In this work, we focus our attention on the study of the interplay between the data distribution and Q-learning-based algorithms with function approximation. We provide a theoretical and empirical analysis as to why different properties of the data distribution can contribute to regulating sources of algorithmic instability. First, we revisit theoretical bounds on the performance of approximate dynamic programming algorithms. Second, we provide a novel four-state MDP that highlights the impact of the data distribution in the performance of a Q-learning algorithm with function approximation, both in online and offline settings. Finally, we experimentally assess the impact of the data distribution properties in the performance of an offline deep Q-network algorithm. Our results show that: (i) the data distribution needs to possess certain properties in order to robustly learn in an offline setting, namely low distance to the distributions induced by optimal policies of the MDP and high coverage over the state-action space; and (ii) high entropy data distributions can contribute to mitigating sources of algorithmic instability.
相關內容
Most modern deep reinforcement learning (RL) algorithms are motivated by either the general policy improvement (GPI) or trust-region learning (TRL) frameworks. However, algorithms that strictly respect these theoretical frameworks have proven unscalable. Surprisingly, the only known scalable algorithms violate the GPI/TRL assumptions, e.g. due to required regularisation or other heuristics. The current explanation of their empirical success is essentially by "analogy": they are deemed approximate adaptations of theoretically sound methods. Unfortunately, studies have shown that in practice these algorithms differ greatly from their conceptual ancestors. In contrast, in this paper, we introduce a novel theoretical framework, named Mirror Learning, which provides theoretical guarantees to a large class of algorithms, including TRPO and PPO. While the latter two exploit the flexibility of our framework, GPI and TRL fit in merely as pathologically restrictive or impractical corner cases thereof. This suggests that the empirical performance of state-of-the-art methods is a direct consequence of their theoretical properties, rather than of aforementioned approximate analogies. Mirror learning sets us free to boldly explore novel, theoretically sound RL algorithms, a thus far uncharted wonderland.
Classical reinforcement learning (RL) aims to optimize the expected cumulative rewards. In this work, we consider the RL setting where the goal is to optimize the quantile of the cumulative rewards. We parameterize the policy controlling actions by neural networks and propose a novel policy gradient algorithm called Quantile-Based Policy Optimization (QPO) and its variant Quantile-Based Proximal Policy Optimization (QPPO) to solve deep RL problems with quantile objectives. QPO uses two coupled iterations running at different time scales for simultaneously estimating quantiles and policy parameters and is shown to converge to the global optimal policy under certain conditions. Our numerical results demonstrate that the proposed algorithms outperform the existing baseline algorithms under the quantile criterion.
Obtaining first-order regret bounds -- regret bounds scaling not as the worst-case but with some measure of the performance of the optimal policy on a given instance -- is a core question in sequential decision-making. While such bounds exist in many settings, they have proven elusive in reinforcement learning with large state spaces. In this work we address this gap, and show that it is possible to obtain regret scaling as $\mathcal{O}(\sqrt{V_1^\star K})$ in reinforcement learning with large state spaces, namely the linear MDP setting. Here $V_1^\star$ is the value of the optimal policy and $K$ is the number of episodes. We demonstrate that existing techniques based on least squares estimation are insufficient to obtain this result, and instead develop a novel robust self-normalized concentration bound based on the robust Catoni mean estimator, which may be of independent interest.
Motivated by applications in reinforcement learning (RL), we study a nonlinear stochastic approximation (SA) algorithm under Markovian noise, and establish its finite-sample convergence bounds under various stepsizes. Specifically, we show that when using constant stepsize (i.e., $\alpha_k\equiv \alpha$), the algorithm achieves exponential fast convergence to a neighborhood (with radius $O(\alpha\log(1/\alpha))$) around the desired limit point. When using diminishing stepsizes with appropriate decay rate, the algorithm converges with rate $O(\log(k)/k)$. Our proof is based on Lyapunov drift arguments, and to handle the Markovian noise, we exploit the fast mixing of the underlying Markov chain. To demonstrate the generality of our theoretical results on Markovian SA, we use it to derive the finite-sample bounds of the popular $Q$-learning with linear function approximation algorithm, under a condition on the behavior policy. Importantly, we do not need to make the assumption that the samples are i.i.d., and do not require an artificial projection step in the algorithm to maintain the boundedness of the iterates. Numerical simulations corroborate our theoretical results.
We propose nonparametric open-end sequential testing procedures that can detect all types of changes in the contemporary distribution function of multivariate observations. Their asymptotic properties are theoretically investigated under stationarity and under alternatives to stationarity. Monte Carlo experiments reveal their good finite-sample behavior in the case of continuous univariate observations. A short data example concludes the work.
We study the offline meta-reinforcement learning (OMRL) problem, a paradigm which enables reinforcement learning (RL) algorithms to quickly adapt to unseen tasks without any interactions with the environments, making RL truly practical in many real-world applications. This problem is still not fully understood, for which two major challenges need to be addressed. First, offline RL usually suffers from bootstrapping errors of out-of-distribution state-actions which leads to divergence of value functions. Second, meta-RL requires efficient and robust task inference learned jointly with control policy. In this work, we enforce behavior regularization on learned policy as a general approach to offline RL, combined with a deterministic context encoder for efficient task inference. We propose a novel negative-power distance metric on bounded context embedding space, whose gradients propagation is detached from the Bellman backup. We provide analysis and insight showing that some simple design choices can yield substantial improvements over recent approaches involving meta-RL and distance metric learning. To the best of our knowledge, our method is the first model-free and end-to-end OMRL algorithm, which is computationally efficient and demonstrated to outperform prior algorithms on several meta-RL benchmarks.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great successes. Unfortunately, the understanding on how it works remains unclear. It has the central importance to lay down the theoretic foundation for deep learning. In this work, we give a geometric view to understand deep learning: we show that the fundamental principle attributing to the success is the manifold structure in data, namely natural high dimensional data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it. We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty to be learned. Then we show for any deep neural network with fixed architecture, there exists a manifold that cannot be learned by the network. Finally, we propose to apply optimal mass transportation theory to control the probability distribution in the latent space.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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