Stephen Tu

Charline Le Lan, Stephen Tu, Adam Oberman et al. · deepmind

In reinforcement learning, state representations are used to tractably deal with large problem spaces. State representations serve both to approximate the value function with few parameters, but also to generalize to newly encountered states. Their features may be learned implicitly (as part of a neural network) or explicitly (for example, the successor representation of \citet{dayan1993improving}). While the approximation properties of representations are reasonably well-understood, a precise characterization of how and when these representations generalize is lacking. In this work, we address this gap and provide an informative bound on the generalization error arising from a specific state representation. This bound is based on the notion of effective dimension which measures the degree to which knowing the value at one state informs the value at other states. Our bound applies to any state representation and quantifies the natural tension between representations that generalize well and those that approximate well. We complement our theoretical results with an empirical survey of classic representation learning methods from the literature and results on the Arcade Learning Environment, and find that the generalization behaviour of learned representations is well-explained by their effective dimension.

Charline Le Lan, Stephen Tu, Mark Rowland et al. · deepmind

In reinforcement learning (RL), state representations are key to dealing with large or continuous state spaces. While one of the promises of deep learning algorithms is to automatically construct features well-tuned for the task they try to solve, such a representation might not emerge from end-to-end training of deep RL agents. To mitigate this issue, auxiliary objectives are often incorporated into the learning process and help shape the learnt state representation. Bootstrapping methods are today's method of choice to make these additional predictions. Yet, it is unclear which features these algorithms capture and how they relate to those from other auxiliary-task-based approaches. In this paper, we address this gap and provide a theoretical characterization of the state representation learnt by temporal difference learning (Sutton, 1988). Surprisingly, we find that this representation differs from the features learned by Monte Carlo and residual gradient algorithms for most transition structures of the environment in the policy evaluation setting. We describe the efficacy of these representations for policy evaluation, and use our theoretical analysis to design new auxiliary learning rules. We complement our theoretical results with an empirical comparison of these learning rules for different cumulant functions on classic domains such as the four-room domain (Sutton et al, 1999) and Mountain Car (Moore, 1990).

Allen Z. Ren, Anushri Dixit, Alexandra Bodrova et al.

Large language models (LLMs) exhibit a wide range of promising capabilities -- from step-by-step planning to commonsense reasoning -- that may provide utility for robots, but remain prone to confidently hallucinated predictions. In this work, we present KnowNo, which is a framework for measuring and aligning the uncertainty of LLM-based planners such that they know when they don't know and ask for help when needed. KnowNo builds on the theory of conformal prediction to provide statistical guarantees on task completion while minimizing human help in complex multi-step planning settings. Experiments across a variety of simulated and real robot setups that involve tasks with different modes of ambiguity (e.g., from spatial to numeric uncertainties, from human preferences to Winograd schemas) show that KnowNo performs favorably over modern baselines (which may involve ensembles or extensive prompt tuning) in terms of improving efficiency and autonomy, while providing formal assurances. KnowNo can be used with LLMs out of the box without model-finetuning, and suggests a promising lightweight approach to modeling uncertainty that can complement and scale with the growing capabilities of foundation models. Website: https://robot-help.github.io

Stephen Tu, Shivaram Venkataraman, Ashia C. Wilson et al.

Recent work by Nesterov and Stich showed that momentum can be used to accelerate the rate of convergence for block Gauss-Seidel in the setting where a fixed partitioning of the coordinates is chosen ahead of time. We show that this setting is too restrictive, constructing instances where breaking locality by running non-accelerated Gauss-Seidel with randomly sampled coordinates substantially outperforms accelerated Gauss-Seidel with any fixed partitioning. Motivated by this finding, we analyze the accelerated block Gauss-Seidel algorithm in the random coordinate sampling setting. Our analysis captures the benefit of acceleration with a new data-dependent parameter which is well behaved when the matrix sub-blocks are well-conditioned. Empirically, we show that accelerated Gauss-Seidel with random coordinate sampling provides speedups for large scale machine learning tasks when compared to non-accelerated Gauss-Seidel and the classical conjugate-gradient algorithm.

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.

Xuesu Xiao, Tingnan Zhang, Krzysztof Choromanski et al.

