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113-413: For any finite Markov decision process , Q -learning finds an optimal policy in the sense of maximizing the expected value of the total reward over any and all successive steps, starting from the current state. Q -learning can identify an optimal action-selection policy for any given finite Markov decision process, given infinite exploration time and a partly random policy. "Q" refers to the function that

226-424: A Q {\displaystyle Q} table) applies only to discrete action and state spaces. Discretization of these values leads to inefficient learning, largely due to the curse of dimensionality . However, there are adaptations of Q-learning that attempt to solve this problem such as Wire-fitted Neural Network Q-Learning. Markov decision process Markov decision process ( MDP ), also called

339-405: A { r t } {\displaystyle \{r_{t}\}} in such a way that their lifetime expected utility is maximized: The expectation E {\displaystyle \mathbb {E} } is taken with respect to the appropriate probability measure given by Q on the sequences of r ' s. Because r is governed by a Markov process, dynamic programming simplifies

452-401: A {\displaystyle a} in state x {\displaystyle x} is F ( x , a ) {\displaystyle F(x,a)} . Finally, we assume impatience, represented by a discount factor 0 < β < 1 {\displaystyle 0<\beta <1} . Under these assumptions, an infinite-horizon decision problem takes

565-438: A 0 {\displaystyle a_{0}} , knowing that our choice will cause the time 1 state to be x 1 = T ( x 0 , a 0 ) {\displaystyle x_{1}=T(x_{0},a_{0})} . That new state will then affect the decision problem from time 1 on. The whole future decision problem appears inside the square brackets on the right. So far it seems we have only made

678-471: A ) {\displaystyle (s,a)} pairs (together with the outcome s ′ {\displaystyle s'} ; that is, "I was in state s {\displaystyle s} and I tried doing a {\displaystyle a} and s ′ {\displaystyle s'} happened"). Thus, one has an array Q {\displaystyle Q} and uses experience to update it directly. This

791-499: A ) {\displaystyle Q(s_{f},a)} is never updated, but is set to the reward value r {\displaystyle r} observed for state s f {\displaystyle s_{f}} . In most cases, Q ( s f , a ) {\displaystyle Q(s_{f},a)} can be taken to equal zero. The learning rate or step size determines to what extent newly acquired information overrides old information. A factor of 0 makes

904-434: A ) {\displaystyle s',r\gets G(s,a)} might denote the action of sampling from the generative model where s {\displaystyle s} and a {\displaystyle a} are the current state and action, and s ′ {\displaystyle s'} and r {\displaystyle r} are the new state and reward. Compared to an episodic simulator,

1017-459: A recursive definition of the value function: This is the Bellman equation. It may be simplified even further if the time subscripts are dropped and the value of the next state is plugged in: The Bellman equation is classified as a functional equation , because solving it means finding the unknown function V {\displaystyle V} , which is the value function . Recall that

1130-461: A stochastic dynamic program or stochastic control problem, is a model for sequential decision making when outcomes are uncertain. Originating from operations research in the 1950s, MDPs have since gained recognition in a variety of fields, including ecology , economics , healthcare , telecommunications and reinforcement learning . Reinforcement learning utilizes the MDP framework to model

1243-412: A Bellman equation is a recursion for expected rewards. For example, the expected reward for being in a particular state s and following some fixed policy π {\displaystyle \pi } has the Bellman equation: This equation describes the expected reward for taking the action prescribed by some policy π {\displaystyle \pi } . The equation for

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1356-423: A Markov decision process is to find a good "policy" for the decision maker: a function π {\displaystyle \pi } that specifies the action π ( s ) {\displaystyle \pi (s)} that the decision maker will choose when in state s {\displaystyle s} . Once a Markov decision process is combined with a policy in this way, this fixes

1469-482: A biologically inspired mechanism that uses a random sample of prior actions instead of the most recent action to proceed. This removes correlations in the observation sequence and smooths changes in the data distribution. Iterative updates adjust Q towards target values that are only periodically updated, further reducing correlations with the target. Because the future maximum approximated action value in Q-learning

1582-404: A bucket. The exact distance of the finger from its starting position (-Infinity to Infinity) is not known, but rather whether it is far away or not (Near, Far). Q -learning was introduced by Chris Watkins in 1989. A convergence proof was presented by Watkins and Peter Dayan in 1992. Watkins was addressing “Learning from delayed rewards”, the title of his PhD thesis. Eight years earlier in 1981

1695-408: A certain point in time involves the position of the finger in space, its velocity, the angle of the stick and the angular velocity of the stick. This yields a four-element vector that describes one state, i.e. a snapshot of one state encoded into four values. The problem is that infinitely many possible states are present. To shrink the possible space of valid actions multiple values can be assigned to

1808-554: A concept developed by the Russian mathematician Andrey Markov . The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property . The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty. A Markov decision process

