Minimization: A Review of Recent Advances in Algorithms and Applications
Xin Liu, Yibing He, and Yufeng Wu
Abstract
Minimization is an important problem in many areas of scientific research, including machine learning, optimization, and computer vision. It involves finding the optimal solution to a problem by minimizing a given objective function. In this paper, we review recent advances in minimization algorithms and their applications. We consider different types of minimization algorithms, such as gradient descent, stochastic gradient descent, Newton’s method, conjugate gradient, and simulated annealing. We also discuss various applications of minimization in machine learning, optimization, and computer vision. We provide a brief overview of the state-of-the-art techniques in each of these areas and discuss the potential of minimization algorithms for solving challenging problems in the future.
Keywords: Minimization, Algorithms, Applications, Machine Learning, Optimization, Computer Vision
1. Introduction
Minimization is an essential problem in many areas of scientific research. It involves finding the optimal solution to a problem by minimizing a given objective function. Minimization algorithms are widely used in machine learning, optimization, and computer vision. In this paper, we review the recent advances in minimization algorithms and their applications.
2. Minimization Algorithms
Many different types of minimization algorithms have been developed over the years. In this section, we discuss some of the most popular algorithms.
2.1 Gradient Descent
Gradient descent is a first-order optimization algorithm that uses the gradient of the objective function to find the local minima. It is an iterative algorithm that starts from an initial guess and gradually moves towards the optimal solution by taking small steps in the direction of the negative gradient.
2.2 Stochastic Gradient Descent
Stochastic gradient descent (SGD) is a variant of gradient descent that uses randomly selected data points to update the parameters of the model. It is a more efficient algorithm than gradient descent since it converges faster, but it is more prone to overfitting.
2.3 Newton’s Method
Newton’s method is a second-order optimization algorithm that uses the Hessian matrix of the objective function to find the local minima. It is a more efficient algorithm than gradient descent since it converges faster, but it is more computationally expensive.
2.4 Conjugate Gradient
Conjugate gradient is an iterative algorithm that uses the conjugate directions to reach the optimal solution. It is a more efficient algorithm than gradient descent since it converges faster, but it is more computationally expensive.
2.5 Simulated Annealing
Simulated annealing is a heuristic optimization algorithm that uses a simulated physical process to find the optimal solution. It is a more robust algorithm than gradient descent since it is less prone to getting stuck in local minima, but it is more computationally expensive.
3. Applications of Minimization
Minimization algorithms have many applications in machine learning, optimization, and computer vision.
3.1 Machine Learning
In machine learning, minimization algorithms are used to train models for different tasks, such as classification, regression, and clustering. Gradient descent and its variants are the most widely used algorithms for training deep neural networks.
3.2 Optimization
In optimization, minimization algorithms are used to find the optimal solutions to various problems, such as linear programming and nonlinear programming. Gradient descent and its variants are the most widely used algorithms for solving these problems.
3.3 Computer Vision
In computer vision, minimization algorithms are used to solve various tasks, such as image segmentation and object detection. Gradient descent and its variants are the most widely used algorithms for these tasks.
4. Conclusion
In this paper, we reviewed recent advances in minimization algorithms and their applications. We discussed different types of algorithms, such as gradient descent, stochastic gradient descent, Newton’s method, conjugate gradient, and simulated annealing. We also discussed various applications of minimization in machine learning, optimization, and computer vision. We provided a brief overview of the state-of-the-art techniques in each of these areas and discussed the potential of minimization algorithms for solving challenging problems in the future.
References
Glorot, X., & Bengio, Y. (2010). Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics (pp. 249-256).
Nocedal, J., & Wright, S. J. (2006). Numerical optimization (Vol. 2). Springer Science & Business Media.
Ruder, S. (2016). An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747.
Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. science, 220(4598), 671-680.
Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems (pp. 1097-1105).