Despite decades of research, existing navigation systems still face real-world challenges when deployed in the wild, e.g., in cluttered home environments or in human-occupied public spaces. To address this, we present a new class of implicit control policies combining the benefits of imitation learning with the robust handling of system constraints from Model Predictive Control (MPC). Our approach, called Performer-MPC, uses a learned cost function parameterized by vision context embeddings provided by Performers -- a low-rank implicit-attention Transformer. We jointly train the cost function and construct the controller relying on it, effectively solving end-to-end the corresponding bi-level optimization problem. We show that the resulting policy improves standard MPC performance by leveraging a few expert demonstrations of the desired navigation behavior in different challenging real-world scenarios. Compared with a standard MPC policy, Performer-MPC achieves >40% better goal reached in cluttered environments and >65% better on social metrics when navigating around humans.

Stephen Tu, Ross Boczar, Benjamin Recht

Given a stable SISO LTI system $G$, we investigate the problem of estimating the $\mathcal{H}_\infty$-norm of $G$, denoted $||G||_\infty$, when $G$ is only accessible via noisy observations. Wahlberg et al. recently proposed a nonparametric algorithm based on the power method for estimating the top eigenvalue of a matrix. In particular, by applying a clever time-reversal trick, Wahlberg et al. implement the power method on the top left $n \times n$ corner $T_n$ of the Toeplitz (convolution) operator associated with $G$. In this paper, we prove sharp non-asymptotic bounds on the necessary length $n$ needed so that $||T_n||$ is an $\varepsilon$-additive approximation of $||G||_\infty$. Furthermore, in the process of demonstrating the sharpness of our bounds, we construct a simple family of finite impulse response (FIR) filters where the number of timesteps needed for the power method is arbitrarily worse than the number of timesteps needed for parametric FIR identification via least-squares to achieve the same $\varepsilon$-additive approximation.

Octavio Narvaez-Aroche, Pierre-Jean Meyer, Stephen Tu et al.

The sit-to-stand movement is a key feature for wide adoption of powered lower limb orthoses for patients with complete paraplegia. In this paper we study the control of the ascending phase of the sit-to-stand movement for a minimally actuated powered lower limb orthosis at the hips. First, we generate a pool of finite horizon Linear Quadratic Regulator feedback gains, designed under the assumption that we can control not only the torque at the hips but also the loads at the shoulders that in reality are applied by the user. Next we conduct reachability analysis to define a performance metric measuring the robustness of each controller against parameter uncertainty, and choose the best controller from the pool with respect to this metric. Then, we replace the presumed shoulder control with an Iterative Learning Control algorithm as a substitute for human experiments. Indeed this algorithm obtains torque and forces at the shoulders that result in successful simulations of the sit-to-stand movement, regardless of parameter uncertainty and factors deliberately introduced to hinder learning. Thus it is reasonable to expect that the superior cognitive skills of real users will enable them to cooperate with the hip torque controller through training.

Sumeet Singh, Stephen Tu, Vikas Sindhwani

A crucial design decision for any robot learning pipeline is the choice of policy representation: what type of model should be used to generate the next set of robot actions? Owing to the inherent multi-modal nature of many robotic tasks, combined with the recent successes in generative modeling, researchers have turned to state-of-the-art probabilistic models such as diffusion models for policy representation. In this work, we revisit the choice of energy-based models (EBM) as a policy class. We show that the prevailing folklore -- that energy models in high dimensional continuous spaces are impractical to train -- is false. We develop a practical training objective and algorithm for energy models which combines several key ingredients: (i) ranking noise contrastive estimation (R-NCE), (ii) learnable negative samplers, and (iii) non-adversarial joint training. We prove that our proposed objective function is asymptotically consistent and quantify its limiting variance. On the other hand, we show that the Implicit Behavior Cloning (IBC) objective is actually biased even at the population level, providing a mathematical explanation for the poor performance of IBC trained energy policies in several independent follow-up works. We further extend our algorithm to learn a continuous stochastic process that bridges noise and data, modeling this process with a family of EBMs indexed by scale variable. In doing so, we demonstrate that the core idea behind recent progress in generative modeling is actually compatible with EBMs. Altogether, our proposed training algorithms enable us to train energy-based models as policies which compete with -- and even outperform -- diffusion models and other state-of-the-art approaches in several challenging multi-modal benchmarks: obstacle avoidance path planning and contact-rich block pushing.

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.

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.

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.

Anthony Liang, Yigit Korkmaz, Jiahui Zhang et al.

