Consistent Model Chasing Is Minimax Optimal: The Exact Value of Scalar Adversarial Adaptive Control under Large Parametric Uncertainty

Abstract:We solve exactly a fundamental problem of adaptive control against adversarial disturbances: regulate the scalar system $x_{t+1} = ax_t + u_t + w_t$, $x_0=0$, $\|w\|_\infty \le 1$, where the constant pole $a \in [-\Delta, \Delta]$ is unknown in sign and magnitude and $\Delta$ is arbitrarily large. Elementary as the system looks, the least worst-case peak $\|x\|_\infty$ that a causal controller can guarantee against an adversarial pair $(a, w)$ (the value of this game) has, to our knowledge, never been determined for any adaptive control problem with parametric uncertainty of arbitrary size under this criterion; existing theory supplies stability certificates, gain bounds, and regret rates, not the value. That value is $\gamma^\star(\Delta) = 1 + \Delta$ for every $\Delta>0$. The summand $1$ is the irreducible price of the disturbance, and $\Delta$ the exact price of a single, unavoidable identification spike. The optimal policy is certainty-equivalent deadbeat control at the midpoint of the set-membership consistent interval, an instance of the robust oracle $\times$ consistent model chasing architecture. The architecture is forced, not merely sufficient: writing $\theta_t := -u_t/x_t$ exhibits every causal controller as an oracle-selector composition, and optimality pins the selector to the midpoint at the critical histories. The standard tools, classical and modern, each fail quantifiably: probing is punished before it pays, commitment is fatal at sub-disturbance excitation once adaptation is necessary, optimism degenerates to tie-breaking or pays asymptotically at least twice the optimum, and regret certificates are blind to the worst-case peak in both directions. The optimal law contains no exploration mechanism, its learning purely passive. These results give the first exact optimality certificate for consistent model chasing as a design principle for adversarial adaptive control.

Systems and Control (eess.SY); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2608.13651 [eess.SY]

(or arXiv:2608.13651v1 [eess.SY] for this version)

https://doi.org/10.48550/arXiv.2608.13651 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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