1921-476: A dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes. The equation applies to algebraic structures with a total ordering; for algebraic structures with a partial ordering, the generic Bellman's equation can be used. The Bellman equation was first applied to engineering control theory and to other topics in applied mathematics, and subsequently became an important tool in economic theory ; though

2034-402: A finite number of samples is, in general, impossible). For the purpose of this section, it is useful to define a further function, which corresponds to taking the action a {\displaystyle a} and then continuing optimally (or according to whatever policy one currently has): While this function is also unknown, experience during learning is based on ( s ,

2147-401: A function h {\displaystyle h} , then V ¯ ∗ {\displaystyle {\bar {V}}^{*}} will be the smallest g {\displaystyle g} satisfying the above equation. In order to find V ¯ ∗ {\displaystyle {\bar {V}}^{*}} , we could use

2260-483: A generative model has the advantage that it can yield data from any state, not only those encountered in a trajectory. These model classes form a hierarchy of information content: an explicit model trivially yields a generative model through sampling from the distributions, and repeated application of a generative model yields an episodic simulator. In the opposite direction, it is only possible to learn approximate models through regression . The type of model available for

2373-441: A guess of the value function . It then iterates, repeatedly computing V i + 1 {\displaystyle V_{i+1}} for all states s {\displaystyle s} , until V {\displaystyle V} converges with the left-hand side equal to the right-hand side (which is the " Bellman equation " for this problem ). Lloyd Shapley 's 1953 paper on stochastic games included as

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2486-413: A learning automata algorithm also has the advantage of solving the problem when probability or rewards are unknown. The difference between learning automata and Q-learning is that the former technique omits the memory of Q-values, but updates the action probability directly to find the learning result. Learning automata is a learning scheme with a rigorous proof of convergence. In learning automata theory,

2599-545: A lower discount factor and increasing it towards its final value accelerates learning. Since Q -learning is an iterative algorithm, it implicitly assumes an initial condition before the first update occurs. High initial values, also known as "optimistic initial conditions", can encourage exploration: no matter what action is selected, the update rule will cause it to have lower values than the other alternative, thus increasing their choice probability. The first reward r {\displaystyle r} can be used to reset

2712-543: A mutually symmetric fashion using separate experiences. The double Q-learning update step is then as follows: Now the estimated value of the discounted future is evaluated using a different policy, which solves the overestimation issue. This algorithm was later modified in 2015 and combined with deep learning , as in the DQN algorithm, resulting in Double DQN, which outperforms the original DQN algorithm. Delayed Q-learning

2825-399: A new state S t + 1 {\displaystyle S_{t+1}} (that may depend on both the previous state S t {\displaystyle S_{t}} and the selected action), and Q {\displaystyle Q} is updated. The core of the algorithm is a Bellman equation as a simple value iteration update , using the weighted average of

2938-429: A particular MDP plays a significant role in determining which solution algorithms are appropriate. For example, the dynamic programming algorithms described in the next section require an explicit model, and Monte Carlo tree search requires a generative model (or an episodic simulator that can be copied at any state), whereas most reinforcement learning algorithms require only an episodic simulator. An example of MDP

3051-451: A set of linear equations. These equations are merely obtained by making s = s ′ {\displaystyle s=s'} in the step two equation. Thus, repeating step two to convergence can be interpreted as solving the linear equations by relaxation . This variant has the advantage that there is a definite stopping condition: when the array π {\displaystyle \pi } does not change in

3164-560: A special case the value iteration method for MDPs, but this was recognized only later on. In policy iteration ( Howard 1960 ) harv error: no target: CITEREFHoward1960 ( help ) , step one is performed once, and then step two is performed once, then both are repeated until policy converges. Then step one is again performed once and so on. (Policy iteration was invented by Howard to optimize Sears catalogue mailing, which he had been optimizing using value iteration. ) Instead of repeating step two to convergence, it may be formulated and solved as

3277-452: A state–action combination: Before learning begins, ⁠ Q {\displaystyle Q} ⁠ is initialized to a possibly arbitrary fixed value (chosen by the programmer). Then, at each time t {\displaystyle t} the agent selects an action A t {\displaystyle A_{t}} , observes a reward R t + 1 {\displaystyle R_{t+1}} , enters

3390-474: A stochastic automaton consists of: Bellman equation A Bellman equation , named after Richard E. Bellman , is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming . It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. This breaks

3503-429: A stochastic optimization problem. Let the interest r follow a Markov process with probability transition function Q ( r , d μ r ) {\displaystyle Q(r,d\mu _{r})} where d μ r {\displaystyle d\mu _{r}} denotes the probability measure governing the distribution of interest rate next period if current interest rate