General-purpose robot reward models are typically trained to predict absolute task progress from expert demonstrations, providing only local, frame-level supervision. While effective for expert demonstrations, this paradigm scales poorly to large-scale robotics datasets where failed and suboptimal trajectories are abundant and assigning dense progress labels is ambiguous. We introduce Robometer, a scalable reward modeling framework that combines intra-trajectory progress supervision with inter-trajectory preference supervision. Robometer is trained with a dual objective: a frame-level progress loss that anchors reward magnitude on expert data, and a trajectory-comparison preference loss that imposes global ordering constraints across trajectories of the same task, enabling effective learning from both real and augmented failed trajectories. To support this formulation at scale, we curate RBM-1M, a reward-learning dataset comprising over one million trajectories spanning diverse robot embodiments and tasks, including substantial suboptimal and failure data. Across benchmarks and real-world evaluations, Robometer learns more generalizable reward functions than prior methods and improves robot learning performance across a diverse set of downstream applications. Code, model weights, and videos at https://robometer.github.io/.

Yichen Zhou, Stephen Tu

There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.

Mohammad Tinati, Stephen Tu

Self-supervised pre-training, where large corpora of unlabeled data are used to learn representations for downstream fine-tuning, has become a cornerstone of modern machine learning. While a growing body of theoretical work has begun to analyze this paradigm, existing bounds leave open the question of how sharp the current rates are, and whether they accurately capture the complex interaction between pre-training and fine-tuning. In this paper, we address this gap by developing an asymptotic theory of pre-training via two-stage M-estimation. A key challenge is that the pre-training estimator is often identifiable only up to a group symmetry, a feature common in representation learning that requires careful treatment. We address this issue using tools from Riemannian geometry to study the intrinsic parameters of the pre-training representation, which we link with the downstream predictor through a notion of orbit-invariance, precisely characterizing the limiting distribution of the downstream test risk. We apply our main result to several case studies, including spectral pre-training, factor models, and Gaussian mixture models, and obtain substantial improvements in problem-specific factors over prior art when applicable.

Zarif Ikram, Arad Firouzkouhi, Stephen Tu et al.

A central challenge in large language model (LLM) editing is capability preservation: methods that successfully change targeted behavior can quietly game the editing proxy and corrupt general capabilities, producing degenerate behaviors reminiscent of proxy/reward hacking. We present CrispEdit, a scalable and principled second-order editing algorithm that treats capability preservation as an explicit constraint, unifying and generalizing several existing editing approaches. CrispEdit formulates editing as constrained optimization and enforces the constraint by projecting edit updates onto the low-curvature subspace of the capability-loss landscape. At the crux of CrispEdit is expressing capability constraint via Bregman divergence, whose quadratic form yields the Gauss-Newton Hessian exactly and even when the base model is not trained to convergence. We make this second-order procedure efficient at the LLM scale using Kronecker-factored approximate curvature (K-FAC) and a novel matrix-free projector that exploits Kronecker structure to avoid constructing massive projection matrices. Across standard model-editing benchmarks, CrispEdit achieves high edit success while keeping capability degradation below 1% on average across datasets, significantly improving over prior editors.

Wei Yang, Jiacheng Pang, Shixuan Li et al.

Multi-agent systems (MAS) built on Large Language Models (LLMs) are being used to approach complex problems and can surpass single model inference. However, their success hinges on navigating a fundamental cognitive tension: the need to balance broad, divergent exploration of the solution space with a principled, convergent synthesis to the optimal solution. Existing paradigms often struggle to manage this duality, leading to premature consensus, error propagation, and a critical credit assignment problem that fails to distinguish between genuine reasoning and superficially plausible arguments. To resolve this core challenge, we propose the Multi-Agent Exploration-Synthesis framework Through Role Orchestration (Maestro), a principled paradigm for collaboration that structurally decouples these cognitive modes. Maestro uses a collective of parallel Execution Agents for diverse exploration and a specialized Central Agent for convergent, evaluative synthesis. To operationalize this critical synthesis phase, we introduce Conditional Listwise Policy Optimization (CLPO), a reinforcement learning objective that disentangles signals for strategic decisions and tactical rationales. By combining decision-focused policy gradients with a list-wise ranking loss over justifications, CLPO achieves clean credit assignment and stronger comparative supervision. Experiments on mathematical reasoning and general problem-solving benchmarks demonstrate that Maestro, coupled with CLPO, consistently outperforms existing state-of-the-art multi-agent approaches, delivering absolute accuracy gains of 6% on average and up to 10% at best.

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.

Anthony Liang, Pavel Czempin, Matthew Hong et al.

