Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control
We study episodic continuous-time control with an unknown vector control gain, scalar state, quadratic action cost, and smooth convex terminal cost. In the scalar Gaussian experiment, let $Δ(H)$ denote the minimax improvement over zero control and set $δ=\sqrt2H^2-1$. We prove the critical law $$ Δ(H)\asymp δ^4\sqrt{\log(1/δ)} \qquad (δ\downarrow0), $$ with a matching lower and upper bound. The lower bound follows from a uniform deficit--energy inequality valid for fully adaptive controls with unbounded amplitudes, while a moving soft-threshold feedback attains the rate. For local parameters $θ=N^{-1/4}h$, $|h|\le H$, we show that the normalized minimax regret over $N$ episodes is within ...
| Authors | Chen Jia |
|---|---|
| Primary category | math.OC |
| Published | 27 September 2026 |
| Last updated at source | 27 September 2026 |
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