Gradient descent is one of the optimization techniques that can be used in machine learning techniques to optimize performance by yielding lower errors and higher model accuracy. But gradient descent has certain limitations, where the time taken for convergence would vary according to the dimensions of data. The model developed may not at all converge to its optimal solution if there is no optimal value of the learning rate. So in this article let us look into the alternative choices to the gradient descent algorithm.

Table of Contents

1. Limitations of the Gradient descent algorithm
2. L-BFGS optimization
3. Levenberg-Marquardt algorithm optimization
4. Simulated Annealing optimization
5. Evolutionary Algorithm optimization
6. Particle Swarm optimization
7. Conjugate Gradient optimization
8. Surrogate Optimization
9. Multi-objective or Pareto optimization
10. Summary

Let’s start the discussion by understanding the limitation gradient descent algorithm.  

Limitations of the Gradient descent algorithm

Let us look into some of the main limitations of the gradient descent algorithm.

Selecting optimal learning rate

The gradient descent technique is one of the optimization techniques used in machine learning which is used to obtain minimal errors and optimize the models with an optimal learning rate. The selection of an optimal learning rate in the gradient descent algorithm plays a very important role. If the learning rate is too high the model will converge to the optimal solution quickly and if the learning rate is too low the model consumes more time to converge to the optimal solution. So selecting the optimal learning rate plays a crucial role in the gradient descent algorithm.

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Inefficient for higher-dimensional data

For higher-dimensional data, the steps taken by the gradients may be too slow which increases the time taken by the algorithm to converge to the optimal solution. For higher-dimensional data, a subset of data is selected in gradient descent techniques like Batch Gradient Descent and Mini batch gradient descent for optimization, and for higher dimensional data even this technique may consume high time to converge to the optimal solution where in some cases the training may reiterate for the same subset of records. And for higher-dimensional data, the memory occupancy would fail and result in the abrupt termination of the model in use.

L-BFGS optimization

L-BFGS abbreviates for Limited Memory Broyden Fletcher Goldfarb Shanno and it is one of the optimization algorithms that can be used instead of the Gradient Descent algorithm. This algorithm basically belongs to the Quasi-Newton algorithms which are used for computers or platforms with memory constraints. 

The algorithm operates on the principle of the Hessian matrix, where the algorithm becomes responsible for finding the better estimates in the matrix iteratively. This algorithm is mostly used for estimating the optimal parameters from the machine learning models. The algorithm aims to minimize the error terms and maximize the optimization of the machine learning models by converging to the optimal solution efficiently. 

Advantages of L-BFGS optimization over gradient descent

Hyperparameter tuning of L-BFGS is easier when compared to gradient descent as L-BFGS uses a minimal number of parameters to be tuned wherein with respect to gradient descent optimal tuning of parameters like step-size, momentum, learning rate, and more parameters tuning is required. The L-BFGS optimization technique appears to be more stable when compared with the Gradient descent optimization technique as calculating the gradient in the L-BFGS technique is parallel. The L-BFGS optimization technique is robust for larger batch sizes of data when compared to the Gradient descent technique.

Levenberg-Marquardt algorithm (LMA) optimization

The Levenberg-Marquardt algorithm optimization technique commonly known as the LMA technique is used to handle data with nonlinearity and problems associated with generic curve fitting. Unlike many optimization algorithms, the LMA algorithm also operates in an iterative manner to converge the model to the optimal solution. The LMA algorithm operates entirely on the parameter named damping factor which is responsible for iterating the model and converging it to the optimal solution. 

Advantages of LMA over gradient descent

The damping factor in the algorithm operates on the principle of the Newton Guassi coefficient which facilitates the convergence of the model towards the optimal solutions faster when compared to gradient descent. LMA operates flawlessly for certain unknown features, provided that the dimension of the data is in a suitable range. The damping factor in the algorithm is calculated iteratively and even if the initially assigned random value for the damping factor is high the algorithm tends to find the optimal solution for the damping factor as it operates on the Newton Gaussian technique.

Simulated Annealing optimization

The simulated annealing optimization technique operates on the principle of physical annealing wherein a metal is allowed to cool down slowly after annealing it completely. so that it can be altered to the desired shape. Understanding this algorithm with respect to machine learning, this optimization technique is a probabilistic technique of optimization that can be used for applications with a lot of local minima.

