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

arXiv: https://arxiv.org/abs/2608.13651 · HTML: https://mlsanity.com/paper/2608.13651/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 29 of its 54 references that are in the map.
- First posted on arXiv 13 Aug 2026.

## Builds on (most cited first, 29 of 29)

- [Naive Exploration is Optimal for Online LQR](https://mlsanity.com/paper/2001.09576/) (2020)
- [Regret Bounds for Robust Adaptive Control of the Linear Quadratic Regulator](https://mlsanity.com/paper/1805.09388/) (2018)
- [Learning Linear-Quadratic Regulators Efficiently with only √T Regret](https://mlsanity.com/paper/1902.06223/) (2019)
- [Online Linear Quadratic Control](https://mlsanity.com/paper/1806.07104/) (2018)
- [The Nonstochastic Control Problem](https://mlsanity.com/paper/1911.12178/) (2019)
- [Black-Box Control for Linear Dynamical Systems](https://mlsanity.com/paper/2007.06650/) (2020)
- [Chasing Nested Convex Bodies Nearly Optimally](https://mlsanity.com/paper/1811.00999/) (2018)
- [Chasing Convex Bodies Optimally](https://mlsanity.com/paper/1905.11968/) (2019)
- [Chasing Convex Bodies with Linear Competitive Ratio](https://mlsanity.com/paper/1905.11877/) (2019)
- [Thompson Sampling for Linear-Quadratic Control Problems](https://mlsanity.com/paper/1703.08972/) (2017)
- [A Nearly-Linear Bound for Chasing Nested Convex Bodies](https://mlsanity.com/paper/1806.08865/) (2018)
- [Competitive Control](https://mlsanity.com/paper/2107.13657/) (2021)
- [A robust adaptive model predictive control framework for nonlinear uncertain systems](https://mlsanity.com/paper/1911.02899/) (2019)
- [On Uninformative Optimal Policies in Adaptive LQR with Unknown B-Matrix](https://mlsanity.com/paper/2011.09288/) (2020)
- [Online Adversarial Stabilization of Unknown Networked Systems](https://mlsanity.com/paper/2203.02630/) (2022)
- [Minimax Adaptive Control for a Finite Set of Linear Systems](https://mlsanity.com/paper/2011.10814/) (2020)
- [Robust Online Voltage Control with an Unknown Grid Topology](https://mlsanity.com/paper/2206.14369/) (2022)
- [Robust Adaptive Model Predictive Control with Worst-Case Cost](https://mlsanity.com/paper/1911.08607/) (2019)
- [Robust Model-Free Learning and Control without Prior Knowledge](https://mlsanity.com/paper/2010.00204/) (2020)
- [Online Stabilization of Unknown Linear Time-Varying Systems](https://mlsanity.com/paper/2304.02878/) (2023)
- [An Online Learning Analysis of Minimax Adaptive Control](https://mlsanity.com/paper/2307.07268/) (2023)
- [Output Feedback Minimax Adaptive Control](https://mlsanity.com/paper/2409.04115/) (2024)
- [Passive-Aggressive Learning and Control](https://mlsanity.com/paper/1812.10910/) (2018)
- [Scalable Robust Adaptive Control from the System Level Perspective](https://mlsanity.com/paper/1904.00077/) (2019)
- [Minimax Adaptive Control for State Matrix with Unknown Sign](https://mlsanity.com/paper/1912.03550/) (2019)
- [Learning-Enabled Robust Control with Noisy Measurements](https://mlsanity.com/paper/2202.08363/) (2022)
- [Minimax dual control with finite-dimensional information state](https://mlsanity.com/paper/2312.05156/) (2023)
- [On Minimax Optimal Dual Control for Fully Actuated Systems](https://mlsanity.com/paper/2511.22753/) (2025)
- [Minimax optimal dual control -- The single input case](https://mlsanity.com/paper/2604.18550/) (2026)

## Abstract (arXiv)

We solve exactly a fundamental problem of adaptive control against adversarial disturbances: regulate the scalar system xt+1 = axt + ut + wt, x0=0, |w|_∞ ≤ 1, where the constant pole a ∈ [-Delta, Delta] is unknown in sign and magnitude and Delta is arbitrarily large. Elementary as the system looks, the least worst-case peak |x|_∞ 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 γ^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 × consistent model chasing architecture. The architecture is forced, not merely sufficient: writing θt := -ut/xt 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.