Title: A Tale of Three Couplings

Abstract: A question throughout this semester has been how in online decision-making, the assumptions we make about uncertainty and constraints influences the resulting algorithms we use (and their performance guarantees). Can we hope for some unified way of understanding these? I will try to summarize some of my thoughts on this via three anecdotes, each revolving around my favorite tool for thinking about these problems -- couplings! Channeling the holiday spirit, I will try to show how coupling based on the past helps in learning from expert advice (based on the classical work of Kalai and Vempala), how coupling based on the present helps in online discrepancy minimization (based on recent work by Alweiss, Liu and Sawhney), and how coupling based on the future helps for online packing (based on work by me and Alberto Vera). In the process, I hope to touch upon some of the main themes that guide me in thinking about these problems.