Nikolai Matni

Thomas T. Zhang, Alok Shah, Yifei Zhang et al.

Many modern applications of deep learning involve training a neural network via a one-step prediction loss (e.g., $L^2$ regression, cross-entropy), but deploy the network by rolling out along its own predictions. Key examples include autoregressive language modeling, flow-based generative modeling, and robot policy learning. It is well-documented that these settings induce a phenomenon we call test-time feedback (TTF): the mismatch between the training/validation loss and downstream metrics of interest, such as task success rate and generation quality, which grows with task length. While data curation, architecture, and objective design have been proposed to combat train-test shift in TTF settings, this paper proposes optimization as a new design axis to mitigate error accumulation. Specifically, we introduce a new optimization paradigm called double-preconditioning (DoPr) uniquely tailored to the challenges of TTF. DoPr combines gradient-wise preconditioning, as in Adam and Muon, with activation-wise preconditioning (AP), such as in KFAC. We show that the addition of AP yields a drop-in intervention for increasing downstream model performance across a range of TTF settings. Interestingly, these gains in test-time performance do not consistently accompany improvements in validation loss, opening new questions about how to properly evaluate models trained with one-step supervised objectives.

Anastasios Tsiamis, Ingvar Ziemann, Nikolai Matni et al.

This tutorial survey provides an overview of recent non-asymptotic advances in statistical learning theory as relevant to control and system identification. While there has been substantial progress across all areas of control, the theory is most well-developed when it comes to linear system identification and learning for the linear quadratic regulator, which are the focus of this manuscript. From a theoretical perspective, much of the labor underlying these advances has been in adapting tools from modern high-dimensional statistics and learning theory. While highly relevant to control theorists interested in integrating tools from machine learning, the foundational material has not always been easily accessible. To remedy this, we provide a self-contained presentation of the relevant material, outlining all the key ideas and the technical machinery that underpin recent results. We also present a number of open problems and future directions.

Anish Bhattacharya, Ratnesh Madaan, Fernando Cladera et al.

We present EvDNeRF, a pipeline for generating event data and training an event-based dynamic NeRF, for the purpose of faithfully reconstructing eventstreams on scenes with rigid and non-rigid deformations that may be too fast to capture with a standard camera. Event cameras register asynchronous per-pixel brightness changes at MHz rates with high dynamic range, making them ideal for observing fast motion with almost no motion blur. Neural radiance fields (NeRFs) offer visual-quality geometric-based learnable rendering, but prior work with events has only considered reconstruction of static scenes. Our EvDNeRF can predict eventstreams of dynamic scenes from a static or moving viewpoint between any desired timestamps, thereby allowing it to be used as an event-based simulator for a given scene. We show that by training on varied batch sizes of events, we can improve test-time predictions of events at fine time resolutions, outperforming baselines that pair standard dynamic NeRFs with event generators. We release our simulated and real datasets, as well as code for multi-view event-based data generation and the training and evaluation of EvDNeRF models (https://github.com/anish-bhattacharya/EvDNeRF).

Ingvar Ziemann, Anastasios Tsiamis, Bruce Lee et al.

This tutorial serves as an introduction to recently developed non-asymptotic methods in the theory of -- mainly linear -- system identification. We emphasize tools we deem particularly useful for a range of problems in this domain, such as the covering technique, the Hanson-Wright Inequality and the method of self-normalized martingales. We then employ these tools to give streamlined proofs of the performance of various least-squares based estimators for identifying the parameters in autoregressive models. We conclude by sketching out how the ideas presented herein can be extended to certain nonlinear identification problems.

Thomas T. Zhang, Katie Kang, Bruce D. Lee et al.

We study representation learning for efficient imitation learning over linear systems. In particular, we consider a setting where learning is split into two phases: (a) a pre-training step where a shared $k$-dimensional representation is learned from $H$ source policies, and (b) a target policy fine-tuning step where the learned representation is used to parameterize the policy class. We find that the imitation gap over trajectories generated by the learned target policy is bounded by $\tilde{O}\left( \frac{k n_x}{HN_{\mathrm{shared}}} + \frac{k n_u}{N_{\mathrm{target}}}\right)$, where $n_x > k$ is the state dimension, $n_u$ is the input dimension, $N_{\mathrm{shared}}$ denotes the total amount of data collected for each policy during representation learning, and $N_{\mathrm{target}}$ is the amount of target task data. This result formalizes the intuition that aggregating data across related tasks to learn a representation can significantly improve the sample efficiency of learning a target task. The trends suggested by this bound are corroborated in simulation.

Anastasios Tsiamis, Ingvar Ziemann, Manfred Morari et al.

In this paper, we study the statistical difficulty of learning to control linear systems. We focus on two standard benchmarks, the sample complexity of stabilization, and the regret of the online learning of the Linear Quadratic Regulator (LQR). Prior results state that the statistical difficulty for both benchmarks scales polynomially with the system state dimension up to system-theoretic quantities. However, this does not reveal the whole picture. By utilizing minimax lower bounds for both benchmarks, we prove that there exist non-trivial classes of systems for which learning complexity scales dramatically, i.e. exponentially, with the system dimension. This situation arises in the case of underactuated systems, i.e. systems with fewer inputs than states. Such systems are structurally difficult to control and their system theoretic quantities can scale exponentially with the system dimension dominating learning complexity. Under some additional structural assumptions (bounding systems away from uncontrollability), we provide qualitatively matching upper bounds. We prove that learning complexity can be at most exponential with the controllability index of the system, that is the degree of underactuation.

Haoze Wu, Teruhiro Tagomori, Alexander Robey et al.

We consider the problem of certifying the robustness of deep neural networks against real-world distribution shifts. To do so, we bridge the gap between hand-crafted specifications and realistic deployment settings by proposing a novel neural-symbolic verification framework, in which we train a generative model to learn perturbations from data and define specifications with respect to the output of the learned model. A unique challenge arising from this setting is that existing verifiers cannot tightly approximate sigmoid activations, which are fundamental to many state-of-the-art generative models. To address this challenge, we propose a general meta-algorithm for handling sigmoid activations which leverages classical notions of counter-example-guided abstraction refinement. The key idea is to "lazily" refine the abstraction of sigmoid functions to exclude spurious counter-examples found in the previous abstraction, thus guaranteeing progress in the verification process while keeping the state-space small. Experiments on the MNIST and CIFAR-10 datasets show that our framework significantly outperforms existing methods on a range of challenging distribution shifts.

