TPE truncated polynomial expansion

TPE, or Truncated Polynomial Expansion, is a technique used in machine learning and optimization problems. It is a regression-based approach that combines polynomial regression with the concept of a surrogate model to approximate the behavior of a complex and computationally expensive objective function.

In many real-world applications, evaluating the objective function can be time-consuming or expensive, especially when it involves simulations or physical experiments. TPE offers a way to approximate this objective function using a polynomial regression model, which is much faster to evaluate.

Here's how TPE works in detail:

1. Sampling: TPE begins by sampling a set of candidate points from the input space. These candidate points represent different configurations or inputs for the objective function. The sampling can be done using various techniques such as Latin hypercube sampling or random sampling.
2. Evaluating the Objective Function: Each candidate point is then evaluated using the computationally expensive objective function. The objective function returns the corresponding output or result for each input configuration. These input-output pairs form the training data for the TPE model.
3. Building the Surrogate Model: TPE constructs a surrogate model using polynomial regression based on the training data. Polynomial regression fits a polynomial equation to the input-output pairs, modeling the relationship between the inputs and outputs. The degree of the polynomial determines the complexity of the model.
4. Pruning and Updating: TPE employs a pruning strategy to discard unpromising candidate points based on the surrogate model's predictions. The pruning process focuses on exploring regions of the input space that are likely to yield better results. The model is continuously updated as new data becomes available.
5. Iterative Improvement: TPE iteratively refines the surrogate model by repeating the sampling, evaluation, model building, and pruning steps. The objective is to progressively improve the surrogate model's accuracy and find the optimal or near-optimal solution to the problem.
6. Exploitation and Exploration: TPE balances exploitation and exploration during the iterative process. Exploitation involves focusing on promising regions of the input space based on the surrogate model's predictions, while exploration involves sampling candidate points from a wider range to avoid getting trapped in local optima.
7. Convergence and Solution: TPE continues the iterative process until convergence criteria are met. Convergence can be determined based on the number of iterations, model accuracy, or a predefined threshold. The final solution is the configuration or input point that yields the best predicted output value according to the surrogate model.

By using TPE, optimization problems with computationally expensive objective functions can be solved more efficiently. TPE reduces the number of evaluations of the expensive objective function by approximating its behavior with a surrogate model based on polynomial regression. This approach allows for faster exploration of the input space and finding near-optimal solutions in a more computationally efficient manner.