TY - JOUR

T1 - A PDE Approach to the Prediction of a Binary Sequence with Advice from Two History-Dependent Experts

AU - Drenska, Nadejda

AU - Kohn, Robert V.

N1 - Publisher Copyright:
© 2022 Wiley Periodicals LLC.

PY - 2022

Y1 - 2022

N2 - The prediction of a binary sequence is a classic example of online machine learning. We like to call it the “stock prediction problem,” viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more “experts,” and strives to minimize her final-time regret with respect to the best-performing expert. Probability plays no role; rather, the market is assumed to be adversarial. We consider the case when there are two history-dependent experts, whose predictions are determined by the d most recent stock moves. Focusing on an appropriate continuum limit and using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market, and we determine associated upper and lower bounds for the investor's final-time regret. When (Formula presented.) our upper and lower bounds coalesce, so the proposed strategies are asymptotically optimal. Compared to other recent applications of partial differential equations to prediction, ours has a new element: there are two timescales, since the recent history changes at every step whereas regret accumulates more slowly.

AB - The prediction of a binary sequence is a classic example of online machine learning. We like to call it the “stock prediction problem,” viewing the sequence as the price history of a stock that goes up or down one unit at each time step. In this problem, an investor has access to the predictions of two or more “experts,” and strives to minimize her final-time regret with respect to the best-performing expert. Probability plays no role; rather, the market is assumed to be adversarial. We consider the case when there are two history-dependent experts, whose predictions are determined by the d most recent stock moves. Focusing on an appropriate continuum limit and using methods from optimal control, graph theory, and partial differential equations, we discuss strategies for the investor and the adversarial market, and we determine associated upper and lower bounds for the investor's final-time regret. When (Formula presented.) our upper and lower bounds coalesce, so the proposed strategies are asymptotically optimal. Compared to other recent applications of partial differential equations to prediction, ours has a new element: there are two timescales, since the recent history changes at every step whereas regret accumulates more slowly.

UR - http://www.scopus.com/inward/record.url?scp=85131234129&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85131234129&partnerID=8YFLogxK

U2 - 10.1002/cpa.22071

DO - 10.1002/cpa.22071

M3 - Article

AN - SCOPUS:85131234129

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -