In this paper, we establish Berry-Esseen-type bounds for federated linear stochastic approximation (LSA). Our results provide the first federated Gaussian approximations for LSA that explicitly capture communication-computation trade-offs and heterogeneity-aware error terms, quantifying the effects of local step size, number of local updates, and heterogeneity on convergence rates. We present results for both (i) constant step size regime and (ii) decreasing step size with an increasing number of local iterations, recovering the recent rates of Bonnerjee et al. [2025] as a special case. As a primary application of our results, we develop an online multiplier bootstrap procedure for inference on the last iterate, which avoids explicit estimation of the asymptotic covariance matrix, and obtain non-asymptotic validity guarantees for this procedure.

In April 2019, Benjamin Burkholder (who is awesome, by the way) published a Medium article showing off a script he wrote that uses SERP result features to infer a user's search intent. The script uses the SerpAPI.com This is one of the coolest ways to estimate search intent, because it uses Google's understanding of search intent (as expressed by the SERP features shown for that search). The one problem with Burkholder's approach is its reliance on the Serp API. If you have a large set of search queries you want to find intent for, you need to pass each query phrase through the API, which then actually does the search and returns the SERP feature results, which Burkholder's script can then classify.

We develop a novel family of algorithms for the online learning setting with regret against any data sequence bounded by the empirical Rademacher complexity of that sequence. To develop a general theory of when this type of adaptive regret bound is achievable we establish a connection to the theory of decoupling inequalities for martingales in Banach spaces. When the hypothesis class is a set of linear functions bounded in some norm, such a regret bound is achievable if and only if the norm satisfies certain decoupling inequalities for martingales. Donald Burkholder's celebrated geometric characterization of decoupling inequalities (1984) states that such an inequality holds if and only if there exists a special function called a Burkholder function satisfying certain restricted concavity properties. Our online learning algorithms are efficient in terms of queries to this function. We realize our general theory by giving novel efficient algorithms for classes including lp norms, Schatten p-norms, group norms, and reproducing kernel Hilbert spaces. The empirical Rademacher complexity regret bound implies --- when used in the i.i.d. setting --- a data-dependent complexity bound for excess risk after online-to-batch conversion. To showcase the power of the empirical Rademacher complexity regret bound, we derive improved rates for a supervised learning generalization of the online learning with low rank experts task and for the online matrix prediction task. In addition to obtaining tight data-dependent regret bounds, our algorithms enjoy improved efficiency over previous techniques based on Rademacher complexity, automatically work in the infinite horizon setting, and are scale-free. To obtain such adaptive methods, we introduce novel machinery, and the resulting algorithms are not based on the standard tools of online convex optimization.