A Cramér Distance perspective on Quantile Regression based Distributional Reinforcement Learning
Amadeus SAS（艾玛迪斯SAS）
Alix Lhéritier, Nicolas Bondoux
备注：
Substantial changes in the experimental part, in particular in the architectures used for our results. Improvements in the presentation of the proof of Lemma 2. Added a Section to show soundness of TD-learning. To be published in AISTATS 2022
Distributional reinforcement learning (DRL) extends the value-based approach by approximating the full distribution over future returns instead of the mean only, providing a richer signal that leads to improved performances. Quantile Regression (QR) based methods like QR-DQN project arbitrary distributions into a parametric subset of staircase distributions by minimizing the 1-Wasserstein distance. However, due to biases in the gradients, the quantile regression loss is used instead for training, guaranteeing the same minimizer and enjoying unbiased gradients. Non-crossing constraints on the quantiles have been shown to improve the performance of QR-DQN for uncertainty-based exploration strategies. The contribution of this work is in the setting of fixed quantile levels and is twofold. First, we prove that the Cramér distance yields a projection that coincides with the 1-Wasserstein one and that, under non-crossing constraints, the squared Cramér and the quantile regression losses yield collinear gradients, shedding light on the connection between these important elements of DRL. Second, we propose a low complexity algorithm to compute the Cramér distance.
Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)'s analysis of the C51 algorithm in terms of the Cramér distance, but their results only apply to the tabular setting and ignore C51's use of a softmax to produce normalized distributions. In this paper we adapt the Cramér distance to deal with arbitrary vectors. From it we derive a new distributional algorithm which is fully Cramér-based and can be combined to linear function approximation, with formal guarantees in …
In this work, we build on recent advances in distributional reinforcement learning to give a generally applicable, flexible, and state-of-the-art distributional variant of DQN. We achieve this by using quantile regression to approximate the full quantile function for the state-action return distribution. By reparameterizing a distribution over the sample space, this yields an implicitly defined return distribution and gives rise to a large class of risk-sensitive policies. We demonstrate improved performance on the 57 Atari 2600 games in the ALE, and use our algorithm's implicitly defined dist…
UC San Diego（加州大学圣迭戈分校）；Microsoft Research（微软研究院）；Tsinghua University（清华大学）
Distributional Reinforcement Learning (RL) differs from traditional RL in that, rather than the expectation of total returns, it estimates distributions and has achieved state-of-the-art performance on Atari Games. The key challenge in practical distributional RL algorithms lies in how to parameterize estimated distributions so as to better approximate the true continuous distribution. Existing distributional RL algorithms parameterize either the probability side or the return value side of the distribution function, leaving the other side uniformly fixed as in C51, QR-DQN or randomly sampled …
Distributional Off-Policy Evaluation with Deep Quantile Process Regression
School of Statistics and Data Science, and Philosophy and Social Sciences Laboratory of Data Science in Finance and Economics at the Ministry of Education, Jiangxi University of Finance and Economics（统计学与数据科学学院，暨教育部金融与经济数据科学与社会科学实验室，江西财经大学）；School of Statistics and Data Science, Shanghai University of Finance and Economics（统计学与数据科学学院，上海财经大学）；School of Artificial Intelligence, and Hubei Key Laboratory of Computational Science, Wuhan University（人工智能学院，湖北省计算科学重点实验室，武汉大学）
Deep Distributional Learning with Non-crossing Quantile Network
The Hong Kong Polytechnic University（香港理工大学）；University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）；ByteDance（字节跳动）；London School of Economics and Political Science（伦敦政治经济学院）
A Spectral Revisit of the Distributional Bellman Operator under the Cramér Metric
School of Electrical and Electronic Engineering, University College Dublin（电子工程学院）；School of Vehicle and Mobility, Tsinghua University（车辆与移动技术学院）；College of Artificial Intelligence, Tsinghua University（人工智能学院）
Shanghai University of Finance and Economics（上海财经大学）
Although distributional reinforcement learning (DRL) has been widely examined in the past few years, there are two open questions people are still trying to address. One is how to ensure the validity of the learned quantile function, the other is how to efficiently utilize the distribution information. This paper attempts to provide some new perspectives to encourage the future in-depth studies in these two fields. We first propose a non-decreasing quantile function network (NDQFN) to guarantee the monotonicity of the obtained quantile estimates and then design a general exploration framework …
