I guess sometimes assumptions are not just about being realistic but rather help to gain insights and to understand

I will be at ICCOPT (USC Los Angeles) next week to present this work. This will be on Wednesday July 23, alongside other nice talks on first-order methods: heavy ball ODE, optimal smoothing and nonsmoothness.

We studied both continuous time and discretized dynamics. The paper also contains other results, on complexity in the convex case, on limit of limit points for discretized set-valued dynamics ...

For instance, if a critical point is flat, it may be more sensible to errors, since the vanishing gradient cannot compensate the perturbations. We thus obtain an estimate (rho) of the fluctuations around the critical set, depending on the coefficients theta and beta.

The idea of the analysis was to quantify how much critical points are flat or sharp. So we relied on KL inequality and a metric subregularity condition. They are satisfied for a large class of functions called "definable" or semialgebraic ones (say, piecewise polynomial).

If it went down, then it must be a definable function, and I know you used a conservative gradient.

I find Coste definitely more accessible to learn the topic, but when it comes to find/cite a specific property I prefer Van den dries!