L-BFGS is a powerful optimization algorithm that accelerates the training process in machine learning applications, particularly for large-scale problems.

Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) is an optimization algorithm widely used in machine learning for solving large-scale problems. It is a quasi-Newton method that approximates the second-order information of the objective function, making it efficient for handling ill-conditioned optimization problems. L-BFGS has been successfully applied to various applications, including tensor decomposition, nonsmooth optimization, and neural network training.

Recent research has focused on improving the performance of L-BFGS in different scenarios. For example, nonlinear preconditioning has been used to accelerate alternating least squares (ALS) methods for tensor decomposition. In nonsmooth optimization, L-BFGS has been compared to full BFGS and other methods, showing that it often performs better when applied to smooth approximations of nonsmooth problems. Asynchronous parallel algorithms have also been developed for stochastic quasi-Newton methods, providing significant speedup and better performance than first-order methods in solving ill-conditioned problems.

Some practical applications of L-BFGS include:

1. Tensor decomposition: L-BFGS has been used to accelerate ALS-type methods for canonical polyadic (CP) and Tucker tensor decompositions, offering substantial improvements in terms of time-to-solution and robustness over state-of-the-art methods.

2. Nonsmooth optimization: L-BFGS has been applied to Nesterov's smooth approximation of nonsmooth functions, demonstrating efficiency in dealing with ill-conditioned problems.

3. Neural network training: L-BFGS has been combined with progressive batching, stochastic line search, and stable quasi-Newton updating to perform well on training logistic regression and deep neural networks.

One company case study involves the use of L-BFGS in large-scale machine learning applications. By adopting a progressive batching approach, the company was able to improve the performance of L-BFGS in training logistic regression and deep neural networks, providing better generalization properties and faster algorithms.

In conclusion, L-BFGS is a versatile and efficient optimization algorithm that has been successfully applied to various machine learning problems. Its ability to handle large-scale and ill-conditioned problems makes it a valuable tool for developers and researchers in the field. As research continues to explore new ways to improve L-BFGS performance, its applications and impact on machine learning are expected to grow.

# L-BFGS

## L-BFGS Further Reading

1.Nonlinearly Preconditioned L-BFGS as an Acceleration Mechanism for Alternating Least Squares, with Application to Tensor Decomposition http://arxiv.org/abs/1803.08849v2 Hans De Sterck, Alexander J. M. Howse2.Behavior of Limited Memory BFGS when Applied to Nonsmooth Functions and their Nesterov Smoothings http://arxiv.org/abs/2006.11336v1 Azam Asl, Michael L. Overton3.Asynchronous Parallel Stochastic Quasi-Newton Methods http://arxiv.org/abs/2011.00667v1 Qianqian Tong, Guannan Liang, Xingyu Cai, Chunjiang Zhu, Jinbo Bi4.On the Acceleration of L-BFGS with Second-Order Information and Stochastic Batches http://arxiv.org/abs/1807.05328v1 Jie Liu, Yu Rong, Martin Takac, Junzhou Huang5.LM-CMA: an Alternative to L-BFGS for Large Scale Black-box Optimization http://arxiv.org/abs/1511.00221v1 Ilya Loshchilov6.Inappropriate use of L-BFGS, Illustrated on frame field design http://arxiv.org/abs/1508.02826v1 Nicolas Ray, Dmitry Sokolov7.A Progressive Batching L-BFGS Method for Machine Learning http://arxiv.org/abs/1802.05374v2 Raghu Bollapragada, Dheevatsa Mudigere, Jorge Nocedal, Hao-Jun Michael Shi, Ping Tak Peter Tang8.An Adaptive Memory Multi-Batch L-BFGS Algorithm for Neural Network Training http://arxiv.org/abs/2012.07434v1 Federico Zocco, Seán McLoone9.Shifted L-BFGS Systems http://arxiv.org/abs/1209.5141v2 Jennifer B. Erway, Vibhor Jain, Roummel F. Marcia10.Fast B-spline Curve Fitting by L-BFGS http://arxiv.org/abs/1201.0070v1 Wenni Zheng, Pengbo Bo, Yang Liu, Wenping Wang## L-BFGS Frequently Asked Questions

## What is the L-BFGS optimization procedure?

The L-BFGS optimization procedure is an iterative method used to find the minimum of a function, typically in the context of machine learning applications. It is a quasi-Newton method that approximates the second-order information of the objective function, making it efficient for handling large-scale and ill-conditioned optimization problems. The procedure involves updating an approximation of the Hessian matrix (the matrix of second-order partial derivatives) using a limited amount of memory, which allows it to scale well for large problems.

## What is the difference between BFGS and L-BFGS?

BFGS (Broyden-Fletcher-Goldfarb-Shanno) and L-BFGS (Limited-memory BFGS) are both quasi-Newton optimization methods. The main difference between them lies in their memory requirements. BFGS requires storing and updating a full Hessian matrix, which can be computationally expensive for large-scale problems. L-BFGS, on the other hand, uses a limited amount of memory to approximate the Hessian matrix, making it more suitable for large-scale optimization problems. This reduced memory requirement allows L-BFGS to be more efficient and scalable compared to the full BFGS method.

## What is the full form of L-BFGS?

L-BFGS stands for Limited-memory Broyden-Fletcher-Goldfarb-Shanno. It is an optimization algorithm widely used in machine learning for solving large-scale problems.

## What is L-BFGS in ML?

In machine learning (ML), L-BFGS is an optimization algorithm used to train models by minimizing a loss function. It is particularly useful for large-scale problems due to its efficient memory usage and ability to handle ill-conditioned optimization problems. L-BFGS has been successfully applied to various ML applications, including tensor decomposition, nonsmooth optimization, and neural network training.

## How does L-BFGS handle large-scale problems?

L-BFGS handles large-scale problems by using a limited amount of memory to approximate the Hessian matrix, which is the matrix of second-order partial derivatives of the objective function. This approximation allows L-BFGS to be more efficient and scalable compared to methods that require storing and updating a full Hessian matrix, such as the full BFGS method. As a result, L-BFGS is well-suited for large-scale optimization problems commonly encountered in machine learning applications.

## What are some practical applications of L-BFGS in machine learning?

Some practical applications of L-BFGS in machine learning include: 1. Tensor decomposition: L-BFGS has been used to accelerate alternating least squares (ALS) methods for canonical polyadic (CP) and Tucker tensor decompositions, offering substantial improvements in terms of time-to-solution and robustness over state-of-the-art methods. 2. Nonsmooth optimization: L-BFGS has been applied to Nesterov's smooth approximation of nonsmooth functions, demonstrating efficiency in dealing with ill-conditioned problems. 3. Neural network training: L-BFGS has been combined with progressive batching, stochastic line search, and stable quasi-Newton updating to perform well on training logistic regression and deep neural networks.

## What are the advantages of using L-BFGS in machine learning?

The advantages of using L-BFGS in machine learning include: 1. Scalability: L-BFGS is well-suited for large-scale optimization problems due to its efficient memory usage and ability to handle ill-conditioned problems. 2. Robustness: L-BFGS has been shown to be robust in various applications, including tensor decomposition and nonsmooth optimization. 3. Performance: L-BFGS often outperforms first-order methods and other optimization algorithms in terms of convergence speed and solution quality, especially for ill-conditioned problems. 4. Versatility: L-BFGS can be applied to a wide range of machine learning problems, making it a valuable tool for developers and researchers in the field.

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