# Updating weights in logistic regression using gradient descent?

The loss function for logistic regression is $$\sigma(z)$$=$$\frac{1}{1+e^{-z}}$$ and $$z=\theta^T.x_i$$ but in order to make our loss function convex we consider this alternative formula: $$-\frac{1}{n}\sum_{i=1}^n[y_i.log(h_\theta x_i) + (1-y_i)(1-h_\theta x_i)]$$

If we take derivative of this loss function w.r.t $$\theta_i$$ then we'd get something like: $$-\frac{1}{n}\sum_{i=1}^n (h_\theta(x) - y_i)*x_j$$

Now the gradient update step will be, $$\theta_i := \theta_i -\alpha.-\frac{1}{n}\sum_{i=1}^n (h_\theta(x) - y_i)*x_j$$

My doubt is what is $$h_\theta(x)$$ here? Is it sigmoid function i.e $$\frac{1}{1+e^{-z}}$$? Another doubt is what is $$x_j$$ term here refer to? Is it our input datapoint?

Is my understanding clear regarding the loss function above. If any mistakes please do correct me? I am following andrew-ng ML course.

My doubt is what is $$hθ(x)$$ here? Is it sigmoid function i.e $$\frac{1}{1+e^{-z}}$$?

$$h_{\theta}(x)$$ is your hypothesis function. Which is given as

$$h_{\theta}(x) = \frac{1}{1+e^{-\theta^{t}x}}$$

So if you sigmoid function $$\sigma(z)$$ is given as

$$\sigma(z) =\frac{1}{1+e^{-z}}$$

Another doubt is what is $$xj$$ term here refer to? Is it our input datapoint?

$$x_{j}$$ is one of the samples.