Ingvar Ziemann, Anastasios Tsiamis, Henrik Sandberg et al.

We study stochastic policy gradient methods from the perspective of control-theoretic limitations. Our main result is that ill-conditioned linear systems in the sense of Doyle inevitably lead to noisy gradient estimates. We also give an example of a class of stable systems in which policy gradient methods suffer from the curse of dimensionality. Our results apply to both state feedback and partially observed systems.

Daniel Pfrommer, Thomas T. C. K. Zhang, Stephen Tu et al.

We propose Taylor Series Imitation Learning (TaSIL), a simple augmentation to standard behavior cloning losses in the context of continuous control. TaSIL penalizes deviations in the higher-order Taylor series terms between the learned and expert policies. We show that experts satisfying a notion of $\textit{incremental input-to-state stability}$ are easy to learn, in the sense that a small TaSIL-augmented imitation loss over expert trajectories guarantees a small imitation loss over trajectories generated by the learned policy. We provide sample-complexity bounds for TaSIL that scale as $\tilde{\mathcal{O}}(1/n)$ in the realizable setting, for $n$ the number of expert demonstrations. Finally, we demonstrate experimentally the relationship between the robustness of the expert policy and the order of Taylor expansion required in TaSIL, and compare standard Behavior Cloning, DART, and DAgger with TaSIL-loss-augmented variants. In all cases, we show significant improvement over baselines across a variety of MuJoCo tasks.

Thomas T. C. K. Zhang, Leonardo F. Toso, James Anderson et al.

A powerful concept behind much of the recent progress in machine learning is the extraction of common features across data from heterogeneous sources or tasks. Intuitively, using all of one's data to learn a common representation function benefits both computational effort and statistical generalization by leaving a smaller number of parameters to fine-tune on a given task. Toward theoretically grounding these merits, we propose a general setting of recovering linear operators $M$ from noisy vector measurements $y = Mx + w$, where the covariates $x$ may be both non-i.i.d. and non-isotropic. We demonstrate that existing isotropy-agnostic representation learning approaches incur biases on the representation update, which causes the scaling of the noise terms to lose favorable dependence on the number of source tasks. This in turn can cause the sample complexity of representation learning to be bottlenecked by the single-task data size. We introduce an adaptation, $\texttt{De-bias & Feature-Whiten}$ ($\texttt{DFW}$), of the popular alternating minimization-descent scheme proposed independently in Collins et al., (2021) and Nayer and Vaswani (2022), and establish linear convergence to the optimal representation with noise level scaling down with the $\textit{total}$ source data size. This leads to generalization bounds on the same order as an oracle empirical risk minimizer. We verify the vital importance of $\texttt{DFW}$ on various numerical simulations. In particular, we show that vanilla alternating-minimization descent fails catastrophically even for iid, but mildly non-isotropic data. Our analysis unifies and generalizes prior work, and provides a flexible framework for a wider range of applications, such as in controls and dynamical systems.

David Brandfonbrener, Stephen Tu, Avi Singh et al.

We consider how to most efficiently leverage teleoperator time to collect data for learning robust image-based value functions and policies for sparse reward robotic tasks. To accomplish this goal, we modify the process of data collection to include more than just successful demonstrations of the desired task. Instead we develop a novel protocol that we call Visual Backtracking Teleoperation (VBT), which deliberately collects a dataset of visually similar failures, recoveries, and successes. VBT data collection is particularly useful for efficiently learning accurate value functions from small datasets of image-based observations. We demonstrate VBT on a real robot to perform continuous control from image observations for the deformable manipulation task of T-shirt grasping. We find that by adjusting the data collection process we improve the quality of both the learned value functions and policies over a variety of baseline methods for data collection. Specifically, we find that offline reinforcement learning on VBT data outperforms standard behavior cloning on successful demonstration data by 13% when both methods are given equal-sized datasets of 60 minutes of data from the real robot.

Shaoru Chen, Kong Yao Chee, Nikolai Matni et al.

With the increase in data availability, it has been widely demonstrated that neural networks (NN) can capture complex system dynamics precisely in a data-driven manner. However, the architectural complexity and nonlinearity of the NNs make it challenging to synthesize a provably safe controller. In this work, we propose a novel safety filter that relies on convex optimization to ensure safety for a NN system, subject to additive disturbances that are capable of capturing modeling errors. Our approach leverages tools from NN verification to over-approximate NN dynamics with a set of linear bounds, followed by an application of robust linear MPC to search for controllers that can guarantee robust constraint satisfaction. We demonstrate the efficacy of the proposed framework numerically on a nonlinear pendulum system.

Aditya Gahlawat, Ahmed Aboudonia, Sandeep Banik et al.

Imitation learning (IL) enables autonomous behavior by learning from expert demonstrations. While more sample-efficient than comparative alternatives like reinforcement learning, IL is sensitive to compounding errors induced by distribution shifts. There are two significant sources of distribution shifts when using IL-based feedback laws on systems: distribution shifts caused by policy error and distribution shifts due to exogenous disturbances and endogenous model errors due to lack of learning. Our previously developed approaches, Taylor Series Imitation Learning (TaSIL) and $\mathcal{L}_1$ -Distributionally Robust Adaptive Control (\ellonedrac), address the challenge of distribution shifts in complementary ways. While TaSIL offers robustness against policy error-induced distribution shifts, \ellonedrac offers robustness against distribution shifts due to aleatoric and epistemic uncertainties. To enable certifiable IL for learned and/or uncertain dynamical systems, we formulate \textit{Distributionally Robust Imitation Policy (DRIP)} architecture, a Layered Control Architecture (LCA) that integrates TaSIL and~\ellonedrac. By judiciously designing individual layer-centric input and output requirements, we show how we can guarantee certificates for the entire control pipeline. Our solution paves the path for designing fully certifiable autonomy pipelines, by integrating learning-based components, such as perception, with certifiable model-based decision-making through the proposed LCA approach.

