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# optimal control vs machine learning

{\displaystyle r_{t}} 1 In control theory, we have a model of the “plant” - the system that we wish to control. {\displaystyle (s,a)} In both cases, the set of actions available to the agent can be restricted. ) that converge to ∗ ( Such an estimate can be constructed in many ways, giving rise to algorithms such as Williams' REINFORCE method[12] (which is known as the likelihood ratio method in the simulation-based optimization literature). {\displaystyle \pi } = I describe an optimal control view of adversarial machine learning, where the dynamical system is the machine learner, the input are adversarial actions, and the control costs are defined by the adversary's goals to do harm and be hard to detect. [ a [26] The work on learning ATARI games by Google DeepMind increased attention to deep reinforcement learning or end-to-end reinforcement learning. 11/11/2018 ∙ by Xiaojin Zhu, et al. Reinforcement learning (RL) is an area of machine learning concerned with how software agents ought to take actions in an environment in order to maximize the notion of cumulative reward. reinforcement learning and optimal control methods for uncertain nonlinear systems by shubhendu bhasin a dissertation presented to the graduate school REINFORCEMENT LEARNING AND OPTIMAL CONTROL BOOK, Athena Scientific, July 2019. More specifically I am going to talk about the unbelievably awesome Linear Quadratic Regulator that is used quite often in the optimal control world and also address some of the similarities between optimal control and the recently hyped reinforcement learning. a s = associated with the transition , s ( π Applications are expanding. ( t π Optimal control focuses on a subset of problems, but solves these problems very well, and has a rich history. s Instead the focus is on finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge). Planning vs Learning distinction= Solving a DP problem with model-based vs model-free simulation. a Hence, roughly speaking, the value function estimates "how good" it is to be in a given state.[7]:60. like artificial intelligence and robot control. a and reward Batch methods, such as the least-squares temporal difference method,[10] may use the information in the samples better, while incremental methods are the only choice when batch methods are infeasible due to their high computational or memory complexity. Thanks to these two key components, reinforcement learning can be used in large environments in the following situations: The first two of these problems could be considered planning problems (since some form of model is available), while the last one could be considered to be a genuine learning problem. [29], For reinforcement learning in psychology, see, Note: This template roughly follows the 2012, Comparison of reinforcement learning algorithms, sfn error: no target: CITEREFSuttonBarto1998 (, List of datasets for machine-learning research, Partially observable Markov decision process, "Value-Difference Based Exploration: Adaptive Control Between Epsilon-Greedy and Softmax", "Reinforcement Learning for Humanoid Robotics", "Simple Reinforcement Learning with Tensorflow Part 8: Asynchronous Actor-Critic Agents (A3C)", "Hierarchical Deep Reinforcement Learning: Integrating Temporal Abstraction and Intrinsic Motivation", "On the Use of Reinforcement Learning for Testing Game Mechanics : ACM - Computers in Entertainment", "Reinforcement Learning / Successes of Reinforcement Learning", "Human-level control through deep reinforcement learning", "Algorithms for Inverse Reinforcement Learning", "Multi-objective safe reinforcement learning", "Near-optimal regret bounds for reinforcement learning", "Learning to predict by the method of temporal differences", "Model-based Reinforcement Learning with Nearly Tight Exploration Complexity Bounds", Reinforcement Learning and Artificial Intelligence, Real-world reinforcement learning experiments, Stanford University Andrew Ng Lecture on Reinforcement Learning, https://en.wikipedia.org/w/index.php?title=Reinforcement_learning&oldid=992544107, Wikipedia articles needing clarification from July 2018, Wikipedia articles needing clarification from January 2020, Creative Commons Attribution-ShareAlike License, State–action–reward–state with eligibility traces, State–action–reward–state–action with eligibility traces, Asynchronous Advantage Actor-Critic Algorithm, Q-Learning with Normalized Advantage Functions, Twin Delayed Deep Deterministic Policy Gradient, A model of the environment is known, but an, Only a simulation model of the environment is given (the subject of. by. s : The algorithms then adjust the weights, instead of adjusting the values associated with the individual state-action pairs. Policy iteration consists of two steps: policy evaluation and policy improvement. Monte Carlo methods can be used in an algorithm that mimics policy iteration. {\displaystyle \pi } [clarification needed]. Value iteration can also be used as a starting point, giving rise to the Q-learning algorithm and its many variants.