We revisit the recently introduced Local Glivenko-Cantelli setting, which studies distribution-dependent uniform convergence rates of the Empirical Mean Estimator (EME). In this work, we investigate generalizations of this setting where arbitrary estimators are allowed rather than just the EME. Can a strictly larger class of measures be learned? Can better risk decay rates be obtained? We provide exhaustive answers to these questions—which are both negative, provided the learner is barred from exploiting some infinite-dimensional pathologies. On the other hand, allowing such exploits does lead to a strictly larger class of learnable measures.

Cohen and Kontorovich (COLT 2023) initiated the study of what we call here the Binomial Empirical Process: the maximal empirical mean deviation for sequences of binary random variables (up to rescaling, the empirical mean of each entry of the random sequence is a binomial hence the naming). They almost fully analyzed the case where the binomials are independent, which corresponds to all random variable entries from the sequence being independent. The remaining gap was closed by Blanchard and Voráček (ALT 2024). In this work, we study the much more general and challenging case with correlations. In contradistinction to Gaussian processes, whose behavior is characterized by the covariance structure, we discover that, at least somewhat surprisingly, for binomial processes covariance does not even characterize convergence. Although a full characterization remains out of reach, we take the first steps with nontrivial upper and lower bounds in terms of covering numbers.

If $\mu$ is a distribution over the $d$-dimensional Boolean cube $\set{0,1}^d$, our goal is to estimate its mean $p\in[0,1]^d$ based on $n$ iid draws from $\mu$. Specifically, we consider the empirical mean estimator $\pn$ and study the expected maximal deviation $\Delta_n=\E\max_{j\in[d]}|\pn(j)-p(j)|$. In the classical Universal Glivenko-Cantelli setting, one seeks distribution-free (i.e., independent of $\mu$) bounds on $\Delta_n$. This regime is well-understood: for all $\mu$, we have $\Delta_n\lesssim\sqrt{\log(d)/n}$ up to universal constants, and the bound is tight.Our present work seeks to establish dimension-free (i.e., without an explicit dependence on $d$) estimates on $\Delta_n$, including those that hold for $d=\infty$. As such bounds must necessarily depend on $\mu$, we refer to this regime as {\em local} Glivenko-Cantelli (also known as $\mu$-GC), and are aware of very few previous bounds of this type — which are either “abstract” or quite sub-optimal. Already the special case of product measures $\mu$ is rather non-trivial. We give necessary and sufficient conditions on $\mu$ for $\Delta_n\to0$, and calculate sharp rates for this decay. Along the way, we discover a novel sub-gamma-type maximal inequality for shifted Bernoullis, of independent interest.

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, minimalistic — weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest. Additionally, we resolve all of the open problems posed by Jürgensen and Matthews, 2010.

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} --- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest.

We present a novel approach to estimating discrete distributions with (potentially) infinite support in the total variation metric. In a departure from the established paradigm, we make no structural assumptions whatsoever on the sampling distribution. In such a setting, distribution-free risk bounds are impossible, and the best one could hope for is a fully empirical data-dependent bound. We derive precisely such bounds, and demonstrate that these are, in a well-defined sense, the best possible. Our main discovery is that the half-norm of the empirical distribution provides tight upper and lower estimates on the empirical risk. Furthermore, this quantity decays at a nearly optimal rate as a function of the true distribution. The optimality follows from a minimax result, of possible independent interest. Additional structural results are provided, including an exact Rademacher complexity calculation and apparently a first connection between the total variation risk and the missing mass.