Exercise 7.14

Take derivative of $E(w)$ w.r.t. $w$, we have $\nabla <!--\; FUNCTION\; APPLICATION\; -->E(w)=(Q^T+Q)(w-w^{\text{∗}})=2Q(w-w^{\text{∗}})=-2Q{w}^{\text{∗}}$ when $w=0$.

The weights that minimize $E(w)$ is $w^{\text{∗}}$ where $E(w^{\text{∗}})=0$. From the calculated $\nabla <!--\; FUNCTION\; APPLICATION\; -->E(w)$, the gradient descent at $w=0$ moves in the same direction as $w^{\text{∗}}$, i.e. $-\nabla <!--\; FUNCTION\; APPLICATION\; -->E(w)=Q{w}^{\text{∗}}$.

The direction should be the same as $w^{\text{∗}}$, we need normalize $\nabla <!--\; FUNCTION\; APPLICATION\; -->E(w)$ to get a unit direction and the size of the step is controlled by the step size. Otherwise, we may overshoot, even if we are moving in the right direction.