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3616-411: A terminal state, or if the agent never reaches one, all environment histories become infinitely long, and utilities with additive, undiscounted rewards generally become infinite. Even with a discount factor only slightly lower than 1, Q -function learning leads to propagation of errors and instabilities when the value function is approximated with an artificial neural network . In that case, starting with

3729-401: A variety of theoretical questions in monetary policy , fiscal policy , taxation , economic growth , search theory , and labor economics . Avinash Dixit and Robert Pindyck showed the value of the method for thinking about capital budgeting . Anderson adapted the technique to business valuation, including privately held businesses. Using dynamic programming to solve concrete problems

3842-504: Is r {\displaystyle r} . In this model the consumer decides their current period consumption after the current period interest rate is announced. Rather than simply choosing a single sequence { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} , the consumer now must choose a sequence { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} for each possible realization of

3955-493: Is a partial differential equation that is called the Hamilton–Jacobi–Bellman equation . In discrete time any multi-stage optimization problem can be solved by analyzing the appropriate Bellman equation. The appropriate Bellman equation can be found by introducing new state variables (state augmentation). However, the resulting augmented-state multi-stage optimization problem has a higher dimensional state space than

4068-412: Is a 4- tuple ( S , A , P a , R a ) {\displaystyle (S,A,P_{a},R_{a})} , where: A policy function π {\displaystyle \pi } is a (potentially probabilistic) mapping from state space ( S {\displaystyle S} ) to action space ( A {\displaystyle A} ). The goal in

4181-461: Is a form of Bellman's equation. Because economic applications of dynamic programming usually result in a Bellman equation that is a difference equation , economists refer to dynamic programming as a "recursive method" and a subfield of recursive economics is now recognized within economics. Nancy Stokey , Robert E. Lucas , and Edward Prescott describe stochastic and nonstochastic dynamic programming in considerable detail, and develop theorems for

4294-437: Is a weighted sum of expected values of the rewards of all future steps starting from the current state. As an example, consider the process of boarding a train, in which the reward is measured by the negative of the total time spent boarding (alternatively, the cost of boarding the train is equal to the boarding time). One strategy is to enter the train door as soon as they open, minimizing the initial wait time for yourself. If

4407-436: Is an alternative implementation of the online Q -learning algorithm, with probably approximately correct (PAC) learning . Greedy GQ is a variant of Q -learning to use in combination with (linear) function approximation. The advantage of Greedy GQ is that convergence is guaranteed even when function approximation is used to estimate the action values. Distributional Q-learning is a variant of Q -learning which seeks to model

4520-511: Is an interdisciplinary area of machine learning and optimal control that has, as main objective, finding an approximately optimal policy for MDPs where transition probabilities and rewards are unknown. Reinforcement learning can solve Markov-Decision processes without explicit specification of the transition probabilities which are instead needed to perform policy iteration. In this setting, transition probabilities and rewards must be learned from experience, i.e. by letting an agent interact with

4633-715: Is called the objective function . Dynamic programming breaks a multi-period planning problem into simpler steps at different points in time. Therefore, it requires keeping track of how the decision situation is evolving over time. The information about the current situation that is needed to make a correct decision is called the "state". For example, to decide how much to consume and spend at each point in time, people would need to know (among other things) their initial wealth. Therefore, wealth ( W ) {\displaystyle (W)} would be one of their state variables , but there would probably be others. The variables chosen at any given point in time are often called

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4746-427: Is called the "Bellman equation". In this approach, the optimal policy in the last time period is specified in advance as a function of the state variable's value at that time, and the resulting optimal value of the objective function is thus expressed in terms of that value of the state variable. Next, the next-to-last period's optimization involves maximizing the sum of that period's period-specific objective function and

4859-456: Is complicated by informational difficulties, such as choosing the unobservable discount rate. There are also computational issues, the main one being the curse of dimensionality arising from the vast number of possible actions and potential state variables that must be considered before an optimal strategy can be selected. For an extensive discussion of computational issues, see Miranda and Fackler, and Meyn 2007. In Markov decision processes ,

4972-582: Is due to Martin Beckmann and Richard Muth . Martin Beckmann also wrote extensively on consumption theory using the Bellman equation in 1959. His work influenced Edmund S. Phelps , among others. A celebrated economic application of a Bellman equation is Robert C. Merton 's seminal 1973 article on the intertemporal capital asset pricing model . (See also Merton's portfolio problem ). The solution to Merton's theoretical model, one in which investors chose between income today and future income or capital gains,

5085-603: Is evaluated using the same Q function as in current action selection policy, in noisy environments Q-learning can sometimes overestimate the action values, slowing the learning. A variant called Double Q-learning was proposed to correct this. Double Q-learning is an off-policy reinforcement learning algorithm, where a different policy is used for value evaluation than what is used to select the next action. In practice, two separate value functions Q A {\displaystyle Q^{A}} and Q B {\displaystyle Q^{B}} are trained in

