Although duality is used extensively in certain fields, such as supervised learning in machine learning, it has been much less explored in others, such as reinforcement learning (RL). In this paper, we show how duality is involved in a variety of RL work, from that which spearheaded the field, such as Richard Bellman's value iteration, to that which was done within just the past few years yet has already had significant impact, such as TRPO, A3C, and GAIL. We show that duality is not uncommon in reinforcement learning, especially when value iteration, or dynamic programming, is used or when first or second order approximations are made to transform initially intractable problems into tractable convex programs.
We review basic concepts of convex duality, focusing on the very general and supremely useful Fenchel-Rockafellar duality. We summarize how this duality may be applied to a variety of reinforcement learning (RL) settings, including policy evaluation or optimization, online or offline learning, and discounted or undiscounted rewards. The derivations yield a number of intriguing results, including the ability to perform policy evaluation and on-policy policy gradient with behavior-agnostic offline data and methods to learn a policy via max-likelihood optimization. Although many of these results …
University of California, Berkeley（加州大学伯克利分校）；OpenAI
Lagrangian methods are widely used algorithms for constrained optimization problems, but their learning dynamics exhibit oscillations and overshoot which, when applied to safe reinforcement learning, leads to constraint-violating behavior during agent training. We address this shortcoming by proposing a novel Lagrange multiplier update method that utilizes derivatives of the constraint function. We take a controls perspective, wherein the traditional Lagrange multiplier update behaves as \emph{integral} control; our terms introduce \emph{proportional} and \emph{derivative} control, achieving f…
Peking University（北京大学）；Beijing International Center for Mathematical Research（北京国际数学研究中心）；Academy for Advanced Interdisciplinary Studies（前沿交叉学科研究院）
In this paper, we consider the linear programming (LP) formulation for deep reinforcement learning. The number of the constraints depends on the size of state and action spaces, which makes the problem intractable in large or continuous environments. The general augmented Lagrangian method suffers the double-sampling obstacle in solving the LP. Namely, the conditional expectations originated from the constraint functions and the quadratic penalties in the augmented Lagrangian function impose difficulties in sampling and evaluation. Motivated from the updates of the multipliers, we overcome the…
Tsinghua-Berkeley Shenzhen Institute, Tsinghua University（清华大学清华-伯克利深圳学院）
This paper addresses the problem of optimizing charging/discharging schedules of electric vehicles (EVs) when participate in demand response (DR). As there exist uncertainties in EVs' remaining energy, arrival and departure time, and future electricity prices, it is quite difficult to make charging decisions to minimize charging cost while guarantee that the EV's battery state-of-the-charge (SOC) is within certain range. To handle with this dilemma, this paper formulates the EV charging scheduling problem as a constrained Markov decision process (CMDP). By synergistically combining the augment…
Safety is essential for reinforcement learning (RL) applied in real-world tasks like autonomous driving. Chance constraints which guarantee the satisfaction of state constraints at a high probability are suitable to represent the requirements in real-world environment with uncertainty. Existing chance constrained RL methods like the penalty method and the Lagrangian method either exhibit periodic oscillations or cannot satisfy the constraints. In this paper, we address these shortcomings by proposing a separated proportional-integral Lagrangian (SPIL) algorithm. Taking a control perspective, w…
Autonomous agents must often deal with conflicting requirements, such as completing tasks using the least amount of time/energy, learning multiple tasks, or dealing with multiple opponents. In the context of reinforcement learning~(RL), these problems are addressed by (i)~designing a reward function that simultaneously describes all requirements or (ii)~combining modular value functions that encode them individually. Though effective, these methods have critical downsides. Designing good reward functions that balance different objectives is challenging, especially as the number of objectives g…
Osaka University（大阪大学）；Graduate School of Engineering Science（工程科学研究生院）
Deep reinforcement learning (DRL) has attracted much attention as an approach to solve optimal control problems without mathematical models of systems. On the other hand, in general, constraints may be imposed on optimal control problems. In this study, we consider the optimal control problems with constraints to complete temporal control tasks. We describe the constraints using signal temporal logic (STL), which is useful for time sensitive control tasks since it can specify continuous signals within bounded time intervals. To deal with the STL constraints, we introduce an extended constraine…
