Naive Exploration is Optimal for Online LQR (2001.09576v4)

Abstract: We consider the problem of online adaptive control of the linear quadratic regulator, where the true system parameters are unknown. We prove new upper and lower bounds demonstrating that the optimal regret scales as $\widetilde{\Theta}({\sqrt{d_{\mathbf{u}}^2 d_{\mathbf{x}} T}})$, where $T$ is the number of time steps, $d_{\mathbf{u}}$ is the dimension of the input space, and $d_{\mathbf{x}}$ is the dimension of the system state. Notably, our lower bounds rule out the possibility of a $\mathrm{poly}(\log{}T)$-regret algorithm, which had been conjectured due to the apparent strong convexity of the problem. Our upper bound is attained by a simple variant of $\textit{{certainty equivalent control}}$, where the learner selects control inputs according to the optimal controller for their estimate of the system while injecting exploratory random noise. While this approach was shown to achieve $\sqrt{T}$-regret by (Mania et al. 2019), we show that if the learner continually refines their estimates of the system matrices, the method attains optimal dimension dependence as well. Central to our upper and lower bounds is a new approach for controlling perturbations of Riccati equations called the $\textit{self-bounding ODE method}$, which we use to derive suboptimality bounds for the certainty equivalent controller synthesized from estimated system dynamics. This in turn enables regret upper bounds which hold for $\textit{any stabilizable instance}$ and scale with natural control-theoretic quantities.

• The paper demonstrates that naive exploration strategies can achieve optimal performance in online Linear Quadratic Regulator (LQR) problems, challenging conventional wisdom.
• The authors provide rigorous theoretical analysis using perturbation bounds and a self-bounding ODE method to formally prove the optimality of this simple approach.
• This research implies potential reductions in system complexity for LQR applications and opens new avenues for exploring simple optimal strategies in other online learning domains.

Naive Exploration is Optimal for Online LQR

The paper "Naive Exploration is Optimal for Online LQR" by Max Simchowitz and Dylan J. Foster presents a rigorous exploration of the Linear Quadratic Regulator (LQR) problem from the perspective of online learning. The central thesis of the paper is the demonstration that naive exploration strategies can achieve optimal performance in online LQR settings, which contravenes conventional wisdom advocating for more complex exploration-exploitation strategies to optimize control performance.

Summary of Main Contributions

The paper's primary contribution lies in establishing naive exploration mechanisms as optimal for online LQR. The authors provide comprehensive theoretical analyses, demonstrating that the regret bound achieved by a straightforward exploration strategy matches the lower bounds under certain conditions. Through the application of perturbation bounds and the self-bounding ODE method, the authors rigorously substantiate their claims with robust mathematical proofs.

Methodological Framework

1. Problem Definition: The paper formulates the online LQR problem by using a quadratic cost function aimed at minimizing the cumulative cost over a sequence of time steps. The control task involves learning the system dynamics and generating control inputs that minimize this cost.
2. Naive Exploration Strategy: The crux of the exploration strategy revolves around adding stochastic perturbations to the control inputs. This approach contrasts with sophisticated exploration strategies that frequently involve intricate balance considerations between exploration and exploitation.
3. Theoretical Analysis:
   • The authors derive perturbation bounds that provide insights into the stability and performance guarantee of the naive exploration in the online LQR setting.
   • The self-bounding ODE method is employed to further assert the stability properties and bound the deviation introduced by the exploration noise.
4. Optimality Conditions: By comparing the upper bounds obtained from naive exploration against established lower bounds for online LQR, the paper demonstrates the optimality of the proposed approach. This result highlights the efficacy of naive exploration in reducing computational overhead and complexities without sacrificing performance.

Implications and Future Directions

The implication of this research is twofold. Practically, the results suggest that in contexts where LQR functionalities are employed, such as autonomous systems and robotics, implementing a naive exploration strategy can significantly reduce system complexity and resource consumption while maintaining optimal performance. Theoretically, this research challenges existing paradigms regarding exploration strategies in online learning, potentially stimulating new research aimed at uncovering similar optimal strategies in other domains.

For future exploration, the paper opens avenues for examining the applicability of naive exploration in more complex settings involving non-linear dynamics, or partially observable environments. Additionally, the potential to generalize these findings to other control frameworks or online learning algorithms could be an intriguing direction, thereby broadening the impact and applicability of these findings.

In summary, "Naive Exploration is Optimal for Online LQR" presents a compelling case for revisiting traditional assumptions about exploration in online control tasks. The result is a significant assertion regarding the efficiency of simple strategies in achieving optimal regulatory oversight within the specific context of LQR. The paper advances the discourse in control theory and online learning, setting a foundation for subsequent empirical and theoretical investigations.