MullOverThings

Useful tips for everyday

# How do you find direction of descent?

## How do you find direction of descent?

A vector ∈ Rn is called a descending direction for f : Rn → R if for all x ∈ Rn we have that ∇f(x)T d < 0. Proposition 1. For a given function f, if its Hessian Hf is positive definite then d = −Hf (xk)−1∇f(xk) is a descent direction.

Gradient descent subtracts the step size from the current value of intercept to get the new value of intercept. This step size is calculated by multiplying the derivative which is -5.7 here to a small number called the learning rate. Usually, we take the value of the learning rate to be 0.1, 0.01 or 0.001.

## What is step size in gradient descent?

Gradient descent can’t tell whether a minimum it has found is local or global. The step size α controls whether the algorithm converges to a minimum quickly or slowly, or whether it diverges. Many real world problems come down to minimizing a function.

## What is steepest descent direction?

The steepest descent method can converge to a local maximum point starting from a point where the gradient of the function is nonzero. 3. Steepest descent directions are orthogonal to each other. 4. Steepest descent direction is orthogonal to the cost surface.

## Is Newton Raphson gradient descent?

Gradient descent maximizes a function using knowledge of its derivative. Newton’s method, a root finding algorithm, maximizes a function using knowledge of its second derivative. That can be faster when the second derivative is known and easy to compute (the Newton-Raphson algorithm is used in logistic regression).

Gradient descent is best used when the parameters cannot be calculated analytically (e.g. using linear algebra) and must be searched for by an optimization algorithm.

Suppose a differentiable, convex function F(x) exists. Then b=a−γ∇F(a) implies that F(b)≤F(a) given γ is chosen properly. The goal is to find the optimal γ at each step.

## How is the step size of gradient descent calculated?

Gradient descent subtracts the step size from the current value of intercept to get the new value of intercept. This step size is calculated by multiplying the derivative which is -5.7 here to a small number called the learning rate. Usually, we take the value of the learning rate to be 0.1, 0.01 or 0.001.

## What’s the difference between vanilla and gradient descent?

A learning rate is a number which stays constant and indicates how quickly you want to reach the minima. Since the learning rate is a hyper-parameter it needs to be chosen carefully. The difference between this method and Vanilla Gradient descent is that this technique considers the previous step before taking the next one.

## When do we stop finding new intercept values in gradient descent?

No, we continue to find new intercept values until the value of step tends to zero (less than 0.001) or even in some cases we predefine the number of steps that are to be taken. In practice, this number can go to 1000 or even greater. Now let us come to the real problem and see how gradient descent optimises slope and intercept simultaneously.

## How is gradient descent used in machine learning?

In machine learning, it refers to minimizing the cost or loss function J (w). Most machine learning algorithms depend on optimization techniques including neural networks, linear regression, k-nearest neighbours, and so on. The gradient descent method is one of the most commonly used optimization techniques when it comes to machine learning.