Enhance the Safety in Reinforcement Learning by ADRC Lagrangian Methods
AI Thrust, The Hong Kong University of Science and Technology (Guangzhou)（人工智能方向，香港科技大学（广州））；School of Artificial Intelligence, Peking University, Beijing, China（人工智能学院，北京大学，北京，中国）；Department of XXX, University of YYY, Location, Country（XXX系，YYY大学，地点，国家）；School of ZZZ, Institute of WWW, Location, Country（ZZZ学院，WWW研究所，地点，国家）
Safety is essential for reinforcement learning (RL) applied in the real world. Adding chance constraints (or probabilistic constraints) is a suitable way to enhance RL safety under uncertainty. Existing chance-constrained RL methods like the penalty methods and the Lagrangian methods either exhibit periodic oscillations or learn an over-conservative or unsafe policy. In this paper, we address these shortcomings by proposing a separated proportional-integral Lagrangian (SPIL) algorithm. We first review the constrained policy optimization process from a feedback control perspective, which regard…
Augmented Lagrangian Multiplier Network for State-wise Safety in Reinforcement Learning
Department of Mathematical Sciences, Tsinghua University（清华大学数学科学系）；School of Vehicle and Mobility, Tsinghua University（清华大学车辆与移动系统学院）；School of Vehicle and Mobility & College of AI, Tsinghua University（清华大学车辆与移动系统学院 & 清华大学人工智能学院）
University of Rome ‘Tor Vergata’（罗马第二大学）；INFN（意大利国家核物理研究所）；École Normale Supérieure（巴黎高等师范学院）；The Abdus Salam International Centre for Theoretical Physics（阿卜杜勒·萨拉姆国际理论物理中心）；Istituto dei Sistemi Complessi, CNR（意大利国家研究委员会复杂系统研究所）
We consider the problem of two active particles in 2D complex flows with the multi-objective goals of minimizing both the dispersion rate and the energy consumption of the pair. We approach the problem by means of Multi Objective Reinforcement Learning (MORL), combining scalarization techniques together with a Q-learning algorithm, for Lagrangian drifters that have variable swimming velocity. We show that MORL is able to find a set of trade-off solutions forming an optimal Pareto frontier. As a benchmark, we show that a set of heuristic strategies are dominated by the MORL solutions. We consid…
Syracuse University（雪城大学）；Georgia Institute of Technology（佐治亚理工学院）；University of Bologna（博洛尼亚大学）
This paper explores the potential of Lagrangian duality for learning applications that feature complex constraints. Such constraints arise in many science and engineering domains, where the task amounts to learning optimization problems which must be solved repeatedly and include hard physical and operational constraints. The paper also considers applications where the learning task must enforce constraints on the predictor itself, either because they are natural properties of the function to learn or because it is desirable from a societal standpoint to impose them. This paper demonstrates ex…
We consider the pool-based active learning problem, where only a subset of the training data is labeled, and the goal is to query a batch of unlabeled samples to be labeled so as to maximally improve model performance. We formulate the problem using constrained learning, where a set of constraints bounds the performance of the model on labeled samples. Considering a primal-dual approach, we optimize the primal variables, corresponding to the model parameters, as well as the dual variables, corresponding to the constraints. As each dual variable indicates how significantly the perturbation of t…
Rakuten Institute of Technology, Rakuten Group Inc.（乐天集团乐天技术研究所）
This paper discusses a new approach to the fundamental problem of learning optimal Q-functions. In this approach, optimal Q-functions are formulated as saddle points of a nonlinear Lagrangian function derived from the classic Bellman optimality equation. The paper shows that the Lagrangian enjoys strong duality, in spite of its nonlinearity, which paves the way to a general Lagrangian method to Q-function learning. As a demonstration, the paper develops an imitation learning algorithm based on the duality theory, and applies the algorithm to a state-of-the-art machine translation benchmark. Th…
Non-convex AC optimal power flow (AC-OPF) is a fundamental optimization problem in power system analysis. The computational complexity of conventional solvers is typically high and not suitable for large-scale networks in real-time operation. Hence, deep learning based approaches have gained intensive attention to conduct the time-consuming training process offline. Supervised learning methods may yield a feasible AC-OPF solution with a small optimality gap. However, they often need conventional solvers to generate the training dataset. This paper proposes an end-to-end unsupervised learning b…
How Does the Lagrangian Guide Safe Reinforcement Learning through Diffusion Models?
University College London（伦敦大学学院）；Imperial College London（伦敦帝国学院）；University of California, San Diego（加州大学圣地亚哥分校）；Kyoto University（京都大学）；King's College London（伦敦国王学院）；University of Sheffield（谢菲尔德大学）；Shanghai University of Finance and Economics（上海财经大学）