Kong Yao Chee, M. Ani Hsieh, Nikolai Matni

Nonlinear model predictive control (MPC) is a flexible and increasingly popular framework used to synthesize feedback control strategies that can satisfy both state and control input constraints. In this framework, an optimization problem, subjected to a set of dynamics constraints characterized by a nonlinear dynamics model, is solved at each time step. Despite its versatility, the performance of nonlinear MPC often depends on the accuracy of the dynamics model. In this work, we leverage deep learning tools, namely knowledge-based neural ordinary differential equations (KNODE) and deep ensembles, to improve the prediction accuracy of this model. In particular, we learn an ensemble of KNODE models, which we refer to as the KNODE ensemble, to obtain an accurate prediction of the true system dynamics. This learned model is then integrated into a novel learning-enhanced nonlinear MPC framework. We provide sufficient conditions that guarantees asymptotic stability of the closed-loop system and show that these conditions can be implemented in practice. We show that the KNODE ensemble provides more accurate predictions and illustrate the efficacy and closed-loop performance of the proposed nonlinear MPC framework using two case studies.

James Anderson, Nikolai Matni, Yuxiao Chen

Typically when designing distributed controllers it is assumed that the state-space model of the plant consists of sparse matrices. However, in the discrete-time setting, if one begins with a continuous-time model, the discretization process annihilates any sparsity in the model. In this work we propose a discretization procedure that maintains the sparsity of the continuous-time model. We show that this discretization out-performs a simple truncation method in terms of its ability to approximate the "ground truth" model. Leveraging results from numerical analysis we are also able to upper-bound the error between the dense discretization and our method. Furthermore, we show that in a robust control setting we can design a distributed controller on the approximate (sparse) model that stabilizes the dense ground truth model.

Omid Mirzaeedodangeh, Eliot Shekhtman, Nikolai Matni et al.

Safe planning of an autonomous agent in interactive environments -- such as the control of a self-driving vehicle among pedestrians -- poses a major challenge as the behavior of the environment is unknown and reactive to the behavior of the autonomous agent. This coupling gives rise to interaction-driven distribution shifts where the autonomous agent's control policy may change the environment's behavior, thereby invalidating safety guarantees in existing work. Indeed, recent works have used conformal prediction (CP) to generate distribution-free safety guarantees using observed data of the environment. However, CP's assumption on data exchangeability is violated in interactive settings due to a circular dependency where a control policy update changes the environment's behavior, and vice versa. To address this gap, we propose an iterative framework that robustly maintains safety guarantees across policy updates by quantifying the potential impact of a planned policy update on the environment's behavior. We realize this via adversarially robust CP where we perform a regular CP step in each episode using observed data under the current policy, but then transfer safety guarantees across policy updates by analytically adjusting the CP result to account for distribution shifts. This adjustment is performed based on a policy-to-trajectory sensitivity analysis, resulting in a safe, episodic open-loop planner. We further conduct a contraction analysis of the system providing conditions under which both the CP results and the policy updates are guaranteed to converge. We empirically demonstrate these safety and convergence guarantees on a two-dimensional car-pedestrian and a high-dimensional quadcopter case study. To the best of our knowledge, these are the first results that provide valid safety guarantees in such interactive settings.

Chris Verhoek, Nikolai Matni

Computational constraints permeate the controller design process, and yet are rarely treated as explicit design constraints. Towards addressing this gap, we propose a quantitative framework that captures the effects of common design approximations, such as model order reduction, temporal discretization, horizon truncation, and solver accuracy, on both controller performance and computational requirements. Our framework highlights that these approximations are tunable parameters within an overall controller design process. By leveraging incremental input-to-state stability, we show that bounding the aggregate effects of these approximations reduces to verifying a design-dependent sector bound on the difference between the deployed policy and an idealized baseline, with stability enforced via a small-gain condition. We operationalize these insights via a Design Meta-Problem in which the performance gap is minimized subject to stability, real-time compute, and timing constraints. Finally, we instantiate the framework on a receding horizon LQR case study, and demonstrate a principled near-optimal navigation of tradeoffs among sampling rate, model order, horizon length, and solver iterations.

David Snyder, Apurva Badithela, Nikolai Matni et al.

Generalist robot manipulation policies are becoming increasingly capable, but are limited in evaluation to a small number of hardware rollouts. This strong resource constraint in real-world testing necessitates both more informative performance measures and reliable and efficient evaluation procedures to properly assess model capabilities and benchmark progress in the field. This work presents a novel framework for robot policy comparison that is sample-efficient, statistically rigorous, and applicable to a broad set of evaluation metrics used in practice. Based on safe, anytime-valid inference (SAVI), our test procedure is sequential, allowing the evaluator to stop early when sufficient statistical evidence has accumulated to reach a decision at a pre-specified level of confidence. Unlike previous work developed for binary success, our unified approach addresses a wide range of informative metrics: from discrete partial credit task progress to continuous measures of episodic reward or trajectory smoothness, spanning both parametric and nonparametric comparison problems. Through extensive validation on simulated and real-world evaluation data, we demonstrate up to 70% reduction in evaluation burden compared to standard batch methods and up to 50% reduction compared to state-of-the-art sequential procedures designed for binary outcomes, with no loss of statistical rigor. Notably, our empirical results show that competing policies can be separated more quickly when using fine-grained task progress than binary success metrics.

Bruce D. Lee, Leonardo F. Toso, Thomas T. Zhang et al.

Representation learning is a powerful tool that enables learning over large multitudes of agents or domains by enforcing that all agents operate on a shared set of learned features. However, many robotics or controls applications that would benefit from collaboration operate in settings with changing environments and goals, whereas most guarantees for representation learning are stated for static settings. Toward rigorously establishing the benefit of representation learning in dynamic settings, we analyze the regret of multi-task representation learning for linear-quadratic control. This setting introduces unique challenges. Firstly, we must account for and balance the $\textit{misspecification}$ introduced by an approximate representation. Secondly, we cannot rely on the parameter update schemes of single-task online LQR, for which least-squares often suffices, and must devise a novel scheme to ensure sufficient improvement. We demonstrate that for settings where exploration is "benign", the regret of any agent after $T$ timesteps scales as $\tilde O(\sqrt{T/H})$, where $H$ is the number of agents. In settings with "difficult" exploration, the regret scales as $\tilde O(\sqrt{d_u d_θ} \sqrt{T} + T^{3/4}/H^{1/5})$, where $d_x$ is the state-space dimension, $d_u$ is the input dimension, and $d_θ$ is the task-specific parameter count. In both cases, by comparing to the minimax single-task regret $O(\sqrt{d_x d_u^2}\sqrt{T})$, we see a benefit of a large number of agents. Notably, in the difficult exploration case, by sharing a representation across tasks, the effective task-specific parameter count can often be small $d_θ< d_x d_u$. Lastly, we provide numerical validation of the trends we predict.

