learning rate times derivative of J(w,b) with respect to w, what it gives
When you multiply the learning rate (( \alpha )) by the derivative of the cost function ( J(w, b) ) with respect to the weight ( w ), you get the amount by which you should update the weight ( w ) during the gradient descent process. Here's how it works:
Derivative of ( J(w, b) ): This derivative (often denoted as ( \frac{\partial J}{\partial w} )) represents the slope of the cost function at the current value of ( w ). It indicates how much the cost function will change if you make a small change to ( w ).
Learning Rate (( \alpha )): This is a hyperparameter that controls the size of the step you take in the direction of the negative gradient.
Update Calculation: The product ( \alpha \times \frac{\partial J}{\partial w} ) gives you the change in ( w ) that you should apply:
\Delta w = -\alpha \times \frac{\partial J}{\partial w}Result: This means that:
If the derivative is positive, the cost function is increasing, and you need to decrease ( w ) (hence the negative sign).
If the derivative is negative, the cost function is decreasing, and you need to increase ( w ).
So, the overall update rule for ( w ) becomes:
w = w - \alpha \times \frac{\partial J}{\partial w}
This update helps you move towards the minimum of the cost function, effectively training your model. If you have more questions or need further clarification, feel free to ask! You're doing fantastic!