In this paper, we study how to perform statistical inference for quantile temporal difference learning (QTD) in distributional reinforcement learning. Assuming access to a generative model, we first establish functional central limit theorems for both synchronous and asynchronous QTD, which show that the averaged iterates of QTD converge weakly to a rescaled Brownian motion. We next provide online inference methods. Based on random scaling, the inference procedure constructs an asymptotically pivotal statistic for inference by using the information along the whole QTD path. Meanwhile, the proposed statistic can be computed online without storing the entire trajectory of QTD iterates. This substantially reduces the memory requirement and enables efficient statistical inference in distributional reinforcement learning.
Spectral clustering methods for network data are commonly based on a few matrix representations, such as the adjacency matrix and the symmetric Laplacian. We study a continuum of degree-normalized spectral embeddings that includes these commonly used choices as special cases. Under a random dot product graph model, we establish a row-wise central limit theorem for this family of embeddings. The result provides an explicit description of how degree normalization affects both population geometry and the local uncertainty of embedded nodes. We use the limiting distributions to compare different normalizations in two-community stochastic block models through a projected-Gaussian Bayes-error diagnostic. These comparisons show that no single normalization is uniformly preferred. Instead, the favored normalization depends on network density, community imbalance, and block-probability structure. Typically, stronger normalization is favored in lower-density or more imbalanced settings. These results provide a unified distributional understanding of when and why alternative normalizations may improve spectral clustering.
In this article, we analyse convergence of the averaged Adam optimizer to an attracting zero of the Adam vector field. We provide a central limit theorem that, in particular, quantifies exactly the speed of convergence. The order of convergence is $n^{-1/2}$ in the number of steps of the algorithm which coincides with the order observed for classical stochastic approximation algorithms. The covariance in the central limit theorem is given in terms of properties of the Adam algorithm in the state of the attractor.
Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $γ\in(α,1)$; the sub-linear regime is essential), not constant-$β$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.