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SUMMARY:Roberto Cominetti (Adolfo Ibáñez University)
DTSTART:20210208T143000Z
DTEND:20210208T153000Z
DTSTAMP:20260423T021013Z
UID:OWOS/36
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OWOS/36/">Co
 nvergence Rates for Krasnoselskii-Mann Fixed-Point Iterations</a>\nby Robe
 rto Cominetti (Adolfo Ibáñez University) as part of One World Optimizati
 on seminar\n\n\nAbstract\nA popular method to approximate a fixed point of
  a non-expansive map $T : C \\to C$ is the Krasnoselskii-Mann iteration \n
 \n$$(KM)\\ \\ \\ \\hspace{3cm}  x_{n+1}  = (1 − \\alpha_{n+1})  x_n + \\
 alpha_{n+1} T  xn.$$\n\nThis covers a wide range of iterative methods in c
 onvex minimization\, equilibria\, and beyond. In the Euclidean setting\, a
  flexible method to obtain convergence rates for this iteration is the PEP
  methodology introduced by Drori and Teboulle (2012)\, which is based on s
 emi-definite programming. When the underlying norm is no longer Hilbert\, 
 PEP can be substituted by an approach based on recursive estimates obtaine
 d by using optimal transport. This approach can be traced back to early wo
 rk by Baillon and Bruck (1992\, 1996). In this talk we describe this optim
 al transport technique\, and we survey some recent progress that settles t
 wo conjectures by Baillon and Bruck\, and yields the following tight metri
 c estimate for the fixed-point residuals\n\n$$x_n – T x_n = \\frac{diam 
 (C)}{\\pi \\sum_(k=1)^n \\alpha_k  (1-\\alpha_k)}.$$\n\nThe recursive esti
 mates exhibit a very rich structure and induce a very peculiar metric over
  the integers. The analysis exploits an unexpected connection with discret
 e probability and combinatorics\, related to the Gambler’s ruin for sums
  of non-homogeneous Bernoulli trials. If time allows\, we will briefly dis
 cuss the extension to inexact iterations\, and a connection to Markov chai
 ns with rewards.\n\nNote: The talk will be based on joint work with Mario 
 Bravo\, Matías Pavez-Signé\, José Soto\, and José Vaisman. Papers are 
 available at https://sites.google.com/site/cominettiroberto/.\n\nThe addre
 ss and password of the zoom room of the seminar are sent by e-mail on the 
 mailinglist of the seminar one day before each talk.\n
LOCATION:https://researchseminars.org/talk/OWOS/36/
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