A central challenge in reinforcement learning (RL) is to learn models that generalize beyond the tasks on which they are trained, a goal traditionally pursued through multi-task and meta RL. Recently, transformer architectures have emerged as a promising approach, enabling adaptation to new tasks via in-context learning without explicit parameter updates. From a functional perspective, a transformer can be viewed as a functional operator that maps a context to a task-specific function. It is thus fundamental to understand and design this operator to support stronger generalization in RL. In this work, we address this resulting question of generalization from a kernel-based perspective by establishing a connection between non-linear transformers and kernel-based temporal difference learning. By interpreting the transformer as performing regression in a Reproducing Kernel Hilbert Space (RKHS), we show that value functions from different domains can be represented using a shared set of weights, provided they lie within the same RKHS. Experiments on multiple MetaWorld domains support this interpretation, demonstrating convergence of the temporal-difference objective.
Personalized reward models require user-specific preference comparisons, but collecting such feedback is costly and provides only limited supervision for each user. At the same time, large pools of responses can be generated without user identities or preference labels, yet existing personalized reward models leave this data unused. Directly learning from these responses is difficult because pairwise objectives constrain relative scores within a prompt, whereas pseudo-labeling individual responses requires scores that are comparable across prompts for the same user. We propose \textbf{PerSPL} (Personalized Semi-supervised Preference Learning), a backbone-agnostic framework that uses pair-mean score centering to stabilize this cross-prompt score coordinate and iteratively turns unlabeled responses into user-specific training signals. Against seven representative baselines across four benchmarks---PersonalLLM, HelpSteer2, Reddit TLDR, and Chatbot Arena---PerSPL achieves the highest reported mean accuracy among the evaluated methods for both seen and unseen users. The two latter benchmarks provide real-user preference comparisons in summarization and open-domain chatbot interaction, respectively. These results indicate that unlabeled responses can provide additional supervision for personalized reward modeling when per-user preference labels are scarce.