''Q''-learning is a
model-free reinforcement learning
Reinforcement learning (RL) is an area of machine learning concerned with how intelligent agents ought to take actions in an environment in order to maximize the notion of cumulative reward. Reinforcement learning is one of three basic machine ...
algorithm to learn the value of an action in a particular state. It does not require a model of the environment (hence "model-free"), and it can handle problems with stochastic transitions and rewards without requiring adaptations.
For any finite
Markov decision process (FMDP), ''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 FMDP, given infinite exploration time and a partly-random policy.
"Q" refers to the function that the algorithm computes – the expected rewards for an action taken in a given state.
Reinforcement learning
Reinforcement learning involves an
agent
Agent may refer to:
Espionage, investigation, and law
*, spies or intelligence officers
* Law of agency, laws involving a person authorized to act on behalf of another
** Agent of record, a person with a contractual agreement with an insuranc ...
, a set of ''states'' , and a set of ''actions'' per state. By performing an action
, 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 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 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:
* 0 seconds wait time + 15 seconds fight time
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 the departing passengers. Overall, this path has a higher reward than that of the previous day, since the total boarding time is now:
* 5 second wait time + 0 second fight time
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.
Algorithm
![Q-Learning Matrix Initialized and After Training](https://upload.wikimedia.org/wikipedia/commons/e/e0/Q-Learning_Matrix_Initialized_and_After_Training.png)
After
steps into the future the agent will decide some next step. The weight for this step is calculated as
, where
(the ''discount factor'') is a number between 0 and 1 (
) and has the effect of valuing rewards received earlier higher than those received later (reflecting the value of a "good start").
may also be interpreted as the probability to succeed (or survive) at every step
.
The algorithm, therefore, has a function that calculates the quality of a state–action combination:
:
.
Before learning begins, is initialized to a possibly arbitrary fixed value (chosen by the programmer). Then, at each time
the agent selects an action
, observes a reward
, enters a new state
(that may depend on both the previous state
and the selected action), and
is updated. The core of the algorithm is a
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 ...
as a simple
value iteration update, using the weighted average of the current value and the new information:
:
where ''
'' is the reward received when moving from the state
to the state
, and
is the
learning rate
In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly ...
.
Note that
is the sum of three factors:
*
: the current value (weighted by one minus the learning rate)
*
: the reward
to obtain if action
is taken when in state
(weighted by learning rate)
*
: the maximum reward that can be obtained from state
(weighted by learning rate and discount factor)
An episode of the algorithm ends when state
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
,
is never updated, but is set to the reward value
observed for state
. In most cases,
can be taken to equal zero.
Influence of variables
Learning rate
The
learning rate
In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly ...
or ''step size'' determines to what extent newly acquired information overrides old information. A factor of 0 makes 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
is optimal. When the problem is
stochastic, the algorithm converges under some technical conditions on the learning rate that require it to decrease to zero. In practice, often a constant learning rate is used, such as
for all
.
Discount factor
The discount factor determines the importance of future rewards. A factor of 0 will make the agent "myopic" (or short-sighted) by only considering current rewards, i.e.
(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 , without 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
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
. In that case, starting with a lower discount factor and increasing it towards its final value accelerates learning.
Initial conditions (''Q''0)
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
can be used to reset the initial conditions.
According to this idea, the first time an action is taken the reward is used to set the value of
. 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.
Implementation
''Q''-learning at its simplest stores data in tables. This approach falters with increasing numbers of states/actions since the likelihood of the agent visiting a particular state and performing a particular action is increasingly small.
Function approximation
''Q''-learning can be combined with
function approximation
In general, a function approximation problem asks us to select a function among a that closely matches ("approximates") a in a task-specific way. The need for function approximations arises in many branches of applied mathematics, and comput ...
. 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
Artificial neural networks (ANNs), usually simply called neural networks (NNs) or neural nets, are computing systems inspired by the biological neural networks that constitute animal brains.
An ANN is based on a collection of connected unit ...
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 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.
Quantization
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 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 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).
History
''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 the same problem, under the name of “Delayed reinforcement learning”, was solved by Bozinovski's Crossbar Adaptive Array (CAA).
The memory matrix
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 the paper, in each iteration performs the following computation:
* In state perform action ;
* Receive consequence state ;
* Compute state evaluation ;
* Update crossbar value
.
The term “secondary reinforcement” is borrowed from animal learning theory, to model state values via
backpropagation
In machine learning, backpropagation (backprop, BP) is a widely used algorithm for training feedforward artificial neural networks. Generalizations of backpropagation exist for other artificial neural networks (ANNs), and for functions gener ...
: the state value 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 the state evaluation function. This learning system was a forerunner of the Q-learning algorithm.
In 2014,
Google DeepMind
DeepMind Technologies is a British artificial intelligence subsidiary of Alphabet Inc. and research laboratory founded in 2010. DeepMind was acquired by Google in 2014 and became a wholly owned subsidiary of Alphabet Inc, after Google's restru ...
patented an application of Q-learning to
deep learning, titled "deep reinforcement learning" or "deep Q-learning" that can play
Atari 2600
The Atari 2600, initially branded as the Atari Video Computer System (Atari VCS) from its release until November 1982, is a home video game console developed and produced by Atari, Inc. Released in September 1977, it popularized microprocesso ...
games at expert human levels.
Variants
Deep Q-learning
The DeepMind system used a deep
convolutional neural network
In deep learning, a convolutional neural network (CNN, or ConvNet) is a class of artificial neural network (ANN), most commonly applied to analyze visual imagery. CNNs are also known as Shift Invariant or Space Invariant Artificial Neural Netwo ...
, with layers of tiled
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
al filters to mimic 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,'' 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.
Double Q-learning
Because the future maximum approximated action value in Q-learning 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 are trained in a mutually symmetric fashion using separate experiences,
and
. The double Q-learning update step is then as follows:
:
, and
:
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.
Others
Delayed Q-learning 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 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.
Limitations
The standard Q-learning algorithm (using a
table) applies only to discrete action and state spaces.
Discretization of these values leads to inefficient learning, largely due to the
curse of dimensionality
The curse of dimensionality refers to various phenomena that arise when analyzing and organizing data in high-dimensional spaces that do not occur in low-dimensional settings such as the three-dimensional physical space of everyday experience. T ...
. However, there are adaptations of Q-learning that attempt to solve this problem such as Wire-fitted Neural Network Q-Learning.
See also
*
Reinforcement learning
Reinforcement learning (RL) is an area of machine learning concerned with how intelligent agents ought to take actions in an environment in order to maximize the notion of cumulative reward. Reinforcement learning is one of three basic machine ...
*
Temporal difference learning
Temporal difference (TD) learning refers to a class of model-free reinforcement learning methods which learn by bootstrapping from the current estimate of the value function. These methods sample from the environment, like Monte Carlo methods, a ...
*
SARSA
*
Iterated prisoner's dilemma
The Prisoner's Dilemma is an example of a game analyzed in game theory. It is also a thought experiment that challenges two completely rational agents to a dilemma: cooperate with their partner for mutual reward, or betray their partner ("def ...
*
Game theory
References
External links
Watkins, C.J.C.H. (1989). Learning from Delayed Rewards. PhD thesis, Cambridge University, Cambridge, England.Strehl, Li, Wiewiora, Langford, Littman (2006). PAC model-free reinforcement learningby Richard Sutton and Andrew S. Barto, an online textbook. Se
Piqle: a Generic Java Platform for Reinforcement LearningReinforcement Learning Maze a demonstration of guiding an ant through a maze using ''Q''-learning
{{Differentiable computing
Machine learning algorithms
Reinforcement learning