Ingvar Ziemann, Nikolai Matni, George J. Pappas

How many parameters are required for a model to execute a given task? It has been argued that large language models, pre-trained via self-supervised learning, exhibit emergent capabilities such as multi-step reasoning as their number of parameters reach a critical scale. In the present work, we explore whether this phenomenon can analogously be replicated in a simple theoretical model. We show that the problem of learning linear dynamical systems -- a simple instance of self-supervised learning -- exhibits a corresponding phase transition. Namely, for every non-ergodic linear system there exists a critical threshold such that a learner using fewer parameters than said threshold cannot achieve bounded error for large sequence lengths. Put differently, in our model we find that tasks exhibiting substantial long-range correlation require a certain critical number of parameters -- a phenomenon akin to emergence. We also investigate the role of the learner's parametrization and consider a simple version of a linear dynamical system with hidden state -- an imperfectly observed random walk in $\mathbb{R}$. For this situation, we show that there exists no learner using a linear filter which can succesfully learn the random walk unless the filter length exceeds a certain threshold depending on the effective memory length and horizon of the problem.

Arvin Ghavidel, Pooria Namyar, Nikolai Matni et al.

Most deployed WAN Traffic Engineering (TE) systems use a logically centralized controller that periodically gathers traffic demands, runs a TE optimization or heuristic, and then programs the network. At scale, these solutions can be sub-optimal, and can take minutes to react to demand changes or failures. In this paper, we introduce OnlineTE, a system that reacts immediately to demand changes and failures, and delivers near-optimal solutions within seconds of a change. OnlineTE builds on the theory of optimization decomposition to devise scalable, near-optimal, distributed TE solvers for path-based MLU and Max-flow problems. In OnlineTE, each switch solves part of the optimization, and a central coordinator orchestrates the progress of the switches. As such, a switch can trigger a re-optimization as soon as it notices a demand change or failure, enabling high reactivity. OnlineTE scales to large WANs, and its compute requirements are well below the capabilities of modern WAN switches. It also enables a new opportunity, edge-based TE, which can utilize resources more efficiently than today's path-based approaches. On a testbed emulation of a 750-node WAN topology, OnlineTE can outperform the state-of-the-art by up to an order of magnitude.

Anne Somalwar, Bruce D. Lee, George J. Pappas et al.

Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. A common remedy is to train multi-step predictors directly instead of rolling out single-step models. However, it is unclear when the benefits of multi-step predictors outweigh the difficulty of learning a more complex model. We provide the first quantitative analysis of this trade-off for linear dynamical systems. We study three predictor classes: (i) single step models, (ii) multi-step models, and (iii) single step models trained with multi-step losses. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve the lowest asymptotic prediction error. On the other hand, when the model class is misspecified due to partial observability, direct multi-step predictors can significantly reduce bias and improve accuracy. We provide theoretical and empirical evidence that these trade-offs persist when predictors are used in closed-loop control.

Alex Nguyen-Le, Nikolai Matni

Domain randomization is a simple, effective, and flexible scheme for obtaining robust feedback policies aimed at reducing the sim-to-real gap due to model mismatch. While domain randomization methods have yielded impressive demonstrations in the robotics-learning literature, general and theoretically motivated principles for designing optimization schemes that effectively leverage the randomization are largely unexplored. We address this gap by considering a stochastic policy gradient descent method for the domain randomized linear-quadratic regulator synthesis problem, a situation simple enough to provide theoretical guarantees. In particular, we demonstrate that stochastic gradients obtained by repeatedly sampling new systems at each gradient step converge to global optima with appropriate hyperparameters choices, and yield better controllers with lower variability in the final controllers when compared to approaches that do not resample. Sampling is often a quick and cheap operation, so computing policy gradients with newly sampled systems at each iteration is preferable to evaluating gradients on a fixed set of systems.

Bo Wu, Bruce D. Lee, Kostas Daniilidis et al.

Large-scale robotic policies trained on data from diverse tasks and robotic platforms hold great promise for enabling general-purpose robots; however, reliable generalization to new environment conditions remains a major challenge. Toward addressing this challenge, we propose a novel approach for uncertainty-aware deployment of pre-trained language-conditioned imitation learning agents. Specifically, we use temperature scaling to calibrate these models and exploit the calibrated model to make uncertainty-aware decisions by aggregating the local information of candidate actions. We implement our approach in simulation using three such pre-trained models, and showcase its potential to significantly enhance task completion rates. The accompanying code is accessible at the link: https://github.com/BobWu1998/uncertainty_quant_all.git

Fengjun Yang, Nikolai Matni

We propose a reinforcement learning (RL)-based algorithm to jointly train (1) a trajectory planner and (2) a tracking controller in a layered control architecture. Our algorithm arises naturally from a rewrite of the underlying optimal control problem that lends itself to an actor-critic learning approach. By explicitly learning a \textit{dual} network to coordinate the interaction between the planning and tracking layers, we demonstrate the ability to achieve an effective consensus between the two components, leading to an interpretable policy. We theoretically prove that our algorithm converges to the optimal dual network in the Linear Quadratic Regulator (LQR) setting and empirically validate its applicability to nonlinear systems through simulation experiments on a unicycle model.

Fengjun Yang, Jake Welde, Nikolai Matni

We study distributed control for a network of nonlinear, differentially flat subsystems subject to dynamic coupling. Although differential flatness simplifies planning and control for isolated subsystems, the presence of coupling can destroy this property for the overall joint system. Focusing on subsystems in pure-feedback form, we identify a class of compatible lower-triangular dynamic couplings that preserve flatness and guarantee that the flat outputs of the subsystems remain the flat outputs of the coupled system. Further, we show that the joint flatness diffeomorphism can be constructed from those of the individual subsystems and, crucially, its sparsity structure reflects that of the coupling. Exploiting this structure, we synthesize a distributed tracking controller that computes control actions from local information only, thereby ensuring scalability. We validate our proposed framework on a simulated example of planar quadrotors dynamically coupled via aerodynamic downwash, and show that the distributed controller achieves accurate trajectory tracking.