Learning robot policies using imitation learning requires collecting large amounts of costly action-labeled expert demonstrations, which fundamentally limits the scale of training data. A promising approach to address this bottleneck is to harness the abundance of unlabeled observations-e.g., from video demonstrations-to learn latent action labels in an unsupervised way. However, we find that existing methods struggle when applied to complex robot tasks requiring fine-grained motions. We design continuous latent action models (CLAM) which incorporate two key ingredients we find necessary for learning to solve complex continuous control tasks from unlabeled observation data: (a) using continuous latent action labels instead of discrete representations, and (b) jointly training an action decoder to ensure that the latent action space can be easily grounded to real actions with relatively few labeled examples. Importantly, the labeled examples can be collected from non-optimal play data, enabling CLAM to learn performant policies without access to any action-labeled expert data. We demonstrate on continuous control benchmarks in DMControl (locomotion) and MetaWorld (manipulation), as well as on a real WidowX robot arm that CLAM significantly outperforms prior state-of-the-art methods, remarkably with a 2-3x improvement in task success rate compared to the best baseline. Videos and code can be found at clamrobot.github.io.

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.

Hesameddin Mohammadi, Mohammad Tinati, Stephen Tu et al.

The symmetric low-rank matrix factorization serves as a building block in many learning tasks, including matrix recovery and training of neural networks. However, despite a flurry of recent research, the dynamics of its training via non-convex factorized gradient-descent-type methods is not fully understood especially in the over-parameterized regime where the fitted rank is higher than the true rank of the target matrix. To overcome this challenge, we characterize equilibrium points of the gradient flow dynamics and examine their local and global stability properties. To facilitate a precise global analysis, we introduce a nonlinear change of variables that brings the dynamics into a cascade connection of three subsystems whose structure is simpler than the structure of the original system. We demonstrate that the Schur complement to a principal eigenspace of the target matrix is governed by an autonomous system that is decoupled from the rest of the dynamics. In the over-parameterized regime, we show that this Schur complement vanishes at an $O(1/t)$ rate, thereby capturing the slow dynamics that arises from excess parameters. We utilize a Lyapunov-based approach to establish exponential convergence of the other two subsystems. By decoupling the fast and slow parts of the dynamics, we offer new insight into the shape of the trajectories associated with local search algorithms and provide a complete characterization of the equilibrium points and their global stability properties. Such an analysis via nonlinear control techniques may prove useful in several related over-parameterized problems.

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.

Sungje Park, Stephen Tu

Backward stochastic differential equation (BSDE)-based deep learning methods provide an alternative to Physics-Informed Neural Networks (PINNs) for solving high-dimensional partial differential equations (PDEs), offering potential algorithmic advantages in settings such as stochastic optimal control, where the PDEs of interest are tied to an underlying dynamical system. However, standard BSDE-based solvers have empirically been shown to underperform relative to PINNs in the literature. In this paper, we identify the root cause of this performance gap as a discretization bias introduced by the standard Euler-Maruyama (EM) integration scheme applied to one-step self-consistency BSDE losses, which shifts the optimization landscape off target. We find that this bias cannot be satisfactorily addressed through finer step-sizes or multi-step self-consistency losses. To properly handle this issue, we propose a Stratonovich-based BSDE formulation, which we implement with stochastic Heun integration. We show that our proposed approach completely eliminates the bias issues faced by EM integration. Furthermore, our empirical results show that our Heun-based BSDE method consistently outperforms EM-based variants and achieves competitive results with PINNs across multiple high-dimensional benchmarks. Our findings highlight the critical role of integration schemes in BSDE-based PDE solvers, an algorithmic detail that has received little attention thus far in the literature.

Amir Ali Ahmadi, Abraar Chaudhry, Vikas Sindhwani et al.

A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. We formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize trajectories. The state of the system must stay within a safety region for a horizon of $T$ time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with information gathered so far. First, we consider safely learning a linear dynamical system involving $n$ states. For the case $T=1$, we present an LP-based algorithm that either safely recovers the true dynamics from at most $n$ trajectories, or certifies that safe learning is impossible. For $T=2$, we give an SDP representation of the set of safe initial conditions and show that $\lceil n/2 \rceil$ trajectories generically suffice for safe learning. For $T = \infty$, we provide SDP-representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. We extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the system has a control input. Second, we consider safely learning a general class of nonlinear dynamical systems. For the case $T=1$, we give an SOCP-based representation of the set of safe initial conditions. For $T=\infty$, we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present some extensions to cases where the measurements are noisy or the dynamical system involves disturbances.