5198-401: Is known as Q-learning . Another application of MDP process in machine learning theory is called learning automata. This is also one type of reinforcement learning if the environment is stochastic. The first detail learning automata paper is surveyed by Narendra and Thathachar (1974), which were originally described explicitly as finite state automata . Similar to reinforcement learning,

5311-464: Is needed. Substituting the calculation of π ( s ) {\displaystyle \pi (s)} into the calculation of V ( s ) {\displaystyle V(s)} gives the combined step : where i {\displaystyle i} is the iteration number. Value iteration starts at i = 0 {\displaystyle i=0} and V 0 {\displaystyle V_{0}} as

5424-412: Is no benefit in taking multiple actions. It is better to take an action only at the time when system is transitioning from the current state to another state. Under some conditions, if our optimal value function V ∗ {\displaystyle V^{*}} is independent of state i {\displaystyle i} , we will have the following inequality: If there exists

5537-667: Is not true, the problem is called a partially observable Markov decision process or POMDP. Constrained Markov decision processes (CMDPS) are extensions to Markov decision process (MDPs). There are three fundamental differences between MDPs and CMDPs. The method of Lagrange multipliers applies to CMDPs. Many Lagrangian-based algorithms have been developed. There are a number of applications for CMDPs. It has recently been used in motion planning scenarios in robotics. In discrete-time Markov Decision Processes, decisions are made at discrete time intervals. However, for continuous-time Markov decision processes , decisions can be made at any time

5650-505: Is possible to construct online planning algorithms that can find an arbitrarily near-optimal policy with no computational complexity dependence on the size of the state space. A Markov decision process is a stochastic game with only one player. The solution above assumes that the state s {\displaystyle s} is known when action is to be taken; otherwise π ( s ) {\displaystyle \pi (s)} cannot be calculated. When this assumption

5763-885: Is the Pole-Balancing model, which comes from classic control theory. In this example, we have Solutions for MDPs with finite state and action spaces may be found through a variety of methods such as dynamic programming . The algorithms in this section apply to MDPs with finite state and action spaces and explicitly given transition probabilities and reward functions, but the basic concepts may be extended to handle other problem classes, for example using function approximation . The standard family of algorithms to calculate optimal policies for finite state and action MDPs requires storage for two arrays indexed by state: value V {\displaystyle V} , which contains real values, and policy π {\displaystyle \pi } , which contains actions. At

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5876-469: Is the capital accumulation/law of motion specified by the problem, while the second constraint is a transversality condition that the consumer does not carry debt at the end of their life. The Bellman equation is Alternatively, one can treat the sequence problem directly using, for example, the Hamiltonian equations . Now, if the interest rate varies from period to period, the consumer is faced with

5989-415: Is the set of actions available to be taken at state x t {\displaystyle x_{t}} . It is also assumed that the state changes from x {\displaystyle x} to a new state T ( x , a ) {\displaystyle T(x,a)} when action a {\displaystyle a} is taken, and that the current payoff from taking action

6102-546: Is the sum of three factors: An episode of the algorithm ends when state S t + 1 {\displaystyle S_{t+1}} is a final or terminal state . However, Q -learning can also learn in non-episodic tasks (as a result of the property of convergent infinite series). If the discount factor is lower than 1, the action values are finite even if the problem can contain infinite loops. For all final states s f {\displaystyle s_{f}} , Q ( s f ,

6215-488: Is the system control vector we try to find. f ( ⋅ ) {\displaystyle f(\cdot )} shows how the state vector changes over time. The Hamilton–Jacobi–Bellman equation is as follows: We could solve the equation to find the optimal control a ( t ) {\displaystyle a(t)} , which could give us the optimal value function V ∗ {\displaystyle V^{*}} Reinforcement learning

6328-553: The control variables . For instance, given their current wealth, people might decide how much to consume now. Choosing the control variables now may be equivalent to choosing the next state; more generally, the next state is affected by other factors in addition to the current control. For example, in the simplest case, today's wealth (the state) and consumption (the control) might exactly determine tomorrow's wealth (the new state), though typically other factors will affect tomorrow's wealth too. The dynamic programming approach describes

6441-404: The principle of optimality , we will consider the first decision separately, setting aside all future decisions (we will start afresh from time 1 with the new state x 1 {\displaystyle x_{1}} ). Collecting the future decisions in brackets on the right, the above infinite-horizon decision problem is equivalent to: subject to the constraints Here we are choosing

6554-482: The D-LP is said to be an optimal solution if for all feasible solution y ( i , a ) {\displaystyle y(i,a)} to the D-LP. Once we have found the optimal solution y ∗ ( i , a ) {\displaystyle y^{*}(i,a)} , we can use it to establish the optimal policies. In continuous-time MDP, if the state space and action space are continuous,

