We consider the problem of sequential hypothesis testing using $e$-processes. For a rich 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 asymptotically and almost surely instance-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 a more general case. Furthermore, we derive matching lower and upper bounds on the expected rejection time in the high-confidence regime 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 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.

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