Statistical guarantees for denoising reflected diffusion models

In recent years, denoising diffusion models have become a crucial area of research due to their abundance in the rapidly expanding field of generative AI. While recent statistical advances have delivered explanations for the generation ability of idealised denoising diffusion models for high-dimensional target data, implementations introduce thresholding procedures for the generating process to overcome issues arising from the unbounded state space of such models. This mismatch between theoretical design and implementation of diffusion models has been addressed empirically by using a \emph{reflected} diffusion process as the driver of noise instead. In this paper, we study statistical guarantees of these denoising reflected diffusion models. In particular, we establish minimax optimal rates of convergence in total variation, up to a polylogarithmic factor, under Sobolev smoothness assumptions. Our main contributions include the statistical analysis of this novel class of denoising reflected diffusion models and a refined score approximation method in both time and space, leveraging spectral decomposition and rigorous neural network analysis.

Data-driven rules for multidimensional reflection problems

Over the recent past data-driven algorithms for solving stochastic optimal control problems in face of model uncertainty have become an increasingly active area of research. However, for singular controls and underlying diffusion dynamics the analysis has so far been restricted to the scalar case. In this paper we fill this gap by studying a multivariate singular control problem for reversible diffusions with controls of reflection type. Our contributions are threefold. We first explicitly determine the long-run average costs as a domain-dependent functional, showing that the control problem can be equivalently characterized as a shape optimization problem. For given diffusion dynamics, assuming the optimal domain to be strongly star-shaped, we then propose a gradient descent algorithm based on polytope approximations to numerically determine a cost-minimizing domain. Finally, we investigate data-driven solutions when the diffusion dynamics are unknown to the controller. Using techniques from nonparametric statistics for stochastic processes, we construct an optimal domain estimator, whose static regret is bounded by the minimax optimal estimation rate of the unreflected process' invariant density. In the most challenging situation, when the dynamics must be learned simultaneously to controlling the process, we develop an episodic learning algorithm to overcome the emerging exploration-exploitation dilemma and show that given the static regret as a baseline, the loss in its sublinear regret per time unit is of natural order compared to the one-dimensional case.

Covariate shift in nonparametric regression with Markovian design

Covariate shift in regression problems and the associated distribution mismatch between training and test data is a commonly encountered phenomenon in machine learning. In this paper, we extend recent results on nonparametric convergence rates for i.i.d. data to Markovian dependence structures. We demonstrate that under H\"older smoothness assumptions on the regression function, convergence rates for the generalization risk of a Nadaraya-Watson kernel estimator are determined by the similarity between the invariant distributions associated to source and target Markov chains. The similarity is explicitly captured in terms of a bandwidth-dependent similarity measure recently introduced in Pathak, Ma and Wainwright [ICML, 2022]. Precise convergence rates are derived for the particular cases of finite Markov chains and spectral gap Markov chains for which the similarity measure between their invariant distributions grows polynomially with decreasing bandwidth. For the latter, we extend the notion of a distribution transfer exponent from Kpotufe and Martinet [Ann. Stat., 49(6), 2021] to kernel transfer exponents of uniformly ergodic Markov chains in order to generate a rich class of Markov kernel pairs for which convergence guarantees for the covariate shift problem can be formulated.

Concentration analysis of multivariate elliptic diffusion processes

We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $\pi$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $\pi(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.

Learning to reflect: A unifying approach for data-driven stochastic control strategies

Stochastic optimal control problems have a long tradition in applied probability, with the questions addressed being of high relevance in a multitude of fields. Even though theoretical solutions are well understood in many scenarios, their practicability suffers from the assumption of known dynamics of the underlying stochastic process, raising the statistical challenge of developing purely data-driven strategies. For the mathematically separated classes of continuous diffusion processes and L\'evy processes, we show that developing efficient strategies for related singular stochastic control problems can essentially be reduced to finding rate-optimal estimators with respect to the sup-norm risk of objects associated to the invariant distribution of ergodic processes which determine the theoretical solution of the control problem. From a statistical perspective, we exploit the exponential $\beta$-mixing property as the common factor of both scenarios to drive the convergence analysis, indicating that relying on general stability properties of Markov processes is a sufficiently powerful and flexible approach to treat complex applications requiring statistical methods. We show moreover that in the L\'evy case $-$ even though per se jump processes are more difficult to handle both in statistics and control theory $-$ a fully data-driven strategy with regret of significantly better order than in the diffusion case can be constructed.