Modular Bilevel Framework (HBS)

• Modular Bilevel Framework (HBS) is a composable optimization approach that decouples inner and outer solvers into interchangeable modules, enhancing flexibility and convergence.
• It enables plug-and-play integration of diverse computational routines including meta-learning, hyperparameter optimization, and quantum-inspired algorithms, improving practical performance.
• Empirical benchmarks demonstrate competitive convergence rates and scalability, while relaxing stringent conditions such as strong convexity required by traditional methods.

A Modular Bilevel Framework (HBS) refers to a class of algorithmic paradigms for bilevel optimization in which the components for inner and outer optimization are designed as interchangeable modules with clearly specified interfaces. This architecture is motivated by both the structural complexity intrinsic to bilevel programs (BLOs) and the diversity of practical computation subroutines used in modern applications, including meta-learning, hyperparameter optimization, stochastic optimization, and hybrid quantum-classical combinatorial optimization. The modular approach permits the combination, extension, and replacement of solver components while ensuring global convergence and preserving desirable complexity properties, even beyond legacy assumptions such as strong convexity or lower-level singletonness. This framework traces major contributions to Bi-level Descent Aggregation (BDA) and its descendants (Liu et al., 2021, Liu et al., 2020, Dagréou et al., 2022), stochastic plug-and-play extensions (Chu et al., 2 May 2025), and hardware-hybrid variants for applied domains (Heese et al., 5 Feb 2026).

1. Formal Problem Statement and Unified Bilevel Formulation

A bilevel optimization problem involves two levels of decision-making:

$$\min_{x\in\mathcal X}\;\varphi(x), \quad \text{where} \quad \varphi(x)\;=\;\inf_{y\in\,\mathcal Y\cap S(x)}F(x,y),$$

with

$S(x)=\arg\min_{y\in\mathcal Y}f(x,y),$

where $x\in\mathcal X\subset\mathbb R^n$ are upper-level (UL) variables and $y\in\mathcal Y\subset\mathbb R^m$ are lower-level (LL) variables, and $F,f: \mathcal X\times\mathcal Y\to\mathbb R$ are smooth objective functions (not necessarily convex in $x$). The goal is to optimize $F$ with respect to $x$, given that for each $x$, the variable $y$ minimizes $f(x, y)$.

The modular framework, such as BDA, is motivated by the limitations of classical approaches that demand $S(x)$ be a singleton (lower-level singleton condition, LLS), typically enforced by strong convexity of $f(x, \cdot)$, which is rarely satisfied in modern large-scale, nonconvex, or discrete settings (Liu et al., 2021, Liu et al., 2020). To overcome this, optimistic bilevel formulations—which treat $\varphi(x)$ as the infimum of $F(x, y)$ over all $y \in S(x)$, even if $S(x)$ is multi-valued—serve as the central abstraction.

2. Modular Algorithmic Architecture

Modular bilevel frameworks decompose the optimization process into composable subroutines for the upper and lower levels, each of which can utilize a variety of descent, projection, or stochastic updating modules. The architecture is typically realized as nested loops (classical two-loop) or tightly coupled single-loop dynamic systems (for stochastic or variance-reduced settings).

Example: Bi-level Descent Aggregation (BDA)

For each outer iteration $t$:

• Inner level (LL) update: For a fixed $x^t$, generate a sequence

$y_{k+1}(x^t) = \Proj_{\mathcal Y} \left[ y_k(x^t) - \mu \alpha_k d_k^F - (1-\mu)\beta_k d_k^f \right],$

with $d_k^F = s_u \nabla_y F(x^t, y_k)$ and $d_k^f = s_\ell \nabla_y f(x^t, y_k)$, and aggregation weights $\alpha_k, \beta_k$ (Liu et al., 2021).

• Outer level (UL) update: After $K$ inner steps,

$x^{t+1} = \Proj_{\mathcal X}\left[ x^t - \lambda \nabla \varphi_K(x^t) \right],$

where $\varphi_K(x) = F(x, y_K(x))$ and differentiation is performed via automatic or finite-difference methods.

The framework "plugs" any suitable LL solver module (gradient descent, accelerated, stochastic, prox-type, etc.) as long as it satisfies specified convergence criteria.

Further Abstractions and Stochastic Extensions

Single-loop variants and stochastic generalizations (e.g., (Dagréou et al., 2022, Chu et al., 2 May 2025)) introduce additional variables and update directions, framing the bilevel program as coupled dynamics in the triple $(x, y, z)$ (UL, LL, and implicit Hessian-inverse), with modular estimators for each direction (SGD, SAGA, SARAH, etc.), and decoupled or coupled variance reduction.

Hybrid frameworks (e.g., for QUBO minimization in logistics) introduce hybrid quantum-classical modules and online hyperparameter tuning at the outer level, with each sub-solver (e.g., QAOA, belief propagation, CACm) being an independent module whose parameters are updated in an outer-level bilevel step (Heese et al., 5 Feb 2026).

3. Mathematical and Convergence Properties

Convergence analysis is organized around proving two critical properties for any combination of modular subroutines:

• LL Solution Property: For every $\epsilon > 0$, there exists $K_0$ such that for all $K > K_0$,

$$\sup_{x \in X} \mathrm{dist}(y_K(x), S(x)) \le \epsilon.$$

• UL Objective Consistency: The sequence $\varphi_K(x) \to \varphi(x)$ uniformly on $X$ as $K \to \infty$.

