learned jointly with the $W_d$, to allow flexible sharing across tasks and depths. This architecture enables the learning of layers that are used in different ways at different depths for different tasks.](figures/soft_ordering-crop.pdf){width="\linewidth" height="1.1in"}

Deep Multitask Learning
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MTL [@Caruana:1998] exploits relationships across problems to increase overall performance. The underlying idea is that if multiple tasks are related, the optimal models for those tasks will be related as well. In the convex optimization setting, this idea has been implemented via various regularization penalties on shared parameter matrices [@Argyriou:2008; @Evgeniou:2004; @Kang:2011; @Kumar:2012]. Evolutionary methods have also had success in MTL, especially in sequential decision-making domains [@Huizinga:2016; @Kelly:2017; @Jaskowski:2008; @Schrum:2016; @Snel:2010].

Deep MTL has extended these ideas to domains where deep learning thrives, including vision [@Bilen:2017; @Kaiser:2017; @Lu:2016; @Misra:2016; @Ranjan:2016; @Rebuffi:2017; @Yang:2017; @Zhang:2014], speech [@Huang:2013; @Huang:2015; @Kaiser:2017; @Seltzer:2013; @Wu:2015], natural language processing [@Collobert:2008; @Dong:2015; @Hashimoto:2016; @Kaiser:2017; @Liu:2015; @Luong:2016; @Zhang:2016], and reinforcement learning [@Devin:2016; @Jaderberg:2016; @Teh:2017]. The key design decision in constructing a deep multitask network is deciding how parameters such as convolutional kernels or weight matrices are shared across tasks. Designing a deep neural network for a single task is already a high-dimensional open-ended optimization problem; having to design a network for multiple 