In conjugate gradient method for solving symmetric positive definite linear system $Ax=b$, which can also be regarded as a convex optimization problem $\dfrac{1}{2} x'Ax - x'b$, the $A$-norm of the error $\|x_k-x_\star\|_A$ at iterate $k$ is monotonically decreasing and converges linearly and we have the classical bound (or more sophisticated analysis)

$$\|x_k-x_\star\|_A \leq 2 \left(\dfrac{\sqrt{\kappa} - 1}{\sqrt{\kappa} + 1}\right)^k \|x_0-x_\star\|_A$$

where $x_\star$ is the solution of the linear system $Ax_{\star}=b$ and $\kappa$ is the condition number of $A$. However, the norm of residuals $r_k = b-A x_k$ is not necessarily monotonically decreasing but it still converges to zero.

My question is: Is there a convergence bound or some qualitative characterization of $\|r_k\|$?