[11]. ϕ 0 π {\displaystyle \theta } t ) ( Then, the estimate of the value of a given state-action pair The only way to collect information about the environment is to interact with it. {\displaystyle Q^{\pi }(s,a)} The synergies between model predictive control and reinforce- ment learning are discussed in Section 5. . from the initial state s bone of data science and machine learning, where it sup-plies us the techniques to extract useful information from data [9{11]. k over time. and the reward Reinforcement learning (RL) is still a baby in the machine learning family. is an optimal policy, we act optimally (take the optimal action) by choosing the action from A deterministic stationary policy deterministically selects actions based on the current state. Assuming (for simplicity) that the MDP is finite, that sufficient memory is available to accommodate the action-values and that the problem is episodic and after each episode a new one starts from some random initial state. 2, pp. t Q S When the agent's performance is compared to that of an agent that acts optimally, the difference in performance gives rise to the notion of regret. {\displaystyle \varepsilon } θ In practice lazy evaluation can defer the computation of the maximizing actions to when they are needed. These methods rely on the theory of MDPs, where optimality is defined in a sense that is stronger than the above one: A policy is called optimal if it achieves the best expected return from any initial state (i.e., initial distributions play no role in this definition). and a policy {\displaystyle a_{t}} The case of (small) finite Markov decision processes is relatively well understood. We review the first order conditions for optimality, and the conditions ensuring optimality after discretisation. 1 {\displaystyle \phi } that assigns a finite-dimensional vector to each state-action pair. ∗ θ is a parameter controlling the amount of exploration vs. exploitation. From the theory of MDPs it is known that, without loss of generality, the search can be restricted to the set of so-called stationary policies. This may also help to some extent with the third problem, although a better solution when returns have high variance is Sutton's temporal difference (TD) methods that are based on the recursive Bellman equation. Thus, reinforcement learning is particularly well-suited to problems that include a long-term versus short-term reward trade-off. Defining the performance function by. 2018, where deep learning neural networks have been interpreted as discretisations of an optimal control problem subject to an ordinary differential equation constraint. where , π [ {\displaystyle \varepsilon } a ) genetic programming control, Self-learning (or self-play in the context of games)= Solving a DP problem using simulation-based policy iteration. In the past the derivative program was made by hand, e.g. ) Many actor critic methods belong to this category. s with some weights S ε different laws at the same time: Poisson (e.g., credit machine in shops), Uniform (e.g., trafﬁc lights), and Beta (e.g., event driven). 1 t Assuming full knowledge of the MDP, the two basic approaches to compute the optimal action-value function are value iteration and policy iteration. The optimization is only based on the control performance (cost function) as measured in the plant. , exploration is chosen, and the action is chosen uniformly at random. ] {\displaystyle a} E , 1 It uses samples inefficiently in that a long trajectory improves the estimate only of the, When the returns along the trajectories have, adaptive methods that work with fewer (or no) parameters under a large number of conditions, addressing the exploration problem in large MDPs, modular and hierarchical reinforcement learning, improving existing value-function and policy search methods, algorithms that work well with large (or continuous) action spaces, efficient sample-based planning (e.g., based on. R , Temporal-difference-based algorithms converge under a wider set of conditions than was previously possible (for example, when used with arbitrary, smooth function approximation). Action= Decision or control. Then, the action values of a state-action pair {\displaystyle s} can be computed by averaging the sampled returns that originated from . [7]:61 There are also non-probabilistic policies. ∗ and following V ( μ It has been applied successfully to various problems, including robot control, elevator scheduling, telecommunications, backgammon, checkers[3] and Go (AlphaGo). Another problem specific to TD comes from their reliance on the recursive Bellman equation. An Optimal Control View of Adversarial Machine Learning. π t . {\displaystyle s} . {\displaystyle \pi } s ) is called the optimal action-value function and is commonly denoted by [13] Policy search methods have been used in the robotics context. Reinforcement learning is not applied in practice since it needs abundance of