A Compositional Framework for First-Order Optimization (2403.05711v1)

Abstract: Optimization decomposition methods are a fundamental tool to develop distributed solution algorithms for large scale optimization problems arising in fields such as machine learning and optimal control. In this paper, we present an algebraic framework for hierarchically composing optimization problems defined on hypergraphs and automatically generating distributed solution algorithms that respect the given hierarchical structure. The central abstractions of our framework are operads, operad algebras, and algebra morphisms, which formalize notions of syntax, semantics, and structure preserving semantic transformations respectively. These abstractions allow us to formally relate composite optimization problems to the distributed algorithms that solve them. Specifically, we show that certain classes of optimization problems form operad algebras, and a collection of first-order solution methods, namely gradient descent, Uzawa's algorithm (also called gradient ascent-descent), and their subgradient variants, yield algebra morphisms from these problem algebras to algebras of dynamical systems. Primal and dual decomposition methods are then recovered by applying these morphisms to certain classes of composite problems. Using this framework, we also derive a novel sufficient condition for when a problem defined by compositional data is solvable by a decomposition method. We show that the minimum cost network flow problem satisfies this condition, thereby allowing us to automatically derive a hierarchical dual decomposition algorithm for finding minimum cost flows on composite flow networks. We implement our operads, algebras, and algebra morphisms in a Julia package called AlgebraicOptimization.jl and use our implementation to empirically demonstrate that hierarchical dual decomposition outperforms standard dual decomposition on classes of flow networks with hierarchical structure.

• The paper presents an algebraic framework using operads and algebra morphisms to compose and solve optimization problems.
• It demonstrates that methods such as gradient descent, Uzawa’s algorithm, and dual decomposition follow the proposed compositional structure.
• Numerical experiments, like the minimum cost network flow problem, validate the framework’s efficiency and scalability.

An Algebraic Framework for Compositional Optimization

The paper presents an innovative algebraic framework for first-order optimization techniques, particularly aimed at dealing with large-scale problems that require distributed solutions. The main contribution is the formalism for composing optimization problems using constructs from category theory such as operads, operad algebras, and algebra morphisms. This abstract mathematical framework unlocks automated solutions to hierarchical optimization problems that are common in machine learning, control systems, and operational research.

Mathematical Foundations and Abstractions

The central abstractions are derived from category theory. The paper introduces undirected wiring diagrams (UWDs), which are illustrative hypergraphs with structured compositional semantics. The real innovation lies in using operads and algebra morphisms to relate these compositional structures with first-order optimization strategies.

1. Operads and Operad Algebras: Operads give a syntax for composition, and the authors demonstrate how optimization problems can be framed as operad algebras. This enables an algebraic construction of complex problems by combining simpler subproblem structures.
2. Algebra Morphisms: Through algebra morphisms, the paper establishes mappings from optimization problems to solution procedures. For instance, it is shown that gradient descent creates an algebra morphism from the operad algebra of optimization problems to an algebra of dynamical systems, capturing both the problem and solution dynamics.

Technical Contributions

The paper rigorously develops numerical and theoretical results across several dimensions of optimization:

• Gradient Descent as an Algebra Morphism: It is established that gradient descent is an algebra morphism for differentiable minimization problems, showing cohesiveness between composite optimization problems and distributed algorithms based on their operadic formulation.
• Uzawa's and Dual Decomposition: Extending previous frameworks, the authors demonstrate that Uzawa’s algorithm—a method for handling equality constraints—and dual decomposition methods respect hierarchical structures, thus unifying previously discrete algorithms under a singular algebraic framework.
• Compositional Data Condition: The authors introduce a theoretical condition that provides a principled way to determine when decomposing a problem via its data structure is valid. This can guide automated derivational pathways from problem definition to solution algorithm construction.

Numerical Experiments and Real-World Applications

The framework is implemented in a Julia package, lending computational efficiency and practical applicability. This is showcased through the minimum cost network flow problem, where a hierarchical composition of the network results in faster computational solutions compared to traditional decomposition methods. Numerical experiments validate the effectiveness of exploiting compositional structure, particularly highlighting performance gains in problems that naturally align with hierarchical decomposition.

Implications and Future Directions

This framework bridges discrete distributed optimization algorithms into a continuous, coherent methodology by leveraging category theory. It is a compelling step towards more automated and efficient problem-solving paradigms in optimization.

Future directions might include extending this framework to more complex decomposition methods, such as the alternating direction method of multipliers (ADMM), and developing asynchronous algorithms that exploit the algebraic properties identified. Additionally, incorporating step-size optimization into the morphism framework holds potential for a broader class of optimization problems.

Overall, this work represents a conceptual and practical progression in how algebraic and categorical methods can systematically simplify and solve distributed optimization problems.