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

• URL: http://arxiv.org/abs/2608.13651v1
• Date: Thu, 13 Aug 2026 18:00:03 GMT
• Title: Consistent Model Chasing Is Minimax Optimal: The Exact Value of Scalar Adversarial Adaptive Control under Large Parametric Uncertainty
• Abstract summary: We solve a fundamental problem of adaptive control against adversarial disturbances.<n>We find the least worst-case peak that a causal controller can guarantee against an adversarial pair.<n>The optimal policy is certainty-equivalent deadbeat control at the midpoint of the set-membership consistent interval.
• Score: 0.15229257192293197
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $a \in [-Δ, Δ]$ is unknown in sign and magnitude and $Δ$ 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 $γ^\star(Δ) = 1 + Δ$ for every $Δ>0$. The summand $1$ is the irreducible price of the disturbance, and $Δ$ 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 $θ_t := -u_t/x_t$ 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.

Related papers

• Sharp Rates and a One-Line Correction for Spectral Representation Learning [8.921166277011347]
  Self-supervised encoder is trained once, frozen, and reused through lightweight probes on tasks nobody named at training time.<n> Canonical correlation analysis, HGR maximal correlation, and the population optimum of the spectral contrastive loss all return the top-$k$ singular subspace of a cross-view dependence operator.<n>We show isotropy is the wrong hypothesis. The prior enters the transfer risk only through the task covariance $=mathbbE[top]$, and only through its compression onto the operator's leading singular directions.
  arXiv  Detail & Related papers  (2026-09-14T16:26:55Z)
• Observation Design for Certified Control Authority: Projection--Estimability Separation and Active-Face Equivalence [2.7954473524479275]
  A gate executes only actions it can certify, and certifies only what its observations support.<n>This paper asks how observations should be designed to maximize the set of actions that can be safely admitted.
  arXiv  Detail & Related papers  (2026-09-05T07:57:04Z)
• Does 1/2-Tsallis-INF Also Work Well for Best-Arm Identification? [11.223878908981725]
  $1/2$-Tsallis-INF is a canonical best-of-both-worlds FTRL algorithm.<n>It achieves logarithmic pseudo-regret in bandits while retaining minimax-optimal regret in adversarial bandits.<n>This raises a natural question: can the same algorithm, without additional exploration, also identify the best arm reliably?
  arXiv  Detail & Related papers  (2026-08-15T18:38:36Z)
• Parameter-Free Heavy-Tailed Bandits [36.983245230853434]
  We study adaptation to the moment bound $u$ for a fixed tail exponent $$.<n>We prove that every algorithm unaware of $u$, or of any upper bound on it, must obey a sharp trade-off between its distribution-dependent and distribution-free regret guarantees.
  arXiv  Detail & Related papers  (2026-07-31T14:27:28Z)
• Bayesian Best-Arm Identification with Abstention: A Polynomial-to-Exponential Phase Transition [51.253617466579335]
  We study the Bayesian fixed-budget best-arm identification problem in which a learner can abstain from making a terminal recommendation.<n>We analyze the probability of undetected error--the risk of recommending a suboptimal arm without abstaining.
  arXiv  Detail & Related papers  (2026-06-28T05:07:27Z)
• Prism: Structural Symmetry Scanning via Duality-Constrained Laplacian Projection [0.0]
  We introduce textbfPrism, a framework for structural symmetry diagnosis in complex networks.<n>Prism computes the emphduality defect $(L,P) = |LP - PL|_F / |L|_F$ -- a scalar measuring how far the network deviates from structural self-consistency.
  arXiv  Detail & Related papers  (2026-05-18T04:20:05Z)
• Reinforcement Learning from Multi-Source Imperfect Preferences: Best-of-Both-Regimes Regret [71.69884486156359]
  We study episodic RL from emphmulti-source imperfect preferences through a cumulative imperfection budget.<n>We propose a unified algorithm with regret $tildeO(sqrtK/M+)$, which exhibits a best-of-both-regimes behavior.
  arXiv  Detail & Related papers  (2026-03-20T19:34:53Z)
• Time Transfer: On Optimal Learning Rate and Batch Size In The Infinite Data Limit [1.8337746049048673]
  We show an intricate dependence of optimal $eta$ scaling on the pretraining token budget $T$, $B$ and its relation to the critical batch size $B_mathrmcrit$.<n>Surprisingly, our results demonstrate that the observed optimal $eta$ and $B$ dynamics are preserved with $mu$P model scaling.
  arXiv  Detail & Related papers  (2024-10-08T09:06:34Z)
• Robust Causal Bandits for Linear Models [20.028245872662843]
  Sequential design of experiments for optimizing a reward function in causal systems can be effectively modeled by the sequential design of interventions in causal bandits (CBs) This paper addresses the robustness of CBs to such model fluctuations. Cumulative regret is adopted as the design criteria, based on which the objective is to design a sequence of interventions that incur the smallest cumulative regret with respect to an oracle aware of the entire causal model and its fluctuations.
  arXiv  Detail & Related papers  (2023-10-30T17:58:01Z)
• Linear Contextual Bandits with Adversarial Corruptions [91.38793800392108]
  We study the linear contextual bandit problem in the presence of adversarial corruption. We present a variance-aware algorithm that is adaptive to the level of adversarial contamination $C$.
  arXiv  Detail & Related papers  (2021-10-25T02:53:24Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)
• Naive Exploration is Optimal for Online LQR [49.681825576239355]
  We show that the optimal regret scales as $widetildeTheta(sqrtd_mathbfu2 d_mathbfx T)$, where $T$ is the number of time steps, $d_mathbfu$ is the dimension of the input space, and $d_mathbfx$ is the dimension of the system state. Our lower bounds rule out the possibility of a $mathrmpoly(logT)$-regret algorithm, which had been
  arXiv  Detail & Related papers  (2020-01-27T03:44:54Z)
• Improper Learning for Non-Stochastic Control [78.65807250350755]
  We consider the problem of controlling a possibly unknown linear dynamical system with adversarial perturbations, adversarially chosen convex loss functions, and partially observed states. Applying online descent to this parametrization yields a new controller which attains sublinear regret vs. a large class of closed-loop policies. Our bounds are the first in the non-stochastic control setting that compete with emphall stabilizing linear dynamical controllers.
  arXiv  Detail & Related papers  (2020-01-25T02:12:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.