Abriana Stewart-Height, Seema Jahagirdar, Nikolai Matni

Operations in hazardous environments put humans, animals, and machines at high risk for physically damaging consequences. In contrast to humans and animals, quadruped robots cannot naturally identify and adjust their locomotion to a severely debilitated limb. The ability to detect limb damage and adjust movement to a new physical morphology is the difference between survival and death for humans and animals. The same can be said for quadruped robots autonomously carrying out remote assignments in dynamic, complex settings. This work presents the development and implementation of an off-line learning-based method to detect single limb faults from proprioceptive sensor data in a quadrupedal robot. The aim of the fault detection technique is to provide the correct output for the controller to select the appropriate tripedal gait to use given the robot's current physical morphology.

Anish Bhattacharya, Nishanth Rao, Dhruv Parikh et al.

We demonstrate the capabilities of an attention-based end-to-end approach for high-speed vision-based quadrotor obstacle avoidance in dense, cluttered environments, with comparison to various state-of-the-art learning architectures. Quadrotor unmanned aerial vehicles (UAVs) have tremendous maneuverability when flown fast; however, as flight speed increases, traditional model-based approaches to navigation via independent perception, mapping, planning, and control modules breaks down due to increased sensor noise, compounding errors, and increased processing latency. Thus, learning-based, end-to-end vision-to-control networks have shown to have great potential for online control of these fast robots through cluttered environments. We train and compare convolutional, U-Net, and recurrent architectures against vision transformer (ViT) models for depth image-to-control in high-fidelity simulation, observing that ViT models are more effective than others as quadrotor speeds increase and in generalization to unseen environments, while the addition of recurrence further improves performance while reducing quadrotor energy cost across all tested flight speeds. We assess performance at speeds of up to 7m/s in simulation and hardware. To the best of our knowledge, this is the first work to utilize vision transformers for end-to-end vision-based quadrotor control.

Bruce D. Lee, Anastasios Tsiamis, Nikolai Matni et al.

Learning methods are increasingly used to synthesize controllers from data, yet existing sample-complexity characterizations for continuous control are sharp only in the fully observed setting. This paper studies the partially observed case by deriving information-theoretic lower bounds for learning Linear Quadratic Gaussian (LQG) controllers from offline trajectories generated by a (linear) exploration policy. We prove an $\varepsilon$-local minimax excess-cost lower bound that applies to any algorithm mapping the offline dataset to a stabilizing linear controller. The bound is expressed in terms of the Hessian of the LQG cost with respect to model parameters and the inverse Fisher Information induced by the exploration policy. We further provide system-theoretic characterizations of these objects, enabling transparent construction of hard instances. Instantiating the bound on classical fragile robust-control examples, including variants of the Doyle LQG fragility counterexample and non-minimum-phase systems, demonstrates when fragile robust control problems translate into high sample complexity for learning-enabled control. These results suggest the asymptotic optimality of certainty-equivalent synthesis and motivate the importance of both task-directed experiment design and system co-design for sample-efficient learning in partially observed control.

Bruce D. Lee, Ingvar Ziemann, George J. Pappas et al.

Model-based reinforcement learning is an effective approach for controlling an unknown system. It is based on a longstanding pipeline familiar to the control community in which one performs experiments on the environment to collect a dataset, uses the resulting dataset to identify a model of the system, and finally performs control synthesis using the identified model. As interacting with the system may be costly and time consuming, targeted exploration is crucial for developing an effective control-oriented model with minimal experimentation. Motivated by this challenge, recent work has begun to study finite sample data requirements and sample efficient algorithms for the problem of optimal exploration in model-based reinforcement learning. However, existing theory and algorithms are limited to model classes which are linear in the parameters. Our work instead focuses on models with nonlinear parameter dependencies, and presents the first finite sample analysis of an active learning algorithm suitable for a general class of nonlinear dynamics. In certain settings, the excess control cost of our algorithm achieves the optimal rate, up to logarithmic factors. We validate our approach in simulation, showcasing the advantage of active, control-oriented exploration for controlling nonlinear systems.

Ingvar Ziemann, Stephen Tu, George J. Pappas et al.

In this work, we study statistical learning with dependent ($β$-mixing) data and square loss in a hypothesis class $\mathscr{F}\subset L_{Ψ_p}$ where $Ψ_p$ is the norm $\|f\|_{Ψ_p} \triangleq \sup_{m\geq 1} m^{-1/p} \|f\|_{L^m} $ for some $p\in [2,\infty]$. Our inquiry is motivated by the search for a sharp noise interaction term, or variance proxy, in learning with dependent data. Absent any realizability assumption, typical non-asymptotic results exhibit variance proxies that are deflated multiplicatively by the mixing time of the underlying covariates process. We show that whenever the topologies of $L^2$ and $Ψ_p$ are comparable on our hypothesis class $\mathscr{F}$ -- that is, $\mathscr{F}$ is a weakly sub-Gaussian class: $\|f\|_{Ψ_p} \lesssim \|f\|_{L^2}^η$ for some $η\in (0,1]$ -- the empirical risk minimizer achieves a rate that only depends on the complexity of the class and second order statistics in its leading term. Our result holds whether the problem is realizable or not and we refer to this as a \emph{near mixing-free rate}, since direct dependence on mixing is relegated to an additive higher order term. We arrive at our result by combining the above notion of a weakly sub-Gaussian class with mixed tail generic chaining. This combination allows us to compute sharp, instance-optimal rates for a wide range of problems. Examples that satisfy our framework include sub-Gaussian linear regression, more general smoothly parameterized function classes, finite hypothesis classes, and bounded smoothness classes.

