EvDNeRF: Reconstructing Event Data with Dynamic Neural Radiance Fields

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

A Tutorial on the Non-Asymptotic Theory of System Identification

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.

Safety Filter Design for Neural Network Systems via Convex Optimization

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

Meta-Learning Operators to Optimality from Multi-Task Non-IID Data

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

Learning Safe and Stable Motion Plans with Neural Ordinary Differential Equations

A learning-based modular motion planning pipeline is presented that is compliant, safe, and reactive to perturbations at task execution. A nominal motion plan, defined as a nonlinear autonomous dynamical system (DS), is learned offline from kinesthetic demonstrations using a Neural Ordinary Differential Equation (NODE) model. To ensure both stability and safety during inference, a novel approach is proposed which selects a target point at each time step for the robot to follow, using a time-varying target trajectory generated by the learned NODE. A correction term to the NODE model is computed online by solving a Quadratic Program that guarantees stability and safety using Control Lyapunov Functions and Control Barrier Functions, respectively. Our approach outperforms baseline DS learning techniques on the LASA handwriting dataset and is validated on real-robot experiments where it is shown to produce stable motions, such as wiping and stirring, while being robust to physical perturbations and safe around humans and obstacles.

A Data-Driven Approach to Synthesizing Dynamics-Aware Trajectories for Underactuated Robotic Systems

We consider joint trajectory generation and tracking control for under-actuated robotic systems. A common solution is to use a layered control architecture, where the top layer uses a simplified model of system dynamics for trajectory generation, and the low layer ensures approximate tracking of this trajectory via feedback control. While such layered control architectures are standard and work well in practice, selecting the simplified model used for trajectory generation typically relies on engineering intuition and experience. In this paper, we propose an alternative data-driven approach to dynamics-aware trajectory generation. We show that a suitable augmented Lagrangian reformulation of a global nonlinear optimal control problem results in a layered decomposition of the overall problem into trajectory planning and feedback control layers. Crucially, the resulting trajectory optimization is dynamics-aware, in that, it is modified with a tracking penalty regularizer encoding the dynamic feasibility of the generated trajectory. We show that this tracking penalty regularizer can be learned from system rollouts for independently-designed low layer feedback control policies, and instantiate our framework in the context of a unicycle and a quadrotor control problem in simulation. Further, we show that our approach handles the sim-to-real gap through experiments on the quadrotor hardware platform without any additional training. For both the synthetic unicycle example and the quadrotor system, our framework shows significant improvements in both computation time and dynamic feasibility in simulation and hardware experiments.

The noise level in linear regression with dependent data

We derive upper bounds for random design linear regression with dependent ($\beta$-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.

The Power of Learned Locally Linear Models for Nonlinear Policy Optimization

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.

Multi-Task Imitation Learning for Linear Dynamical Systems

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.