ARS (accelerated random search)

Accelerated Random Search (ARS) is an optimization algorithm that belongs to the family of derivative-free optimization methods. In contrast to derivative-based optimization methods, ARS does not use the gradient information of the objective function, making it suitable for optimizing non-differentiable or discontinuous functions.

ARS was first introduced by Andrew W. Moore and Leslie P. Kaelbling in a paper published in 1999. The algorithm is based on the idea of sampling the search space randomly and iteratively refining the search by selecting the best samples from previous iterations.

In this article, we will provide a detailed explanation of how ARS works and its advantages and disadvantages compared to other optimization algorithms.

The Algorithm

ARS consists of the following steps:

1. Initialization: Initialize a set of N random points in the search space. This set is called the basis.
2. Search direction: Choose a random direction in the search space.
3. Step size: Choose a step size along the chosen direction.
4. Sampling: Generate a new set of N+1 points by adding a new point to the basis along the chosen direction with the chosen step size.
5. Evaluation: Evaluate the objective function at each point in the new set.
6. Selection: Select the N best points from the new set as the basis for the next iteration.
7. Termination: Terminate the algorithm if a stopping criterion is met (e.g., the maximum number of iterations is reached or the objective function value is below a certain threshold).

These steps are repeated until the termination condition is met.

Advantages

One of the main advantages of ARS is that it does not require the gradient information of the objective function, making it suitable for optimizing non-differentiable or discontinuous functions. In addition, ARS is relatively easy to implement and requires only a few parameters to be tuned.

ARS is also a stochastic algorithm, which means that it can escape from local optima by exploring the search space randomly. This feature makes ARS suitable for optimizing complex and high-dimensional functions that may have multiple local optima.

Disadvantages

One of the main disadvantages of ARS is that it requires a large number of function evaluations to converge to a good solution. This is because ARS explores the search space randomly and may sample regions of the search space that are far from the optimal solution.

In addition, the convergence rate of ARS is generally slower than that of gradient-based optimization algorithms. This is because gradient-based methods can exploit the gradient information to move towards the optimal solution more efficiently.

Finally, ARS may suffer from a lack of diversity in the generated samples, which may result in premature convergence to a suboptimal solution. This issue can be partially mitigated by using a diverse set of initial points or by adding diversity-promoting mechanisms to the algorithm.

Extensions

Several extensions of ARS have been proposed to improve its performance or to address its limitations. Some of these extensions include:

• Adaptive step size: Instead of using a fixed step size, the step size is adjusted based on the past performance of the algorithm.
• Restart strategies: The algorithm is restarted periodically with a different set of initial points to promote diversity and escape from local optima.
• Constraint handling: The algorithm is modified to handle constraints on the search space or the objective function.
• Multi-objective optimization: The algorithm is extended to optimize multiple objectives simultaneously.

Conclusion

Accelerated Random Search (ARS) is a derivative-free optimization algorithm that is suitable for optimizing non-differentiable or discontinuous functions. ARS works by iteratively sampling the search space randomly and refining the search by selecting the best samples from previous iterations.

ARS has several advantages, such as its ability to escape from local optima and its ease of implementation. However, it also has several disadvantages, such as its slow convergence rate and the large number of function evaluations required to converge to a good solution.

Despite its limitations, ARS has been successfully applied to various optimization problems in different fields, such as robotics, control systems, and machine learning. Moreover, several extensions of ARS have been proposed to improve its performance or to address its limitations.