Analysis of many Riemannian optimization algorithms assume bounded iterates and then they show convergence and rates. This is compatible with the algorithms diverging. Rates usually depend on the diameter of the set where your iterates lie. Quantifying this diameter and making sure it’s small has been largely overlooked.
I gave a talk at the SIAM LA24 conference in Paris about different general techniques that I developed along with my coauthors in order to design Riemannian optimization algorithms without strong assumptions and with quantified small geometric constants in their rates.
There was some interest in this, so I’m posting the slides here. I now added references for each technique.
I may expand this post into a proper blog post in the future.