Running nonlinear RNNs for T steps takes O(T) time. Our construction, called LDStack, approximately runs them in O(log T) parallel time, and obtains arbitrarily low error via repetition. First, we show nonlinear RNNs can be approximated by a stack of multiple-input, multiple-output (MIMO) LDS. This replaces nonlinearity across time with nonlinearity along depth. Next, we show that MIMO LDS can be approximated by an average or a concatenation of single-input, multiple-output (SIMO) LDS. Finally, we present an algorithm for running (and differentiating) SIMO LDS in O(log T) parallel time. On long sequences, LDStack is much faster than traditional RNNs, yet it achieves similar accuracy in our experiments. Furthermore, LDStack is amenable to linear systems theory. Therefore, it improves not only speed, but also interpretability and mathematical tractability.

We use the statistical quantity of margin --- the distance between a decision boundary and a classified point, or the gap between two scores --- to formalize the principle of equal opportunity --- the chance to improve one's outcome, regardless of group status. This leads to a better definition of opportunity which recognizes, for example, that a strongly rejected individual was offered less recourse than a weakly rejected one, despite the shared outcome. It also leads to simpler algorithms, since real-valued margins are easier to analyze and optimize than discrete outcomes. We formalize two ways that a protected group may be guaranteed equal opportunity: (1) (social) mobility: acceptance should be within reach for the group (conversely, the general population shouldn't be cushioned from rejection), and (2) contrast: within the group, good candidates should get substantially higher scores than bad candidates, preventing the so-called 'token' effect. A simple linear classifier seems to offer roughly equal opportunity both experimentally and mathematically.

Classical linear/shallow learning is relatively easy to analyze and understand, but the power of deep learning is often desirable. I am developing a hybrid approach in order to obtain learning algorithms that are both trustworthy and accurate. My research has mostly focused on learning from corrupted or inconsistent training data (`agnostic learning'). Recently, I, as well as independent researchers, have found these same techniques could help make algorithms more fair.