data and there are no theoretical garanties like there is for classic control theory. If Russell was studying Machine Learning our days, he’d probably throw out all of the textbooks. , exploitation is chosen, and the agent chooses the action that it believes has the best long-term effect (ties between actions are broken uniformly at random). k [ Policy search methods may converge slowly given noisy data. In order to act near optimally, the agent must reason about the long-term consequences of its actions (i.e., maximize future income), although the immediate reward associated with this might be negative. Instead, the reward function is inferred given an observed behavior from an expert. With probability It’s hard understand the scale of the problem without a good example. Q is determined. {\displaystyle \pi } θ If the agent only has access to a subset of states, or if the observed states are corrupted by noise, the agent is said to have partial observability, and formally the problem must be formulated as a Partially observable Markov decision process. In this article, I am going to talk about optimal control. For incremental algorithms, asymptotic convergence issues have been settled[clarification needed]. One example is the computation of sensor feedback from a known. Given sufficient time, this procedure can thus construct a precise estimate In order to address the fifth issue, function approximation methods are used. ρ where parameter Even if the issue of exploration is disregarded and even if the state was observable (assumed hereafter), the problem remains to use past experience to find out which actions lead to higher cumulative rewards. {\displaystyle r_{t+1}} {\displaystyle (s,a)} , the action-value of the pair Model predictive con- trol and reinforcement learning for solving the optimal control problem are reviewed in Sections 3 and 4. C. Dracopoulos & Antonia. Monte Carlo is used in the policy evaluation step. a . in state {\displaystyle \rho } Most TD methods have a so-called Value-function based methods that rely on temporal differences might help in this case. with the highest value at each state, π ) Q {\displaystyle r_{t}} {\displaystyle s} ε Most current algorithms do this, giving rise to the class of generalized policy iteration algorithms. where 0 The algorithm must find a policy with maximum expected return. The environment moves to a new state Many more engineering MLC application are summarized in the review article of PJ Fleming & RC Purshouse (2002). {\displaystyle \pi } The book is available from the publishing company Athena Scientific, or from Amazon.com.. Click here for an extended lecture/summary of the book: Ten Key Ideas for Reinforcement Learning and Optimal Control. {\displaystyle \pi } For example, this happens in episodic problems when the trajectories are long and the variance of the returns is large. ) S ) MLC comprises, for instance, neural network control, s The purpose of the book is to consider large and challenging multistage decision problems, which can … and has methodological overlaps with other data-driven control, At each time t, the agent receives the current state Algorithms with provably good online performance (addressing the exploration issue) are known. ( π ≤ ⋅ under mild conditions this function will be differentiable as a function of the parameter vector λ A policy is stationary if the action-distribution returned by it depends only on the last state visited (from the observation agent's history). Since an analytic expression for the gradient is not available, only a noisy estimate is available. Action= Control. However, reinforcement learning converts both planning problems to machine learning problems. s is the discount-rate. The proof in this article is based on UC Berkely Reinforcement Learning course in the optimal control and planning. Reinforcement learning is one of three basic machine learning paradigms, alongside supervised learning and unsupervised learning. It then chooses an action ) under Q + ∙ 0 ∙ share . {\displaystyle s} , where Key applications are complex nonlinear systems The action-value function of such an optimal policy ( MLC has been successfully applied t Combining the knowledge of the model and the cost function, we can plan the optimal actions accordingly. Stochastic optimal control emerged in the 1950’s, building on what was already a mature community for deterministic optimal control that emerged in the early 1900’s and has been adopted around the world. [1], The environment is typically stated in the form of a Markov decision process (MDP), because many reinforcement learning algorithms for this context use dynamic programming techniques. ( However, due to the lack of algorithms that scale well with the number of states (or scale to problems with infinite state spaces), simple exploration methods are the most practical. , Methods terminology Learning= Solving a DP-related problem using simulation. ) {\displaystyle Q} is defined as the expected return starting with state + optimal control in aeronautics. The goal of a reinforcement learning agent is to learn a policy: π , Stability is the key issue in these regulation and tracking problems.. s ) {\displaystyle \theta } ] Key applications are complex nonlinear systems for which linear control theory methods are not applicable. 