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.

Stephen Tu, Roy Frostig, Mahdi Soltanolkotabi

We initiate a study of supervised learning from many independent sequences ("trajectories") of non-independent covariates, reflecting tasks in sequence modeling, control, and reinforcement learning. Conceptually, our multi-trajectory setup sits between two traditional settings in statistical learning theory: learning from independent examples and learning from a single auto-correlated sequence. Our conditions for efficient learning generalize the former setting--trajectories must be non-degenerate in ways that extend standard requirements for independent examples. Notably, we do not require that trajectories be ergodic, long, nor strictly stable. For linear least-squares regression, given $n$-dimensional examples produced by $m$ trajectories, each of length $T$, we observe a notable change in statistical efficiency as the number of trajectories increases from a few (namely $m \lesssim n$) to many (namely $m \gtrsim n$). Specifically, we establish that the worst-case error rate of this problem is $Θ(n / m T)$ whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp) lower bound of $Ω(n^2 / m^2 T)$ on the worst-case error rate, realized by a simple, marginally unstable linear dynamical system. A key upshot is that, in domains where trajectories regularly reset, the error rate eventually behaves as if all of the examples were independent, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.

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.

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.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits applications to very specific classes of systems. We introduce a novel nonparametric adaptive algorithm that estimates an infinite-dimensional density over parameters online to learn an unknown dynamics in a reproducing kernel Hilbert space. Surprisingly, the resulting control input admits an analytical expression that enables its implementation despite its underlying infinite-dimensional structure. While this adaptive input is rich and expressive - subsuming, for example, traditional linear parameterizations - its computational complexity grows linearly with time, making it comparatively more expensive than its parametric counterparts. Leveraging the theory of random Fourier features, we provide an efficient randomized implementation that recovers the complexity of classical parametric methods while provably retaining the expressivity of the nonparametric input. In particular, our explicit bounds only depend polynomially on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. As an illustration of the method, we demonstrate the ability of the randomized approximation algorithm to learn a predictive model of a 60-dimensional system consisting of ten point masses interacting through Newtonian gravitation. By reinterpretation as a gradient flow on a specific loss, we conclude with a natural extension of our kernel-based adaptive algorithms to deep neural networks. We show empirically that the extra expressivity afforded by deep representations can lead to improved performance at the expense of closed-loop stability that is rigorously guaranteed and consistently observed for kernel machines.

Stephen Tu, Alexander Robey, Tingnan Zhang et al.

We study the following question in the context of imitation learning for continuous control: how are the underlying stability properties of an expert policy reflected in the sample-complexity of an imitation learning task? We provide the first results showing that a surprisingly granular connection can be made between the underlying expert system's incremental gain stability, a novel measure of robust convergence between pairs of system trajectories, and the dependency on the task horizon $T$ of the resulting generalization bounds. In particular, we propose and analyze incremental gain stability constrained versions of behavior cloning and a DAgger-like algorithm, and show that the resulting sample-complexity bounds naturally reflect the underlying stability properties of the expert system. As a special case, we delineate a class of systems for which the number of trajectories needed to achieve $\varepsilon$-suboptimality is sublinear in the task horizon $T$, and do so without requiring (strong) convexity of the loss function in the policy parameters. Finally, we conduct numerical experiments demonstrating the validity of our insights on both a simple nonlinear system for which the underlying stability properties can be easily tuned, and on a high-dimensional quadrupedal robotic simulation.

Alexander Robey, Lars Lindemann, Stephen Tu et al.

The need for robust control laws is especially important in safety-critical applications. We propose robust hybrid control barrier functions as a means to synthesize control laws that ensure robust safety. Based on this notion, we formulate an optimization problem for learning robust hybrid control barrier functions from data. We identify sufficient conditions on the data such that feasibility of the optimization problem ensures correctness of the learned robust hybrid control barrier functions. Our techniques allow us to safely expand the region of attraction of a compass gait walker that is subject to model uncertainty.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

We study the problem of adaptively controlling a known discrete-time nonlinear system subject to unmodeled disturbances. We prove the first finite-time regret bounds for adaptive nonlinear control with matched uncertainty in the stochastic setting, showing that the regret suffered by certainty equivalence adaptive control, compared to an oracle controller with perfect knowledge of the unmodeled disturbances, is upper bounded by $\widetilde{O}(\sqrt{T})$ in expectation. Furthermore, we show that when the input is subject to a $k$ timestep delay, the regret degrades to $\widetilde{O}(k \sqrt{T})$. Our analysis draws connections between classical stability notions in nonlinear control theory (Lyapunov stability and contraction theory) and modern regret analysis from online convex optimization. The use of stability theory allows us to analyze the challenging infinite-horizon single trajectory setting.

