual Learning, a proposed bilevel gradient method, claims exact convergence under state-dependent analog-hardware bias with rate O~(kappa1*kappa2^4*sigma^2/(mu*K)).

  4. Hypergradient-based Bilevel Reinforcement Learning with Improved Sample Complexity

    cs.LG 2026-07 reject novelty 5.0

    A bilevel RL algorithm using Boltzmann-policy optimality achieves Õ(ε⁻²) sample complexity for first-order stationarity, removing the outer-level PL condition but relying on a non-realizability error bound that is not proven.

  5. Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation

    math.OC 2024-12 unverdicted novelty 5.0

    AGILS is an alternating gradient algorithm for bilevel optimization that uses Moreau envelope reformulation to handle inexact lower-level solves, with convergence to stationary points proven under stated assumptions.