Distributional Soft Bellman Operator under the Cramér Geometry
School of Electrical and Electronic Engineering, University College Dublin（都柏林大学学院电气与电子工程学院）；School of Vehicle and Mobility, Tsinghua University（清华大学车辆与运载学院）；College of Artificial Intelligence, Tsinghua University（清华大学人工智能学院）
DeepMind；University of Cambridge（剑桥大学）；Google Brain（谷歌大脑）
In reinforcement learning an agent interacts with the environment by taking actions and observing the next state and reward. When sampled probabilistically, these state transitions, rewards, and actions can all induce randomness in the observed long-term return. Traditionally, reinforcement learning algorithms average over this randomness to estimate the value function. In this paper, we build on recent work advocating a distributional approach to reinforcement learning in which the distribution over returns is modeled explicitly instead of only estimating the mean. That is, we examine methods…
National Tsing Hua University（台湾清华大学）；NVIDIA Corporation（英伟达公司）
In fully cooperative multi-agent reinforcement learning (MARL) settings, the environments are highly stochastic due to the partial observability of each agent and the continuously changing policies of the other agents. To address the above issues, we integrate distributional RL and value function factorization methods by proposing a Distributional Value Function Factorization (DFAC) framework to generalize expected value function factorization methods to their DFAC variants. DFAC extends the individual utility functions from deterministic variables to random variables, and models the quantile …
Ruhr West University of Applied Sciences（鲁尔西应用科学大学）
In recent years distributional reinforcement learning has produced many state of the art results. Increasingly sample efficient Distributional algorithms for the discrete action domain have been developed over time that vary primarily in the way they parameterize their approximations of value distributions, and how they quantify the differences between those distributions. In this work we transfer three of the most well-known and successful of those algorithms (QR-DQN, IQN and FQF) to the continuous action domain by extending two powerful actor-critic algorithms (TD3 and SAC) with distribution…
Quantile-Coupled Flow Matching for Distributional Reinforcement Learning
Oxford Robotics Institute, University of Oxford（牛津大学机器人研究所）；Bristol Robotics Laboratory, University of the West of England（布里斯托尔机器人实验室，西英格兰大学）；Dyson Institute of Engineering & Technology（戴森工程与技术研究院）
We study distributional robustness in the context of Extreme Value Theory (EVT). We provide a data-driven method for estimating extreme quantiles in a manner that is robust against incorrect model assumptions underlying the application of the standard Extremal Types Theorem. Typical studies in distributional robustness involve computing worst case estimates over a model uncertainty region expressed in terms of the Kullback-Leibler discrepancy. We go beyond standard distributional robustness in that we investigate different forms of discrepancies, and prove rigorous results which are helpful fo…
The distributional perspective on reinforcement learning (RL) has given rise to a series of successful Q-learning algorithms, resulting in state-of-the-art performance in arcade game environments. However, it has not yet been analyzed how these findings from a discrete setting translate to complex practical applications characterized by noisy, high dimensional and continuous state-action spaces. In this work, we propose Quantile QT-Opt (Q2-Opt), a distributional variant of the recently introduced distributed Q-learning algorithm for continuous domains, and examine its behaviour in a series of …
Many reinforcement learning (RL) problems in practice are offline, learning purely from observational data. A key challenge is how to ensure the learned policy is safe, which requires quantifying the risk associated with different actions. In the online setting, distributional RL algorithms do so by learning the distribution over returns (i.e., cumulative rewards) instead of the expected return; beyond quantifying risk, they have also been shown to learn better representations for planning. We propose Conservative Offline Distributional Actor Critic (CODAC), an offline RL algorithm suitable fo…
University of Alberta（阿尔伯塔大学）；Noah’s Ark Lab, Huawei（华为诺亚方舟实验室）；Auburn University（奥本大学）
In this paper, we propose the Quantile Option Architecture (QUOTA) for exploration based on recent advances in distributional reinforcement learning (RL). In QUOTA, decision making is based on quantiles of a value distribution, not only the mean. QUOTA provides a new dimension for exploration via making use of both optimism and pessimism of a value distribution. We demonstrate the performance advantage of QUOTA in both challenging video games and physical robot simulators.