6667-403: The MDP for a given number of steps. Both on a theoretical and on a practical level, effort is put in maximizing the sample efficiency, i.e. minimimizing the number of samples needed to learn a policy whose performance is ε − {\displaystyle \varepsilon -} close to the optimal one (due to the stochastic nature of the process, learning the optimal policy with

6780-471: The Markov property, it can be shown that the optimal policy is a function of the current state, as assumed above. In many cases, it is difficult to represent the transition probability distributions, P a ( s , s ′ ) {\displaystyle P_{a}(s,s')} , explicitly. In such cases, a simulator can be used to model the MDP implicitly by providing samples from

6893-666: The above optimal control problem. However, the Bellman Equation is often the most convenient method of solving stochastic optimal control problems. For a specific example from economics, consider an infinitely-lived consumer with initial wealth endowment a 0 {\displaystyle {\color {Red}a_{0}}} at period 0 {\displaystyle 0} . They have an instantaneous utility function u ( c ) {\displaystyle u(c)} where c {\displaystyle c} denotes consumption and discounts

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7006-412: The action for each state and the resulting combination behaves like a Markov chain (since the action chosen in state s {\displaystyle s} is completely determined by π ( s ) {\displaystyle \pi (s)} ). The objective is to choose a policy π {\displaystyle \pi } that will maximize some cumulative function of

7119-404: The advantage of being a human-readable knowledge representation form. Function approximation may speed up learning in finite problems, due to the fact that the algorithm can generalize earlier experiences to previously unseen states. Another technique to decrease the state/action space quantizes possible values. Consider the example of learning to balance a stick on a finger. To describe a state at

7232-423: The agent "myopic" (or short-sighted) by only considering current rewards, i.e. r t {\displaystyle r_{t}} (in the update rule above), while a factor approaching 1 will make it strive for a long-term high reward. If the discount factor meets or exceeds 1, the action values may diverge. For ⁠ γ = 1 {\displaystyle \gamma =1} ⁠ , without

7345-448: The agent learn nothing (exclusively exploiting prior knowledge), while a factor of 1 makes the agent consider only the most recent information (ignoring prior knowledge to explore possibilities). In fully deterministic environments, a learning rate of α t = 1 {\displaystyle \alpha _{t}=1} is optimal. When the problem is stochastic , the algorithm converges under some technical conditions on

7458-411: The agent transitions from state to state. Executing an action in a specific state provides the agent with a reward (a numerical score). The goal of the agent is to maximize its total reward. It does this by adding the maximum reward attainable from future states to the reward for achieving its current state, effectively influencing the current action by the potential future reward. This potential reward

7571-400: The algorithm (there were large changes in V {\displaystyle V} or π {\displaystyle \pi } around those states recently) or based on use (those states are near the starting state, or otherwise of interest to the person or program using the algorithm). Algorithms for finding optimal policies with time complexity polynomial in the size of

7684-431: The algorithm computes – the expected rewards for an action taken in a given state. Reinforcement learning involves an agent , a set of states S {\displaystyle {\mathcal {S}}} , and a set A {\displaystyle {\mathcal {A}}} of actions per state. By performing an action a ∈ A {\displaystyle a\in {\mathcal {A}}} ,

7797-473: The algorithm will eventually arrive at the correct solution. In value iteration ( Bellman 1957 ) harv error: no target: CITEREFBellman1957 ( help ) , which is also called backward induction , the π {\displaystyle \pi } function is not used; instead, the value of π ( s ) {\displaystyle \pi (s)} is calculated within V ( s ) {\displaystyle V(s)} whenever it

7910-449: The basic concepts of dynamic programming are prefigured in John von Neumann and Oskar Morgenstern 's Theory of Games and Economic Behavior and Abraham Wald 's sequential analysis . The term "Bellman equation" usually refers to the dynamic programming equation (DPE) associated with discrete-time optimization problems. In continuous-time optimization problems, the analogous equation

8023-402: The best value obtainable depends on the initial situation. The dynamic programming method breaks this decision problem into smaller subproblems. Bellman's principle of optimality describes how to do this: Principle of Optimality: An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to

8136-489: The course of applying step 1 to all states, the algorithm is completed. Policy iteration is usually slower than value iteration for a large number of possible states. In modified policy iteration ( van Nunen 1976 ; Puterman & Shin 1978 ), step one is performed once, and then step two is repeated several times. Then step one is again performed once and so on. In this variant, the steps are preferentially applied to states which are in some way important – whether based on