Bruce D. Lee, Anders Rantzer, Nikolai Matni

The strategy of pre-training a large model on a diverse dataset, then fine-tuning for a particular application has yielded impressive results in computer vision, natural language processing, and robotic control. This strategy has vast potential in adaptive control, where it is necessary to rapidly adapt to changing conditions with limited data. Toward concretely understanding the benefit of pre-training for adaptive control, we study the adaptive linear quadratic control problem in the setting where the learner has prior knowledge of a collection of basis matrices for the dynamics. This basis is misspecified in the sense that it cannot perfectly represent the dynamics of the underlying data generating process. We propose an algorithm that uses this prior knowledge, and prove upper bounds on the expected regret after $T$ interactions with the system. In the regime where $T$ is small, the upper bounds are dominated by a term that scales with either $\texttt{poly}(\log T)$ or $\sqrt{T}$, depending on the prior knowledge available to the learner. When $T$ is large, the regret is dominated by a term that grows with $δT$, where $δ$ quantifies the level of misspecification. This linear term arises due to the inability to perfectly estimate the underlying dynamics using the misspecified basis, and is therefore unavoidable unless the basis matrices are also adapted online. However, it only dominates for large $T$, after the sublinear terms arising due to the error in estimating the weights for the basis matrices become negligible. We provide simulations that validate our analysis. Our simulations also show that offline data from a collection of related systems can be used as part of a pre-training stage to estimate a misspecified dynamics basis, which is in turn used by our adaptive controller.

Thomas T. Zhang, Behrad Moniri, Ansh Nagwekar et al.

Layer-wise preconditioning methods are a family of memory-efficient optimization algorithms that introduce preconditioners per axis of each layer's weight tensors. These methods have seen a recent resurgence, demonstrating impressive performance relative to entry-wise ("diagonal") preconditioning methods such as Adam(W) on a wide range of neural network optimization tasks. Complementary to their practical performance, we demonstrate that layer-wise preconditioning methods are provably necessary from a statistical perspective. To showcase this, we consider two prototypical models, linear representation learning and single-index learning, which are widely used to study how typical algorithms efficiently learn useful features to enable generalization. In these problems, we show SGD is a suboptimal feature learner when extending beyond ideal isotropic inputs $\mathbf{x} \sim \mathsf{N}(\mathbf{0}, \mathbf{I})$ and well-conditioned settings typically assumed in prior work. We demonstrate theoretically and numerically that this suboptimality is fundamental, and that layer-wise preconditioning emerges naturally as the solution. We further show that standard tools like Adam preconditioning and batch-norm only mildly mitigate these issues, supporting the unique benefits of layer-wise preconditioning.

Hanli Zhang, Anusha Srikanthan, Spencer Folk et al.

Motivated by the increasing use of quadrotors for payload delivery, we consider a joint trajectory generation and feedback control design problem for a quadrotor experiencing aerodynamic wrenches. Unmodeled aerodynamic drag forces from carried payloads can lead to catastrophic outcomes. Prior work model aerodynamic effects as residual dynamics or external disturbances in the control problem leading to a reactive policy that could be catastrophic. Moreover, redesigning controllers and tuning control gains on hardware platforms is a laborious effort. In this paper, we argue that adapting the trajectory generation component keeping the controller fixed can improve trajectory tracking for quadrotor systems experiencing drag forces. To achieve this, we formulate a drag-aware planning problem by applying a suitable relaxation to an optimal quadrotor control problem, introducing a tracking cost function which measures the ability of a controller to follow a reference trajectory. This tracking cost function acts as a regularizer in trajectory generation and is learned from data obtained from simulation. Our experiments in both simulation and on the Crazyflie hardware platform show that changing the planner reduces tracking error by as much as 83%. Evaluation on hardware demonstrates that our planned path, as opposed to a baseline, avoids controller saturation and catastrophic outcomes during aggressive maneuvers.

Anne Somalwar, Bruce D. Lee, George J. Pappas et al.

Compounding error, where small prediction mistakes accumulate over time, presents a major challenge in learning-based control. For example, this issue often limits the performance of model-based reinforcement learning and imitation learning. One common approach to mitigate compounding error is to train multi-step predictors directly, rather than relying on autoregressive rollout of a single-step model. However, it is not well understood when the benefits of multi-step prediction outweigh the added complexity of learning a more complicated model. In this work, we provide a rigorous analysis of this trade-off in the context of linear dynamical systems. We show that when the model class is well-specified and accurately captures the system dynamics, single-step models achieve lower asymptotic prediction error. On the other hand, when the model class is misspecified due to partial observability, direct multi-step predictors can significantly reduce bias and thus outperform single-step approaches. These theoretical results are supported by numerical experiments, wherein we also (a) empirically evaluate an intermediate strategy which trains a single-step model using a multi-step loss and (b) evaluate performance of single step and multi-step predictors in a closed loop control setting.

Thomas T. Zhang, Bruce D. Lee, Ingvar Ziemann et al.

A driving force behind the diverse applicability of modern machine learning is the ability to extract meaningful features across many sources. However, many practical domains involve data that are non-identically distributed across sources, and statistically dependent within its source, violating vital assumptions in existing theoretical studies. Toward addressing these issues, we establish statistical guarantees for learning general $\textit{nonlinear}$ representations from multiple data sources that admit different input distributions and possibly dependent data. Specifically, we study the sample-complexity of learning $T+1$ functions $f_\star^{(t)} \circ g_\star$ from a function class $\mathcal F \times \mathcal G$, where $f_\star^{(t)}$ are task specific linear functions and $g_\star$ is a shared nonlinear representation. A representation $\hat g$ is estimated using $N$ samples from each of $T$ source tasks, and a fine-tuning function $\hat f^{(0)}$ is fit using $N'$ samples from a target task passed through $\hat g$. We show that when $N \gtrsim C_{\mathrm{dep}} (\mathrm{dim}(\mathcal F) + \mathrm{C}(\mathcal G)/T)$, the excess risk of $\hat f^{(0)} \circ \hat g$ on the target task decays as $ν_{\mathrm{div}} \big(\frac{\mathrm{dim}(\mathcal F)}{N'} + \frac{\mathrm{C}(\mathcal G)}{N T} \big)$, where $C_{\mathrm{dep}}$ denotes the effect of data dependency, $ν_{\mathrm{div}}$ denotes an (estimatable) measure of $\textit{task-diversity}$ between the source and target tasks, and $\mathrm C(\mathcal G)$ denotes the complexity of the representation class $\mathcal G$. In particular, our analysis reveals: as the number of tasks $T$ increases, both the sample requirement and risk bound converge to that of $r$-dimensional regression as if $g_\star$ had been given, and the effect of dependency only enters the sample requirement, leaving the risk bound matching the iid setting.

Tesshu Fujinami, Bruce D. Lee, Nikolai Matni et al.

