We study estimation and inference for online sequential procedures, with a primary focus on Adam-type stochastic optimization and an ongoing investigation of nonstationary bandit problems. 

 

For Adam-type stochastic approximation applied to minimizing an expected loss, we establish convergence results for the expected suboptimality and asymptotic distribution theory in a strongly convex and smooth regime. Under mild moment conditions on the gradient noise, we derive a recursion implying the t^{-\gamma} rate up to logarithmic factors, together with a corresponding expected regret bound. We further analyze Polyak-Juditsky averaging and obtain a \sqrt{T}-central limit theorem with sandwich-form limiting covariance, yielding asymptotically valid Wald-type confidence intervals and hypothesis tests for parameters learned through Adam-type updates. 

Motivated by broader questions in sequential decision making, we also discuss ongoing work on nonstationary bandits, focusing on how time variation in the underlying reward model changes the regret behavior and adaptability of classical bandit methods.

Together, these directions aim to develop a unified perspective on learning, inference, and decision making for online sequential data.