Gradient descent

In mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. It is particularly useful in machine learning for minimizing the cost or loss function. Gradient descent should not be confused with local search algorithms, although both are iterative methods for optimization. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. A simple extension of gradient descent, stochastic gradient descent, serves as the most basic algorithm used for training most deep networks today. Gradient descent is based on the observation that if the multi-variable function is defined and differentiable in a neighborhood of a point , then decreases fastest if one goes from in the direction of the negative gradient of at . It follows that, if for a small enough step size or learning rate , then . In other words, the term is subtracted from because we want to move against the gradient, toward the local minimum. With this observation in mind, one starts with a guess for a local minimum of , and considers the sequence such that We have a monotonic sequence so, hopefully, the sequence converges to the desired local minimum. Note that the value of the step size is allowed to change at every iteration. With certain assumptions on the function (for example, convex and Lipschitz) and particular choices of (e.