Consider betting against a sequence of data in $[0,1]$, where one is allowed to make any bet that is fair if the data have a conditional mean $m_0 \in (0,1)$. Cover's universal portfolio algorithm delivers a worst-case regret of $O(\ln n)$ compared to the best constant bet in hindsight, and this bound is unimprovable against adversarially generated data. In this work, we present a novel mixture betting strategy that combines insights from Robbins and Cover, and exhibits a different behavior: it eventually produces a regret of $O(\ln \ln n)$ on \emph{almost} all paths (a measure-one set of paths if each conditional mean equals $m_0$ and intrinsic variance increases to $\infty$), but has an $O(\log n)$ regret on the complement (a measure zero set of paths). Our paper appears to be the first to point out the value in hedging two very different strategies to achieve a best-of-both-worlds adaptivity to stochastic data and protection against adversarial data. We contrast our results to those in~\cite{agrawal2025regret} for a sub-Gaussian mixture on unbounded data: their worst-case regret has to be unbounded, but a similar hedging delivers both an optimal betting growth-rate and an almost sure $\ln\ln n$ regret on stochastic data. Finally, our strategy witnesses a sharp game-theoretic upper law of the iterated logarithm, analogous to~\cite{shafer2005probability}.

We study stopping rules for stochastic gradient descent (SGD) for convex optimization from the perspective of anytime-valid confidence sequences. Classical analyses of SGD provide convergence guarantees in expectation or at a fixed horizon, but offer no statistically valid way to assess, at an arbitrary time, how close the current iterate is to the optimum. We develop an anytime-valid, data-dependent upper confidence sequence for the weighted average suboptimality of projected SGD, constructed via nonnegative supermartingales and requiring no smoothness or strong convexity. This confidence sequence yields a simple stopping rule that is provably $\varepsilon$-optimal with probability at least $1-α$, with explicit bounds on the stopping time under standard stochastic approximation stepsizes. To the best of our knowledge, these are the first rigorous, time-uniform performance guarantees and finite-time $\varepsilon$-optimality certificates for projected SGD with general convex objectives, based solely on observable trajectory quantities.

We propose a practical sequential test for auditing differential privacy guarantees of black-box mechanisms. The test processes streams of mechanisms' outputs providing anytime-valid inference while controlling Type I error, overcoming the fixed sample size limitation of previous batch auditing methods. Experiments show this test detects violations with sample sizes that are orders of magnitude smaller than existing methods, reducing this number from 50K to a few hundred examples, across diverse realistic mechanisms. Notably, it identifies DP-SGD privacy violations in \textit{under} one training run, unlike prior methods needing full model training.

A.1 Proof of Theorem 3.1 First we set up some notation. All algorithms we are considering, if not discrete, induce a density w.r.t. the Lebesgue measure. The only difference between Theorem 3.1 and this theorem is that a privacy filter halts at a random The same argument can be used to bound the other direction of the divergence. Since we run batch gradient descent and not SGD as in the library example, we tune all hyperparameters from scratch. We think of the minimum of an empty set as .

Reinforcement learning has shown promising results in learning neural network policies for complicated control tasks. However, the lack of formal guarantees about the behavior of such policies remains an impediment to their deployment.