A hallmark of adaptation in humans and other animals is our ability to control how we think and behave across different settings. Research has characterized the various forms cognitive control can take—including enhancement of goal-relevant information, suppression of goal-irrelevant information, and overall inhibition of potential responses—and has identified computations and neural circuits that underpin this multitude of control types. Studies have also identified a wide range of situations that elicit adjustments in control allocation (e.g., those eliciting signals indicating an error or increased processing conflict), but the rules governing when a given situation will give rise to a given control adjustment remain poorly understood. Significant progress has recently been made on this front by casting the allocation of control as a decision-making problem. This approach has developed unifying and normative models that prescribe when and how a change in incentives and task demands will result in changes in a given form of control. Despite their successes, these models, and the experiments that have been developed to test them, have yet to face their greatest challenge: deciding how to select among the multiplicity of configurations that control can take at any given time. Here, we will lay out the complexities of the inverse problem inherent to cognitive control allocation, and their close parallels to inverse problems within motor control (e.g., choosing between redundant limb movements). We discuss existing solutions to motor control's inverse problems drawn from optimal control theory, which have proposed that effort costs act to regularize actions and transform motor planning into a well-posed problem. These same principles may help shed light on how our brains optimize over complex control configuration, while providing a new normative perspective on the origins of mental effort.

To behave adaptively in environments that are noisy and nonstationary, humans and other animals must monitor feedback from their environment and adjust their predictions and actions accordingly. An understudied approach for modeling these adaptive processes comes from the engineering field of control theory, which provides general principles for regulating dynamical systems, often without requiring a generative model. The proportional–integral–derivative (PID) controller is one of the most popular models of industrial process control. The proportional term is analogous to the “delta rule” in psychology, adjusting estimates in proportion to each error in prediction. The integral and derivative terms augment this update to simultaneously improve accuracy and stability. Here, we tested whether the PID algorithm can describe how people sequentially adjust their predictions in response to new information. Across three experiments, we found that the PID controller was an effective model of participants' decisions in noisy, changing environments. In Experiment 1, we reanalyzed a change-point detection experiment and showed that participants' behavior incorporated elements of PID updating. In Experiments 2–3, we developed a task with gradual transitions that we optimized to detect PID-like adjustments. In both experiments, the PID model offered better descriptions of behavioral adjustments than both the classical delta-rule model and its more sophisticated variant, the Kalman filter. We further examined how participants weighted different PID terms in response to salient environmental events, finding that these control terms were modulated by reward, surprise, and outcome entropy. These experiments provide preliminary evidence that adaptive learning in dynamic environments resembles PID control.