8249-674: The current value and the new information: where R t + 1 {\displaystyle R_{t+1}} is the reward received when moving from the state S t {\displaystyle S_{t}} to the state S t + 1 {\displaystyle S_{t+1}} , and α {\displaystyle \alpha } is the learning rate ( 0 < α ≤ 1 ) {\displaystyle (0<\alpha \leq 1)} . Note that Q n e w ( S t , A t ) {\displaystyle Q^{new}(S_{t},A_{t})}

8362-415: The decision maker chooses. In comparison to discrete-time Markov decision processes, continuous-time Markov decision processes can better model the decision-making process for a system that has continuous dynamics , i.e., the system dynamics is defined by ordinary differential equations (ODEs). These kind of applications raise in queueing systems , epidemic processes, and population processes . Like

8475-405: The decision maker to favor taking actions early, rather than postpone them indefinitely. Another possible, but strictly related, objective that is commonly used is the H − {\displaystyle H-} step return. This time, instead of using a discount factor   γ   {\displaystyle \ \gamma \ } , the agent is interested only in

8588-426: The departing passengers. Overall, this path has a higher reward than that of the previous day, since the total boarding time is now: Through exploration, despite the initial (patient) action resulting in a larger cost (or negative reward) than in the forceful strategy, the overall cost is lower, thus revealing a more rewarding strategy. After Δ t {\displaystyle \Delta t} steps into

8701-459: The discrete-time Markov decision processes, in continuous-time Markov decision processes the agent aims at finding the optimal policy which could maximize the expected cumulated reward. The only difference with the standard case stays in the fact that, due to the continuous nature of the time variable, the sum is replaced by an integral: where 0 ≤ γ < 1. {\displaystyle 0\leq \gamma <1.} If

8814-460: The distribution of returns rather than the expected return of each action. It has been observed to facilitate estimate by deep neural networks and can enable alternative control methods, such as risk-sensitive control. Q-learning has been proposed in the multi-agent setting (see Section 4.1.2 in ). One approach consists in pretending the environment is passive. Littman proposes the minimax Q learning algorithm. The standard Q-learning algorithm (using

8927-403: The effect of valuing rewards received earlier higher than those received later (reflecting the value of a "good start"). γ {\displaystyle \gamma } may also be interpreted as the probability to succeed (or survive) at every step Δ t {\displaystyle \Delta t} . The algorithm, therefore, has a function that calculates the quality of

9040-540: The effects of receptive fields. Reinforcement learning is unstable or divergent when a nonlinear function approximator such as a neural network is used to represent Q. This instability comes from the correlations present in the sequence of observations, the fact that small updates to Q may significantly change the policy of the agent and the data distribution, and the correlations between Q and the target values. The method can be used for stochastic search in various domains and applications. The technique used experience replay,

9153-450: The end of the algorithm, π {\displaystyle \pi } will contain the solution and V ( s ) {\displaystyle V(s)} will contain the discounted sum of the rewards to be earned (on average) by following that solution from state s {\displaystyle s} . The algorithm has two steps, (1) a value update and (2) a policy update, which are repeated in some order for all

9266-551: The existence of solutions to problems meeting certain conditions. They also describe many examples of modeling theoretical problems in economics using recursive methods. This book led to dynamic programming being employed to solve a wide range of theoretical problems in economics, including optimal economic growth , resource extraction , principal–agent problems , public finance , business investment , asset pricing , factor supply, and industrial organization . Lars Ljungqvist and Thomas Sargent apply dynamic programming to study

9379-625: The first H {\displaystyle H} steps of the process, with each reward having the same weight. where   H   {\displaystyle \ H\ } is the time horizon. Compared to the previous objective, the latter one is more used in Learning Theory . A policy that maximizes the function above is called an optimal policy and is usually denoted π ∗ {\displaystyle \pi ^{*}} . A particular MDP may have multiple distinct optimal policies. Because of

9492-447: The following form: subject to the constraints Notice that we have defined notation V ( x 0 ) {\displaystyle V(x_{0})} to denote the optimal value that can be obtained by maximizing this objective function subject to the assumed constraints. This function is the value function . It is a function of the initial state variable x 0 {\displaystyle x_{0}} , since

9605-419: The following linear programming model: y ( i , a ) {\displaystyle y(i,a)} is a feasible solution to the D-LP if y ( i , a ) {\displaystyle y(i,a)} is nonnative and satisfied the constraints in the D-LP problem. A feasible solution y ∗ ( i , a ) {\displaystyle y^{*}(i,a)} to

9718-512: The future the agent will decide some next step. The weight for this step is calculated as γ Δ t {\displaystyle \gamma ^{\Delta t}} , where γ {\displaystyle \gamma } (the discount factor ) is a number between 0 and 1 ( 0 ≤ γ ≤ 1 {\displaystyle 0\leq \gamma \leq 1} ). Assuming γ < 1 {\displaystyle \gamma <1} , it has

