The ODE Method for Stochastic Approximation and Reinforcement Learning with Markovian Noise

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a stochastic approximation algorithm is to establish its stability, i.e., to show that the stochastic vector iterates are bounded almost surely. In this paper, we extend the celebrated Borkar-Meyn theorem for stability from the Martingale difference noise setting to the Markovian noise setting, which greatly improves its applicability in reinforcement learning, especially in those off-policy reinforcement learning algorithms with linear function approximation and eligibility traces. Central to our analysis is the diminishing asymptotic rate of change of a few functions, which is implied by both a form of strong law of large numbers and a commonly used V4 Lyapunov drift condition and trivially holds if the Markov chain is finite and irreducible.

GRID: A Platform for General Robot Intelligence Development

Developing machine intelligence abilities in robots and autonomous systems is an expensive and time consuming process. Existing solutions are tailored to specific applications and are harder to generalize. Furthermore, scarcity of training data adds a layer of complexity in deploying deep machine learning models. We present a new platform for General Robot Intelligence Development (GRID) to address both of these issues. The platform enables robots to learn, compose and adapt skills to their physical capabilities, environmental constraints and goals. The platform addresses AI problems in robotics via foundation models that know the physical world. GRID is designed from the ground up to be extensible to accommodate new types of robots, vehicles, hardware platforms and software protocols. In addition, the modular design enables various deep ML components and existing foundation models to be easily usable in a wider variety of robot-centric problems. We demonstrate the platform in various aerial robotics scenarios and demonstrate how the platform dramatically accelerates development of machine intelligent robots.

Is Imitation All You Need? Generalized Decision-Making with Dual-Phase Training

We introduce DualMind, a generalist agent designed to tackle various decision-making tasks that addresses challenges posed by current methods, such as overfitting behaviors and dependence on task-specific fine-tuning. DualMind uses a novel "Dual-phase" training strategy that emulates how humans learn to act in the world. The model first learns fundamental common knowledge through a self-supervised objective tailored for control tasks and then learns how to make decisions based on different contexts through imitating behaviors conditioned on given prompts. DualMind can handle tasks across domains, scenes, and embodiments using just a single set of model weights and can execute zero-shot prompting without requiring task-specific fine-tuning. We evaluate DualMind on MetaWorld and Habitat through extensive experiments and demonstrate its superior generalizability compared to previous techniques, outperforming other generalist agents by over 50$\%$ and 70$\%$ on Habitat and MetaWorld, respectively. On the 45 tasks in MetaWorld, DualMind achieves over 30 tasks at a 90$\%$ success rate.

PACT: Perception-Action Causal Transformer for Autoregressive Robotics Pre-Training

Robotics has long been a field riddled with complex systems architectures whose modules and connections, whether traditional or learning-based, require significant human expertise and prior knowledge. Inspired by large pre-trained language models, this work introduces a paradigm for pre-training a general purpose representation that can serve as a starting point for multiple tasks on a given robot. We present the Perception-Action Causal Transformer (PACT), a generative transformer-based architecture that aims to build representations directly from robot data in a self-supervised fashion. Through autoregressive prediction of states and actions over time, our model implicitly encodes dynamics and behaviors for a particular robot. Our experimental evaluation focuses on the domain of mobile agents, where we show that this robot-specific representation can function as a single starting point to achieve distinct tasks such as safe navigation, localization and mapping. We evaluate two form factors: a wheeled robot that uses a LiDAR sensor as perception input (MuSHR), and a simulated agent that uses first-person RGB images (Habitat). We show that finetuning small task-specific networks on top of the larger pretrained model results in significantly better performance compared to training a single model from scratch for all tasks simultaneously, and comparable performance to training a separate large model for each task independently. By sharing a common good-quality representation across tasks we can lower overall model capacity and speed up the real-time deployment of such systems.

