Timeline for Asymptotic behavior of gradient descent on a smooth, convex, non-negative function with no finite minimum
Current License: CC BY-SA 3.0
18 events
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| Mar 9, 2018 at 12:32 | comment | added | Pietro Majer | But if you want an analytic counterexample, you can approximate $u$ and $v$ with decreasing real analytic functions $\mathbb{R}\to\mathbb{R}$ in such a way that their ratio at any $x$ stays between half and twice the ratio of $u(x)$ and $v(x)$ (apply to $u'$ and $v'$ Carleman's theorem quoted here mathoverflow.net/questions/26243/…). Also, you can build directly $u$ and $v$ as entire functions by means of conveniently defined lacunary series. | |
| Mar 9, 2018 at 12:22 | comment | added | Daniel Soudry | Ah, yes, these concepts are not the same. Sorry, I was confused. Thanks for the quick answer! | |
| Mar 9, 2018 at 12:18 | comment | added | Pietro Majer | Where in the question, or in the answer, it is mentioned analicity? The functions here are smooth, i.e. $C^\infty$. No obstruction to make them locally constant in some interval as required. | |
| Mar 9, 2018 at 12:09 | comment | added | Daniel Soudry | I just noticed a small issue: the function $f$ you defined is not analytic (in $C^{\infty}$, as required in the question. Explanation: since $u$ and $v$ are not globally constant yet are constant on open non-empty segments, they cannot be analytic; therefore their integral is also not analytic. However, I think this could be corrected though, right? | |
| Oct 7, 2017 at 15:53 | comment | added | Pietro Majer | PS: I think a variant that makes the computation simpler should be, replacing the sequence $n!$ with a generic strictly increasing diverging sequence $r(n)$. Then $y_n/x_n$ can be made arbitrarily close to $0$ and to $+\infty$ along subsequences, just defining $r(n)$ suitably, by induction. | |
| Oct 7, 2017 at 15:08 | comment | added | Daniel Soudry | If it is good question, then perhaps it should also be on MO. So here it is: mathoverflow.net/q/282934/44790 | |
| Oct 7, 2017 at 14:18 | comment | added | Pietro Majer | Good question and nice conjecture! | |
| Oct 7, 2017 at 14:11 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 7, 2017 at 14:05 | comment | added | Daniel Soudry | Thanks, very nice! I wonder if there are some additional ("light") assumptions which one can add to $f$ so that $x_n/||x_n||$ would converge. Maybe that the Jacobian has a bounded eigenvalue ratio? | |
| Oct 7, 2017 at 13:58 | vote | accept | Daniel Soudry | ||
| Oct 7, 2017 at 12:51 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 7, 2017 at 6:50 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 7, 2017 at 6:39 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 6, 2017 at 18:21 | comment | added | Pietro Majer | Yes, sorry, I should add more details. | |
| Oct 6, 2017 at 18:16 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 6, 2017 at 18:10 | comment | added | Daniel Soudry | Thanks! It is not clear to me why such $\phi,\psi$ (or $u,v$) exist (or how to find them), and why the inequalities imply the result in the end. | |
| Oct 6, 2017 at 17:09 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Oct 6, 2017 at 17:03 | history | answered | Pietro Majer | CC BY-SA 3.0 |