Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ Let us use stochastic gradient descent to solve this problem. This yields a stochastic sequence : $$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega, x_i , y_i), \omega_0 = 0$$ where $f(\omega, x_i, y_i) = \frac{1}{2}|y_i − \omega^T x_i|^2 $and $\eta$ is the step size. Formulate an appropriate range of $\eta$ so that $\omega_i$ will become an ergodic process. Also find mean and covariance under the equilibrium distribution.

So, I tried to calculate it and got another form of the expression:

$$\omega_{i+1}-\omega^*=(I-\eta x_i x_i^T)(\omega_i-\omega^*)+\eta\xi_ix_i$$

which could help to directly express $\omega_i$ by a successive multiplication of $(I-\eta x_i x_i^T)$ and $\omega^*$. Then I wanted to calculate $\frac{\sum \omega_i}{n}$ to get the ergodicity, but I didn't make it.

Suppose we have i.i.d. samples $x_i\sim N(0,\Sigma)$ and $y_i\sim x_i^T\omega^*+\xi_i,\xi_i\sim N(0,1)$ where $\omega^*$ is the fixed point of: $$\omega_{i+1} = \omega_i − \eta\nabla_\omega f(\omega_i, x_i , y_i), \quad \omega_0 = 0$$ where $f(\omega, x_i, y_i) = \frac{1}{2}|y_i − \omega^T x_i|^2 $and $\eta$ is the step size. Formulate an appropriate range of $\eta$ so that $\omega_i$ will become an ergodic process. Also find mean and covariance under the equilibrium distribution.

So, I tried to calculate it and got another form of the expression:

$$\omega_{i+1}-\omega^*=(I-\eta x_i x_i^T)(\omega_i-\omega^*)+\eta\xi_ix_i$$

which could help to directly express $\omega_i$ by a successive multiplication of $(I-\eta x_i x_i^T)$ and $\omega^*$. Then I wanted to calculate $\frac{\sum \omega_i}{n}$ to get the ergodicity, but I didn't make it.