  We investigate the convergence theory of several known as well as new
heuristic parameter choice rules for convex Tikhonov regularisation. The
success of such methods is dependent on whether certain restrictions on the
noise are satisfied. In the linear theory, such conditions are well understood
and hold for typically irregular noise. In this paper, we extend the
convergence analysis of heuristic rules using noise restrictions to the convex
setting and prove convergence of the aforementioned methods therewith. The
convergence theory is exemplified for the case of an ill-posed problem with a
diagonal forward operator in $\ell^q$ spaces. Numerical examples also provide
further insight.