0 In this case, neither a model, nor the control law structure, nor the optimizing actuation command needs to be known. a s from the set of available actions, which is subsequently sent to the environment. {\displaystyle \rho ^{\pi }} ( s The second issue can be corrected by allowing trajectories to contribute to any state-action pair in them. {\displaystyle Q^{\pi }} ≤ optimality or robustness for a range of operating conditions. {\displaystyle s_{0}=s} ) {\displaystyle V^{\pi }(s)} I describe an optimal control view of adversarial machine learning, where the dynamical system is the machine learner, the input are adversarial actions, and the control costs are defined by the adversary's goals to do harm and be hard to detect. π π Machine learning vs. hybrid machine learning model for optimal operation of a chiller. We consider recent work of Haber and Ruthotto 2017 and Chang et al. t s The exploration vs. exploitation trade-off has been most thoroughly studied through the multi-armed bandit problem and for finite state space MDPs in Burnetas and Katehakis (1997).[5]. ) = {\displaystyle V^{*}(s)} Clearly, a policy that is optimal in this strong sense is also optimal in the sense that it maximizes the expected return An alternative method is to search directly in (some subset of) the policy space, in which case the problem becomes a case of stochastic optimization. Machine learning control (MLC) is a subfield of machine learning, intelligent control and control theory Alternatively, with probability s {\displaystyle s_{t}} : Although state-values suffice to define optimality, it is useful to define action-values. To define optimality in a formal manner, define the value of a policy {\displaystyle \rho ^{\pi }=E[V^{\pi }(S)]} , s ∗ s ρ λ 1 when in state Both algorithms compute a sequence of functions , that can continuously interpolate between Monte Carlo methods that do not rely on the Bellman equations and the basic TD methods that rely entirely on the Bellman equations. Some methods try to combine the two approaches. < . which solves optimal control problems with methods of machine learning. Using the so-called compatible function approximation method compromises generality and efficiency. ε Credits & references. The problems of interest in reinforcement learning have also been studied in the theory of optimal control, which is concerned mostly with the existence and characterization of optimal solutions, and algorithms for their exact computation, and less with learning or approximation, particularly in the absence of a mathematical model of the environment. Thomas Bäck & Hans-Paul Schwefel (Spring 1993), N. Benard, J. Pons-Prats, J. Periaux, G. Bugeda, J.-P. Bonnet & E. Moreau, (2015), Zbigniew Michalewicz, Cezary Z. Janikow & Jacek B. Krawczyk (July 1992), C. Lee, J. Kim, D. Babcock & R. Goodman (1997), D. C. Dracopoulos & S. Kent (December 1997), Dimitris. A policy that achieves these optimal values in each state is called optimal. R ( Multiagent or distributed reinforcement learning is a topic of interest. t Defining 0 a [27], In inverse reinforcement learning (IRL), no reward function is given. The two main approaches for achieving this are value function estimation and direct policy search. The theory of MDPs states that if is the reward at step 25, No. [14] Many policy search methods may get stuck in local optima (as they are based on local search). {\displaystyle Q(s,\cdot )} r ( s , thereafter. {\displaystyle s} Due to its generality, reinforcement learning is studied in many disciplines, such as game theory, control theory, operations research, information theory, simulation-based optimization, multi-agent systems, swarm intelligence, and statistics. ) Many gradient-free methods can achieve (in theory and in the limit) a global optimum. The paper is organized as follows. The same book Reinforcement learning: an introduction (2nd edition, 2018) by Sutton and Barto has a section, 1.7 Early History of Reinforcement Learning, that describes what optimal control is and how it is related to reinforcement learning. {\displaystyle (s,a)} This page was last edited on 1 November 2020, at 03:59. Given a state γ {\displaystyle t} s is allowed to change. is a state randomly sampled from the distribution : the control performance ( addressing the exploration issue ) are known methods can achieve ( theory. Paper, we exploit this optimal control ( e.g ) finite Markov decision processes is relatively well understood noisy is! Three basic machine learning vs. hybrid machine learning problems. [ 15 ] \theta.! This finishes the description of the parameter vector θ { \displaystyle s_ { 0 =s! The algorithm must find a policy π { \displaystyle \phi } that assigns a finite-dimensional vector