Thanks Yaroslav, visualizations are great in Mathematica ... here also performance is very nice, should be taken to python but it's above my patience, students were supposed to do it but it's hard to count on them - in any case I would gladly collaborate ... indeed low dimensional performance might no longer be true for high dimensions and nasty landscapes (?)

Regarding "3D optimizer", the demonstration contains only 2D, for 3D I only use the same type of evaluation: start with a lattice of initial positions, perform some number of steps for each, sort final values (for 0 as global minimum) - getting ~empirical distributions for various methods.

I see you worked on K-FAC like the Google team - I think they recently switched to shampoo, are there 2nd order methods used outside research? Here are my slides gathering various 2nd order: https://www.dropbox.com/s/54v8cwqyp7uvddk/SGD.pdf . Gauss-Newton, K-FAC, L-BFGS use this positive defined H>0 approximation - that very nonconvex function is convex, I am not able to trust. Also K-FAC still uses huge Hessian models.

These two Hessian approximations I use seem very practical for high dimension: "d" as diagonal handled as vector (separate parabola model in each canonical direction), and "s" full Hessian in locally dominating e.g. d=10 dim. subspace (e.g. online PCA of gradients). My OGR approach is the closest to quasi Newton like L-BFGS: Hessian model from sequence of gradients - with advantages: real Hessian (no H>0 positive defined approximation), much lower computational cost (mainly EMA update of averages), more local (exponentially weakening weights), flexible (e.g. allowing to slowly rotate the considered subspace), and good control (working on interpretable means, variance, covariances).

Thanks, indeed it might be difficult to get successful second order methods for training, but maybe it is still worth to search. For example the basic approach (cdOGR) in diagram below turned out similar to ADAM-like: also derived division by mean squared gradient coordinates. But additionally in nominator there is similar for position, what is natural when looking at shown "position vs gradient plot" - to find the g=0 minimum, we should divide gradient by width in g, but also multiply by width in theta.

Regarding other applications, I though about about continuous, metalearning - to hold beside the actual model, also model of Hessian there - e.g. in these diagonal/subspace approximations, to better exploit new scarce gradients.

Regarding saddle attraction, sure I know and cite saddle-free Newton, and use its |lambda|. However, there is this nice paper: https://arxiv.org/pdf/1902.02366 , calculating 3Mx3M Hessian, and suggesting asymmetry - different behavior for negative curvatures. Also, a few first dominant eigenvectors have rather positive curvature - so e.g. in subspace "s" OGR variants we can restrict to such directions.

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