Optimal control with learning on the fly: A toy problem

Charles L. Fefferman, Bernat Guillen Pegueroles, Clarence W. Rowley, Melanie Weber

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We exhibit optimal control strategies for a simple toy problem in which the underlying dynamics depend on a parameter that is initially unknown and must be learned.We consider a cost function posed over a finite time interval, in contrast to much previous work that considers asymptotics as the time horizon tends to infinity. We study several different versions of the problem, including Bayesian control, in which we assume a prior distribution on the unknown parameter; and "agnostic" control, in which we assume nothing about the unknown parameter. For the agnostic problems, we compare our performance with that of an opponent who knows the value of the parameter. This comparison gives rise to several notions of "regret", and we obtain strategies that minimize the "worst-case regret" arising from the most unfavorable choice of the unknown parameter. In every case, the optimal strategy turns out to be a Bayesian strategy or a limit of Bayesian strategies.

Original languageEnglish (US)
Pages (from-to)175-187
Number of pages13
JournalRevista Matematica Iberoamericana
Issue number1
StatePublished - 2022

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Adaptive control
  • Agnostic control
  • Competitive ratio
  • Fuel tax regret
  • Regret


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