This paper investigates distributionally robust filtering for state-space models subject to exogenous disturbances in state evolution and observation processes. The joint probability distribution of the disturbance process over an arbitrary horizon is unknown but is assumed to reside within a Wasserstein-2 ambiguity set centered around a nominal distribution. We seek to develop a causal estimator that minimizes the worst-case mean squared error (MSE) among all distributions in this ambiguity set. Unlike prior works, our framework accommodates disturbances with arbitrary temporal correlations. In the finite-horizon setting, this problem reduces to a semi-definite program (SDP) whose complexity scales with the time horizon. Consequently, we focus on the infinite-horizon case, where we derive the optimal linear time-invariant (LTI) filter using the Karush-Kuhn-Tucker (KKT) conditions and propose an efficient frequency-domain algorithm to compute it. While the optimal LTI filter generally has a non-rational transfer function, we provide a minimax rational approximation method to approximate the optimal non-rational filter, in the H∞-norm, with a near-optimal finite-order state-space estimator. This approach avoids the computational challenges associated with horizon-dependent scaling in the finite-horizon case. Numerical simulations demonstrate the effectiveness of the proposed method.

We study the scalar quantization of random information sources with unknown probability distributions through the lens of distributional robustness. Assuming the true source distribution resides within an ambiguity set of plausible distributions, we formulate the problem of distributionally robust scalar quantization (DRQ) as finding a minimax optimal quantizer that minimizes the mean-squared quantization error (MSQE) under the least favorable distribution in the ambiguity set. Specifically, we define the ambiguity set using the Wasserstein-2 ($\mathrm{W}_{2}$) distance, a statistical measure of distribution shift, as a statistical ball centered around a nominal source distribution. We derive a tractable dual reformulation and establish the necessary optimality conditions for$\mathrm{W}_{2}$-DRQ. Building on these results, we propose a novel fixedpoint iteration algorithm, inspired by the classical Lloyd-Max algorithm, to compute locally optimal DR quantizers. We prove the monotonic convergence of the algorithm and demonstrate its effectiveness through numerical simulations. Our results highlight the performance advantages of the proposed approach under distributional uncertainty, particularly in comparison to the classical Lloyd-Max algorithm.

We study the infinite-horizon distributionally robust (DR) control of linear systems with quadratic costs, where disturbances have unknown, possibly time-correlated distribution within a Wasserstein-2 ambiguity set. We aim to minimize the worst-case expected regret—the excess cost of a causal policy compared to a non-causal one with access to future disturbance. Though the optimal policy lacks a finite-order state-space realization (i.e., it is non-rational), it can be characterized by a finite-dimensional parameter. Leveraging this, we develop an efficient frequency-domain algorithm to compute this optimal control policy and present a convex optimization method to construct a near-optimal state-space controller that approximates the optimal non-rational controller in the $\mathit{H}_\infty$-norm. This approach avoids solving a computationally expensive semi-definite program (SDP) that scales with the time horizon in the finite-horizon setting.

This paper studies uncertainty set estimation for unknown linear systems. Uncertainty sets are crucial for the quality of robust control since they directly influence the conservativeness of the control design. Departing from the confidence region analysis of least squares estimation, this paper focuses on set membership estimation (SME). Though good numerical performances have attracted applications of SME in the control literature, the non-asymptotic convergence rate of SME for linear systems remains an open question. This paper provides the first convergence rate bounds for SME and discusses variations of SME under relaxed assumptions. We also provide numerical results demonstrating SME’s practical promise.

We consider the rate-distortion function for lossy source compression, as well as the channel capacity for error cor-rection, through the lens of distributional robustness. We assume that the distribution of the source or of the additive channel noise is unknown and lies within a Wasserstein-2 ambiguity set of a given radius centered around a specified nominal distribution, and we look for the worst-case asymptotically optimal coding rate over such an ambiguity set. Varying the radius of the ambiguity set allows us to interpolate between the worst-case and stochastic scenarios using probabilistic tools. Our problem setting fits into the paradigm of compound source / channel models introduced by Sakrison [1] and Blackwell [2], respectively. This paper shows that if the nominal distribution is Gaussian, then so is the worst-case source / noise distribution, and the compound rate-distortion / channel capacity functions admit convex formulations with Linear Matrix Inequality (LMI) constraints. These formulations yield simple closed-form expressions in the scalar case, offering insights into the behavior of Shannon limits with the changing radius of the Wasserstein-2 ambiguity set.

The stochastic mirror descent (SMD) algorithm is a general class of training algorithms that utilizes a mirror potential to influence the implicit bias of the training algorithm and includes stochastic gradient descent (SGD) as a special case. In this paper, we explore the performance of the SMD on mean-field ensemble models and generalize earlier results obtained for SGD. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial differential equation (PDE) to which the continuous time process converges in the asymptotic of large networks. The impact of the mirror potential appears through a multiplicative term that is equal to the inverse of its Hessian and defines a gradient flow over an appropriate Riemannian manifold. We provide numerical simulations which allow us to study and characterize the effect of the mirror potential on the performance of networks trained with SMD for some binary classification problems.

Thompson Sampling (TS) is an efficient method for decision-making under uncertainty, where an action is sampled from a carefully prescribed distribution which is updated based on the observed data. In this work, we study the problem of adaptive control of stabilizable linear-quadratic regulators (LQRs) using TS, where the system dynamics are unknown. Previous works have established that $\tilde{O}(\sqrt{T})$ frequentist regret is optimal for the adaptive control of LQRs. However, the existing methods either work only in restrictive settings, require a priori known stabilizing controllers or utilize computationally intractable approaches. We propose an efficient TS algorithm for the adaptive control of LQRs, TS-based Adaptive Control, TSAC, that attains $\tilde{O}(\sqrt{T})$ regret, even for multidimensional systems, thereby solving the open problem posed in Abeille and Lazaric (2018). TSAC does not require a priori known stabilizing controller and achieves fast stabilization of the underlying system by effectively exploring the environment in the early stages. Our result hinges on developing a novel lower bound on the probability that the TS provides an optimistic sample. By carefully prescribing an early exploration strategy and a policy update rule, we show that TS achieves order-optimal regret in adaptive control of multidimensional stabilizable LQRs. We empirically demonstrate the performance and the efficiency of the proposed algorithm in several adaptive control tasks.