Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm in engineering, computer science, economics, and neuroscience. It studies linear dynamical systems with imperfect observations and additive noise, with the goal of minimizing a quadratic cost in the state and control variables. In this work, we consider a distributionally robust generalization of discrete-time LQG control, where the noise distributions are unknown and belong to Wasserstein ambiguity sets centered at nominal Gaussian distributions. The objective is to minimize the worst-case cost over all distributions in the ambiguity set, including non-Gaussian distributions. Despite this added complexity, we prove that a control policy that is linear in the observations remains optimal, as in classical LQG control, and that the least-favorable distribution is Gaussian. We also extend the structural results to an infinite-horizon average-cost formulation, where stationary linear policies remain optimal and the worst-case distribution is time-invariant and Gaussian. Finally, we propose an efficient numerical method based on Frank-Wolfe iterations: each iteration identifies a least-favorable distribution within the ambiguity set and computes the corresponding optimal controller via Kalman filtering and dynamic programming.
The free energy principle from computational neuroscience proposes that adaptive behaviors emerge from minimizing variational free energy—a bound on the surprise in the data. The principle has been advanced as a unifying account across computational neuroscience, machine learning, and decision-making, and is increasingly invoked in control. In this talk I cast the free energy principle as an optimal control problem in the space of densities. From here, I introduce the Distributionally Robust Free Energy principle. This control-theoretic framework finds optimal policies that minimize the variational free energy across an ambiguity set centered around a nominal, trained model, providing robustness guarantees against training-environment mismatches. Remarkably, despite the apparent abstract setting of the control formulation, we prove it admits a closed-form solution and provide methods to provably compute optimal policies. I will illustrate these ideas with concrete learning and control examples. Across all benchmarks, our approach enables the agents to complete the task even when, in contrast, state-of-the-art methods fail.
We study the probabilistic performance of first-order methods on convex optimization problems drawn from an unknown distribution accessible only through samples. By combining performance estimation (PEP) with Wasserstein distributionally robust optimization (DRO), we formulate the analysis as a tractable semidefinite program. Our approach unifies worst-case and average-case analyses, producing probabilistic, data-driven guarantees on the expectation or conditional value-at-risk of the chosen performance metric. The same framework also enables principled algorithm design: by optimizing hyperparameters against the DRO objective, we interpolate between learning to optimize (L2O) and worst-case optimal design via PEP, recovering each as the robustness radius vanishes or grows. Experiments on smooth convex minimization, logistic regression, and Lasso show that our method sharply reduces the conservatism of classical worst-case bounds and yields algorithms with strong out-of-sample performance and certifiable robustness guarantees.
We study distributionally robust online learning, where a risk-averse learner updates decisions sequentially to guard against worst-case distributions drawn from a Wasserstein ambiguity set centered at past observations. While this paradigm is well understood in the offline setting through Wasserstein Distributionally Robust Optimization (DRO), its online extension poses significant challenges in both convergence and computation. In this paper, we address these challenges. First, we formulate the problem as an online saddle-point stochastic game between a decision maker and an adversary selecting worst-case distributions, and propose a general framework that converges to a robust Nash equilibrium coinciding with the solution of the corresponding offline Wasserstein DRO problem. Second, we address the main computational bottleneck, which is the repeated solution of worst-case expectation problems. For the important class of piecewise concave loss functions, we propose a tailored algorithm that exploits problem geometry to achieve substantial speedups over state-of-the-art solvers such as Gurobi. The key insight is a novel connection between the worst-case expectation problem, an inherently infinite-dimensional optimization problem, and a classical and tractable budget allocation problem, which is of independent interest.
We discuss distributionally robust games where each agent is allowed to choose their risk aversion with respect to heterogeneous uncertainties. The key idea is that heterogeneous Wasserstein ball constraints on each distribution can be enforced through a penalty function leveraging a Lagrangian formulation. Distributionally robust games can be then formulated as finite-dimensional monotone variational inequality problems. Nevertheless, due to the inner maximization problem, we discuss approximate distributionally robust equilibrium seeking and study convergence of the average regret, thus learning an equilibrium up to an a priori specified accuracy.
In many game-theoretic settings, agents are challenged with taking decisions against the uncertain behavior exhibited by others. Often, this uncertainty arises from multiple sources, e.g., incomplete information, limited computation, bounded rationality. While it may be possible to guide the agents’ decisions by modeling each source, their joint presence makes this task particularly daunting. Toward this goal, it is natural for agents to seek protection against deviations around the emergent behavior itself, which is ultimately impacted by all the above sources of uncertainty. To do so, we propose that each agent takes decisions in face of the worst-case behavior contained in an ambiguity set of tunable size, centered at the emergent behavior implicitly defined. This gives rise to a novel equilibrium notion, which we call strategically robust equilibrium. Building on its definition we show that, when judiciously operationalized via optimal transport, strategically robust equilibria (i) interpolate between Nash and security strategies; (ii) come at no additional computational cost compared to Nash equilibria; (iii) often lead to better decisions and higher payoffs. Through a variety of experiments including bi-matrix games, congestion games, and Cournot competition, we show that strategic robustness protects against uncertainty in the opponents’ behavior and, surprisingly, results in higher equilibrium payoffs---an effect we refer to as coordination via robustification.
Data centers are significant contributors to carbon emissions and can strain power systems due to their high electricity consumption. To mitigate this impact and to participate in demand response programs, cloud computing companies strive to balance and optimize operations across their global fleets by making strategic decisions about when and where to place compute jobs for execution. In this paper, we introduce a load shaping scheme which reacts to time-varying grid signals by leveraging both temporal and spatial flexibility of compute jobs to provide risk-aware management guidelines and job placement with provable performance guarantees based on distributionally robust optimization. Our approach divides the problem into two key components: (i) day-ahead planning, which generates an optimal scheduling strategy based on historical load data, and (ii) real-time job placement and (time) scheduling, which dynamically tracks the optimal strategy generated in (i). We validate our method in simulation using normalized load profiles from randomly selected Google clusters, incorporating time-varying grid signals. We can demonstrate significant reductions in carbon cost and peak power with our approach compared to myopic greedy policies, while maintaining computational efficiency and abiding to system and grid constraints.
This talk presents techniques for robot motion control in dynamic environments using onboard sensing and distributionally robust constraints to ensure safety. Our method introduces distributionally robust control barrier functions (DR-CBFs) constructed directly from noisy sensor measurements and uncertain state estimates to define safety constraints. We show that DR-CBF constraints can be imposed on a virtual dynamical system, called reference governor, which decouples control tracking and safety objectives and simplifies the design of safe stabilizing control in dynamic environments. We demonstrate applications to mobile robots and robot manipulators accounting for modeling errors, changing shape, uncertain states and real-time operation.