The ODE Method for Asymptotic Statistics in Stochastic Approximation and Reinforcement Learning

The paper concerns convergence and asymptotic statistics for stochastic approximation driven by Markovian noise: $$ \theta_{n+1}= \theta_n + \alpha_{n + 1} f(\theta_n, \Phi_{n+1}) \,,\quad n\ge 0, $$ in which each $\theta_n\in\Re^d$, $ \{ \Phi_n \}$ is a Markov chain on a general state space X with stationary distribution $\pi$, and $f:\Re^d\times \text{X} \to\Re^d$. In addition to standard Lipschitz bounds on $f$, and conditions on the vanishing step-size sequence $\{\alpha_n\}$, it is assumed that the associated ODE is globally asymptotically stable with stationary point denoted $\theta^*$, where $\bar f(\theta)=E[f(\theta,\Phi)]$ with $\Phi\sim\pi$. Moreover, the ODE@$\infty$ defined with respect to the vector field, $$ \bar f_\infty(\theta):= \lim_{r\to\infty} r^{-1} \bar f(r\theta) \,,\qquad \theta\in\Re^d, $$ is asymptotically stable. The main contributions are summarized as follows: (i) The sequence $\theta$ is convergent if $\Phi$ is geometrically ergodic, and subject to compatible bounds on $f$. The remaining results are established under a stronger assumption on the Markov chain: A slightly weaker version of the Donsker-Varadhan Lyapunov drift condition known as (DV3). (ii) A Lyapunov function is constructed for the joint process $\{\theta_n,\Phi_n\}$ that implies convergence of $\{ \theta_n\}$ in $L_4$. (iii) A functional CLT is established, as well as the usual one-dimensional CLT for the normalized error $z_n:= (\theta_n-\theta^*)/\sqrt{\alpha_n}$. Moment bounds combined with the CLT imply convergence of the normalized covariance, $$ \lim_{n \to \infty} E [ z_n z_n^T ] = \Sigma_\theta, $$ where $\Sigma_\theta$ is the asymptotic covariance appearing in the CLT. (iv) An example is provided where the Markov chain $\Phi$ is geometrically ergodic but it does not satisfy (DV3). While the algorithm is convergent, the second moment is unbounded.

Tracking Fast Neural Adaptation by Globally Adaptive Point Process Estimation for Brain-Machine Interface

Brain-machine interfaces (BMIs) help the disabled restore body functions by translating neural activity into digital commands to control external devices. Neural adaptation, where the brain signals change in response to external stimuli or movements, plays an important role in BMIs. When subjects purely use neural activity to brain-control a prosthesis, some neurons will actively explore a new tuning property to accomplish the movement task. The prediction of this neural tuning property can help subjects adapt more efficiently to brain control and maintain good decoding performance. Existing prediction methods track the slow change of the tuning property in the manual control, which is not suitable for the fast neural adaptation in brain control. In order to identify the active neurons in brain control and track their tuning property changes, we propose a globally adaptive point process method (GaPP) to estimate the neural modulation state from spike trains, decompose the states into the hyper preferred direction and reconstruct the kinematics in a dual-model framework. We implement the method on real data from rats performing a two-lever discrimination task under manual control and brain control. The results show our method successfully predicts the neural modulation state and identifies the neurons that become active in brain control. Compared to existing methods, ours tracks the fast changes of the hyper preferred direction from manual control to brain control more accurately and efficiently and reconstructs the kinematics better and faster.

