The interaction is usually modeled as Markov Decision Processes (MDPs). Research on MDPs can be broadly divided into two lines based on the reward generation mechanism. The first line of work [Jaksch et al., 2010, Azar et al., 2013, 2017, He et al., 2021] considers the

Functional bilevel optimization (FBO) provides a powerful framework for hierarchical learning in function spaces, yet current methods are limited to static offline settings and perform suboptimally in online, non-stationary scenarios. We propose SmoothFBO, the first algorithm for non-stationary FBO with both theoretical guarantees and practical scalability. SmoothFBO introduces a time-smoothed stochastic hypergradient estimator that reduces variance through a window parameter, enabling stable outer-loop updates with sublinear regret. Importantly, the classical parametric bilevel case is a special reduction of our framework, making SmoothFBO a natural extension to online, non-stationary settings. Empirically, SmoothFBO consistently outperforms existing FBO methods in non-stationary hyperparameter optimization and model-based reinforcement learning, demonstrating its practical effectiveness. Together, these results establish SmoothFBO as a general, theoretically grounded, and practically viable foundation for bilevel optimization in online, non-stationary scenarios.

Adapting to distribution shifts is a critical challenge in modern machine learning, especially as data in many real-world applications accumulate continuously in the form of streams. We investigate the problem of sequentially adapting a model to non-stationary environments, where the data distribution is continuously shifting and only a small amount of unlabeled data are available each time. Continual test-time adaptation methods have shown promising results by using reliable pseudo-labels, but they still fall short in exploring representation alignment with the source domain in non-stationary environments. In this paper, we propose to leverage non-stationary representation learning to adaptively align the unlabeled data stream, with its changing distributions, to the source data representation using a sketch of the source data. To alleviate the data scarcity in non-stationary representation learning, we propose a novel adaptive representation alignment algorithm called Ada-ReAlign. This approach employs a group of base learners to explore different lengths of the unlabeled data stream, which are adaptively combined by a meta learner to handle unknown and continuously evolving data distributions. The proposed method comes with nice theoretical guarantees under convexity assumptions. Experiments on both benchmark datasets and a real-world application validate the effectiveness and adaptability of our proposed algorithm.

We study episodic linear mixture MDPs with the unknown transition and adversarial rewards under full-information feedback, employing *dynamic regret* as the performance measure. We start with in-depth analyses of the strengths and limitations of the two most popular methods: occupancy-measure-based and policy-based methods. We observe that while the occupancy-measure-based method is effective in addressing non-stationary environments, it encounters difficulties with the unknown transition. In contrast, the policy-based method can deal with the unknown transition effectively but faces challenges in handling non-stationary environments. Building on this, we propose a novel algorithm that combines the benefits of both methods. Specifically, it employs (i) an *occupancy-measure-based global optimization* with a two-layer structure to handle non-stationary environments; and (ii) a *policy-based variance-aware value-targeted regression* to tackle the unknown transition.

One of the key challenges in deploying RL to real-world applications is to adapt to variations of unknown environment contexts, such as changing terrains in robotic tasks and fluctuated bandwidth in congestion control. Existing works on adaptation to unknown environment contexts either assume the contexts are the same for the whole episode or assume the context variables are Markovian. However, in many real-world applications, the environment context usually stays stable for a stochastic period and then changes in an abrupt and unpredictable manner within an episode, resulting in a segment structure, which existing works fail to address.