We consider the infinite-horizon, discrete-time fullinformation control problem. Motivated by learning theory, as a criterion for controller design we focus on regret, defined as the difference between the LQR cost of a causal controller (that has only access to past and current disturbances) and the LQR cost of a clairvoyant one (that has also access to future disturbances). In the full-information setting, there is a unique optimal non-causal controller that in terms of LQR cost dominates all other controllers, and we focus on the regret compared to this particular controller. Since the regret itself is a function of the disturbances, we consider the worst-case regret over all possible bounded energy disturbances, and propose to find a causal controller that minimizes this worst-case regret. The resulting controller has the interpretation of guaranteeing the smallest possible regret compared to the best non-causal controller that can see the future, no matter what the disturbances are. We show that the regretoptimal control problem can be reduced to a Nehari extension problem, i.e., to approximate an anticausal operator with a causal one in the operator norm. In the state-space setting we obtain explicit formulas for the optimal regret and for the regret-optimal controller (in both the causal and the strictly causal settings). The regret-optimal controller is the sum of the classical H2 state-feedback law and an n-th order controller (where n is the state dimension of the plant) obtained from the Nehari problem. The controller construction simply requires the solution to the standard LQR Riccati equation, in addition to two Lyapunov equations. Simulations over a range of plants demonstrates that the regret-optimal controller interpolates nicely between the H2 and the H∞ optimal controllers, and generally has H2 and H∞ costs that are simultaneously close to their optimal values. The regret-optimal controller thus presents itself as a viable option for control system design.
Stabilizing the unknown dynamics of a control system and minimizing regret in control of an unknown system are among the main goals in control theory and reinforcement learning. In this work, we pursue both these goals for adaptive control of linear quadratic regulators (LQR).Prior works accomplish either one of these goals at the cost of the other one. The algorithms that are guaranteed to find a stabilizing controller suffer from high regret, whereas algorithms that focus on achieving low regret assume the presence of a stabilizing controller at the early stages of agent-environment interaction. In the absence of such stabilizing controller, at the early stages, the lack of reasonable model estimates needed for (i) strategic exploration and (ii) design of controllers that stabilize the system, results in regret that scales exponentially in the problem dimensions. We propose a framework for adaptive control that exploits the characteristics of linear dynamical systems and deploys additional exploration in the early stages of agent-environment interaction to guarantee sooner design of stabilizing controllers. We show that for the classes of controllable and stabilizable LQRs, where the latter is a generalization of prior work, these methods achieve Õ( √ T ) regret with a polynomial dependence in the problem dimensions.
We consider the infinite-horizon, discrete-time fullinformation control problem. Motivated by learning theory, as a criterion for controller design we focus on regret, defined as the difference between the LQR cost of a causal controller (that has only access to past and current disturbances) and the LQR cost of a clairvoyant one (that has also access to future disturbances). In the full-information setting, there is a unique optimal non-causal controller that in terms of LQR cost dominates all other controllers, and we focus on the regret compared to this particular controller. Since the regret itself is a function of the disturbances, we consider the worst-case regret over all possible bounded energy disturbances, and propose to find a causal controller that minimizes this worst-case regret. The resulting controller has the interpretation of guaranteeing the smallest possible regret compared to the best non-causal controller that can see the future, no matter what the disturbances are. We show that the regretoptimal control problem can be reduced to a Nehari extension problem, i.e., to approximate an anticausal operator with a causal one in the operator norm. In the state-space setting we obtain explicit formulas for the optimal regret and for the regret-optimal controller (in both the causal and the strictly causal settings). The regret-optimal controller is the sum of the classical H2 state-feedback law and an n-th order controller (where n is the state dimension of the plant) obtained from the Nehari problem. The controller construction simply requires the solution to the standard LQR Riccati equation, in addition to two Lyapunov equations. Simulations over a range of plants demonstrates that the regret-optimal controller interpolates nicely between the H2 and the H∞ optimal controllers, and generally has H2 and H∞ costs that are simultaneously close to their optimal values. The regret-optimal controller thus presents itself as a viable option for control system design.
Most modern learning problems are highly overparameterized, i.e., have many more model parameters than the number of training data points. As a result, the training loss may have infinitely many global minima (parameter vectors that perfectly "interpolate" the training data). It is therefore imperative to understand which interpolating solutions we converge to, how they depend on the initialization and learning algorithm, and whether they yield different test errors. In this article, we study these questions for the family of stochastic mirror descent (SMD) algorithms, of which stochastic gradient descent (SGD) is a special case. Recently, it has been shown that for overparameterized linear models, SMD converges to the closest global minimum to the initialization point, where closeness is in terms of the Bregman divergence corresponding to the potential function of the mirror descent. With appropriate initialization, this yields convergence to the minimum-potential interpolating solution, a phenomenon referred to as implicit regularization. On the theory side, we show that for sufficientlyoverparameterized nonlinear models, SMD with a (small enough) fixed step size converges to a global minimum that is "very close" (in Bregman divergence) to the minimum-potential interpolating solution, thus attaining approximate implicit regularization. On the empirical side, our experiments on the MNIST and CIFAR-10 datasets consistently confirm that the above phenomenon occurs in practical scenarios. They further indicate a clear difference in the generalization performances of different SMD algorithms: experiments on the CIFAR-10 dataset with different regularizers, 1 to encourage sparsity, 2 (SGD) to encourage small Euclidean norm, and ∞ to discourage large components, surprisingly show that the ∞ norm consistently yields better generalization performance than SGD, which in turn generalizes better than the 1 norm.
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