This paper presents local minimax regret lower bounds for adaptively controlling linear-quadratic-Gaussian (LQG) systems. We consider smoothly parametrized instances and provide an understanding of when logarithmic regret is impossible which is both instance specific and flexible enough to take problem structure into account. This understanding relies on two key notions: That of local-uninformativeness; when the optimal policy does not provide sufficient excitation for identification of the optimal policy, and yields a degenerate Fisher information matrix; and that of information-regretboundedness, when the small eigenvalues of a policy-dependent information matrix are boundable in terms of the regret of that policy. Combined with a reduction to Bayesian estimation and application of Van Trees' inequality, these two conditions are sufficient for proving regret bounds on order of magnitude √ T in the time horizon, T . This method yields lower bounds that exhibit tight dimensional dependencies and scale naturally with control-theoretic problem constants. For instance, we are able to prove that systems operating near marginal stability are fundamentally hard to learn to control. We further show that large classes of systems satisfy these conditions, among them any state-feedback system with both A-and B-matrices unknown. Most importantly, we also establish that a nontrivial class of partially observable systems, essentially those that are over-actuated, satisfy these conditions, thus providing a √ T lower bound also valid for partially observable systems. Finally, we turn to two simple examples which demonstrate that our lower bound captures classical control-theoretic intuition: our lower bounds diverge for systems operating near marginal stability or with large filter gain -these can be arbitrarily hard to (learn to) control.

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We present lower bounds for the regret of adaptive control of the linear quadratic regulator. These are given in terms of problem specific expected regret lower bounds valid for unbiased policies linear in the state. Our approach is based on the insight that the adaptive control problem can, given our assumptions, be reduced to a sequential estimation problem. This enables the use of the Cramér-Rao information inequality which yields a scaling limit lower bound of logarithmic order. The bound features both information-theoretic and control-theoretic quantities. By leveraging existing results, we are able to show that the bound is tight in a special case.

We analyze the impact of sensor attacks on a linear state estimation problem subject to variance and sparsity constraints. We show that the maximum impact in a leaderfollower game where the attacker first chooses the distribution of an adversarial perturbation and the defender follows by choosing an estimator is characterized by a minimum Fisher information principle. In general, this is a nonlinear variational problem, but we show that it can be reduced to a finitedimensional mixed integer SDP.