Single-loop approaches to nonsmooth bilevel optimisation

Abstract: We study bilevel optimisation problems in which the inner problem is represented as a set-valued, parametric constraint. We develop relevant optimistic and pessimistic calculus rules, derive corresponding optimality conditions, and formulate nonsmooth adjoint inclusions based on both the Fréchet and limiting coderivatives. Founded on these results, we propose a single-loop algorithm that accommodates a wide range of inner and adjoint steps, including those of primal-dual methods. We prove its convergence. Numerical experiments on total variation regularised inverse problems demonstrate the practicality of the approach.

• The paper introduces single-loop methods that overcome nested iterations in bilevel optimization by leveraging set-valued adjoint calculus.
• It develops new Fréchet subdifferential and coderivative calculus rules to handle nonsmooth, infinite-dimensional inner problems.
• The method demonstrates convergence in TV-regularized inverse problems, offering computational efficiency and strong optimality conditions.

Single-Loop Methods for Nonsmooth Bilevel Optimization: Set-Valued Adjoint Calculus and Algorithmic Implications

Problem Formulation and Context

The work addresses bilevel optimization problems of the form

$$\min_{u, x} J(u) + R(x) \quad \text{subject to} \quad 0 \in G(u, x),$$

where $u \in U, x \in X$ are elements of normed spaces, $J$ and $R$ are proper lower semicontinuous functionals, and $G$ is a (possibly nonsmooth and set-valued) constraint encoding the parametric optimality conditions for the inner problem. Unlike the dominant literature that requires strong smoothness assumptions (e.g., differentiability and second-order growth) on $G$, this work permits $G$ to be set-valued and $g$ (the inner objective) to be nonsmooth, only requiring metric subregularity.

Conventional bilevel algorithms often involve nested loops (an inner optimization for each outer candidate), which is computationally expensive, especially in nonsmooth and high-dimensional settings. Motivated by recent advances [suonpera2022bilevel, suonpera2024general, dizonvalkonen2024tracking], the paper proposes and analyzes single-loop methods which take only one or a fixed number of inner (and adjoint) steps per outer iteration. The authors provide substantial theoretical and algorithmic advancements for nonsmooth, set-valued bilevel problems, including rigorous calculus rules for subdifferentials and coderivatives, general optimality conditions, and a convergence guarantee for their single-loop method.

Subdifferential and Coderivative Calculus

The paper develops new calculus rules for the Fréchet subdifferential and coderivative in the context of set-valued mappings and nonsmooth functionals. Chain and sum rules are established for compositions and sums involving set-valued solution mappings. The framework distinguishes between "optimistic" and "pessimistic" variants (i.e., using infimum or supremum over selections), which correspond to alternative formulations in bilevel optimization.

Key results:

• Chain rules (Theorem 2.3) establish inclusion relationships between subdifferentials/coderivatives of compositions $J \circ S$ and those of $J$ and $u \in U, x \in X$0 under inner (semi-)Lipschitz continuity assumptions.
• Sum rules (Theorem 2.4) extend subdifferential calculus to $u \in U, x \in X$1 with weaker conditions, allowing for convex, proper, locally Lipschitz $u \in U, x \in X$2 possibly with nontrivial domains (including indicator functions).
• Novelty: These are among the few results treating Fréchet subdifferentials/coderivatives rigorously for compositions involving set-valued mappings in infinite-dimensional settings.

Generalized Optimality Conditions for Nonsmooth Bilevel Problems

Theoretical developments yield new necessary and sufficient optimality conditions using either Fréchet or limiting/Mordukhovich coderivatives for bilevel problems with nonsmooth inner structure. The analysis clarifies the connection between coderivative inclusions, non-classical KKT-type systems, and the differentiability properties of the solution mapping.

Key claims:

• Detailed Fermat rule generalizations show that any bilevel solution $u \in U, x \in X$3 must satisfy a sum-inclusion involving subdifferentials/coderivatives of the composed cost and the regularizer.
• The "optimistic" (min-inf) and "pessimistic" (min-sup) settings yield different inclusion directions; necessity/sufficiency is modulated by coderivative choice and (semi-)Lipschitz assumptions.
• Adjoint inclusion (Lemma 3.6): For both coderivative types, it characterizes how (non)smooth (generalized) Lagrange multipliers must be chosen so that

$u \in U, x \in X$4

for $u \in U, x \in X$5 given by functional gradients, and $u \in U, x \in X$6 an adjoint variable.

• Full-system optimality (Theorems 3.10, 3.13): Bilevel stationarity is reduced to finding $u \in U, x \in X$7 solving the system:

$u \in U, x \in X$8

where $u \in U, x \in X$9 denotes either the Fréchet or limiting coderivative.