Amir Ali Ahmadi, Abraar Chaudhry, Vikas Sindhwani et al.

A fundamental challenge in learning to control an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a given safety region under the (possibly repeated) action of all dynamical systems that are consistent with the information gathered so far. For our first two results, we consider the setting of safely learning linear dynamics. We present a linear programming-based algorithm that either safely recovers the true dynamics from trajectories of length one, or certifies that safe learning is impossible. We also give an efficient semidefinite representation of the set of initial conditions whose resulting trajectories of length two are guaranteed to stay in the safety region. For our final result, we study the problem of safely learning a nonlinear dynamical system. We give a second-order cone programming based representation of the set of initial conditions that are guaranteed to remain in the safety region after one application of the system dynamics.

Lars Lindemann, Haimin Hu, Alexander Robey et al.

Motivated by the lack of systematic tools to obtain safe control laws for hybrid systems, we propose an optimization-based framework for learning certifiably safe control laws from data. In particular, we assume a setting in which the system dynamics are known and in which data exhibiting safe system behavior is available. We propose hybrid control barrier functions for hybrid systems as a means to synthesize safe control inputs. Based on this notion, we present an optimization-based framework to learn such hybrid control barrier functions from data. Importantly, we identify sufficient conditions on the data such that feasibility of the optimization problem ensures correctness of the learned hybrid control barrier functions, and hence the safety of the system. We illustrate our findings in two simulations studies, including a compass gait walker.

Nicholas M. Boffi, Stephen Tu, Nikolai Matni et al.

Many existing tools in nonlinear control theory for establishing stability or safety of a dynamical system can be distilled to the construction of a certificate function that guarantees a desired property. However, algorithms for synthesizing certificate functions typically require a closed-form analytical expression of the underlying dynamics, which rules out their use on many modern robotic platforms. To circumvent this issue, we develop algorithms for learning certificate functions only from trajectory data. We establish bounds on the generalization error - the probability that a certificate will not certify a new, unseen trajectory - when learning from trajectories, and we convert such generalization error bounds into global stability guarantees. We demonstrate empirically that certificates for complex dynamics can be efficiently learned, and that the learned certificates can be used for downstream tasks such as adaptive control.

Nicholas M. Boffi, Stephen Tu, Jean-Jacques E. Slotine

Recent numerical experiments have demonstrated that the choice of optimization geometry used during training can impact generalization performance when learning expressive nonlinear model classes such as deep neural networks. These observations have important implications for modern deep learning but remain poorly understood due to the difficulty of the associated nonconvex optimization problem. Towards an understanding of this phenomenon, we analyze a family of pseudogradient methods for learning generalized linear models under the square loss - a simplified problem containing both nonlinearity in the model parameters and nonconvexity of the optimization which admits a single neuron as a special case. We prove non-asymptotic bounds on the generalization error that sharply characterize how the interplay between the optimization geometry and the feature space geometry sets the out-of-sample performance of the learned model. Experimentally, selecting the optimization geometry as suggested by our theory leads to improved performance in generalized linear model estimation problems such as nonlinear and nonconvex variants of sparse vector recovery and low-rank matrix sensing.

Alexander Robey, Haimin Hu, Lars Lindemann et al.

Inspired by the success of imitation and inverse reinforcement learning in replicating expert behavior through optimal control, we propose a learning based approach to safe controller synthesis based on control barrier functions (CBFs). We consider the setting of a known nonlinear control affine dynamical system and assume that we have access to safe trajectories generated by an expert - a practical example of such a setting would be a kinematic model of a self-driving vehicle with safe trajectories (e.g., trajectories that avoid collisions with obstacles in the environment) generated by a human driver. We then propose and analyze an optimization-based approach to learning a CBF that enjoys provable safety guarantees under suitable Lipschitz smoothness assumptions on the underlying dynamical system. A strength of our approach is that it is agnostic to the parameterization used to represent the CBF, assuming only that the Lipschitz constant of such functions can be efficiently bounded. Furthermore, if the CBF parameterization is convex, then under mild assumptions, so is our learning process. We end with extensive numerical evaluations of our results on both planar and realistic examples, using both random feature and deep neural network parameterizations of the CBF. To the best of our knowledge, these are the first results that learn provably safe control barrier functions from data.

