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### Naive Exploration is Optimal for Online LQR

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 has been conjectured due to the apparent strong convexity of the problem. Our upper bounds are attained by a simple variant of \emph{certainty equivalence 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 Ricatti equations, which we call the \emph{self-bounding ODE method}. The approach enables regret upper bounds which hold for \emph{any stabilizable instance}, require no foreknowledge of the system except for a single stabilizing controller, and scale with natural control-theoretic quantities.

### Prediction in latent factor regression: Adaptive PCR and beyond

This work is devoted to the finite sample prediction risk analysis of a class of linear predictors of a response $Y\in \mathbb{R}$ from a high-dimensional random vector $X\in \mathbb{R}^p$ when $(X,Y)$ follows a latent factor regression model generated by a unobservable latent vector $Z$ of dimension less than $p$. Our primary contribution is in establishing finite sample risk bounds for prediction with the ubiquitous Principal Component Regression (PCR) method, under the factor regression model, with the number of principal components adaptively selected from the data---a form of theoretical guarantee that is surprisingly lacking from the PCR literature. To accomplish this, we prove a master theorem that establishes a risk bound for a large class of predictors, including the PCR predictor as a special case. This approach has the benefit of providing a unified framework for the analysis of a wide range of linear prediction methods, under the factor regression setting. In particular, we use our main theorem to recover known risk bounds for the minimum-norm interpolating predictor, which has received renewed attention in the past two years, and a prediction method tailored to a subclass of factor regression models with identifiable parameters. This model-tailored method can be interpreted as prediction via clusters with latent centers. To address the problem of selecting among a set of candidate predictors, we analyze a simple model selection procedure based on data-splitting, providing an oracle inequality under the factor model to prove that the performance of the selected predictor is close to the optimal candidate. We conclude with a detailed simulation study to support and complement our theoretical results.

### Improper Learning for Non-Stochastic Control

We consider the problem of controlling a possibly unknown linear dynamical system with adversarial perturbations, adversarially chosen convex loss functions, and partially observed states, known as non-stochastic control. We introduce a controller parametrization based on the denoised observations, and prove that applying online gradient descent to this parametrization yields a new controller which attains sublinear regret vs. a large class of closed-loop policies. In the fully-adversarial setting, our controller attains an optimal regret bound of $\sqrt{T}$-when the system is known, and, when combined with an initial stage of least-squares estimation, $T^{2/3}$ when the system is unknown; both yield the first sublinear regret for the partially observed setting. Our bounds are the first in the non-stochastic control setting that compete with \emph{all} stabilizing linear dynamical controllers, not just state feedback. Moreover, in the presence of semi-adversarial noise containing both stochastic and adversarial components, our controller attains the optimal regret bounds of $\mathrm{poly}(\log T)$ when the system is known, and $\sqrt{T}$ when unknown. To our knowledge, this gives the first end-to-end $\sqrt{T}$ regret for online Linear Quadratic Gaussian controller, and applies in a more general setting with adversarial losses and semi-adversarial noise.

### Efficient Learning of Distributed Linear-Quadratic Controllers

In this work, we propose a robust approach to design distributed controllers for unknown-but-sparse linear and time-invariant systems. By leveraging modern techniques in distributed controller synthesis and structured linear inverse problems as applied to system identification, we show that near-optimal distributed controllers can be learned with sub-linear sample complexity and computed with near-linear time complexity, both measured with respect to the dimension of the system. In particular, we provide sharp end-to-end guarantees on the stability and the performance of the designed distributed controller and prove that for sparse systems, the number of samples needed to guarantee robust and near optimal performance of the designed controller can be significantly smaller than the dimension of the system. Finally, we show that the proposed optimization problem can be solved to global optimality with near-linear time complexity by iteratively solving a series of small quadratic programs.

### Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data

The imperfectness of data acquisition processes, however, presents several common yet critical challenges: (1) random noise: which reflects the uncertainty of the environment and/or the measurement processes; (2) outliers: which represent a sort of corruption that exhibits abnormal behavior; and (3) incomplete data, namely, we might only get to observe a fraction of the matrix entries. Low-rank matrix estimation algorithms aimed at addressing these challenges have been extensively studied under the umbrella of robust principal component analysis (Robust PCA) [CSPW11, CLMW11], a terminology popularized by the seminal work [CLMW11]. To formulate the above-mentioned problem in a more precise manner, imagine that we seek to estimate an unknown low-rank matrix L null R n n . 1 What we can obtain is a collection of partially observed and corrupted entries as follows M ij L null ij S null ij E ij, (i,j) Ω obs, (1.1) where S null [ S null ij] 1 i,j n is a matrix consisting of outliers, E [ E ij] 1 i,j n represents the random noise, and we only observe entries over an index subset Ω obs [n ] [n ] with [n ]: {1, 2, ···,n }. The current paper assumes that S null is a relatively sparse matrix whose nonzero entries might have arbitrary magnitudes. This assumption has been commonly adopted in prior work to model gross outliers, while enabling reliable disentanglement of the outlier component and the low-rank component [CSPW11,CLMW11,CJSC13,Li13]. In addition, we suppose that the entries {E ij} are independent zero-mean sub-Gaussian random variables, as commonly assumed in the statistics literature to model a large family of random noise. The aim is to reliably estimate L null given the grossly corrupted and possibly incomplete data (1.1). Ideally, this task should be accomplished without knowing the locations and magnitudes of the outliers S null . 1 To avoid cluttered notation, this paper works with square matrices of size n by n. Our results and analysis can be extended to accommodate rectangular matrices.