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.
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 $L^{2+\epsilon}$ 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.
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 \emph{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.
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. They 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 this problem is $\Theta(n / m T)$ whenever $m \gtrsim n$. Meanwhile, when $m \lesssim n$, we establish a (sharp) lower bound of $\Omega(n^2 / m^2 T)$ 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 altogether, drawn from their marginals. As a corollary of our analysis, we also improve guarantees for the linear system identification problem.
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.
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.
This paper addresses learning safe control laws from expert demonstrations. We assume that appropriate models of the system dynamics and the output measurement map are available, along with corresponding error bounds. 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 present an optimization problem to learn ROCBFs from expert demonstrations that exhibit safe system behavior, e.g., data collected from a human operator. Along with the optimization problem, we provide verifiable conditions that guarantee validity of the obtained ROCBF. These conditions are stated in terms of the density of the data and on Lipschitz and boundedness constants of the learned function and the models of the system dynamics and the output measurement map. When the parametrization of the ROCBF is linear, then, under mild assumptions, the optimization problem is convex. We validate our findings in the autonomous driving simulator CARLA and show how to learn safe control laws from RGB camera images.
A key assumption in the theory of adaptive control for nonlinear systems 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, verifying it in practice can be difficult, particularly for complex systems. Here we leverage connections between reproducing kernel Hilbert spaces, random Fourier features, and universal approximation theory to propose a computationally tractable algorithm for both adaptive control and adaptive prediction that does not rely on a linearly parameterized unknown. Specifically, we approximate the unknown dynamics with a finite expansion in $\textit{random}$ basis functions, and provide an explicit guarantee on the number of random features needed to track a desired trajectory with high probability. Remarkably, our explicit bounds only depend $\textit{polynomially}$ on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. We study a setting where the unknown dynamics splits into a component that can be modeled through available physical knowledge of the system and a component that lives in a reproducing kernel Hilbert space. Our algorithms simultaneously adapt over parameters for physical basis functions and random features to learn both components of the dynamics online.
Commonly used optimization-based control strategies such as model-predictive and control Lyapunov/barrier function based controllers often enjoy provable stability, robustness, and safety properties. However, implementing such approaches requires solving optimization problems online at high-frequencies, which may not be possible on resource-constrained commodity hardware. Furthermore, how to extend the safety guarantees of such approaches to systems that use rich perceptual sensing modalities, such as cameras, remains unclear. In this paper, we address this gap by treating safe optimization-based control strategies as experts in an imitation learning problem, and train a learned policy that can be cheaply evaluated at run-time and that provably satisfies the same safety guarantees as the expert. In particular, we propose Constrained Mixing Iterative Learning (CMILe), a novel on-policy robust imitation learning algorithm that integrates ideas from stochastic mixing iterative learning, constrained policy optimization, and nonlinear robust control. Our approach allows us to control errors introduced by both the learning task of imitating an expert and by the distribution shift inherent to deviating from the original expert policy. The value of using tools from nonlinear robust control to impose stability constraints on learned policies is shown through sample-complexity bounds that are independent of the task time-horizon. We demonstrate the usefulness of CMILe through extensive experiments, including training a provably safe perception-based controller using a state-feedback-based expert.
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.