5.2 Gradient Descent with Momentum

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

$x_{i+1} = x_i -\gamma_i((\bigtriangledown f)(x_i))^T+\alpha \bigtriangleup x_i \\
\bigtriangleup x_i=x_{i}-x_{i-1}=\alpha \bigtriangleup x_{i-1}-\gamma_i((\bigtriangledown f)(x_{i-1}))^T$

where $\alpha \in [0, 1]$. Sometimes we will only know the gradient approximately.In such cases, the momentum term is useful since it averages out different noisy estimates of the gradient. One particularly useful way to obtain an approximate gradient is by using a stochastic approximation,

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Mathematics for Machine Learning- CST 284 - KTU Minor Notes - Dr Binu V P

Introduction About Me Syllabus Course Outcomes and Model Question Paper University Question Papers and Evaluation Scheme -Mathematics for Machine learning CST 284 KTU Overview of Machine Learning What is Machine Learning (video) Learn the Seven Steps in Machine Learning (video) Linear Algebra in Machine Learning Module I- Linear Algebra 1.Geometry of Linear Equations (video-Gilbert Strang) 2.Elimination with Matrices (video-Gilbert Strang) 3.Solving System of equations using Gauss Elimination Method 4.Row Echelon form and Reduced Row Echelon Form -Python Code 5.Solving system of equations Python code 6. Practice problems Gauss Elimination ( contact) 7.Finding Inverse using Gauss Jordan Elimination (video) 8.Finding Inverse using Gauss Jordan Elimination-Python code Vectors in Machine Learning- Basics 9.Vector spaces and sub spaces 10.Linear Independence 11.Linear Independence, Basis and Dimension (video) 12.Generating set basis and span 13.Rank of a Matrix 14.Linear Mapping...

4.3 Sum Rule, Product Rule, and Bayes’ Theorem

We think of probability theory as an extension to logical reasoning Probabilistic modeling provides a principled foundation for designing machine learning methods. Once we have defined probability distributions corresponding to the uncertainties of the data and our problem, it turns out that there are only two fundamental rules, the sum rule and the product rule. Let $p(x,y)$ is the joint distribution of the two random variables $x, y$. The distributions $p(x)$ and $p(y)$ are the corresponding marginal distributions, and $p(y |x)$ is the conditional distribution of $y$ given $x$. Sum Rule The addition rule states the probability of two events is the sum of the probability that either will happen minus the probability that both will happen. The addition rule is: $P(A∪B)=P(A)+P(B)−P(A∩B)$ Suppose $A$ and $B$ are disjoint, their intersection is empty. Then the probability of their intersection is zero. In symbols: $P(A∩B)=0$ The addition law then simplifies to: $P(...

5.1 Optimization using Gradient Descent

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...

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

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