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Hom-PGD+: Fast Reparameterized Optimization over Non-convex Ball-Homeomorphic Set

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.6Download Report PDF
homeomorphismreparameterizationnon-convex constraintprojectioninvertible nerual network
Abstract:

Optimization over general non-convex constraint sets poses significant computational challenges due to their inherent complexity. In this paper, we focus on optimization problems over non-convex constraint sets that are homeomorphic to a ball, which encompasses important problem classes such as star-shaped sets that frequently arise in machine learning and engineering applications. We propose \textbf{Hom-PGD+^+}, a fast, \textit{learning-based} and \textit{projection-efficient} first-order method that efficiently solves such optimization problems without requiring expensive projection or optimization oracles. Our approach leverages an invertible neural network (INN) to learn the homeomorphism between the non-convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained optimization problem. This transformation enables fast projection-efficient optimization while preserving the fundamental structure of the original problem. We establish that Hom-PGD+^+ achieves an O(ϵ−2)\mathcal{O}(\epsilon^{-2}) convergence rate to obtain an ϵ+O(ϵinn)\epsilon + \mathcal{O}(\sqrt{\epsilon_{\rm inn}})-approximate stationary solution, where ϵinn\epsilon_{\rm inn} denotes the homeomorphism learning error. This convergence rate represents a significant improvement over existing methods for optimization over non-convex sets. Moreover, Hom-PGD+^+ maintains a per-iteration computational complexity of O(W)\mathcal{O}(W), where WW is the number of INN parameters. Extensive numerical experiments, including chance-constrained optimization popular in power systems, demonstrate that Hom-PGD+^+ achieves convergence rates comparable to state-of-the-art methods while delivering speedups of up to one order of magnitude.

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This report is AI-GENERATED using Large Language Models and WisPaper (A scholar search engine). It analyzes academic papers' tasks and contributions against retrieved prior work. While this system identifies POTENTIAL overlaps and novel directions, ITS COVERAGE IS NOT EXHAUSTIVE AND JUDGMENTS ARE APPROXIMATE. These results are intended to assist human reviewers and SHOULD NOT be relied upon as a definitive verdict on novelty.
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Overview

Overall Novelty Assessment

The paper proposes Hom-PGD+, a learning-based first-order method for optimization over non-convex constraint sets homeomorphic to a ball. It sits within the Neural Network-Based Homeomorphic Projection leaf, which contains six papers including the original work. This leaf represents a focused research direction exploring invertible neural networks to learn homeomorphisms that transform constrained problems into ball-constrained equivalents. The taxonomy shows this is a relatively sparse area compared to broader ball-constrained optimization branches, suggesting the neural learning approach is less explored than analytical or direct optimization methods.

The taxonomy reveals neighboring directions that provide context. The sibling leaf Analytical Homeomorphic Transformations contains two papers using closed-form mappings rather than learned transformations, highlighting a methodological divide. Adjacent branches include Ball-Constrained Optimization Algorithms (nine papers across three sub-categories) focusing on direct iterative methods without homeomorphic transformations, and General Non-Convex Optimization Methods addressing broader non-convex settings. The scope notes clarify that homeomorphic projection methods explicitly construct mappings to enable projection-based optimization, distinguishing them from ball oracle methods or majorization-minimization frameworks that operate directly on original constraint sets.

Among thirty candidates examined, the core algorithmic contribution shows one refutable candidate from ten examined, indicating some prior work on projection-based methods for ball-homeomorphic sets. The convergence rate guarantee incorporating homeomorphism learning error examined ten candidates with none clearly refuting, suggesting this theoretical analysis accounting for neural network approximation error may be less directly addressed in prior work. The computational complexity analysis with INN parameters similarly found no refutations among ten candidates. The limited search scope means these statistics reflect top-semantic matches rather than exhaustive coverage, and the single refutable pair suggests partial but not complete overlap with existing projection-based approaches.

Based on the limited thirty-candidate search, the work appears to occupy a niche intersection of neural learning and homeomorphic projection for constrained optimization. The taxonomy structure shows this is a less crowded direction compared to direct ball-constrained methods or analytical transformations. The contribution-level statistics suggest the algorithmic framework has some precedent while the theoretical treatment of learning error may be more distinctive, though definitive assessment would require broader literature coverage beyond top-semantic matches.

