Universal Log-Optimality for General Classes of e-processes and Sequential Hypothesis Tests

We consider the problem of sequential hypothesis testing by betting. For a general class of composite testing problems -- which include bounded mean testing, equal mean testing for bounded random tuples, and some key ingredients of two-sample and independence testing as special cases -- we show that any $e$-process satisfying a certain sublinear regret bound is adaptively, asymptotically, and almost surely log-optimal for a composite alternative. This is a strong notion of optimality that has not previously been established for the aforementioned problems and we provide explicit test supermartingales and $e$-processes satisfying this notion in the more general case. Furthermore, we derive matching lower and upper bounds on the expected rejection time for the resulting sequential tests in all of these cases. The proofs of these results make weak, algorithm-agnostic moment assumptions and rely on a general-purpose proof technique involving the aforementioned regret and a family of numeraire portfolios. Finally, we discuss how all of these theorems hold in a distribution-uniform sense, a notion of log-optimality that is stronger still and seems to be new to the literature.