Faculty of Electrical Engineering, Technion（以色列理工学院电气工程学院）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
We devise a distributional variant of gradient temporal-difference (TD) learning. Distributional reinforcement learning has been demonstrated to outperform the regular one in the recent study \citep{bellemare2017distributional}. In the policy evaluation setting, we design two new algorithms called distributional GTD2 and distributional TDC using the Cram{é}r distance on the distributional version of the Bellman error objective function, which inherits advantages of both the nonlinear gradient TD algorithms and the distributional RL approach. In the control setting, we propose the distributiona…
Conformal prediction is a technique for constructing prediction intervals that attain valid coverage in finite samples, without making distributional assumptions. Despite this appeal, existing conformal methods can be unnecessarily conservative because they form intervals of constant or weakly varying length across the input space. In this paper we propose a new method that is fully adaptive to heteroscedasticity. It combines conformal prediction with classical quantile regression, inheriting the advantages of both. We establish a theoretical guarantee of valid coverage, supplemented by extens…
In Cooperative Multi-Agent Reinforcement Learning (MARL) and under the setting of Centralized Training with Decentralized Execution (CTDE), agents observe and interact with their environment locally and independently. With local observation and random sampling, the randomness in rewards and observations leads to randomness in long-term returns. Existing methods such as Value Decomposition Network (VDN) and QMIX estimate the value of long-term returns as a scalar that does not contain the information of randomness. Our proposed model QR-MIX introduces quantile regression, modeling joint state-a…
The multi-agent setting is intricate and unpredictable since the behaviors of multiple agents influence one another. To address this environmental uncertainty, distributional reinforcement learning algorithms that incorporate uncertainty via distributional output have been integrated with multi-agent reinforcement learning (MARL) methods, achieving state-of-the-art performance. However, distributional MARL algorithms still rely on the traditional $ε$-greedy, which does not take cooperative strategy into account. In this paper, we present a risk-based exploration that leads to collaboratively o…
The newsvendor problem is a popular inventory management problem in supply chain management and logistics. Solutions to the newsvendor problem determine optimal inventory levels. This model is typically fully determined by a purchase and sale prices and a distribution of random market demand. From a statistical point of view, this problem is often considered as a quantile estimation of a critical fractile which maximizes anticipated profit. The distribution of demand is a random variable and is often estimated on historic data. In an ideal situation, when the probability distribution of the de…
Diurnal temperature range is an important variable in climate science that can provide information regarding climate variability and climate change. Changes in diurnal temperature range can have implications for hydrology, human health and ecology, among others. Yet, the statistical literature on modelling diurnal temperature range is lacking. In this paper we propose to model the distribution of diurnal temperature range using the five-parameter lambda (FPL) distribution. Additionally, in order to model diurnal temperature range with explanatory variables, we propose a distributional quantile…
In this paper, we devise a distributional framework on actor-critic as a solution to distributional instability, action type restriction, and conflation between samples and statistics. We propose a new method that minimizes the Cramér distance with the multi-step Bellman target distribution generated from a novel Sample-Replacement algorithm denoted SR($λ$), which learns the correct value distribution under multiple Bellman operations. Parameterizing a value distribution with Gaussian Mixture Model further improves the efficiency and the performance of the method, which we name GMAC. We empiri…
In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is…
University of Alberta（阿尔伯塔大学）；Huawei Noah’s Ark（华为诺亚方舟实验室）；University of Oxford（牛津大学）；Huawei Hi-Silicon（华为海思）；University of Waterloo（滑铁卢大学）
In distributional reinforcement learning (RL), the estimated distribution of value function models both the parametric and intrinsic uncertainties. We propose a novel and efficient exploration method for deep RL that has two components. The first is a decaying schedule to suppress the intrinsic uncertainty. The second is an exploration bonus calculated from the upper quantiles of the learned distribution. In Atari 2600 games, our method outperforms QR-DQN in 12 out of 14 hard games (achieving 483 \% average gain across 49 games in cumulative rewards over QR-DQN with a big win in Venture). We a…
We study the multi-step off-policy learning approach to distributional RL. Despite the apparent similarity between value-based RL and distributional RL, our study reveals intriguing and fundamental differences between the two cases in the multi-step setting. We identify a novel notion of path-dependent distributional TD error, which is indispensable for principled multi-step distributional RL. The distinction from the value-based case bears important implications on concepts such as backward-view algorithms. Our work provides the first theoretical guarantees on multi-step off-policy distributi…
Department of Mathematics, Linnæus University（林奈大学数学系）
In this work we continue to build upon recent advances in reinforcement learning for finite Markov processes. A common approach among previous existing algorithms, both single-actor and distributed, is to either clip rewards or to apply a transformation method on Q-functions to handle a large variety of magnitudes in real discounted returns. We theoretically show that one of the most successful methods may not yield an optimal policy if we have a non-deterministic process. As a solution, we argue that distributional reinforcement learning lends itself to remedy this situation completely. By th…
Post-processing of ensemble photovoltaic power forecasts with distributional and quantile regression methods
University of Debrecen（德布勒森大学）；Marburg University（马尔堡大学）；Heidelberg Institute for Theoretical Studies（海德堡理论研究所）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Harbin Institute of Technology（哈尔滨工业大学）
In traditional reinforcement learning, an agent maximizes the reward collected during its interaction with the environment by approximating the optimal policy through the estimation of value functions. Typically, given a state s and action a, the corresponding value is the expected discounted sum of rewards. The optimal action is then chosen to be the action a with the largest value estimated by value function. However, recent developments have shown both theoretical and experimental evidence of superior performance when value function is replaced with value distribution in context of deep Q l…
