Distributed Nonsmooth Optimization with Momentum and Frank-Wolfe
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Solution Overview
Problem
Existing distributed optimization methods struggle to efficiently solve nonsmooth composite optimization problems with constraints and stochastic factors, particularly in high-dimensional scenarios, due to high computational complexity and storage overhead, and are not suitable for distributed environments.
Innovation Solution
A distributed stochastic nonsmooth optimization method using smoothing and momentum techniques, involving weighted averaging, stochastic gradient estimation with recursive momentum, and Frank-Wolfe steps, to minimize an average objective function across multi-agent networks.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If projection operation is used in distributed subgradient or proximal gradient descent algorithms, then convergence to optimal solution is guaranteed, but computational complexity increases significantly in high-dimensional problems
Solution Approach 1:
The patent extracts and eliminates the projection operation from the optimization algorithm by replacing it with a conditional gradient approach. Instead of projecting onto constraint sets, the method solves linear minimization subproblems to obtain conditional gradients, which avoids the computationally expensive projection step while maintaining convergence guarantees for convex optimization problems.
Solution Approach 2:
The patent introduces an intermediary variable y_k^i that tracks the average gradient across all agents. This intermediary gradient estimate serves as a mediator to coordinate updates across the distributed network, enabling convergence without requiring direct projection operations on constraint sets. The variable y_k^i facilitates information exchange and coordination while avoiding complex projections.
2Device complexity
If centralized Frank-Wolfe algorithm is used to avoid projection computation, then computational complexity is reduced, but the method cannot be directly applied to distributed environments and convergence rate is suboptimal
Solution Approach 1:
The patent segments the centralized optimization problem into distributed subproblems across multiple agents. Each agent i maintains local variables x_k^i and computes local conditional gradients independently. The global optimization is achieved through distributed coordination using weight matrices W and averaging operations, enabling the Frank-Wolfe method to operate in distributed environments without requiring centralized control.
Solution Approach 2:
The patent creates a universal distributed optimization framework that combines the computational efficiency of Frank-Wolfe with the flexibility of distributed computing. The algorithm handles both smooth and nonsmooth objective functions, works with stochastic gradients, and adapts to various network topologies through the weight matrix W, making it universally applicable to diverse distributed optimization scenarios.
3Device complexity
If stochastic gradient methods are used to handle large-scale data, then computational complexity is reduced, but storage overhead and communication requirements increase in distributed settings
Solution Approach 1:
The patent applies partial action by having each agent compute stochastic gradients based on random subsets of data rather than processing all data. Each agent i maintains only local gradient estimates and partial Hessian information, significantly reducing storage requirements compared to centralized methods. The distributed averaging of these partial computations achieves the desired optimization without requiring all agents to store and process the entire dataset.
Data Source
AI summary
A distributed stochastic nonsmooth optimization method based on smoothing and momentum techniques is provided. The distributed stochastic nonsmooth optimization method solves nonsmooth composite optimization problems with constraints and stochastic factors in a distributed manner and achieves a higher convergence rate, lower computational complexity, and lower storage overhead. The method includes: using a distributed algorithm based on smoothing and momentum techniques to process all the agents in a loop, and providing initial states, step sizes, recommended ranges of smoothing parameters, and so on of the algorithm; and specifying performance metrics of the algorithm, and depicting clustering comparison results according to the metrics. The method is applicable to composite optimization problems with nonsmooth terms, including but not limited to clustering problems, least absolute shrinkage and selection operator (LASSO) regression in machine learning and compressed sensing problems in sensor networks, and is applicable to large-scale distributed nonsmooth optimization scenarios with high-dimensional complex constraints.