Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control
Abstract
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=√2H2−1. We prove the critical law Δ(H)asymp δ4√(log(1/δ)) (δ↓0), 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−1/4h, |h|≤ H, we show that the normalized minimax regret over N episodes is within O(N−1/2) of a fixed-horizon Gaussian sequential control problem, without an additional dimension factor. The terminal task enters the limit only through cg=Var(g(Z)). The Gaussian problem exhibits an exact no-learning phase transition: CdT(H)=TH2/2 ⟺ H4T≤ d2/2, yielding the asymptotic task boundary cgH4=d2/2.
Related
- Minimax and Adaptive Transfer Learning for Sparse Canonical Correlation Analysis
- Learning Sparse Support under Differential Privacy: Adaptive Algorithms and Minimax Limits
- Development of approximate scheduling-adaptive controllers for multi-products continuous chemical processes using deep learning techniques and model predictive control
- Minimax estimation of the pseudo-spectral gap of continuous Markov chains, with applications to empirical concentration bounds
- Minimax-Optimal Early Stopping for Continuous-Time SGD via the Discrepancy Principle
BibTeX
@misc{jia2026sharp,
title = {{Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control}},
author = {Chen Jia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33154},
eprint = {2609.33154},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33154},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Critical Minimax Laws and No-Learning Thresholds in Continuous-Time Adaptive Control
AU - Chen Jia
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33154
UR - https://arxiv.org/abs/2609.33154
AB - 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=√2H^2−1. We prove the critical law Δ(H)asymp δ^4√(log(1/δ)) (δ↓0), 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^−1/4h, |h|≤ H, we show that the normalized minimax regret over N episodes is within O(N^−1/2) of a fixed-horizon Gaussian sequential control problem, without an additional dimension factor. The terminal task enters the limit only through c_g=Var(g(Z)). The Gaussian problem exhibits an exact no-learning phase transition: C_d^T(H)=TH^2/2 ⟺ H^4T≤ d^2/2, yielding the asymptotic task boundary c_gH^4=d^2/2.
ER -