9831-631: The initial conditions. According to this idea, the first time an action is taken the reward is used to set the value of Q {\displaystyle Q} . This allows immediate learning in case of fixed deterministic rewards. A model that incorporates reset of initial conditions (RIC) is expected to predict participants' behavior better than a model that assumes any arbitrary initial condition (AIC). RIC seems to be consistent with human behaviour in repeated binary choice experiments. Q -learning at its simplest stores data in tables. This approach falters with increasing numbers of states/actions since

9944-481: The interaction between a learning agent and its environment. In this framework, the interaction is characterized by states, actions, and rewards. The MDP framework is designed to provide a simplified representation of key elements of artificial intelligence challenges. These elements encompass the understanding of cause and effect , the management of uncertainty and nondeterminism, and the pursuit of explicit goals. The name comes from its connection to Markov chains ,

10057-430: The learning rate that require it to decrease to zero. In practice, often a constant learning rate is used, such as α t = 0.1 {\displaystyle \alpha _{t}=0.1} for all t {\displaystyle t} . The discount factor ⁠ γ {\displaystyle \gamma } ⁠ determines the importance of future rewards. A factor of 0 will make

10170-525: The likelihood of the agent visiting a particular state and performing a particular action is increasingly small. Q -learning can be combined with function approximation . This makes it possible to apply the algorithm to larger problems, even when the state space is continuous. One solution is to use an (adapted) artificial neural network as a function approximator. Another possibility is to integrate Fuzzy Rule Interpolation (FRI) and use sparse fuzzy rule-bases instead of discrete Q-tables or ANNs, which has

10283-560: The next period utility at a rate of 0 < β < 1 {\displaystyle 0<\beta <1} . Assume that what is not consumed in period t {\displaystyle t} carries over to the next period with interest rate r {\displaystyle r} . Then the consumer's utility maximization problem is to choose a consumption plan { c t } {\displaystyle \{{\color {OliveGreen}c_{t}}\}} that solves subject to and The first constraint

10396-421: The next state and reward given any state and action. (Note that this is a different meaning from the term generative model in the context of statistical classification.) In algorithms that are expressed using pseudocode , G {\displaystyle G} is often used to represent a generative model. For example, the expression s ′ , r ← G ( s ,

10509-405: The objective, written as a function of the state, is called the value function . Bellman showed that a dynamic optimization problem in discrete time can be stated in a recursive , step-by-step form known as backward induction by writing down the relationship between the value function in one period and the value function in the next period. The relationship between these two value functions

10622-492: The optimal criterion could be found by solving Hamilton–Jacobi–Bellman (HJB) partial differential equation . In order to discuss the HJB equation, we need to reformulate our problem D ( ⋅ ) {\displaystyle D(\cdot )} is the terminal reward function, s ( t ) {\displaystyle s(t)} is the system state vector, a ( t ) {\displaystyle a(t)}

10735-503: The optimal decision rule is the one that achieves the best possible value of the objective. For example, if someone chooses consumption, given wealth, in order to maximize happiness (assuming happiness H can be represented by a mathematical function, such as a utility function and is something defined by wealth), then each level of wealth will be associated with some highest possible level of happiness, H ( W ) {\displaystyle H(W)} . The best possible value of

10848-439: The optimal plan by finding a rule that tells what the controls should be, given any possible value of the state. For example, if consumption ( c ) depends only on wealth ( W ), we would seek a rule c ( W ) {\displaystyle c(W)} that gives consumption as a function of wealth. Such a rule, determining the controls as a function of the states, is called a policy function . Finally, by definition,

10961-410: The optimal value of the future objective function, giving that period's optimal policy contingent upon the value of the state variable as of the next-to-last period decision. This logic continues recursively back in time, until the first period decision rule is derived, as a function of the initial state variable value, by optimizing the sum of the first-period-specific objective function and the value of

11074-661: The original multi-stage optimization problem - an issue that can potentially render the augmented problem intractable due to the “ curse of dimensionality ”. Alternatively, it has been shown that if the cost function of the multi-stage optimization problem satisfies a "backward separable" structure, then the appropriate Bellman equation can be found without state augmentation. To understand the Bellman equation, several underlying concepts must be understood. First, any optimization problem has some objective: minimizing travel time, minimizing cost, maximizing profits, maximizing utility, etc. The mathematical function that describes this objective

11187-615: The paper, in each iteration performs the following computation: The term “secondary reinforcement” is borrowed from animal learning theory, to model state values via backpropagation : the state value ⁠ v ( s ′ ) {\displaystyle v(s')} ⁠ of the consequence situation is backpropagated to the previously encountered situations. CAA computes state values vertically and actions horizontally (the "crossbar"). Demonstration graphs showing delayed reinforcement learning contained states (desirable, undesirable, and neutral states), which were computed by

