Should I remove correlations between random effects before removing some of them? I have read that a good approach to mixed modeling is to try and fit the maximally complex random effects structure, and then simplify it until it converges.
I would like to be principled about this and have a procedure for simplifying the model. Is this sensible?
If it does not converge will:
1) Increase the number of iterations to 1,000,000.
If this does not work, or the fit is singular, I will:
2) Remove correlation among random effects parameters, beginning with item effects.
3) Remove random effects parameters with variance 0, and/or highest-order effects with lowest variance.
I'm wondering if it makes sense to remove the correlation between terms before trying to simplify the random effects structure. Or if the latter should come first?
 A: The approach of fitting mixed models with maximally complex random effects structure gained a lot of traction following the article by Barr and colleagues in 2013: "Random effects structure for confirmatory hypothesis testing: Keep it maximal", which argued, through a simulation, that such an approach will help to avoid anti-conservative conclusions.
This has since received strong criticism, notably from Douglas Bates, one of the world's foremost experts on mixed models (being the author of many papers, a book and several R & Juia packages). Instead, Bates and collaborators suggest the following approach:

We proposed (1) to use PCA to determine the dimensionality of the variance-covariance matrix of the random-effect structure, (2) to initially constrain correlation parameters to zero, especially when an initial attempt to fit a maximal model does not converge, and (3) to drop non-significant variance components and their associated correlation parameters from the model

In the same paper, they also note:

Importantly, failure to converge is not due to defects of the estimation algorithm, but is a straightforward consequence of attempting to fit a model that is too complex to be properly supported by the data.

And:

maximal models are not necessary to protect against anti-conservative
  conclusions. This protection is fully provided by comprehensive models that are guided by realistic expectations about the complexity that the data can support. In statistics, as elsewhere in science, parsimony is a virtue, not a vice.
  Bates et al (2015)

Shravan Vasishth, writing on R-Bloggers, also argues against this approach and instead favours running high power studies
References:
Barr, D.J., Levy, R., Scheepers, C. and Tily, H.J., 2013. Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of memory and language, 68(3), pp.255-278.
Bates, D., Kliegl, R., Vasishth, S. and Baayen, H., 2015. Parsimonious mixed models. arXiv preprint arXiv:1506.04967.
