We prove that conditional diffusion models whose reverse kernels are finite Gaussian mixtures with ReLU-network logits can approximate suitably regular target distributions arbitrarily well in context-averaged conditional KL divergence, up to an irreducible terminal mismatch that typically vanishes with increasing diffusion horizon. A path-space decomposition reduces the output error to this mismatch plus per-step reverse-kernel errors; assuming each reverse kernel factors through a finite-dimensional feature map, each step becomes a static conditional density approximation problem, solved by composing Norets' Gaussian-mixture theory with quantitative ReLU bounds. Under exact terminal matching the resulting neural reverse-kernel class is dense in conditional KL.

We study stochastic convex optimization (SCO) with heavy-tailed gradients under pure epsilon-differential privacy (DP). Instead of assuming a bound on the worst-case Lipschitz parameter of the loss, we assume only a bounded k-th moment. This assumption allows for unbounded, heavy-tailed stochastic gradient distributions, and can yield sharper excess risk bounds. The minimax optimal rate for approximate (epsilon, delta)-DP SCO is known in this setting, but the pure epsilon-DP case has remained open. We characterize the minimax optimal excess-risk rate for pure epsilon-DP heavy-tailed SCO up to logarithmic factors. Our algorithm achieves this rate in polynomial time with high probability. Moreover, it runs in polynomial time with probability 1 when the worst-case Lipschitz parameter is polynomially bounded. For important structured problem classes - including hinge/ReLU-type and absolute-value losses on Euclidean balls, ellipsoids, and polytopes - we achieve the same excess-risk guarantee in polynomial time with probability 1 even when the worst-case Lipschitz parameter is infinite. Our approach is based on a novel framework for privately optimizing Lipschitz extensions of the empirical loss. We complement our excess risk upper bound with a novel high probability lower bound.

Bayesian neural networks (BNNs) offer a natural probabilistic formulation for inference in deep learning models. Despite their popularity, their optimality has received limited attention through the lens of statistical decision theory. In this paper, we study decision rules induced by deep, fully connected feedforward ReLU BNNs in the normal location model under quadratic loss. We show that, for fixed prior scales, the induced Bayes decision rule is not minimax. We then propose a hyperprior on the effective output variance of the BNN prior that yields a superharmonic square-root marginal density, establishing that the resulting decision rule is simultaneously admissible and minimax. We further extend these results from the quadratic loss setting to the predictive density estimation problem with Kullback--Leibler loss. Finally, we validate our theoretical findings numerically through simulation.

While the mathematical foundations of score-based generative models are increasingly well understood for unconstrained Euclidean spaces, many practical applications involve data restricted to bounded domains. This paper provides a statistical analysis of reflected diffusion models on the hypercube $[0,1]^D$ for target distributions supported on $d$-dimensional linear subspaces. A primary challenge in this setting is the absence of Gaussian transition kernels, which play a central role in standard theory in $\mathbb{R}^D$. By employing an easily implementable infinite series expansion of the transition densities, we develop analytic tools to bound the score function and its approximation by sparse ReLU networks. For target densities with Sobolev smoothness $α$, we establish a convergence rate in the $1$-Wasserstein distance of order $n^{-\frac{α+1-δ}{2α+d}}$ for arbitrarily small $δ> 0$, demonstrating that the generative algorithm fully adapts to the intrinsic dimension $d$. These results confirm that the presence of reflecting boundaries does not degrade the fundamental statistical efficiency of the diffusion paradigm, matching the almost optimal rates known for unconstrained settings.

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by $n$-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) $n$-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.

The global Lipschitz smoothness condition underlies most convergence and complexity analyses via two key consequences: the descent lemma and the gradient Lipschitz continuity. How to study the performance of optimization algorithms in the absence of Lipschitz smoothness remains an active area. The relative smoothness framework from Bauschke-Bolte-Teboulle (2017) and Lu-Freund-Nesterov (2018) provides an extended descent lemma, ensuring convergence of Bregman-based proximal gradient methods and their vanilla stochastic counterparts. However, many widely used techniques (e.g., momentum schemes, random reshuffling, and variance reduction) additionally require the Lipschitz-type bound for gradient deviations, leaving their analysis under relative smoothness an open area. To resolve this issue, we introduce the dual kernel conditioning (DKC) regularity condition to regulate the local relative curvature of the kernel functions. Combined with the relative smoothness, DKC provides a dual Lipschitz continuity for gradients: even though the gradient mapping is not Lipschitz in the primal space, it preserves Lipschitz continuity in the dual space induced by a mirror map. We verify that DKC is widely satisfied by popular kernels and is closed under affine composition and conic combination. With these novel tools, we establish the first complexity bounds as well as the iterate convergence of random reshuffling mirror descent for constrained nonconvex relative smooth problems.

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This paper studies the problem of entropy identity testing: given sample access to a distribution p and a fully described distribution q (both discrete distributions over a domain of size k), and the promise that either p = q or |H (p) H (q)| ε, where H () denotes the Shannon entropy, a tester needs to distinguish between the two cases with high probability.