Core Concepts

A novel regularized Barzilai-Borwein (RBB) stepsize is developed to efficiently solve challenging optimization problems. The RBB stepsize includes a class of valid stepsizes and can overcome the instability of the original Barzilai-Borwein (BB) stepsize in ill-conditioned optimization problems.

Abstract

The authors develop a novel stepsize called the regularized Barzilai-Borwein (RBB) stepsize for solving challenging optimization problems efficiently. The RBB stepsize is based on the Barzilai-Borwein (BB) method, but includes a regularization term to improve the stability of the BB method.
The key highlights and insights are:
The RBB stepsize is the solution to a ℓ2-regularized least squares problem. When the regularization term vanishes, the RBB stepsize reduces to the original BB stepsize.
The global convergence of the RBB algorithm is proved for solving strictly convex quadratic optimization problems.
An adaptive two-step parameter scheme is proposed for generating the regularization parameters, which helps control the fluctuation scale of the stepsize and improves the stability of the RBB method.
An enhanced RBB stepsize is used for solving quadratic and general unconstrained optimization problems more efficiently.
The RBB stepsize can overcome the instability of the BB stepsize in many ill-conditioned optimization problems and is more robust than the BB stepsize in numerical experiments.

Stats

The RBB stepsize satisfies the following properties:
0 < λ1 ≤ αRBB
k ≤ λn
The reciprocal of the RBB stepsize is the close solution to a ℓ2-regularized least squares problem.

Quotes

"The RBB stepsize includes a class of valid stepsizes, such as another version of Barzilai-Borwein stepsize."
"RBB stepsize could overcome the instability of BB stepsize in many ill-conditioned optimization problems."
"RBB stepsize is more robust than BB stepsize in numerical experiments."

Key Insights Distilled From

by Congpei An,X... at **arxiv.org** 04-17-2024

Deeper Inquiries

The Regularized Barzilai-Borwein (RBB) method can be extended to solve constrained optimization problems by incorporating constraints into the optimization process. One common approach is to use a constrained optimization algorithm, such as the projected gradient method or the augmented Lagrangian method, in conjunction with the RBB method.
In the projected gradient method, the RBB step is taken in the feasible direction by projecting the update onto the feasible set at each iteration. This ensures that the iterates generated by the RBB method satisfy the constraints imposed on the optimization problem.
Alternatively, the augmented Lagrangian method can be used to handle constraints by introducing penalty terms into the objective function. The RBB method can then be applied to solve the augmented Lagrangian problem, where the regularization parameter helps control the trade-off between the original objective and the penalty terms associated with the constraints.
Overall, by integrating the RBB method with constraint-handling techniques, it is possible to effectively solve constrained optimization problems while benefiting from the efficiency and stability of the RBB method.

The theoretical convergence guarantees of the Regularized Barzilai-Borwein (RBB) method for non-convex optimization problems can be analyzed using appropriate convergence criteria and mathematical proofs.
For non-convex optimization problems, the RBB method may not converge to a global minimum, but rather to a local minimum or a stationary point. The convergence analysis typically involves showing that the sequence of iterates generated by the RBB method converges to a critical point of the objective function.
To establish convergence guarantees for non-convex problems, one approach is to analyze the properties of the objective function, such as the existence of stationary points, saddle points, and local minima. By considering the behavior of the RBB method in the vicinity of these critical points, one can derive convergence results under certain conditions.
Additionally, techniques such as Lyapunov functions, subsequence analysis, and convergence rate analysis can be employed to prove convergence properties of the RBB method for non-convex optimization problems. These theoretical guarantees provide insights into the behavior of the algorithm and its ability to find satisfactory solutions in non-convex optimization scenarios.

Adapting the Regularized Barzilai-Borwein (RBB) method to leverage parallel and distributed computing architectures can significantly enhance its scalability and efficiency for large-scale optimization tasks. Here are some strategies to achieve this adaptation:
Parallelization of Iterations: Divide the iterations of the RBB method into independent tasks that can be executed concurrently on multiple processors or nodes. This parallelization can reduce the overall computation time and speed up the optimization process.
Distributed Data Processing: Distribute the data involved in the optimization problem across multiple nodes in a distributed computing environment. Each node can perform computations on its subset of data and communicate with other nodes to exchange information and update the optimization variables.
Asynchronous Updates: Implement asynchronous updates in the RBB method to allow different nodes to update their variables independently and asynchronously. This approach can improve the convergence speed and efficiency of the optimization algorithm in a distributed setting.
Communication Optimization: Optimize the communication overhead between nodes by minimizing the data transfer and synchronization requirements. Techniques such as batching updates, compressing data, and reducing network latency can help improve the performance of the RBB method in distributed environments.
By incorporating these strategies, the RBB method can effectively harness the computational power of parallel and distributed computing architectures to tackle large-scale optimization problems efficiently and effectively.

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