Continuous diffusion models have demonstrated remarkable generative performance across diverse domains but are often constrained by the computational cost of simulating reverse Ornstein–Uhlenbeck processes via SDE/ODE solvers. Existing theoretical results typically establish query complexities that scale polynomially with both the dimension $d$ and the error tolerance $\epsilon$ (e.g., $\tilde{\mathcal{O}}(d/\epsilon)$). This mirrors the limitations of unadjusted Langevin algorithm, where standard first-order score solvers lack access to zeroth-order density information, precluding natural error-correction mechanisms and thus preventing the fast $\ln(1/\epsilon)$ convergence attainable by Metropolis-adjusted methods. In this paper, we develop an improved generative modeling method by introducing Quantized Transition Diffusion (QTD), a framework that reformulates continuous diffusion into a discrete generation problem through spatial quantization and the parameterization of zeroth-order information (e.g., density ratios). To sample from this discrete target, we propose a truncated uniformization algorithm that simulates the underlying continuous-time Markov chain of the discrete diffusion process without discretization error, while eliminating the restrictive bounded-score assumption required by prior uniformization-based approaches. Consequently, QTD attains $\epsilon$-accuracy in total variation distance with a query complexity of $\mathcal{O}(d \ln^2(d/\epsilon))$, yielding a notable improvement in $\epsilon$-dependence compared to existing continuous diffusion samplers. Crucially, our analysis capitalizes on a novel proof technique based on the infinitesimal chain rule of KL divergence, providing a fresh perspective on unifying continuous and discrete diffusion paradigms.

We consider the problem of sampling from a $d$-dimensional log-concave distribution $\pi(\theta) \propto \exp(-f(\theta))$ for $L$-Lipschitz $f$, constrained to a convex body (described by $n$ hyperplanes) equipped with a barrier function, contained in a ball of radius $R$ with a $w$-warm start. We propose a \emph{robust} sampling framework that computes spectral approximations to the Hessian of the barrier functions in each iteration. We prove that for the polytope constraints, sampling with the Lee-Sidford barrier function mixes within $\widetilde O((d^2+dL^2R^2)\log(w/\delta))$ steps with a per step cost of $\widetilde O(nd^{\omega-1})$, where $\omega\approx 2.37$ is the fast matrix multiplication exponent. Compared to the prior work of Mangoubi and Vishnoi, our approach gives faster mixing time as we are able to design a generalized soft-threshold Dikin walk beyond log-barrier. We further extend our result to show how to sample from a $d$-dimensional spectrahedron, the constrained set of a semidefinite program, specified by the set $\{x\in \mathbb{R}^d: \sum_{i=1}^d x_i A_i \succeq C \}$ where $A_1,\ldots,A_d, C$ are $n\times n$ real symmetric matrices. We design a walk that mixes in $\widetilde O((nd+dL^2R^2)\log(w/\delta))$ steps with a per iteration cost of $\widetilde O(n^\omega+n^2d^{3\omega-5})$. We improve the mixing time bound of prior best Dikin walk due to Narayanan and Rakhlin that mixes in $\widetilde O((n^2d^3+n^2dL^2R^2)\log(w/\delta))$ steps.

Thompson sampling (TS) is widely used in sequential decision making due to its ease of use and appealing empirical performance. However, many existing analytical and empirical results for TS rely on restrictive assumptions on reward distributions, such as belonging to conjugate families, which limits their applicability in realistic scenarios. Moreover, sequential decision making problems are often carried out in a batched manner, either due to the inherent nature of the problem or to serve the purpose of reducing communication and computation costs. In this work, we jointly study these problems in two popular settings, namely, stochastic multi-armed bandits (MABs) and infinite-horizon reinforcement learning (RL), where TS is used to learn the unknown reward distributions and transition dynamics, respectively. We propose batched *Langevin Thompson Sampling* algorithms that leverage MCMC methods to sample from approximate posteriors with only logarithmic communication costs in terms of batches. Our algorithms are computationally efficient and maintain the same order-optimal regret guarantees of $\mathcal{O}(\log T)$ for stochastic MABs, and $\mathcal{O}(\sqrt{T})$ for RL. We complement our theoretical findings with experimental results.

Recent studies in reinforcement learning (RL) have made significant progress by leveraging function approximation to alleviate the sample complexity hurdle for better performance. Despite the success, existing provably efficient algorithms typically rely on the accessibility of immediate feedback upon taking actions. The failure to account for the impact of delay in observations can significantly degrade the performance of real-world systems due to the regret blow-up. In this work, we tackle the challenge of delayed feedback in RL with linear function approximation by employing posterior sampling, which has been shown to empirically outperform the popular UCB algorithms in a wide range of regimes. We first introduce \textit{Delayed-PSVI}, an optimistic value-based algorithm that effectively explores the value function space via noise perturbation with posterior sampling. We provide the first analysis for posterior sampling algorithms with delayed feedback in RL and show our algorithm achieves $\widetilde{O}(\sqrt{d^3H^3 T} + d^2H^2 \mathbb{E}[\tau])$ worst-case regret in the presence of unknown stochastic delays. Here $\mathbb{E}[\tau]$ is the expected delay. To further improve its computational efficiency and to expand its applicability in high-dimensional RL problems, we incorporate a gradient-based approximate sampling scheme via Langevin dynamics for \textit{Delayed-LPSVI}, which maintains the same order-optimal regret guarantee with $\widetilde{O}(dHK)$ computational cost. Empirical evaluations are performed to demonstrate the statistical and computational efficacy of our algorithms.

The diversity of intrinsic qualities of multimedia entities tends to impede their effective retrieval. In a Self-Learning Search Engine architecture, the subtle nuances of human perceptions and deep knowledge are taught and captured through unsupervised reinforcement learning, where the degree of reinforcement may be suitably calibrated. Such architectural paradigm enables indexes to evolve naturally while accommodating the dynamic changes of user interests. It operates by continuously constructing indexes over time, while injecting progressive improvement in search performance. For search operations to be effective, convergence of index learning is of crucial importance to ensure efficiency and robustness. In this paper, we develop a Self-Learning Search Engine architecture based on reinforcement learning using a Markov Decision Process framework. The balance between exploration and exploitation is achieved through evolutionary exploration Strategies. The evolutionary index learning behavior is then studied and formulated using stochastic analysis. Experimental results are presented which corroborate the steady convergence of the index evolution mechanism.

In the search and retrieval of multimedia objects, it is impractical to either manually or automatically extract the contents for indexing since most of the multimedia contents are not machine extractable, while manual extraction tends to be highly laborious and time-consuming. However, by systematically capturing and analyzing the feedback patterns of human users, vital information concerning the multimedia contents can be harvested for effective indexing and subsequent search. By learning from the human judgment and mental evaluation of users, effective search indices can be gradually developed and built up, and subsequently be exploited to find the most relevant multimedia objects. To avoid hovering around a local maximum, we apply the -greedy method to systematically explore the search space. Through such methodic exploration, we show that the proposed approach is able to guarantee that the most relevant objects can always be discovered, even though initially it may have been overlooked or not regarded as relevant. The search behavior of the present approach is quantitatively analyzed, and closed-form expressions are obtained for the performance of two variants of the -greedy algorithm, namely EGSE-A and EGSE-B. Simulations and experiments on real data set have been performed which show good agreement with the theoretical findings. The present method is able to leverage exploration in an effective way to significantly raise the performance of multimedia information search, and enables the certain discovery of relevant objects which may be otherwise undiscoverable.