Maximizing the Value of Predictions in Control: Accuracy Is Not Enough

We study the value of stochastic predictions in online optimal control with random disturbances. Prior work provides performance guarantees based on prediction error but ignores the stochastic dependence between predictions and disturbances. We introduce a general framework modeling their joint distribution and define "prediction power" as the control cost improvement from the optimal use of predictions compared to ignoring the predictions. In the time-varying Linear Quadratic Regulator (LQR) setting, we derive a closed-form expression for prediction power and discuss its mismatch with prediction accuracy and connection with online policy optimization. To extend beyond LQR, we study general dynamics and costs. We establish a lower bound on prediction power under two sufficient conditions that generalize the properties of the LQR setting, characterizing the fundamental benefit of incorporating stochastic predictions. We apply this lower bound to nonquadratic costs and show that even weakly dependent predictions yield significant performance gains.

Consider a fixed predictor parameter θ. For each time step t, let I t (θ) := (W 0:t-1 , V 0:t (θ)) denote the history of past disturbances and predictions, and let F t (θ) := σ(I t (θ)) 1 . A predictive policy that applies to the predictor with parameter θ is a sequence of functions π 0:T -1 , where π t maps a state/history pair to a control action.

Given a fixed predictive policy sequence π = π 0:T -1 for a predictor parameter θ, we evaluate its performance via the expected total cost over Ξ:

), for t = 0, . . . , T -1. The optimal cost under θ is defined as J * (θ) = min π J π (θ), where the minimum is over all predictive policies that use the predictor parameter θ.