#gradient-noise

For $f(\mathbf{x}) = x_1^2 + 2x_2^2$, take the per-example loss $f(\mathbf{x}, \mathbf{w}) = (x_1 - w_1)^2 + 2(x_2 - w_2)^2$ with $\mathbf{w} \sim \mathcal{N}(\mathbf{0}, I)$. Its gradient $\nabla_{\mathbf{x}} f(\mathbf{x}, \mathbf{w})$ is $\nabla f(\mathbf{x})$ plus zero-mean Gaussian noise, and $E_{\mathbf{w}}[f(\mathbf{x}, \mathbf{w})] = x_1^2 + 2x_2^2 + c$. What is $c$?