Continuous Control with Contexts, Provably

A fundamental challenge in artificial intelligence is to build an agent that generalizes and adapts to unseen environments. A common strategy is to build a decoder that takes the context of the unseen new environment as input and generates a policy accordingly. The current paper studies how to build a decoder for the fundamental continuous control task, linear quadratic regulator (LQR), which can model a wide range of real-world physical environments. We present a simple algorithm for this problem, which uses upper confidence bound (UCB) to refine the estimate of the decoder and balance the exploration-exploitation trade-off. Theoretically, our algorithm enjoys a $\widetilde{O}\left(\sqrt{T}\right)$ regret bound in the online setting where $T$ is the number of environments the agent played. This also implies after playing $\widetilde{O}\left(1/\epsilon^2\right)$ environments, the agent is able to transfer the learned knowledge to obtain an $\epsilon$-suboptimal policy for an unseen environment. To our knowledge, this is first provably efficient algorithm to build a decoder in the continuous control setting. While our main focus is theoretical, we also present experiments that demonstrate the effectiveness of our algorithm.

Is a Good Representation Sufficient for Sample Efficient Reinforcement Learning?

Modern deep learning methods provide an effective means to learn good representations. However, is a good representation itself sufficient for efficient reinforcement learning? This question is largely unexplored, and the extant body of literature mainly focuses on conditions which permit efficient reinforcement learning with little understanding of what are necessary conditions for efficient reinforcement learning. This work provides strong negative results for reinforcement learning methods with function approximation for which a good representation (feature extractor) is known to the agent, focusing on natural representational conditions relevant to value-based learning and policy-based learning. For value-based learning, we show that even if the agent has a highly accurate linear representation, the agent still needs to sample exponentially many trajectories in order to find a near-optimal policy. For policy-based learning, we show even if the agent's linear representation is capable of perfectly representing the optimal policy, the agent still needs to sample exponentially many trajectories in order to find a near-optimal policy. These lower bounds highlight the fact that having a good (value-based or policy-based) representation in and of itself is insufficient for efficient reinforcement learning. In particular, these results provide new insights into why the existing provably efficient reinforcement learning methods rely on further assumptions, which are often model-based in nature. Additionally, our lower bounds imply exponential separations in the sample complexity between 1) value-based learning with perfect representation and value-based learning with a good-but-not-perfect representation, 2) value-based learning and policy-based learning, 3) policy-based learning and supervised learning and 4) reinforcement learning and imitation learning.

Efficient Symmetric Norm Regression via Linear Sketching

We provide efficient algorithms for overconstrained linear regression problems with size $n \times d$ when the loss function is a symmetric norm (a norm invariant under sign-flips and coordinate-permutations). An important class of symmetric norms are Orlicz norms, where for a function $G$ and a vector $y \in \mathbb{R}^n$, the corresponding Orlicz norm $\|y\|_G$ is defined as the unique value $\alpha$ such that $\sum_{i=1}^n G(|y_i|/\alpha) = 1$. When the loss function is an Orlicz norm, our algorithm produces a $(1 + \varepsilon)$-approximate solution for an arbitrarily small constant $\varepsilon > 0$ in input-sparsity time, improving over the previously best-known algorithm which produces a $d \cdot \mathrm{polylog} n$-approximate solution. When the loss function is a general symmetric norm, our algorithm produces a $\sqrt{d} \cdot \mathrm{polylog} n \cdot \mathrm{mmc}(\ell)$-approximate solution in input-sparsity time, where $\mathrm{mmc}(\ell)$ is a quantity related to the symmetric norm under consideration. To the best of our knowledge, this is the first input-sparsity time algorithm with provable guarantees for the general class of symmetric norm regression problem. Our results shed light on resolving the universal sketching problem for linear regression, and the techniques might be of independent interest to numerical linear algebra problems more broadly.

Solving Discounted Stochastic Two-Player Games with Near-Optimal Time and Sample Complexity

In this paper, we settle the sampling complexity of solving discounted two-player turn-based zero-sum stochastic games up to polylogarithmic factors. Given a stochastic game with discount factor $\gamma\in(0,1)$ we provide an algorithm that computes an $\epsilon$-optimal strategy with high-probability given $\tilde{O}((1 - \gamma)^{-3} \epsilon^{-2})$ samples from the transition function for each state-action-pair. Our algorithm runs in time nearly linear in the number of samples and uses space nearly linear in the number of state-action pairs. As stochastic games generalize Markov decision processes (MDPs) our runtime and sample complexities are optimal due to Azar et al (2013). We achieve our results by showing how to generalize a near-optimal Q-learning based algorithms for MDP, in particular Sidford et al (2018), to two-player strategy computation algorithms. This overcomes limitations of standard Q-learning and strategy iteration or alternating minimization based approaches and we hope will pave the way for future reinforcement learning results by facilitating the extension of MDP results to multi-agent settings with little loss.