Taxonomy

Core-task Taxonomy Papers
28
3
Claimed Contributions
30
Contribution Candidate Papers Compared
1
Refutable Paper

Research Landscape Overview

Core task: Optimization over non-convex ball-homeomorphic constraint sets. The field addresses optimization problems where feasible regions are homeomorphic to a ball but not necessarily convex, requiring specialized techniques beyond standard convex methods. The taxonomy reveals five main branches: Homeomorphic Projection and Reparameterization Methods develop projection operators and transformations that exploit the topological ball structure; Ball-Constrained Optimization Algorithms focus on iterative schemes tailored to ball-shaped feasible sets (including indefinite quadratic balls and Lp-balls); Distributionally Robust Optimization with Wasserstein Balls tackles uncertainty quantification using Wasserstein ambiguity sets; General Non-Convex Optimization Methods provide broader algorithmic frameworks applicable to various non-convex constraints; and Application-Specific Optimization addresses domain problems such as robust beamforming, portfolio optimization, and shape design. Works like Homeomorphic Projection Feasibility[2] and Efficient Bisection Projection[4] illustrate how projection-based techniques can be made computationally tractable, while Wasserstein Ball Optimization[6] and Portfolio Wasserstein Ball[9] demonstrate the practical relevance of ball-constrained formulations in robust decision-making. Recent activity highlights a tension between computational efficiency and theoretical guarantees. Many studies explore low-complexity projection schemes (e.g., Low Complexity Homeomorphic[7]) or leverage special structure (e.g., Constant Trace Property[10]) to avoid expensive subroutines, while others pursue convergence guarantees for general non-convex settings (e.g., Ball-Proximal Point[5], Strongly Convex Majorization[3]). Hom-PGD[0] sits within the Homeomorphic Projection and Reparameterization Methods branch, specifically targeting neural network-based homeomorphic projections. Compared to neighbors like Efficient Bisection Projection[4], which emphasizes bisection-based feasibility restoration, and Probabilistic Homeomorphic Projection[19], which incorporates stochastic elements, Hom-PGD[0] focuses on leveraging neural architectures to learn or approximate projection mappings. This positions it among a small cluster of works exploring data-driven or parameterized projection strategies, contrasting with classical analytic projection methods and offering potential scalability advantages for high-dimensional or complex constraint geometries.

Claimed Contributions

Hom-PGD+ algorithm for optimization over ball-homeomorphic sets

The authors introduce Hom-PGD+, a first-order optimization method that uses an invertible neural network to learn a homeomorphism between non-convex ball-homeomorphic constraint sets and a unit ball, enabling efficient optimization without expensive projection operations.

10 retrieved papers
Can Refute
Convergence rate guarantee with homeomorphism learning error

The authors prove that their method achieves an O(ε−2) convergence rate to reach an approximate stationary solution, with the approximation quality depending on both the optimization tolerance and the homeomorphism learning error. This represents an improvement over existing methods for non-convex constrained optimization.

10 retrieved papers
Computational complexity analysis with INN parameters

The authors establish that their algorithm has linear per-iteration computational complexity in terms of the number of invertible neural network parameters, making it computationally efficient for practical implementation.

10 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Hom-PGD+ algorithm for optimization over ball-homeomorphic sets

The authors introduce Hom-PGD+, a first-order optimization method that uses an invertible neural network to learn a homeomorphism between non-convex ball-homeomorphic constraint sets and a unit ball, enabling efficient optimization without expensive projection operations.

Contribution

Convergence rate guarantee with homeomorphism learning error

The authors prove that their method achieves an O(ε−2) convergence rate to reach an approximate stationary solution, with the approximation quality depending on both the optimization tolerance and the homeomorphism learning error. This represents an improvement over existing methods for non-convex constrained optimization.

Contribution

Computational complexity analysis with INN parameters

The authors establish that their algorithm has linear per-iteration computational complexity in terms of the number of invertible neural network parameters, making it computationally efficient for practical implementation.

Table of Contents