Consistent model chasing proven minimax optimal: exact value is 1+Δ

Dimitar Ho's new paper "Consistent Model Chasing Is Minimax Optimal: The Exact Value of Scalar Adversarial Adaptive Control under Large Parametric Uncertainty" (arXiv:2608.13651) solves a decades-open problem in adaptive control. Ho considers a simple scalar system x_{t+1} = ax_t + u_t + w_t, where the pole a is unknown in sign and magnitude within [-Δ, Δ] for arbitrarily large Δ, and disturbance w is adversarial with |w_t| ≤ 1. The question: what is the least worst-case peak |x|_∞ any causal controller can guarantee? His answer: exactly γ*(Δ) = 1 + Δ. The '1' is the irreducible price of disturbance, while the 'Δ' is the exact cost of a single unavoidable identification spike.

Ho shows the optimal policy is certainty-equivalent deadbeat control at the midpoint of the set-membership consistent interval—an instance of the "robust oracle × consistent model chasing" architecture. He proves this architecture is forced, not just sufficient: every causal controller can be written as an oracle-selector composition, and optimality pins the selector to the midpoint at critical histories. Classical and modern tools each fail quantifiably: probing is punished before it pays, commitment is fatal at sub-disturbance excitation, optimism degenerates to tie-breaking or pays asymptotically at least twice the optimum, and regret bounds are blind to worst-case peak. Notably, the optimal law contains no exploration mechanism—learning is purely passive, overturning intuitions that adaptive control must probe to identify the unknown parameter.