Abstract
Many compound properties depend directly on the dissociation constants of its acidic and basic groups. Significant effort has been invested in computational models to predict these constants. For linear regression models, compounds are often divided into chemically motivated classes, with a separate model for each class. However, sometimes too few measurements are available for a class to build a reasonable model, e.g., when investigating a new compound series. If data for related classes are available, we show that multi-task learning can be used to improve predictions by utilizing data from these other classes. We investigate performance of linear Gaussian process regression models (single task, pooling, and multi-task models) in the low sample size regime, using a published data set (n = 698, mostly monoprotic, in aqueous solution) divided beforehand into 15 classes. A multi-task regression model using the intrinsic model of co-regionalization and incomplete Cholesky decomposition performed best in 85 % of all experiments. The presented approach can be applied to estimate other molecular properties where few measurements are available.
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Notes
As opposed to parametric approaches, where the information from the training data are summarized in the parameters of a distribution, non-parametric approaches require the training data for later predictions. This distinction does not prevent non-parametric approaches from having parameters, here the regression weights \(\varvec{\alpha}\) and hyper-parameters \(\varvec{\theta}\). Parameters \(\varvec{\alpha}\), which directly belong to the model itself, are computed from the data by solving an optimization problem. Hyper-parameters \(\varvec{\theta}\) parameterize the kernel, and can be estimated via gradient-based optimization by maximizing the marginal likelihood.
Predictions are technically equivalent to those of kernel ridge regression [28], a regularized form of ordinary regression. Here, we do not use additional features of GPs like predictive variance. However, the used GP MTL methods do make use of Bayesian aspects of GPs.
Technically, \({\mathbf{K}^\mathbf{t} \otimes \mathbf{K}^\mathbf{x} \in {\mathbb{R}}^{MN \times MN}}\). In our setting, each sample (compound) occurs in one task only. After removing (marginalizing out) rows and columns corresponding to combinations of compounds and tasks that don’t occur, the resulting matrix is N × N. In practice, it is not necessary to construct the MN × MN matrix explicitly.
Task similarity matrices are positive definite. Their entries thus correspond to evaluations of an inner product in some Hilbert space, which can be converted to Euclidean distance by using \(||\mathbf{x}-\mathbf{z}||_{2}^{2}= \sum_{i=1}^d |x_i-z_i|^2 = \; <\!\mathbf{x}-\mathbf{z},\mathbf{x}-\mathbf{z}\!> =<\!\mathbf{x},\mathbf{x}\!> -2 <\!\mathbf{x},\mathbf{z}\!> + <\!\mathbf{z},\mathbf{z}\!>\).
Comparison is based on Table S2 of the supplement of Ref. [20], using column R’ and third lines from each row of the common tasks.
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Acknowledgments
The authors thank Klaus-Robert Müller, Gisbert Schneider, Tiago Rodrigues, and an anonymous referee for helpful suggestions, and David Manallack for the provision of data. M. Rupp and K. Hansen acknowledge partial support by FP7-ICT programme of the European Community (PASCAL2) and DFG (grant MU 987/4-2). M. Rupp acknowledges partial support by FP7 programme of the European Community (Marie Curie IEF 273039). G. Sanguinetti and G. Skolidis acknowledge support from the Engineering and Physical Sciences Research Council (EPSRC, grant EP/F009461/2). G. Sanguinetti is funded by the Scottish government through the SICSA initiative.
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Skolidis, G., Hansen, K., Sanguinetti, G. et al. Multi-task learning for pKa prediction. J Comput Aided Mol Des 26, 883–895 (2012). https://doi.org/10.1007/s10822-012-9582-x
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DOI: https://doi.org/10.1007/s10822-012-9582-x