Mathematics for Machine Learning with Python- CST284 KTU Minor - Dr. Binu V P -9847390760

Computing the gradient can be very time consuming. However, often it is possible to find a “cheap” approximation of the gradient. Approximating the gradient is still useful as long as it points in roughly the same direction as the true gradient.  Stochastic gradient descent (often shortened as SGD) is a stochastic approximation of the gradient descent method for minimizing an objective function that is written as a sum of differentiable functions. The word stochastic here refers to the fact that we acknowledge that we do not know the gradient precisely, but  instead only know a noisy approximation to it. By constraining the probability distribution of the approximate gradients, we can still theoretically guarantee that SGD will converge. In machine learning, given $n = 1,\ldots,N$ data points, we often consider objective functions that are the sum of the losses $L(\theta)$ incurred by each example $n$. In mathematical notation, we have the form $L(\theta)=\sum_{n=1}^N L_n(\theta)$ 

The convergence of gradient descent may be very slow if the curvature of the optimization surface is such that there are regions that are poorly scaled. The curvature is such that the gradient descent steps hops between the walls of the valley and approaches the optimum in small steps. The proposed tweak to improve convergence is to give gradient descent some memory. Gradient descent with momentum (Rumelhart et al., 1986) is a method that introduces an additional term to remember what happened in the previous iteration. This memory dampens oscillations and smoothes out the gradient updates. Continuing the ball analogy, the momentum term emulates the phenomenon of a heavy ball that is reluctant to change directions.The idea is to have a gradient update with memory to implement a moving average. The momentum-based method remembers the update $\bigtriangledown x_i$ at each iteration $i$ and determines the next update as a linear combination of the current and previous gradients $x_{i+1}

Since machine learning algorithms are implemented on a computer, the mathematical formulations are expressed as numerical optimization methods.Training a machine learning model often boils down to finding a good set of parameters. The notion of “good” is determined by the objective function or the probabilistic model. Given an objective function, finding the best value is done using optimization algorithms. There are two main branches of continuous optimization constrained and unconstrained. By convention, most objective functions in machine learning are intended to be minimized, that is, the best value is the minimum value. Intuitively finding the best value is like finding the valleys of the objective function, and the gradients point us uphill. The idea is to move downhill (opposite to the gradient) and hope to find the deepest point. For unconstrained optimization, this is the only concept we need,but there are several design choices. For constrained optimization, we need to intr