BFGS is a powerful optimization algorithm for solving unconstrained optimization problems in machine learning and other fields.

The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is a widely used optimization method for solving unconstrained optimization problems in various fields, including machine learning. It is a quasi-Newton method that iteratively updates an approximation of the Hessian matrix to find the optimal solution. BFGS has been proven to be globally convergent and superlinearly convergent under certain conditions, making it an attractive choice for many optimization tasks.

Recent research has focused on improving the BFGS algorithm in various ways. For example, a modified BFGS algorithm has been proposed that dynamically chooses the coefficient of the convex combination in each iteration, resulting in global convergence to a stationary point and superlinear convergence when the Hessian is strongly positive definite. Another development is the Block BFGS method, which updates the Hessian matrix in blocks and has been shown to converge globally and superlinearly under the same convexity assumptions as the standard BFGS.

In addition to these advancements, researchers have explored the performance of BFGS in the presence of noise and nonsmooth optimization problems. The Secant Penalized BFGS (SP-BFGS) method has been introduced to handle noisy gradient measurements by smoothly interpolating between updating the inverse Hessian approximation and not updating it. This approach allows for better resistance to the destructive effects of noise and can cope with negative curvature measurements. Furthermore, the Limited-Memory BFGS (L-BFGS) method has been analyzed for its behavior on nonsmooth convex functions, shedding light on its performance in such scenarios.

Practical applications of the BFGS algorithm can be found in various machine learning tasks, such as training neural networks, logistic regression, and support vector machines. One company that has successfully utilized BFGS is Google, which employed the L-BFGS algorithm to train large-scale deep neural networks for speech recognition.

In conclusion, the BFGS algorithm is a powerful and versatile optimization method that has been extensively researched and improved upon. Its ability to handle a wide range of optimization problems, including those with noise and nonsmooth functions, makes it an essential tool for machine learning practitioners and researchers alike.

# BFGS

## BFGS Further Reading

1.A Globally and Superlinearly Convergent Modified BFGS Algorithm for Unconstrained Optimization http://arxiv.org/abs/1212.5929v1 Yaguang Yang2.Block BFGS Methods http://arxiv.org/abs/1609.00318v3 Wenbo Gao, Donald Goldfarb3.Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood http://arxiv.org/abs/2202.10538v2 Qiujiang Jin, Alec Koppel, Ketan Rajawat, Aryan Mokhtari4.Rescaling nonsmooth optimization using BFGS and Shor updates http://arxiv.org/abs/1802.06453v1 Jiayi Guo, Adrian S. Lewis5.Secant Penalized BFGS: A Noise Robust Quasi-Newton Method Via Penalizing The Secant Condition http://arxiv.org/abs/2010.01275v2 Brian Irwin, Eldad Haber6.BV-Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of (W-) Strings http://arxiv.org/abs/hep-th/9512032v1 Peter Bouwknegt, Jim Mccarthy, Krzysztof Pilch7.A variational derivation of a class of BFGS-like methods http://arxiv.org/abs/1712.00680v3 Michele Pavon8.On the W-gravity spectrum and its G-structure http://arxiv.org/abs/hep-th/9311137v2 P. Bouwknegt, J. Mccarthy, K. Pilch9.Analysis of the BFGS Method with Errors http://arxiv.org/abs/1901.09063v1 Yuchen Xie, Richard Byrd, Jorge Nocedal10.Analysis of Limited-Memory BFGS on a Class of Nonsmooth Convex Functions http://arxiv.org/abs/1810.00292v2 Azam Asl, Michael L. Overton## BFGS Frequently Asked Questions

## What is the BFGS algorithm?

The Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm is a widely used optimization method for solving unconstrained optimization problems in various fields, including machine learning. It is a quasi-Newton method that iteratively updates an approximation of the Hessian matrix to find the optimal solution. BFGS has been proven to be globally convergent and superlinearly convergent under certain conditions, making it an attractive choice for many optimization tasks.

## What is the difference between BFGS and Newton's method?

Newton's method is an optimization algorithm that uses the second-order derivative information (the Hessian matrix) to find the optimal solution. However, computing the Hessian matrix can be computationally expensive, especially for high-dimensional problems. BFGS is a quasi-Newton method that approximates the Hessian matrix using gradient information, making it more computationally efficient than Newton's method while still maintaining good convergence properties.

## What are the disadvantages of BFGS?

Some disadvantages of the BFGS algorithm include: 1. Memory requirements: BFGS requires storing and updating the full Hessian matrix approximation, which can be memory-intensive for large-scale problems. 2. Sensitivity to noise: BFGS can be sensitive to noise in the gradient information, which may lead to poor convergence or divergence. 3. Limited applicability: BFGS is designed for unconstrained optimization problems and may not be directly applicable to constrained optimization problems without modifications.

## What are the benefits of BFGS?

The benefits of the BFGS algorithm include: 1. Superlinear convergence: BFGS has been proven to converge superlinearly under certain conditions, making it an efficient optimization method. 2. Lower computational cost: BFGS approximates the Hessian matrix using gradient information, reducing the computational cost compared to methods that require the exact Hessian matrix, such as Newton's method. 3. Versatility: BFGS can be applied to a wide range of optimization problems, including those with noise and nonsmooth functions, making it a valuable tool for machine learning practitioners and researchers.

## How is the Limited-Memory BFGS (L-BFGS) different from the standard BFGS?

The Limited-Memory BFGS (L-BFGS) is a variant of the BFGS algorithm that addresses the memory requirements of the standard BFGS. Instead of storing the full Hessian matrix approximation, L-BFGS maintains a limited number of past gradient updates to approximate the Hessian matrix. This approach significantly reduces the memory requirements, making L-BFGS more suitable for large-scale optimization problems.

## In what machine learning applications is BFGS commonly used?

BFGS is commonly used in various machine learning tasks, such as training neural networks, logistic regression, and support vector machines. For example, Google employed the L-BFGS algorithm to train large-scale deep neural networks for speech recognition.

## How has recent research improved the BFGS algorithm?

Recent research has focused on improving the BFGS algorithm in various ways, such as modifying the algorithm to dynamically choose the coefficient of the convex combination in each iteration, resulting in global convergence to a stationary point and superlinear convergence when the Hessian is strongly positive definite. Other developments include the Block BFGS method, which updates the Hessian matrix in blocks, and the Secant Penalized BFGS (SP-BFGS) method, which handles noisy gradient measurements by smoothly interpolating between updating the inverse Hessian approximation and not updating it.

## Explore More Machine Learning Terms & Concepts