4/5/2023 0 Comments Spiral dynamics calculatorInstead of just one point moving in a spiral motion towards a fixed center, spiral dynamics optimization uses multiple points. If you examine the example carefully, you'll see the pattern. For dim = 3 you compute like so (omitting the theta): To spiral in higher dimensions you must compose multiple R matrices. This rotation matrix is called R12 in research literature, which means it spirals a point in dimensions 1 and 2. This is not at all obvious and can be considered a magic equation for spiral dynamics optimization. Given an angle theta and a shrink factor r, it is possible to compute a new location of a point at (x, y) by multiplying the point by the rotation matrix R given by: Large values of r, close to 1, don't move a point towards the center as quickly as smaller values of r such as 0.80. The shrink factor r controls how quickly the spiral moves towards the center. Large values of theta give a more boxy, rectangular shape. Small values of theta give a very smooth curve. The angle theta controls how curved the spiral is. The net effect is that the point spirals in towards the center. On each iteration, the point moves by an angle theta and then towards the center. The Rosenbrock function is challenging because it's easy to get close to the minimum but then the surface becomes very flat and it's difficult to make progress. The graph in Figure 2 shows the Rosenbrock function for dim = 2 where the minimum value is 0.0 at (1, 1). The Rosenbrock function can be defined for dimension = 2 or higher. The Rosenbrock function is a standard benchmark problem for optimization algorithms. The demo program has no significant dependencies so any relatively recent version of Python 3 will work fine. The demo program was developed on Windows 10 using the Anaconda 2020.02 64-bit distribution (which contains Python 3.7.6). To run the demo program, you must have Python installed on your machine. All normal error checking has been removed to keep the main ideas as clear as possible. The complete source code for the demo program is presented in this article and is also available in the accompanying file download. The demo program is implemented using Python but you should have no trouble refactoring to another language such as C# or JavaScript if you wish. This article assumes you have an intermediate or better familiarity with a C-family programming language. The demo program uses SDO to solve the Rosenbrock function in three dimensions. A good way to see where this article is headed is to take a look at the screenshot of a demo program in Figure 1. This article explains how to implement a geometry-inspired optimization technique called spiral dynamics optimization (SDO). Examples include bio-inspired optimization techniques such as genetic algorithms and particle swarm optimization, and geometry-inspired techniques such as Nelder-Mead (also known as simplex, or amoeba method). But there are dozens of alternative optimization techniques that don't use gradients. All SGD-based optimization algorithms use the Calculus derivative (gradient) of an error function. SGD has many variations including Adam (adaptive momentum estimation), Adagrad (adaptive gradient), and so on. Put another way, training a neural network is the process of using an optimization algorithm of some sort to find values for the weights and biases that minimize the error between the network's computed output values and the known correct output values from the training data.īy far the most common form of optimization for neural network training is stochastic gradient descent (SGD). Training a neural network is the process of finding good values for the network's weights and biases.
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