Ingvar Ziemann

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.

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.

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.

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.

Ingvar Ziemann, Stephen Tu

We study square loss in a realizable time-series framework with martingale difference noise. Our main result is a fast rate excess risk bound which shows that whenever a trajectory hypercontractivity condition holds, the risk of the least-squares estimator on dependent data matches the iid rate order-wise after a burn-in time. In comparison, many existing results in learning from dependent data have rates where the effective sample size is deflated by a factor of the mixing-time of the underlying process, even after the burn-in time. Furthermore, our results allow the covariate process to exhibit long range correlations which are substantially weaker than geometric ergodicity. We call this phenomenon learning with little mixing, and present several examples for when it occurs: bounded function classes for which the $L^2$ and $L^{2+ε}$ norms are equivalent, ergodic finite state Markov chains, various parametric models, and a broad family of infinite dimensional $\ell^2(\mathbb{N})$ ellipsoids. By instantiating our main result to system identification of nonlinear dynamics with generalized linear model transitions, we obtain a nearly minimax optimal excess risk bound after only a polynomial burn-in time.

Ingvar Ziemann

We present a simple proof for bounding the smallest eigenvalue of the empirical covariance in a causal Gaussian process. Along the way, we establish a one-sided tail inequality for Gaussian quadratic forms using a causal decomposition. Our proof only uses elementary facts about the Gaussian distribution and the union bound. We conclude with an example in which we provide a performance guarantee for least squares identification of a vector autoregression.

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.

Ingvar Ziemann

In this note, we give a short information-theoretic proof of the consistency of the Gaussian maximum likelihood estimator in linear auto-regressive models. Our proof yields nearly optimal non-asymptotic rates for parameter recovery and works without any invocation of stability in the case of finite hypothesis classes.

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.

Nicholas M. Boffi, Arthur Jacot, Stephen Tu et al.

Diffusion-based generative models provide a powerful framework for learning to sample from a complex target distribution. The remarkable empirical success of these models applied to high-dimensional signals, including images and video, stands in stark contrast to classical results highlighting the curse of dimensionality for distribution recovery. In this work, we take a step towards understanding this gap through a careful analysis of learning diffusion models over the Barron space of single layer neural networks. In particular, we show that these shallow models provably adapt to simple forms of low dimensional structure, thereby avoiding the curse of dimensionality. We combine our results with recent analyses of sampling with diffusion models to provide an end-to-end sample complexity bound for learning to sample from structured distributions. Importantly, our results do not require specialized architectures tailored to particular latent structures, and instead rely on the low-index structure of the Barron space to adapt to the underlying distribution.

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.

Charis Stamouli, Ingvar Ziemann, George J. Pappas

We study the quadratic prediction error method -- i.e., nonlinear least squares -- for a class of time-varying parametric predictor models satisfying a certain identifiability condition. While this method is known to asymptotically achieve the optimal rate for a wide range of problems, there have been no non-asymptotic results matching these optimal rates outside of a select few, typically linear, model classes. By leveraging modern tools from learning with dependent data, we provide the first rate-optimal non-asymptotic analysis of this method for our more general setting of nonlinearly parametrized model classes. Moreover, we show that our results can be applied to a particular class of identifiable AutoRegressive Moving Average (ARMA) models, resulting in the first optimal non-asymptotic rates for identification of ARMA models.

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.

Uday Kiran Reddy Tadipatri, Benjamin D. Haeffele, Joshua Agterberg et al.

Low-order linear System IDentification (SysID) addresses the challenge of estimating the parameters of a linear dynamical system from finite samples of observations and control inputs with minimal state representation. Traditional approaches often utilize Hankel-rank minimization, which relies on convex relaxations that can require numerous, costly singular value decompositions (SVDs) to optimize. In this work, we propose two nonconvex reformulations to tackle low-order SysID (i) Burer-Monterio (BM) factorization of the Hankel matrix for efficient nuclear norm minimization, and (ii) optimizing directly over system parameters for real, diagonalizable systems with an atomic norm style decomposition. These reformulations circumvent the need for repeated heavy SVD computations, significantly improving computational efficiency. Moreover, we prove that optimizing directly over the system parameters yields lower statistical error rates, and lower sample complexities that do not scale linearly with trajectory length like in Hankel-nuclear norm minimization. Additionally, while our proposed formulations are nonconvex, we provide theoretical guarantees of achieving global optimality in polynomial time. Finally, we demonstrate algorithms that solve these nonconvex programs and validate our theoretical claims on synthetic data.

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.

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.

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).

Ingvar Ziemann, Henrik Sandberg

TWe establish regret lower bounds for adaptively controlling an unknown linear Gaussian system with quadratic costs. We combine ideas from experiment design, estimation theory and a perturbation bound of certain information matrices to derive regret lower bounds exhibiting scaling on the order of magnitude $\sqrt{T}$ in the time horizon $T$. Our bounds accurately capture the role of control-theoretic parameters and we are able to show that systems that are hard to control are also hard to learn to control; when instantiated to state feedback systems we recover the dimensional dependency of earlier work but with improved scaling with system-theoretic constants such as system costs and Gramians. Furthermore, we extend our results to a class of partially observed systems and demonstrate that systems with poor observability structure also are hard to learn to control.

Ingvar Ziemann, Henrik Sandberg

This paper presents local asymptotic minimax regret lower bounds for adaptive Linear Quadratic Regulators (LQR). We consider affinely parametrized $B$-matrices and known $A$-matrices and aim to understand when logarithmic regret is impossible even in the presence of structural side information. After defining the intrinsic notion of an uninformative optimal policy in terms of a singularity condition for Fisher information we obtain local minimax regret lower bounds for such uninformative instances of LQR by appealing to van Trees' inequality (Bayesian Cramér-Rao) and a representation of regret in terms of a quadratic form (Bellman error). It is shown that if the parametrization induces an uninformative optimal policy, logarithmic regret is impossible and the rate is at least order square root in the time horizon. We explicitly characterize the notion of an uninformative optimal policy in terms of the nullspaces of system-theoretic quantities and the particular instance parametrization.