In this problem, we will try to understand the loss in Passive-Aggressive (PA) Perceptron algorithm. The passive-aggressive (PA) algorithm (without offset) responds to a labeled training example (x,y) by finding that minimizes /2||(k)||^2+Lossh(yx) where (k) is the current setting of the parameters prior to encountering (x,y) and Lossh(yx)=max{0,1yx} is the hinge loss. We could replace the loss function with something else (e.g., the zero-one loss). The form of the update is similar to the perceptron algorithm, i.e., (k+1)=(k)+yx but the real-valued step-size parameter is no longer equal to one; it now depends on both (k) and the training example (x,y).

Suppose Lossh(y(k+1)x)>0 after the update. Express the value of in terms of in this case. (Hint: you can simplify the loss function in this case).