Xingyou Song, Yiding Jiang, Stephen Tu et al.

A major component of overfitting in model-free reinforcement learning (RL) involves the case where the agent may mistakenly correlate reward with certain spurious features from the observations generated by the Markov Decision Process (MDP). We provide a general framework for analyzing this scenario, which we use to design multiple synthetic benchmarks from only modifying the observation space of an MDP. When an agent overfits to different observation spaces even if the underlying MDP dynamics is fixed, we term this observational overfitting. Our experiments expose intriguing properties especially with regards to implicit regularization, and also corroborate results from previous works in RL generalization and supervised learning (SL).

Nikolai Matni, Stephen Tu

We provide a brief tutorial on the use of concentration inequalities as they apply to system identification of state-space parameters of linear time invariant systems, with a focus on the fully observed setting. We draw upon tools from the theories of large-deviations and self-normalized martingales, and provide both data-dependent and independent bounds on the learning rate.

Nikolai Matni, Alexandre Proutiere, Anders Rantzer et al.

Machine and reinforcement learning (RL) are increasingly being applied to plan and control the behavior of autonomous systems interacting with the physical world. Examples include self-driving vehicles, distributed sensor networks, and agile robots. However, when machine learning is to be applied in these new settings, the algorithms had better come with the same type of reliability, robustness, and safety bounds that are hallmarks of control theory, or failures could be catastrophic. Thus, as learning algorithms are increasingly and more aggressively deployed in safety critical settings, it is imperative that control theorists join the conversation. The goal of this tutorial paper is to provide a starting point for control theorists wishing to work on learning related problems, by covering recent advances bridging learning and control theory, and by placing these results within an appropriate historical context of system identification and adaptive control.

Karl Krauth, Stephen Tu, Benjamin Recht

We study the sample complexity of approximate policy iteration (PI) for the Linear Quadratic Regulator (LQR), building on a recent line of work using LQR as a testbed to understand the limits of reinforcement learning (RL) algorithms on continuous control tasks. Our analysis quantifies the tension between policy improvement and policy evaluation, and suggests that policy evaluation is the dominant factor in terms of sample complexity. Specifically, we show that to obtain a controller that is within $\varepsilon$ of the optimal LQR controller, each step of policy evaluation requires at most $(n+d)^3/\varepsilon^2$ samples, where $n$ is the dimension of the state vector and $d$ is the dimension of the input vector. On the other hand, only $\log(1/\varepsilon)$ policy improvement steps suffice, resulting in an overall sample complexity of $(n+d)^3 \varepsilon^{-2} \log(1/\varepsilon)$. We furthermore build on our analysis and construct a simple adaptive procedure based on $\varepsilon$-greedy exploration which relies on approximate PI as a sub-routine and obtains $T^{2/3}$ regret, improving upon a recent result of Abbasi-Yadkori et al.

Horia Mania, Stephen Tu, Benjamin Recht

We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system and the cost incurred by using the optimal LQ controller enjoys a fast statistical rate, scaling as the square of the parameter error. To the best of our knowledge, our result is the first sub-optimality guarantee in the partially observed Linear Quadratic Gaussian (LQG) setting. Furthermore, in the fully observed Linear Quadratic Regulator (LQR), our result improves upon recent work by Dean et al. (2017), who present an algorithm achieving a sub-optimality gap linear in the parameter error. A key part of our analysis relies on perturbation bounds for discrete Riccati equations. We provide two new perturbation bounds, one that expands on an existing result from Konstantinov et al. (1993), and another based on a new elementary proof strategy.

Stephen Tu, Benjamin Recht

The effectiveness of model-based versus model-free methods is a long-standing question in reinforcement learning (RL). Motivated by recent empirical success of RL on continuous control tasks, we study the sample complexity of popular model-based and model-free algorithms on the Linear Quadratic Regulator (LQR). We show that for policy evaluation, a simple model-based plugin method requires asymptotically less samples than the classical least-squares temporal difference (LSTD) estimator to reach the same quality of solution; the sample complexity gap between the two methods can be at least a factor of state dimension. For policy evaluation, we study a simple family of problem instances and show that nominal (certainty equivalence principle) control also requires several factors of state and input dimension fewer samples than the policy gradient method to reach the same level of control performance on these instances. Furthermore, the gap persists even when employing commonly used baselines. To the best of our knowledge, this is the first theoretical result which demonstrates a separation in the sample complexity between model-based and model-free methods on a continuous control task.

