the-problem-in-general.md

🔎 The Problem in General

Given a dataset like:

$$[(x^{1},y^{1}), (x^{2},y^{2}), ...., (x^{m},y^{m})]$$

We want:

$$\hat{y}^{(i)} \approx y^{(i)}$$

📚 Basic Concepts and Notations

Concept	Description
m	Number of examples in dataset
$$x^{(i)}$$	ith example in the dataset
ŷ	Predicted output
Loss Function 𝓛(ŷ, y)	A function to compute the error for a single training example
Cost Function 𝙹(w, b)	The average of the loss functions of the entire training set
Convex Function	A function that has one local value
Non-Convex Function	A function that has lots of different local values
Gradient Descent	An iterative optimization method that we use to converge to the global optimum of Cost Function

In other words: The Cost Function measures how well our parameters w and b are doing on the training set, so the best w and b are the values that minimize 𝙹(w, b) as possible

📉 Gradient Descent

General Formula:

$$w:=w-\alpha\frac{dJ(w,b)}{dw}$$

$$b:=b-\alpha\frac{dJ(w,b)}{dw}$$

α (alpha) is the Learning Rate

🥽 Learning Rate

It is a positive scalar determining the size of the step of each iteration of gradient descent due to the corresponded estimated error each time the model weights are updated, so, it controls how quickly or slowly a neural network model learns a problem.

🎀 Good Learning Rate

💢 Bad Learning Rate

🧐 References

• Introduction to Artificial Neural Networks (ANN)
• More on Learning Rate