11300-509: The problem representation exist for finite MDPs. Thus, decision problems based on MDPs are in computational complexity class P . However, due to the curse of dimensionality , the size of the problem representation is often exponential in the number of state and action variables, limiting exact solution techniques to problems that have a compact representation. In practice, online planning techniques such as Monte Carlo tree search can find useful solutions in larger problems, and, in theory, it

11413-422: The problem significantly. Then the Bellman equation is simply: Under some reasonable assumption, the resulting optimal policy function g ( a , r ) is measurable . For a general stochastic sequential optimization problem with Markovian shocks and where the agent is faced with their decision ex-post , the Bellman equation takes a very similar form The first known application of a Bellman equation in economics

11526-409: The problem uglier by separating today's decision from future decisions. But we can simplify by noticing that what is inside the square brackets on the right is the value of the time 1 decision problem, starting from state x 1 = T ( x 0 , a 0 ) {\displaystyle x_{1}=T(x_{0},a_{0})} . Therefore, the problem can be rewritten as

11639-668: The random rewards, typically the expected discounted sum over a potentially infinite horizon: where   γ   {\displaystyle \ \gamma \ } is the discount factor satisfying 0 ≤   γ   ≤   1 {\displaystyle 0\leq \ \gamma \ \leq \ 1} , which is usually close to 1 {\displaystyle 1} (for example, γ = 1 / ( 1 + r ) {\displaystyle \gamma =1/(1+r)} for some discount rate r {\displaystyle r} ). A lower discount factor motivates

11752-480: The same problem, under the name of “Delayed reinforcement learning”, was solved by Bozinovski's Crossbar Adaptive Array (CAA). The memory matrix W = ‖ w ( a , s ) ‖ {\displaystyle W=\|w(a,s)\|} was the same as the eight years later Q-table of Q-learning. The architecture introduced the term “state evaluation” in reinforcement learning. The crossbar learning algorithm, written in mathematical pseudocode in

11865-488: The second period's value function, which gives the value for all the future periods. Thus, each period's decision is made by explicitly acknowledging that all future decisions will be optimally made. Let x t {\displaystyle x_{t}} be the state at time t {\displaystyle t} . For a decision that begins at time 0, we take as given the initial state x 0 {\displaystyle x_{0}} . At any time,

11978-456: The set of possible actions depends on the current state; we express this as a t ∈ Γ ( x t ) {\displaystyle a_{t}\in \Gamma (x_{t})} , where a particular action a t {\displaystyle a_{t}} represents particular values for one or more control variables, and Γ ( x t ) {\displaystyle \Gamma (x_{t})}

12091-419: The state evaluation function. This learning system was a forerunner of the Q-learning algorithm. In 2014, Google DeepMind patented an application of Q-learning to deep learning , titled "deep reinforcement learning" or "deep Q-learning" that can play Atari 2600 games at expert human levels. The DeepMind system used a deep convolutional neural network , with layers of tiled convolutional filters to mimic

12204-493: The state resulting from the first decision. (See Bellman, 1957, Chap. III.3.) In computer science, a problem that can be broken apart like this is said to have optimal substructure . In the context of dynamic game theory , this principle is analogous to the concept of subgame perfect equilibrium , although what constitutes an optimal policy in this case is conditioned on the decision-maker's opponents choosing similarly optimal policies from their points of view. As suggested by

12317-421: The state space and action space are finite, we could use linear programming to find the optimal policy, which was one of the earliest approaches applied. Here we only consider the ergodic model, which means our continuous-time MDP becomes an ergodic continuous-time Markov chain under a stationary policy . Under this assumption, although the decision maker can make a decision at any time in the current state, there

12430-401: The states until no further changes take place. Both recursively update a new estimation of the optimal policy and state value using an older estimation of those values. Their order depends on the variant of the algorithm; one can also do them for all states at once or state by state, and more often to some states than others. As long as no state is permanently excluded from either of the steps,

12543-411: The train is crowded, however, then you will have a slow entry after the initial action of entering the door as people are fighting you to depart the train as you attempt to board. The total boarding time, or cost, is then: On the next day, by random chance (exploration), you decide to wait and let other people depart first. This initially results in a longer wait time. However, less time is spent fighting

12656-444: The transition distributions. One common form of implicit MDP model is an episodic environment simulator that can be started from an initial state and yields a subsequent state and reward every time it receives an action input. In this manner, trajectories of states, actions, and rewards, often called episodes may be produced. Another form of simulator is a generative model , a single step simulator that can generate samples of

12769-473: The value function describes the best possible value of the objective, as a function of the state x {\displaystyle x} . By calculating the value function, we will also find the function a ( x ) {\displaystyle a(x)} that describes the optimal action as a function of the state; this is called the policy function . In the deterministic setting, other techniques besides dynamic programming can be used to tackle

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