Machine learning can infer how protein sequence maps to function without requiring a detailed understanding of the underlying physical or biological mechanisms. It's challenging to apply existing supervised learning frameworks to large-scale experimental data generated by deep mutational scanning (DMS) and related methods. DMS data often contain high dimensional and correlated sequence variables, experimental sampling error and bias, and the presence of missing data. Importantly, most DMS data do not contain examples of negative sequences, making it challenging to directly estimate how sequence a↵ects function. Here, we develop a positive-unlabeled (PU) learning framework to infer sequence-function relationships from largescale DMS data. Our PU learning method displays excellent predictive performance across ten large-scale sequence-function data sets, representing proteins of di↵erent folds, functions, and library types. The estimated parameters pinpoint key residues that dictate protein structure and function. Finally, we apply our statistical sequence-function model to design highly stabilized enzymes.

In various real-world problems, we are presented with classification problems with positive and unlabeled data, referred to as presence-only responses. In this paper, we study variable selection in the context of presence only responses where the number of features or covariates p is large. The combination of presence-only responses and high dimensionality presents both statistical and computational challenges. In this paper, we develop the PUlasso algorithm for variable selection and classification with positive and unlabeled responses. Our algorithm involves using the majorization-minimization (MM) framework which is a generalization of the well-known expectation-maximization (EM) algorithm. In particular to make our algorithm scalable, we provide two computational speed-ups to the standard EM algorithm. We provide a theoretical guarantee where we first show that our algorithm converges to a stationary point, and then prove that any stationary point within a local neighborhood of the true parameter achieves the minimax optimal mean-squared error under both strict sparsity and group sparsity assumptions. We also demonstrate through simulations that our algorithm outperforms state-of-the-art algorithms in the moderate p settings in terms of classification performance. Finally, we demonstrate that our PUlasso algorithm performs well on a biochemistry example.1 -sparsity or 1 / 2 -group sparsity penalty and provide two speed-ups. Firstly we use a quadratic majorizer of the logistic regression objective, and secondly we use techniques in linear algebra to exploit sparsity of the design matrix X which commonly arises in the applications we are dealing with. Our PUlasso algorithm fits into the majorizationminimization (MM) framework (see e.g. Lange et al. (2000); Ortega and Rheinboldt (2000)) for which the EM algorithm is a special case.

Machine learning can infer how protein sequence maps to function without requiring a detailed understanding of the underlying physical or biological mechanisms. It's challenging to apply existing supervised learning frameworks to large-scale experimental data generated by deep mutational scanning (DMS) and related methods. DMS data often contain high dimensional and correlated sequence variables, experimental sampling error and bias, and the presence of missing data. Importantly, most DMS data do not contain examples of negative sequences, making it challenging to directly estimate how sequence affects function. Here, we develop a positive-unlabeled (PU) learning framework to infer sequence-function relationships from large-scale DMS data. Our PU learning method displays excellent predictive performance across ten large-scale sequence-function data sets, representing proteins of different folds, functions, and library types. The estimated parameters pinpoint key residues that dictate protein structure and function. Finally, we apply our statistical sequence-function model to design highly stabilized enzymes.

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