Single-Loop Algorithmic Framework and Convergence

Building on the "tracking theory" of [dizonvalkonen2024tracking] and the above optimality system, the paper presents a general algorithmic structure for single-loop bilevel optimization involving three working variables: inner $J$0, adjoint $J$1, and outer $J$2.

Algorithmic regime:

• Inner algorithm: One or a fixed number of steps toward updating $J$3 (e.g., PDPS, FB iteration) for $J$4 fixed.
• Adjoint algorithm: One or a fixed number of steps to approximate $J$5 in the adjoint inclusion.
• Outer iteration: Forward-backward (proximal gradient) step in $J$6 using an estimated generalized derivative.

Convergence theory:

• Formal "tracking inequalities" are established for the inexactness in updating $J$7 and $J$8, quantifying how single (or multiple) steps approximate the required fixed points or inclusions (Assumption 5.2, Theorems 5.4 and 5.7).
• The key technical result, Theorem 5.14, proves the convergence of the method to generalized criticality, i.e., the sum-inclusion

$J$9

• The work outlines the exact requirements (e.g., set-valued descent inequalities, Aubin/inner Lipschitz regularity) for the theoretical guarantees, providing a basis for their satisfaction in relevant classes of nonsmooth problems and in infinite-dimensional spaces.

Figure 1: Outer function value $R$0 convergence versus computational resources for single-loop and reference methods in the denoising problem.

Example Applications: TV-Regularized Inverse Problems

The algorithm is specialized and analyzed in detail for nonsmooth TV-regularized inverse imaging problems:

• Denoising: Tuning the data-fidelity parameter for TV denoising under nonsmooth constraints.
• Deblurring: Learning convolution kernel parameters along with regularization in a TV-regularized deconvolution setting.

Specialized adjoint inclusions and coderivative calculations are derived for the structure of the TV norm (see Section 6.2, Lemmas 6.7 and 6.8), revealing that the adjoint problem can be efficiently solved via constrained quadratic programming due to the separable nonsmooth structure of the TV term.

Figure 2: Visualization of the deblurring kernel structure (left) and example images for the deblurring and denoising applications (right), demonstrating the tractability of the set-valued adjoint system.

Numerical experiments (Section 7) on realistic imaging problems demonstrate:

• Reliability and computational efficiency of the single-loop method, often outperforming reference methods requiring highly accurate inner subproblem solutions.
• Empirically verified convergence of the generalized differential criterion and outer function value, even under high computational constraints and for large imaging problems.

Theoretical and Practical Implications

Theoretical:

The paper demonstrates that bilevel optimization for a broad class of nonsmooth and set-valued inner problems is amenable to first-order, single-loop optimization schemes. It systematically extends the generalized differentiation and coderivative calculus to compositions central in bilevel learning, even in infinite-dimensional settings and without requiring strong constraints or smoothness. The analysis clarifies how different types of coderivatives (Fréchet, limiting) correspond to various criticality notions in bilevel programs.

Practical:

The developed methods enable the practical solution of challenging inverse problems (e.g., with TV or similar nonsmooth regularizers) where conventional nested or exact adjoint-based approaches fail due to computational intractability or ill-posedness of derivatives. In particular, the reduction to tractable system solves (often convex quadratic programs, even under set-valued structure) and the convergence of the single-loop algorithm make these methods highly attractive for large-scale applications—such as imaging and variational parameter estimation—where automation of hyperparameter selection is crucial.

Future directions:

• Tightening sufficiency/necessity: Further work may address sufficient conditions for global convergence to global optima in nonconvex settings and clarify the tightness of optimality conditions under weaker regularity.
• Extensions: The approach can likely be generalized to vector-valued or infinite-dimensional outer variables, more general inner constraints, or settings where $R$1 has more complicated set-valued nonconvex structure.
• Acceleration/stochasticity: Investigation into acceleration, variance reduction, and stochastic versions may significantly improve the practical performance and scalability of these single-loop methods for high-dimensional machine learning and signal processing tasks.

Conclusion

This paper rigorously extends the theoretical, algorithmic, and empirical state of bilevel optimization for problems where the inner problem is set-valued and nonsmooth, fundamentally expanding the class of problems for which first-order, single-loop methods can be proved to converge to criticality. The contributions—including new calculus rules for generalized derivatives, the explicit set-valued optimality systems, and the convergence theory for generalized single-loop algorithms—provide a foundation for both practical algorithms and further research in nonsmooth variational optimization.

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References:

• E. Suonperä, T. Valkonen, "Single-loop approaches to nonsmooth bilevel optimisation" (2606.19143).
• [suonpera2022bilevel], [suonpera2024general], [dizonvalkonen2024tracking], and others as cited throughout the original manuscript.