Stephen Tu, Ross Boczar, Benjamin Recht

The problem of estimating the $\mathcal{H}_\infty$-norm of an LTI system from noisy input/output measurements has attracted recent attention as an alternative to parameter identification for bounding unmodeled dynamics in robust control. In this paper, we study lower bounds for $\mathcal{H}_\infty$-norm estimation under a query model where at each iteration the algorithm chooses a bounded input signal and receives the response of the chosen signal corrupted by white noise. We prove that when the underlying system is an FIR filter, $\mathcal{H}_\infty$-norm estimation is no more efficient than model identification for passive sampling. For active sampling, we show that norm estimation is at most a factor of $\log{r}$ more sample efficient than model identification, where $r$ is the length of the filter. We complement our theoretical results with experiments which demonstrate that a simple non-adaptive estimator of the norm is competitive with state-of-the-art adaptive norm estimation algorithms.

Sarah Dean, Stephen Tu, Nikolai Matni et al.

We study the constrained linear quadratic regulator with unknown dynamics, addressing the tension between safety and exploration in data-driven control techniques. We present a framework which allows for system identification through persistent excitation, while maintaining safety by guaranteeing the satisfaction of state and input constraints. This framework involves a novel method for synthesizing robust constraint-satisfying feedback controllers, leveraging newly developed tools from system level synthesis. We connect statistical results with cost sub-optimality bounds to give non-asymptotic guarantees on both estimation and controller performance.

Sarah Dean, Horia Mania, Nikolai Matni et al.

We consider adaptive control of the Linear Quadratic Regulator (LQR), where an unknown linear system is controlled subject to quadratic costs. Leveraging recent developments in the estimation of linear systems and in robust controller synthesis, we present the first provably polynomial time algorithm that provides high probability guarantees of sub-linear regret on this problem. We further study the interplay between regret minimization and parameter estimation by proving a lower bound on the expected regret in terms of the exploration schedule used by any algorithm. Finally, we conduct a numerical study comparing our robust adaptive algorithm to other methods from the adaptive LQR literature, and demonstrate the flexibility of our proposed method by extending it to a demand forecasting problem subject to state constraints.

Vikas Sindhwani, Stephen Tu, Mohi Khansari

We propose a new non-parametric framework for learning incrementally stable dynamical systems x' = f(x) from a set of sampled trajectories. We construct a rich family of smooth vector fields induced by certain classes of matrix-valued kernels, whose equilibria are placed exactly at a desired set of locations and whose local contraction and curvature properties at various points can be explicitly controlled using convex optimization. With curl-free kernels, our framework may also be viewed as a mechanism to learn potential fields and gradient flows. We develop large-scale techniques using randomized kernel approximations in this context. We demonstrate our approach, called contracting vector fields (CVF), on imitation learning tasks involving complex point-to-point human handwriting motions.

Max Simchowitz, Horia Mania, Stephen Tu et al.

We prove that the ordinary least-squares (OLS) estimator attains nearly minimax optimal performance for the identification of linear dynamical systems from a single observed trajectory. Our upper bound relies on a generalization of Mendelson's small-ball method to dependent data, eschewing the use of standard mixing-time arguments. Our lower bounds reveal that these upper bounds match up to logarithmic factors. In particular, we capture the correct signal-to-noise behavior of the problem, showing that more unstable linear systems are easier to estimate. This behavior is qualitatively different from arguments which rely on mixing-time calculations that suggest that unstable systems are more difficult to estimate. We generalize our technique to provide bounds for a more general class of linear response time-series.

Stephen Tu, Benjamin Recht

Reinforcement learning (RL) has been successfully used to solve many continuous control tasks. Despite its impressive results however, fundamental questions regarding the sample complexity of RL on continuous problems remain open. We study the performance of RL in this setting by considering the behavior of the Least-Squares Temporal Difference (LSTD) estimator on the classic Linear Quadratic Regulator (LQR) problem from optimal control. We give the first finite-time analysis of the number of samples needed to estimate the value function for a fixed static state-feedback policy to within $\varepsilon$-relative error. In the process of deriving our result, we give a general characterization for when the minimum eigenvalue of the empirical covariance matrix formed along the sample path of a fast-mixing stochastic process concentrates above zero, extending a result by Koltchinskii and Mendelson in the independent covariates setting. Finally, we provide experimental evidence indicating that our analysis correctly captures the qualitative behavior of LSTD on several LQR instances.