Designing protein sequences of both high fitness and novelty is a challenging task in data-efficient protein engineering. Exploration beyond wild-type neighborhoods often leads to biologically implausible sequences or relies on surrogate models that lose fidelity in novel regions. Here, we propose PROSPERO, an active learning framework in which a frozen pre-trained generative model is guided by a surrogate updated from oracle feedback. By integrating fitness-relevant residue selection with biologically-constrained Sequential Monte Carlo sampling, our approach enables exploration beyond wild-type neighborhoods while preserving biological plausibility. We show that our framework remains effective even when the surrogate is misspecified. PROSPERO consistently outperforms or matches existing methods across diverse protein engineering tasks, retrieving sequences of both high fitness and novelty.

Transformer models have become the dominant backbone for sequence modeling, leveraging self-attention to produce contextualized token representations. These are typically aggregated into fixed-size vectors via pooling operations for downstream tasks. While much of the literature has focused on attention mechanisms, the role of pooling remains underexplored despite its critical impact on model behavior. In this paper, we introduce a theoretical framework that rigorously characterizes the expressivity of Transformer-based models equipped with widely used pooling methods by deriving closed-form bounds on their representational capacity and the ability to distinguish similar inputs. Our analysis extends to different variations of attention formulations, demonstrating that these bounds hold across diverse architectural variants. We empirically evaluate pooling strategies across tasks requiring both global and local contextual understanding, spanning three major modalities: computer vision, natural language processing, and time-series analysis. Results reveal consistent trends in how pooling choices affect accuracy, sensitivity, and optimization behavior. Our findings unify theoretical and empirical perspectives, providing practical guidance for selecting or designing pooling mechanisms suited to specific tasks. This work positions pooling as a key architectural component in Transformer models and lays the foundation for more principled model design beyond attention alone.

Mole Fractions indicates mole fractions availability. Arrhenius relationship indicates if the task can be modeled using the Arrhenius equation.

Source codes for reproducing our experimental results are available at https://github.com/ We utilize DQNReplay dataset5 [1] for expert demonstrations on 27 Atari environments [5]. To encourage the size of the dataset to be consistent across multiple environments, we use the number of expert demonstrations N 2{ 20,50}. We provide the size of a dataset for each environment in Table 4. We process input images to grayscale images of 84 84 1, by utilizing Dopamine library6 [9].

Recent developments in reinforcement learning (RL) led to algorithms that surpass human experts in a broad range of tasks [Mnih et al., 2015, Vinyals et al., 2019, Schrittwieser et al., 2020, Wurman et al.,