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The secret behind random search: Why is this method so powerful?
In the field of numerical optimization, Random Search (RS) is a method that has received widespread attention. What is special about this method is that it does not require the gradient of the problem to be optimized, which means that RS can still work effectively even on discontinuous or non-differentiable functions. This type of optimization method is called direct search, derivative-free, or black-box. The power of random search comes from its application in a range of scenarios that do not require complex calculations, making the optimization process more flexible and robust.
The power of random search methods lies in their ability to explore the unknown and show amazing results in a variety of environments.
But how exactly does random search work? As early as 1953, Anderson evaluated methods for finding maximum or minimum values of problems in his review article and described a series of guesses based on a certain order or pattern. In this process, these guesses are stepped through the search space and better guesses are continually refined. The search can be performed via a grid search (full factorial design), a sequential search, or a combination of both. These methods were initially used mainly for screening experimental conditions for chemical reactions and were therefore widely adopted by scientists.
In contemporary applications, random search methods are widely used for hyperparameter optimization of artificial neural networks. The study found that when only 5% of the volume of the search space has good properties, this means that the probability of finding a good configuration is still around 5%. However, after 60 configuration attempts, the probability of finding at least one good configuration is over 95%. This combination greatly improves the search success rate, demonstrating the effectiveness and potential of RS.
After 60 configuration attempts, the probability of finding at least one good configuration is over 95%, making this approach well worth exploring.
Basic Algorithm
The basic process of the random search algorithm is simple and clear. Assume there is a fitness or cost function f: ℝn → ℝ that needs to be minimized, and x ∈ ℝn represents a position or candidate solution in the search space. The basic random search algorithm can be described as follows:
- Randomly initialize the position of x in the search space.
- Until the termination condition is met (such as the number of iterations executed or the fitness reaches a standard), repeat the following operations:
- Sample a new position y from a hypersphere of given radius about the current position x.
- If f(y) < f(x), then move to the new position by setting x = y.
Variants
True random search tends to rely on luck, which can range from very expensive to very lucky, but structured random search is strategic. As the literature has evolved, many variations of random search have emerged, using structured sampling to perform searches:
- Friedman-Savage procedure: sequentially searches each parameter through a set of spatially patterned guesses.
- Fixed-step random search (FSSRS): Sampling is performed within a hypersphere of fixed radius.
- Optimal Step Size Random Search (OSSRS): Theoretically studies how to optimize the radius of the hypersphere to accelerate the convergence to the optimal solution.
- Adaptive Step-Size Random Search (ASSRS): Automatically adjusts the radius by generating two candidate solutions.
- Optimized Relative Step-Size Random Search (ORSSRS): Approximates the optimal step size using a simple exponential decrease.
These variants make the application of random search more diverse and sophisticated, and can better address different optimization challenges.
Various variations of random search demonstrate its flexibility and power in different situations.
In any case, random search is indeed an important method that demonstrates its unique advantages in a series of optimization problems. It is not only attractive in theory, but also demonstrates remarkable effects in practical applications. Random search may become a key component of future optimization methods, especially when computational resources are too demanding or the problem complexity is too large. So, faced with such a variety of optimization strategies, can we find the most appropriate search method to meet future challenges?