If these hold, then under compactness and regularity, cluster points of the sequence of UL iterates converge to global/local/stationary solutions of the original bilevel program (Liu et al., 2021, Liu et al., 2020).

Explicit complexity bounds are available: for non-strongly convex LL, sublinear rates $O(\sqrt{(1+\ln K) / K^{1/4}})$ are achieved for LL gaps; under strong convexity, linear convergence in the LL gap and the UL surrogate gradient error are proven (Liu et al., 2021).

Stochastic extensions achieve convergence rates and sample complexities matching those of optimal single-level methods, e.g., $O(\sqrt{n+m}\ \epsilon^{-1})$ in finite-sum settings (Chu et al., 2 May 2025).

4. Practical Implementation and Modular Extensions

The modular approach supports a wide variety of instantiations:

• Inner-level modules: gradient descent, accelerated descent, stochastic estimators, projection or proximal maps, ODE-based or quantum-inspired updates (e.g., in hybrid solvers) (Liu et al., 2021, Liu et al., 2020, Heese et al., 5 Feb 2026).
• Outer-level modules: gradient, quasi-Newton, population-based search, reinforcement learning for hyperparameter adaptation.
• Composition: Each module is isolated with a defined API, allowing integration of variance reduction (e.g., SAGA (Dagréou et al., 2022)), quantum-classical orchestration (QAOA, CACm, IBP (Heese et al., 5 Feb 2026)), or fragmentation-aware feedback (as in adaptive service mapping (Xie et al., 10 Jul 2025)).

The plug-and-play principle enables variance-reduced, memory-efficient stochastic approaches; for example, global SAGA-style updates are applied simultaneously to all dynamics variables in (Dagréou et al., 2022). Complex bilevel applications (e.g., resource-mapping in networks, supply-chain optimization) exploit the same modular subproblem–feedback–outer search separation (Xie et al., 10 Jul 2025).

5. Applications and Empirical Benchmarks

Modular bilevel frameworks have demonstrated state-of-the-art empirical results for:

• Hyperparameter optimization and meta-learning: BDA and its variants outperform implicit differentiation, reverse hypergradient, and truncated variants in both accuracy and iteration efficiency on tasks such as data hyper-cleaning (MNIST, Fashion-MNIST) and few-shot classification (Omniglot, MiniImageNet) (Liu et al., 2021, Liu et al., 2020).
• Stochastic hyperparameter selection: SABA, a variance-reduced method within the modular HBS, achieves faster convergence in practice and sublinear or linear rates in theory, outperforming alternatives such as SOBA and STORM variants (Dagréou et al., 2022).
• Quantum-classical hybrid optimization: The hybrid HBS in (Heese et al., 5 Feb 2026) combines QAOA, CACm, and IBP to solve large-scale QUBO instances for supply chain optimization, leveraging modularity for improved solution quality and parallel scalability.
• Service mapping in computing power networks: The Adaptive Bilevel Search framework isolates outer (resource proportion) and inner (graph partition + multicommodity flow) modules with global fragmentation-aware feedback, yielding substantial improvements in resource utilization and acceptance rates (Xie et al., 10 Jul 2025).

Summary tables (from (Heese et al., 5 Feb 2026)) exhibit Pareto-front hypervolumes and highlight wall-clock efficiency and solution diversity advantages due to modular parallelism.

6. Comparison with Related Frameworks and Theoretical Landscape

A defining feature is the relaxation of restrictive assumptions pervasive in prior BLO solvers. Classical methods (Reverse Hypergradient, MAML, Implicit Hypergradient, ANIL) depend on lower-level single-valuedness and strong convexity, which modular frameworks avoid by requiring only level-boundedness and continuity for LL objective (Liu et al., 2021, Liu et al., 2020).

Plug-and-play architectures (PnPBO) unify both biased and unbiased stochastic estimators for all variables, supporting moving average control, clipping for stability, and matching the sample complexity of single-level rates (Chu et al., 2 May 2025). Variance-reduced frameworks (e.g., SABA) enable linear or optimal sublinear rates under less stringent smoothness or PL conditions (Dagréou et al., 2022).

The modular paradigm is extensible to hybrid solver ecosystems (quantum-classical, heuristic, learning-based), parallel/distributed implementations, and integration of domain-specific constraints and global metrics (Heese et al., 5 Feb 2026, Xie et al., 10 Jul 2025).

7. Extensibility, Limitations, and Perspective

The modular bilevel framework's extensibility is evidenced by its deployment in domains ranging from hyperparameter optimization to combinatorial logistics to networked resource allocation. Key interfaces (outer optimizer module ↔ subproblem solver ↔ global evaluator) enable method reconfiguration—by swapping modules or altering their analytic properties—without affecting overall pipeline convergence or complexity guarantees (Xie et al., 10 Jul 2025, Heese et al., 5 Feb 2026).

Limitations arise in the computational burden of inner solvers for high-dimensional or implicitly constrained LL problems, and in the choice of surrogate and aggregation parameters shaping the balance between convergence rate and solution fidelity.

A plausible implication is that as new optimization hardware or algorithms (quantum, analog, or neural) mature, modular bilevel frameworks will remain relevant by providing a principled, compositional abstraction for integrating such technologies while maintaining theoretical guarantees seen in their original design (Heese et al., 5 Feb 2026, Liu et al., 2021).