Prediction without probability: a PDE approach to a model problem from the machine learning literature
- Robert Kohn (Courant Institute of Mathematical Sciences, New York University)
In the machine learning literature, one approach to "prediction" assumes that there are two more "experts"; the best prediction in this setting is the one that "minimizes regret", i.e. minimizes the shortfall relative to the best-performing expert. My talk is motivated by a model problem recently considered by Rina Panigraphy, Michael Kapralov, and Alexandr Andoni, involving the prediction of a binary sequence (loosely speaking: a stock whose price is restricted to a binomial tree). I'll discuss joint work with Kangping Zhu about a continuum limit, in which the optimal strategy is associated with a 2nd order PDE. While the PDE is very nonlinear, explicit solutions are available in many cases due to a surprising link to the linear heat equation. The continuum limit being considered here is related to my 2006 work with Sylvia Serfaty on a deterministic-game interpretation of motion by curvature.