Accelerating Optimization and Reinforcement Learning with Quasi-Stochastic Approximation

The ODE method has been a workhorse for algorithm design and analysis since the introduction of the stochastic approximation. It is now understood that convergence theory amounts to establishing robustness of Euler approximations for ODEs, while theory of rates of convergence requires finer analysis. This paper sets out to extend this theory to quasi-stochastic approximation, based on algorithms in which the "noise" is based on deterministic signals. The main results are obtained under minimal assumptions: the usual Lipschitz conditions for ODE vector fields, and it is assumed that there is a well defined linearization near the optimal parameter $\theta^*$, with Hurwitz linearization matrix $A^*$. The main contributions are summarized as follows: (i) If the algorithm gain is $a_t=g/(1+t)^\rho$ with $g>0$ and $\rho\in(0,1)$, then the rate of convergence of the algorithm is $1/t^\rho$. There is also a well defined "finite-$t$" approximation: \[ a_t^{-1}\{\Theta_t-\theta^*\}=\bar{Y}+\Xi^{\mathrm{I}}_t+o(1) \] where $\bar{Y}\in\mathbb{R}^d$ is a vector identified in the paper, and $\{\Xi^{\mathrm{I}}_t\}$ is bounded with zero temporal mean. (ii) With gain $a_t = g/(1+t)$ the results are not as sharp: the rate of convergence $1/t$ holds only if $I + g A^*$ is Hurwitz. (iii) Based on the Ruppert-Polyak averaging of stochastic approximation, one would expect that a convergence rate of $1/t$ can be obtained by averaging: \[ \Theta^{\text{RP}}_T=\frac{1}{T}\int_{0}^T \Theta_t\,dt \] where the estimates $\{\Theta_t\}$ are obtained using the gain in (i). The preceding sharp bounds imply that averaging results in $1/t$ convergence rate if and only if $\bar{Y}=\sf 0$. This condition holds if the noise is additive, but appears to fail in general. (iv) The theory is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.

Explicit Mean-Square Error Bounds for Monte-Carlo and Linear Stochastic Approximation

This paper concerns error bounds for recursive equations subject to Markovian disturbances. Motivating examples abound within the fields of Markov chain Monte Carlo (MCMC) and Reinforcement Learning (RL), and many of these algorithms can be interpreted as special cases of stochastic approximation (SA). It is argued that it is not possible in general to obtain a Hoeffding bound on the error sequence, even when the underlying Markov chain is reversible and geometrically ergodic, such as the M/M/1 queue. This is motivation for the focus on mean square error bounds for parameter estimates. It is shown that mean square error achieves the optimal rate of $O(1/n)$, subject to conditions on the step-size sequence. Moreover, the exact constants in the rate are obtained, which is of great value in algorithm design.

Zap Q-Learning With Nonlinear Function Approximation

The Zap stochastic approximation (SA) algorithm was introduced recently as a means to accelerate convergence in reinforcement learning algorithms. While numerical results were impressive, stability (in the sense of boundedness of parameter estimates) was established in only a few special cases. This class of algorithms is generalized in this paper, and stability is established under very general conditions. This general result can be applied to a wide range of algorithms found in reinforcement learning. Two classes are considered in this paper: (i)The natural generalization of Watkins' algorithm is not always stable in function approximation settings. Parameter estimates may diverge to infinity even in the \textit{linear} function approximation setting with a simple finite state-action MDP. Under mild conditions, the Zap SA algorithm provides a stable algorithm, even in the case of \textit{nonlinear} function approximation. (ii) The GQ algorithm of Maei et.~al.~2010 is designed to address the stability challenge. Analysis is provided to explain why the algorithm may be very slow to converge in practice. The new Zap GQ algorithm is stable even for nonlinear function approximation.

Zap Q-Learning for Optimal Stopping Time Problems

We propose a novel reinforcement learning algorithm that approximates solutions to the problem of discounted optimal stopping in an irreducible, uniformly ergodic Markov chain evolving on a compact subset of $\mathbb R^n$. A dynamic programming approach has been taken by Tsitsikilis and Van Roy to solve this problem, wherein they propose a Q-learning algorithm to estimate the value function, in a linear function approximation setting. The Zap-Q learning algorithm proposed in this work is the first algorithm that is designed to achieve optimal asymptotic variance. We prove convergence of the algorithm using ODE analysis, and the optimal asymptotic variance property is reflected via fast convergence in a finance example.