I am having trouble understanding EM gradient algorithm can someone help? I understand the formula for the EM gradient algorithm but I don't understand what it means by taking one step of Newton's method. Does it mean that it just take one iteration to converge to the conclusion? 
 A: Taking one step in Newton's method means
1) Calculating the gradient $\nabla f$ and hessian $\nabla^2 f$ of the function $f$ being maximized at the current point $x_i$
2) Applying the update rule $x_{i+1} = x_i - (\nabla^2 f(x_i))^{-1} \nabla f (x_i)$ to get a new point.
You stepped from $x_i$ to $x_{i+1}$.
Generally no, you are not done after one step.  You need to check to see if the change in the value of $f$, from $f(x_i)$ to $f(x_{i+1})$, is smaller than your convergence threshold, then you stop.
You are only guaranteed to stop in one step if $f$ is a quadratic function.

Can xi be any arbitrary value? If the f(xi+1) does not meet the convergence threshold, what shall we do?

You do it over again.
In detail.  To begin you choose a point $x_0$ by any means, it could be randomly, you could start at $0$, really anything.  Then you compute $\nabla f (x_0)$ and $\nabla^2 f (x_0)$, and take a step:
$$x_{1} = x_0 - (\nabla^2 f(x_0))^{-1} \nabla f (x_0)$$
No we are at $x_1$, if $| f(x_1) - f(x_0) | < \epsilon$, we stop and return $x_1$ as our maximum.  Otherwise, compute $\nabla f (x_1)$ and $\nabla^2 f (x_1)$, then take another step:
$$x_{2} = x_1 - (\nabla^2 f(x_1))^{-1} \nabla f (x_1)$$
Now we are at $x_2$, if $| f(x_2) - f(x_1) | < \epsilon$  we stop and return $x_2$ as our maximum.  Otherwise, compute $\nabla f (x_2)$ and $\nabla^2 f (x_2),  take another step...   