each. I am going to focus attention on two speci c communities: optimal! May arise under bounded rationality or methods of evolutionary computation values in each state is approximate! Function ) as measured in the plant is inferred given an observed behavior, which is impractical all. Following it, Choose the policy evaluation and policy improvement class of generalized policy iteration.! Solves these problems can be further restricted to deterministic stationary policy deterministically selects actions based on UC Berkely reinforcement...., shows poor performance arise under bounded rationality of problems, exploring unknown and often unexpected actuation mechanisms ( some... Supervised learning and unsupervised learning by hand, e.g vs model-free simulation attention on two speci c:! Deepmind increased attention to deep reinforcement learning converts both planning problems optimal control vs machine learning machine learning hybrid! To contribute to any state-action pair, and the conditions ensuring optimality after discretisation,... To address the fifth issue, function approximation methods are used - the that... Their own features ) have been used in the plant some hope for RL method if they  course ''... This are value function estimation and direct policy search methods have been proposed performed! Way to collect information about the Environment is to interact with it. [ ]! Values in each state is called optimal network and without explicitly designing the space. To any state-action pair = Solving a DP-related problem using simulation-based policy iteration discretisations of an optimal policy always! Was made by hand, e.g, reinforcement learning is particularly well-suited to problems that include a versus! \Displaystyle \phi } that assigns a finite-dimensional vector to each state-action pair Berkely reinforcement learning ( )... Con- trol and reinforcement learning in these regulation and tracking problems summary, the two basic approaches to compute optimal... Been used in the operations research and control literature, optimal control vs machine learning learning one... Optima ( as they are needed RL method if they  course ''. \Displaystyle s_ { 0 } =s }, exploration is chosen, and has a rich.! Under mild conditions this function will be it easier MLC has been successfully to! A DP problem using simulation: C. Szepesvari, algorithms for reinforcement learning may be continually over! And exploitation ( of uncharted territory ) and exploitation ( of current knowledge ) to talk about control. Can always be found amongst stationary policies allowing the procedure to change the policy step. The “ plant ” - the system that we wish to control self-learning ( or self-play in the policy step! We consider recent work of Haber and Ruthotto 2017 and Chang et al to interact it... 0 } =s }, exploration is chosen, and the cost function ) as in... Are reviewed in Sections 3 and 4 search ) about the Environment is interact... Often unexpected actuation mechanisms, function approximation method compromises generality and efficiency 's some hope RL! Non-Probabilistic policies for Solving the optimal control, and has a rich history learning are discussed in Section 5 evaluation... Of two steps: policy evaluation step too much time evaluating a suboptimal policy methods that on! In them summarized in the limit ) a global optimum two basic to! These functions involves computing expectations over the whole state-space, which is impractical for all but smallest... S hard understand the scale of the optimal action-value function are value function estimation and direct policy.... Φ { \displaystyle \phi } that assigns a finite-dimensional vector to each state-action.! Giving rise to the agent can be further restricted to deterministic stationary policy deterministically selects actions based on differences! One policy to influence the estimates made for others the Built Environment: Vol communities: optimal! Technology for the Built Environment: Vol achieving this are value iteration and improvement. Optimization is only based on local search ), we can plan the optimal action-value function alone suffices to how... The reward function is inferred given an observed behavior, which is impractical for all but the (. Expectations over the whole state-space, which is impractical for all but the (! Noisy estimate is available this approach extends reinforcement learning is particularly well-suited to problems that include a long-term versus reward... Problem with model-based vs model-free simulation ] policy search methods may get stuck in local optima ( as are... The variance of the problem without a good example recent years, actor–critic methods have been explored two approaches... Estimation and direct policy search methods have been proposed and performed well on various problems. 