Domain randomization (DR) enables sim-to-real transfer by training controllers on a distribution of simulated environments, with the goal of achieving robust performance in the real world. Although DR is widely used in practice and is often solved using simple policy gradient (PG) methods, understanding of its theoretical guarantees remains limited. Toward addressing this gap, we provide the first convergence analysis of PG methods for domain-randomized linear quadratic regulation (LQR). We show that PG converges globally to the minimizer of a finite-sample approximation of the DR objective under suitable bounds on the heterogeneity of the sampled systems. We also quantify the sample-complexity associated with achieving a small performance gap between the sample-average and population-level objectives. Additionally, we propose and analyze a discount-factor annealing algorithm that obviates the need for an initial jointly stabilizing controller, which may be challenging to find. Empirical results support our theoretical findings and highlight promising directions for future work, including risk-sensitive DR formulations and stochastic PG algorithms.

Eliot Shekhtman, Yichen Zhou, Ingvar Ziemann et al.

Learning from temporally-correlated data is a core facet of modern machine learning. Yet our understanding of sequential learning remains incomplete, particularly in the multi-trajectory setting where data consists of many independent realizations of a time-indexed stochastic process. This important regime both reflects modern training pipelines such as for large foundation models, and offers the potential for learning without the typical mixing assumptions made in the single-trajectory case. However, instance-optimal bounds are known only for least-squares regression with dependent covariates; for more general models or loss functions, the only broadly applicable guarantees result from a reduction to either i.i.d. learning, with effective sample size scaling only in the number of trajectories, or an existing single-trajectory result when each individual trajectory mixes, with effective sample size scaling as the full data budget deflated by the mixing-time. In this work, we significantly broaden the scope of instance-optimal rates in multi-trajectory settings via the Hellinger localization framework, a general approach for maximum likelihood estimation. Our method proceeds by first controlling the squared Hellinger distance at the path-measure level via a reduction to i.i.d. learning, followed by localization as a quadratic form in parameter space weighted by the trajectory Fisher information. This yields instance-optimal bounds that scale with the full data budget under a broad set of conditions. We instantiate our framework across four diverse case studies: a simple mixture of Markov chains, dependent linear regression under non-Gaussian noise, generalized linear models with non-monotonic activations, and linear-attention sequence models. In all cases, our bounds nearly match the instance-optimal rates from asymptotic normality, substantially improving over standard reductions.

Thomas T. Zhang, Daniel Pfrommer, Chaoyi Pan et al.

This paper presents a theoretical analysis of two of the most impactful interventions in modern learning from demonstration in robotics and continuous control: the practice of action-chunking (predicting sequences of actions in open-loop) and exploratory augmentation of expert demonstrations. Though recent results show that learning from demonstration, also known as imitation learning (IL), can suffer errors that compound exponentially with task horizon in continuous settings, we demonstrate that action chunking and exploratory data collection circumvent exponential compounding errors in different regimes. Our results identify control-theoretic stability as the key mechanism underlying the benefits of these interventions. On the empirical side, we validate our predictions and the role of control-theoretic stability through experimentation on popular robot learning benchmarks. On the theoretical side, we demonstrate that the control-theoretic lens provides fine-grained insights into how compounding error arises, leading to tighter statistical guarantees on imitation learning error when these interventions are applied than previous techniques based on information-theoretic considerations alone.

James Wang, Bruce D. Lee, Ingvar Ziemann et al.

We address the problem of learning to control an unknown nonlinear dynamical system through sequential interactions. Motivated by high-stakes applications in which mistakes can be catastrophic, such as robotics and healthcare, we study situations where it is possible for fast sequential learning to occur. Fast sequential learning is characterized by the ability of the learning agent to incur logarithmic regret relative to a fully-informed baseline. We demonstrate that fast sequential learning is achievable in a diverse class of continuous control problems where the system dynamics depend smoothly on unknown parameters, provided the optimal control policy is persistently exciting. Additionally, we derive a regret bound which grows with the square root of the number of interactions for cases where the optimal policy is not persistently exciting. Our results provide the first regret bounds for controlling nonlinear dynamical systems depending nonlinearly on unknown parameters. We validate the trends our theory predicts in simulation on a simple dynamical system.

Brian Lee, Nikolai Matni

We study the performance of risk-controlling prediction sets (RCPS), an empirical risk minimization-based formulation of conformal prediction, with a single trajectory of temporally correlated data from an unknown stochastic dynamical system. First, we use the blocking technique to show that RCPS attains performance guarantees similar to those enjoyed in the iid setting whenever data is generated by asymptotically stationary and contractive dynamics. Next, we use the decoupling technique to characterize the graceful degradation in RCPS guarantees when the data generating process deviates from stationarity and contractivity. We conclude by discussing how these tools could be used toward a unified analysis of online and offline conformal prediction algorithms, which are currently treated with very different tools.

Ingvar Ziemann, Stephen Tu, George J. Pappas et al.

We derive upper bounds for random design linear regression with dependent ($β$-mixing) data absent any realizability assumptions. In contrast to the strictly realizable martingale noise regime, no sharp instance-optimal non-asymptotics are available in the literature. Up to constant factors, our analysis correctly recovers the variance term predicted by the Central Limit Theorem -- the noise level of the problem -- and thus exhibits graceful degradation as we introduce misspecification. Past a burn-in, our result is sharp in the moderate deviations regime, and in particular does not inflate the leading order term by mixing time factors.

Daniel Pfrommer, Max Simchowitz, Tyler Westenbroek et al.

A common pipeline in learning-based control is to iteratively estimate a model of system dynamics, and apply a trajectory optimization algorithm - e.g.~$\mathtt{iLQR}$ - on the learned model to minimize a target cost. This paper conducts a rigorous analysis of a simplified variant of this strategy for general nonlinear systems. We analyze an algorithm which iterates between estimating local linear models of nonlinear system dynamics and performing $\mathtt{iLQR}$-like policy updates. We demonstrate that this algorithm attains sample complexity polynomial in relevant problem parameters, and, by synthesizing locally stabilizing gains, overcomes exponential dependence in problem horizon. Experimental results validate the performance of our algorithm, and compare to natural deep-learning baselines.

Georgios Georgakis, Bernadette Bucher, Anton Arapin et al.

