It seems that I have not figured out the following question after some attempts. I would be very appreciated if you can help me. Thank you! mathoverflow.net/questions/401256/…

The counter example is very nice! I know what the problem is. May I ask a further question? If $F(x,y)$ is monotonically non-decreasing in $y$ given $x$, but $y=y(x)$ may not be bounded, can I obtain the continuity property of $y=y(x)$? Thank you again!

Thank you very much! Does it mean that although $f$ may not be Lipschitz continuous, it must be continuous? Also, how can I prove the remark?

May I consult you further about the stochastic case of heavy ball method? For SGD algorithm, we can prove convergence if one decays the step size gradually. However, I can not prove convergence of heavy ball method using the same technique as in the deterministic case.

Thank you very much!

Thank you very much. I have browsed the paper successfully. This paper proved the convergence of heavy-ball method, but it did not show a convergence rate. Can we show that under $L$ smooth property, after $T$ steps, we can find a point $x$ such that $\|\nabla f(x)\|^2\le O(L/T)$, for example? (This bound applies for traditional gradient descent.)

It seems that the I must spend a lot of money buying the pdf. Does this paper prove the convergence of heavy-ball method?