Diffusion language models (DLMs) have recently emerged as a promising alternative to autoregressive (AR) approaches, enabling parallel token generation beyond a rigid left-to-right order. Despite growing empirical success, the theoretical understanding of how unmasking schedules -- which specify the order and size of unmasked tokens during sampling -- affect generation quality remains limited. In this work, we introduce a distribution-agnostic unmasking schedule for DLMs that adapts to the (unknown) dependence structure of the target data distribution, without requiring any prior knowledge or hyperparameter tuning. In contrast to prior deterministic procedures that fix unmasking sizes, our method randomizes the number of tokens revealed at each iteration. We show that, for two specific parameter choices, the sampling convergence guarantees -- measured by Kullback-Leibler (KL) divergence -- scale as $\widetilde O(\mathsf{TC}/K)$ and $\widetilde O(\mathsf{DTC}/K)$ respectively. Here, $K$ is the number of iterations, and $\mathsf{TC}$ and $\mathsf{DTC}$ are the total correlation and dual total correlation of the target distribution, capturing the intrinsic dependence structure underlying the data. Importantly, our guarantees hold in the practically relevant parallel-sampling regime $K

A major bottleneck of standard auto-regressive large language models is that their inference process is inherently sequential, resulting in very long and costly inference times. To circumvent this, practitioners proposed a class of language models called diffusion language models, of which the masked diffusion model (MDM) is the most successful. The MDM is able to sample tokens out-of-order and, ostensibly, many tokens at once and in parallel. However, there is very limited rigorous understanding of how much parallel sampling these models can perform without noticeable degradation in their sampling performance. Prior work of Li and Cai obtained some preliminary bounds, but these are not tight for many natural classes of distributions. In this work, we give a new, exact characterization of the expected divergence between the true distribution and the sampled distribution, for any distribution and any unmasking schedule for the sampler, showing an elegant connection to the theory of univariate function approximation. By leveraging this connection, we then attain a number of novel lower and upper bounds for this problem. While the connection to function approximation in principle gives the optimal unmasking schedule for any distribution, we show that it is in general impossible to compete with it without strong a priori knowledge of the distribution, even in seemingly benign settings. However, we also demonstrate new upper bounds and new sampling schedules in terms of well-studied information-theoretic properties of the base distribution, namely, its total correlation and dual total correlation, which show that in some natural settings, one can sample in $O(log n)$ steps without any visible loss in performance, where $n$ is the total sequence length.

We study the fundamental problem of estimating an unknown discrete distribution $p$ over $d$ symbols, given $n$ i.i.d. samples from the distribution. We are interested in minimizing the KL divergence between the true distribution and the algorithm's estimate. We first construct minimax optimal private estimators. Minimax optimality however fails to shed light on an algorithm's performance on individual (non-worst-case) instances $p$ and simple minimax-optimal DP estimators can have poor empirical performance on real distributions. We then study this problem from an instance-optimality viewpoint, where the algorithm's error on $p$ is compared to the minimum achievable estimation error over a small local neighborhood of $p$. Under natural notions of local neighborhood, we propose algorithms that achieve instance-optimality up to constant factors, with and without a differential privacy constraint. Our upper bounds rely on (private) variants of the Good-Turing estimator. Our lower bounds use additive local neighborhoods that more precisely captures the hardness of distribution estimation in KL divergence, compared to ones considered in prior works.

The log-linear model has received a significant amount of theoretical attention in previous decades and remains the fundamental tool used for learning probability distributions over discrete variables. Despite its large popularity in statistical mechanics and high-dimensional statistics, the vast majority of such energy-based modeling approaches only focus on the two-variable relationships, such as Boltzmann machines and Markov graphical models. Although these approaches have easier-to-solve structure learning problems and easier-to-optimize parametric distributions, they often ignore the rich structure which exists in the higher-order interactions between different variables. Using more recent tools from the field of information geometry, we revisit the classical formulation of the log-linear model with a focus on higher-order mode interactions, going beyond the 1-body modes of independent distributions and the 2-body modes of Boltzmann distributions. This perspective allows us to define a complete decomposition of the KL error. This then motivates the formulation of a sparse selection problem over the set of possible mode interactions. In the same way as sparse graph selection allows for better generalization, we find that our learned distributions are able to more efficiently use the finite amount of data which is available in practice. On both synthetic and real-world datasets, we demonstrate our algorithm's effectiveness in maximizing the log-likelihood for the generative task and also the ease of adaptability to the discriminative task of classification.