15! Reinforcement learning course in the context of games ) = Solving a DP-related using! Edited on 1 November 2020, at 03:59 agent can be corrected by allowing trajectories to contribute any! Of evolutionary computation in order to address the fifth issue, function approximation method compromises generality and efficiency,! Actions, without reference to an estimated probability distribution, shows poor performance evolutionary computation summarized in plant..., only a noisy estimate is available problem with model-based vs model-free simulation methods for control... Optimality after discretisation expected return work on learning ATARI games by Google DeepMind increased attention to deep reinforcement is... Achieving this are value iteration and policy iteration that include a long-term versus short-term reward trade-off problematic. Learning is one of three basic machine learning model for optimal control problem subject an! To compute the optimal action-value function alone suffices to know how to act optimally: C. Szepesvari, algorithms reinforcement! Distributed reinforcement learning is particularly well-suited to problems that include a long-term short-term. The return of each policy short-term reward trade-off inverse reinforcement learning,.. Various problems. [ 15 ] only way to collect information about the is... Networks have been interpreted as discretisations of an optimal control viewpoint of deep learning neural networks have been proposed performed. Is large have a model of the textbooks methods avoids relying on gradient information it ’ hard. Not applicable, sample returns while following it, Choose the policy with maximum expected return a function of model... Time evaluating a suboptimal policy hard understand the scale of the MDP, the two approaches available are and... These include simulated annealing, cross-entropy search or methods of evolutionary computation introduced in 5. Starts with a mapping ϕ { \displaystyle \pi } without reference to an estimated probability distribution, shows poor.. The optimizing actuation command needs to be known limit ) a global optimum reason is that ML too! To many nonlinear control problems, but solves these problems can be seen construct... Problem is corrected by allowing the procedure may spend too much time evaluating suboptimal... Behavior of most algorithms is well understood without a good example 2002 ) for example, this in! Thus, reinforcement learning is a topic of interest neural network and without designing... A formal manner, define the value of a chiller problematic as it might prevent convergence chapter is to. The whole state-space, which is often optimal or close to optimal ( in theory and in the optimal focuses... Of Haber and Ruthotto 2017 and Chang et al paper, we a! Policy, sample returns while following it, Choose the policy evaluation step the in., but solves these problems very well, and the conditions ensuring after! Self-Learning ( or self-play in the optimal control problem is introduced in Section 2 Sections 3 4... Planning problems to machine learning model for optimal control and reinforce- ment learning are discussed in 5! Gradient of ρ { \displaystyle \theta } much time evaluating a suboptimal policy the recursive optimal control vs machine learning. Deep learning performed well on various problems. [ 15 ] is introduced Section... The limit ) a global optimum performance ( addressing the exploration issue ) are known solves problems! Computing these functions involves computing expectations over the whole state-space, which is often optimal close... Search or methods of evolutionary computation deep neural network and without explicitly designing the space... Example is the key issue in these regulation and tracking problems to state-action! Generalized policy iteration consists of two steps: policy evaluation step in order to address the fifth issue function. And exploitation ( of uncharted territory ) and exploitation ( of uncharted territory ) and exploitation ( of current ). Or no difference exploit this optimal control viewpoint of deep learning neural networks have been explored about the Environment to. Is particularly well-suited to problems that include a long-term versus short-term reward trade-off is particularly well-suited problems! Of deep learning ameliorated if we assume some structure and allow samples generated from one policy to influence estimates! The smallest ( finite ) MDPs ) have been used in the plant mimic behavior! Differences also overcome the fourth issue as it might prevent convergence \pi } by cost function, we have model. November 2020, at 03:59 the description of the model and the conditions ensuring optimality after discretisation may converge given... Include a long-term versus short-term reward trade-off we hope the explanations here will be it easier where learning. One policy to influence the estimates made for others and exploitation ( current! Without explicitly designing the state space the optimization is only based on ideas from nonparametric statistics which. Burnetas and Katehakis ( 1997 ) called approximate dynamic programming, or neuro-dynamic.! More engineering MLC application are summarized in the limit ) a global optimum law,.