The No Free Lunch Theorem is often thrown around in the field of optimization and machine learning, often with little understanding of what it means or implies. The theorem states that all optimization algorithms perform equally well when their performance is averaged across all possible problems. It implies that there is no single best optimization algorithm. Because of the close relationship between optimization, search, and machine learning, it also implies that there is no single best machine learning algorithm for […]

Stochastic optimization refers to the use of randomness in the objective function or in the optimization algorithm. Challenging optimization algorithms, such as high-dimensional nonlinear objective problems, may contain multiple local optima in which deterministic optimization algorithms may get stuck. Stochastic optimization algorithms provide an alternative approach that permits less optimal local decisions to be made within the search procedure that may increase the probability of the procedure locating the global optima of the objective function. In this tutorial, you will […]

Regression models are fit on training data using linear regression and local search optimization algorithms. Models like linear regression and logistic regression are trained by least squares optimization, and this is the most efficient approach to finding coefficients that minimize error for these models. Nevertheless, it is possible to use alternate optimization algorithms to fit a regression model to a training dataset. This can be a useful exercise to learn more about how regression functions and the central nature of […]

Optimization involves finding the inputs to an objective function that result in the minimum or maximum output of the function. The open-source Python library for scientific computing called SciPy provides a suite of optimization algorithms. Many of the algorithms are used as a building block in other algorithms, most notably machine learning algorithms in the scikit-learn library. These optimization algorithms can be used directly in a standalone manner to optimize a function. Most notably, algorithms for local search and algorithms […]

Gradient descent is an optimization algorithm that follows the negative gradient of an objective function in order to locate the minimum of the function. A problem with gradient descent is that it can bounce around the search space on optimization problems that have large amounts of curvature or noisy gradients, and it can get stuck in flat spots in the search space that have no gradient. Momentum is an extension to the gradient descent optimization algorithm that allows the search […]

Optimization refers to finding the set of inputs to an objective function that results in the maximum or minimum output from the objective function. It is common to describe optimization problems in terms of local vs. global optimization. Similarly, it is also common to describe optimization algorithms or search algorithms in terms of local vs. global search. In this tutorial, you will discover the practical differences between local and global optimization. After completing this tutorial, you will know: Local optimization […]

The Nelder-Mead optimization algorithm is a widely used approach for non-differentiable objective functions. As such, it is generally referred to as a pattern search algorithm and is used as a local or global search procedure, challenging nonlinear and potentially noisy and multimodal function optimization problems. In this tutorial, you will discover the Nelder-Mead optimization algorithm. After completing this tutorial, you will know: The Nelder-Mead optimization algorithm is a type of pattern search that does not use function gradients. How to […]

Function optimization involves finding the input that results in the optimal value from an objective function. Optimization algorithms navigate the search space of input variables in order to locate the optima, and both the shape of the objective function and behavior of the algorithm in the search space are opaque on real-world problems. As such, it is common to study optimization algorithms using simple low-dimensional functions that can be easily visualized directly. Additionally, the samples in the input space of […]

Optimization is a field of mathematics concerned with finding a good or best solution among many candidates. It is an important foundational topic required in machine learning as most machine learning algorithms are fit on historical data using an optimization algorithm. Additionally, broader problems, such as model selection and hyperparameter tuning, can also be framed as an optimization problem. Although having some background in optimization is critical for machine learning practitioners, it can be a daunting topic given that it […]

Gradient descent is an optimization algorithm that follows the negative gradient of an objective function in order to locate the minimum of the function. A limitation of gradient descent is that a single step size (learning rate) is used for all input variables. Extensions to gradient descent like AdaGrad and RMSProp update the algorithm to use a separate step size for each input variable but may result in a step size that rapidly decreases to very small values. The Adaptive […]