On the Optimality of Sparse Model-Based Planning for Markov Decision Processes

This work considers the sample complexity of obtaining an $\epsilon$-optimal policy in a discounted Markov Decision Process (MDP), given only access to a generative model. In this model, the learner accesses the underlying transition model via a sampling oracle that provides a sample of the next state, when given any state-action pair as input. In this work, we study the effectiveness of the most natural plug-in approach to model-based planning: we build the maximum likelihood estimate of the transition model in the MDP from observations and then find an optimal policy in this empirical MDP. We ask arguably the most basic and unresolved question in model-based planning: is the na\"ive "plug-in" approach, non-asymptotically, minimax optimal in the quality of the policy it finds, given a fixed sample size? With access to a generative model, we resolve this question in the strongest possible sense: our main result shows that \emph{any} high accuracy solution in the plug-in model constructed with $N$ samples, provides an $\epsilon$-optimal policy in the true underlying MDP. In comparison, all prior (non-asymptotically) minimax optimal results use model-free approaches, such as the Variance Reduced Q-value iteration algorithm (Sidford et al 2018), while the best known model-based results (e.g. Azar et al 2013) require larger sample sample sizes in their dependence on the planning horizon or the state space. Notably, we show that the model-based approach allows the use of \emph{any} efficient planning algorithm in the empirical MDP, which simplifies the algorithm design as this approach does not tie the algorithm to the sampling procedure. The core of our analysis is a novel "absorbing MDP" construction to address the statistical dependency issues that arise in the analysis of model-based planning approaches, a construction which may be helpful more generally.

Reinforcement Learning in Feature Space: Matrix Bandit, Kernels, and Regret Bound

Exploration in reinforcement learning (RL) suffers from the curse of dimensionality when the state-action space is large. A common practice is to parameterize the high-dimensional value and policy functions using given features. However existing methods either have no theoretical guarantee or suffer a regret that is exponential in the planning horizon $H$. In this paper, we propose an online RL algorithm, namely the MatrixRL, that leverages ideas from linear bandit to learn a low-dimensional representation of the probability transition model while carefully balancing the exploitation-exploration tradeoff. We show that MatrixRL achieves a regret bound ${O}\big(H^2d\log T\sqrt{T}\big)$ where $d$ is the number of features. MatrixRL has an equivalent kernelized version, which is able to work with an arbitrary kernel Hilbert space without using explicit features. In this case, the kernelized MatrixRL satisfies a regret bound ${O}\big(H^2\widetilde{d}\log T\sqrt{T}\big)$, where $\widetilde{d}$ is the effective dimension of the kernel space. To our best knowledge, for RL using features or kernels, our results are the first regret bounds that are near-optimal in time $T$ and dimension $d$ (or $\widetilde{d}$) and polynomial in the planning horizon $H$.

Feature-Based Q-Learning for Two-Player Stochastic Games

Consider a two-player zero-sum stochastic game where the transition function can be embedded in a given feature space. We propose a two-player Q-learning algorithm for approximating the Nash equilibrium strategy via sampling. The algorithm is shown to find an $\epsilon$-optimal strategy using sample size linear to the number of features. To further improve its sample efficiency, we develop an accelerated algorithm by adopting techniques such as variance reduction, monotonicity preservation and two-sided strategy approximation. We prove that the algorithm is guaranteed to find an $\epsilon$-optimal strategy using no more than $\tilde{\mathcal{O}}(K/(\epsilon^{2}(1-\gamma)^{4}))$ samples with high probability, where $K$ is the number of features and $\gamma$ is a discount factor. The sample, time and space complexities of the algorithm are independent of original dimensions of the game.

Reinforcement Leaning in Feature Space: Matrix Bandit, Kernels, and Regret Bound

Exploration in reinforcement learning (RL) suffers from the curse of dimensionality when the state-action space is large. A common practice is to parameterize the high-dimensional value and policy functions using given features. However existing methods either have no theoretical guarantee or suffer a regret that is exponential in the planning horizon $H$. In this paper, we propose an online RL algorithm, namely the MatrixRL, that leverages ideas from linear bandit to learn a low-dimensional representation of the probability transition model while carefully balancing the exploitation-exploration tradeoff. We show that MatrixRL achieves a regret bound ${O}\big(H^2d\log T\sqrt{T}\big)$ where $d$ is the number of features. MatrixRL has an equivalent kernelized version, which is able to work with an arbitrary kernel Hilbert space without using explicit features. In this case, the kernelized MatrixRL satisfies a regret bound ${O}\big(H^2\widetilde{d}\log T\sqrt{T}\big)$, where $\widetilde{d}$ is the effective dimension of the kernel space. To our best knowledge, for RL using features or kernels, our results are the first regret bounds that are near-optimal in time $T$ and dimension $d$ (or $\widetilde{d}$) and polynomial in the planning horizon $H$.