We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages on both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results.

We study neural-linear bandits for solving problems where both exploration and representation learning play an important role. Neural-linear bandits leverage the representation power of Deep Neural Networks (DNNs) and combine it with efficient exploration mechanisms designed for linear contextual bandits on top of the last hidden layer. A recent analysis of DNNs in the "infinite-width" regime suggests that when these models are trained with gradient descent the optimal solution is close to the initialization point and the DNN can be viewed as a kernel machine. As a result, it is possible to exploit linear exploration algorithms on top of a DNN via the kernel construction. The problem is that in practice the kernel changes during the learning process and the agent's performance degrades. This can be resolved by recomputing new uncertainty estimations with stored data. Nevertheless, when the buffer's size is limited, a phenomenon called catastrophic forgetting emerges. Instead, we propose a likelihood matching algorithm that is resilient to catastrophic forgetting and is completely online. We perform simulations on a variety of datasets and observe that our algorithm achieves comparable performance to the unlimited memory approach while exhibits resilience to catastrophic forgetting.