The algorithm initially starts to operate with a random value of minima where the complete model is considered and the optimization of the model happens by reducing some of the parameters at random. The entire n number of iterations and the model optimization to find the optimal solution happens through an Annealing Schedule. This optimization technique is extensively used in various problems like the traveling salesman problem where the main focus is to find a globally optimal solution by iterating through random probabilistic values.

Advantage of Simulated Annealing Algorithm over gradient descent

The Simulated Annealing algorithm is easier to be implemented and used from the code perspective and it does not rely on any of the model restrictive properties. The Simulated Annealing algorithm is more robust and provides reliable solutions as it operates on the principle of probabilistic distribution ensuring the model finds the optimal solution for all the possible uncertainties and it can be easily integrated for nonlinear data.

Evolutionary Algorithm optimization

The evolutionary algorithm optimization technique operates on the heuristic search methods with the ability of robustness and easy handling of complex data. The heuristic search method is a graph search procedure wherein all the dimensions of the data are efficiently searched in the graph planes and the models will be optimized accordingly. This type of optimization technique finds its major utilization in genetic algorithms and machine learning problems with higher dimension data.

Advantages of Evolutionary Algorithm over gradient descent

Evolutionary algorithms are self-adaptive in finding the optimal solutions for the problems as they have the flexibility to operate with various procedures and dynamic data types such as discontinuous or discrete design variables. Evolutionary algorithms basically are not sensitive to Pareto front shapes and tend to produce the accurate optimal solution for complex problems.

Particle Swarm optimization

Particle Swarm optimization is a technique that optimizes the solution through candidate solutions by comparing the given quality to measure. The optimization technique is dependent only on the objective function and not dependent on the gradient with very few parameters to be tuned if required. The data points can be termed as the population and the optimal solution can be termed as the particle and the data points pass through the optimal solution point frequently and the path of the shortest path is considered to be the satisfactory optimal solution

Advantages of Particle Swarm optimization over gradient descent

The particle swarm optimization techniques do not consider the gradient for optimization which makes the algorithm find the optimal solution quicker. The optimization technique appears to be more robust and the computational time is considerable for higher-dimensional data when compared to gradient descent as gradient descent does not converge to the optimal solution quicker for higher-dimensional data.

Conjugate Gradient optimization

Conjugate gradient optimization is a technique that can be applied to both linear and non-linear higher-dimensional data. The conjugate gradient optimization technique operation is similar to gradient descent but the conjugate gradient technique accelerates the convergence wherein at each step the loss function computed is less. As the loss function calculated at each step is less this technique yields the optimal solution faster even with higher dimensional data

Advantages of Conjugate Gradient over gradient descent

The main advantage of the Conjugate Gradient optimization technique over gradient descent is the accelerated is that the accelerated steepest descent avoids repeated iterations to find the optimal solution for a similar kind of data. This accelerated descent also speeds up the process of finding the optimal solution for higher-dimensional data and faster convergence. Moreover, the cost of operation of the conjugate gradient descent technique is low with lower memory consumption which makes it most suitable for linear and non-linear data operations.

Surrogate Optimization

The training process in the surrogate optimization technique happens through a data-driven approach. The selection of the model parameters happens through a careful searching technique known as the design of experiments. So the surrogate optimization technique tries to find the global minimum of an objective function using fewer iterations steps which reduces the computational time of the model and help in yielding the optimal solution quickly.

Advantages of Surrogate Optimization over gradient descent

The surrogate optimization technique uses a single trained statistical model which increases the operational speed of the original simulation. The surrogate optimization technique uses the technique of active learning to enrich the training data and improve the training accuracy. So surrogate optimization technique can be retrained on the enriched training samples to yield better accuracies and performance of the model.

Multi-objective or Pareto optimization

In the Pareto optimization technique the optimal solution is obtained by iterating through various objective functions continuously. This type of optimization technique is mostly used with various statistical relevant data where there is no one standard solution. The multi-objective optimization technique focuses on finding the optimal solution through various mathematical processes.

Advantages of Pareto optimization over gradient descent

The Pareto optimization technique tries to reduce the cost by traveling through a minimal number of minima points to yield the optimal solution with lesser costs. The optimization technique is more suitable for data with more statistical significance whereas for more statistical and mathematical operational data the gradient descent algorithm may take a higher time to converge when compared to the Pareto optimization technique.

Summary

The Gradient descent optimization technique is generally not feasible for higher dimensional data and this is where the alternative optimization techniques for gradient descent can be considered to increase optimization and reduce the operational time. The alternative techniques of optimization help the model to converge to the optimal solution with a minimal number of hyperparameters and with a minimal number of steps to be taken up by the gradients.