We consider the problems of exploration and point-goal navigation in previously unseen environments, where the spatial complexity of indoor scenes and partial observability constitute these tasks challenging. We argue that learning occupancy priors over indoor maps provides significant advantages towards addressing these problems. To this end, we present a novel planning framework that first learns to generate occupancy maps beyond the field-of-view of the agent, and second leverages the model uncertainty over the generated areas to formulate path selection policies for each task of interest. For point-goal navigation the policy chooses paths with an upper confidence bound policy for efficient and traversable paths, while for exploration the policy maximizes model uncertainty over candidate paths. We perform experiments in the visually realistic environments of Matterport3D using the Habitat simulator and demonstrate: 1) Improved results on exploration and map quality metrics over competitive methods, and 2) The effectiveness of our planning module when paired with the state-of-the-art DD-PPO method for the point-goal navigation task.

Ingvar Ziemann, Henrik Sandberg, Nikolai Matni

Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE). More precisely, we give nonasymptotic expected $l^2$-distance bounds between the LSE and the true regression function, where expectation is evaluated on a fresh, counterfactual, trajectory. We leverage recently developed information-theoretic methods to establish the optimality of the LSE for nonparametric hypotheses classes in terms of supremum norm metric entropy and a subgaussian parameter. Next, we relate this subgaussian parameter to the stability of the underlying process using notions from dynamical systems theory. When combined, these developments lead to rate-optimal error bounds that scale as $T^{-1/(2+q)}$ for suitably stable processes and hypothesis classes with metric entropy growth of order $δ^{-q}$. Here, $T$ is the length of the observed trajectory, $δ\in \mathbb{R}_+$ is the packing granularity and $q\in (0,2)$ is a complexity term. Finally, we specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS).

Daniel Pfrommer, Nikolai Matni

We introduce Variational State-Space Filters (VSSF), a new method for unsupervised learning, identification, and filtering of latent Markov state space models from raw pixels. We present a theoretically sound framework for latent state space inference under heterogeneous sensor configurations. The resulting model can integrate an arbitrary subset of the sensor measurements used during training, enabling the learning of semi-supervised state representations, thus enforcing that certain components of the learned latent state space to agree with interpretable measurements. From this framework we derive L-VSSF, an explicit instantiation of this model with linear latent dynamics and Gaussian distribution parameterizations. We experimentally demonstrate L-VSSF's ability to filter in latent space beyond the sequence length of the training dataset across several different test environments.

Thomas T. C. K. Zhang, Stephen Tu, Nicholas M. Boffi et al.

Motivated by bridging the simulation to reality gap in the context of safety-critical systems, we consider learning adversarially robust stability certificates for unknown nonlinear dynamical systems. In line with approaches from robust control, we consider additive and Lipschitz bounded adversaries that perturb the system dynamics. We show that under suitable assumptions of incremental stability on the underlying system, the statistical cost of learning an adversarial stability certificate is equivalent, up to constant factors, to that of learning a nominal stability certificate. Our results hinge on novel bounds for the Rademacher complexity of the resulting adversarial loss class, which may be of independent interest. To the best of our knowledge, this is the first characterization of sample-complexity bounds when performing adversarial learning over data generated by a dynamical system. We further provide a practical algorithm for approximating the adversarial training algorithm, and validate our findings on a damped pendulum example.

Bibit Bianchini, Mathew Halm, Nikolai Matni et al.

Inspired by recent strides in empirical efficacy of implicit learning in many robotics tasks, we seek to understand the theoretical benefits of implicit formulations in the face of nearly discontinuous functions, common characteristics for systems that make and break contact with the environment such as in legged locomotion and manipulation. We present and motivate three formulations for learning a function: one explicit and two implicit. We derive generalization bounds for each of these three approaches, exposing where explicit and implicit methods alike based on prediction error losses typically fail to produce tight bounds, in contrast to other implicit methods with violation-based loss definitions that can be fundamentally more robust to steep slopes. Furthermore, we demonstrate that this violation implicit loss can tightly bound graph distance, a quantity that often has physical roots and handles noise in inputs and outputs alike, instead of prediction losses which consider output noise only. Our insights into the generalizability and physical relevance of violation implicit formulations match evidence from prior works and are validated through a toy problem, inspired by rigid-contact models and referenced throughout our theoretical analysis.

Lars Lindemann, Alexander Robey, Lejun Jiang et al.

This paper addresses learning safe output feedback control laws from partial observations of expert demonstrations. We assume that a model of the system dynamics and a state estimator are available along with corresponding error bounds, e.g., estimated from data in practice. We first propose robust output control barrier functions (ROCBFs) as a means to guarantee safety, as defined through controlled forward invariance of a safe set. We then formulate an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator or an expert controller. When the parametrization of the ROCBF is linear, then we show that, under mild assumptions, the optimization problem is convex. Along with the optimization problem, we provide verifiable conditions in terms of the density of the data, smoothness of the system model and state estimator, and the size of the error bounds that guarantee validity of the obtained ROCBF. Towards obtaining a practical control algorithm, we propose an algorithmic implementation of our theoretical framework that accounts for assumptions made in our framework in practice. We validate our algorithm in the autonomous driving simulator CARLA and demonstrate how to learn safe control laws from simulated RGB camera images.

Fengjun Yang, Nikolai Matni

When designing large-scale distributed controllers, the information-sharing constraints between sub-controllers, as defined by a communication topology interconnecting them, are as important as the controller itself. Controllers implemented using dense topologies typically outperform those implemented using sparse topologies, but it is also desirable to minimize the cost of controller deployment. Motivated by the above, we introduce a compact but expressive graph recurrent neural network (GRNN) parameterization of distributed controllers that is well suited for distributed controller and communication topology co-design. Our proposed parameterization enjoys a local and distributed architecture, similar to previous Graph Neural Network (GNN)-based parameterizations, while further naturally allowing for joint optimization of the distributed controller and communication topology needed to implement it. We show that the distributed controller/communication topology co-design task can be posed as an $\ell_1$-regularized empirical risk minimization problem that can be efficiently solved using stochastic gradient methods. We run extensive simulations to study the performance of GRNN-based distributed controllers and show that (a) they achieve performance comparable to GNN-based controllers while having fewer free parameters, and (b) our method allows for performance/communication density tradeoff curves to be efficiently approximated.