Q-learning is a reinforcement learning algorithm that trains an agent to assign values to its possible actions based on its current state, without requiring a model of the environment (model-free). It can handle problems with stochastic transitions and rewards without requiring adaptations. For example, in a grid maze, an agent learns to reach an exit worth 10 points. At a junction, Q-learning might assign a higher value to moving right than left if right gets to the exit faster, improving this choice by trying both directions over time. 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 the algorithm computes: the expected reward—that is, the quality—of an action taken in a given state. == Reinforcement learning == 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}}} , 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 == After Δ t {\displaystyle \Delta t} steps into 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 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 a state–action combination: Q : S × A → R {\displaystyle Q:{\mathcal {S}}\times {\mathcal {A}}\to \mathbb {R} } . 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 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 the current value and the new information: Q n e w ( S t , A t ) ← ( 1 − α ⏟ learning rate ) ⋅ Q ( S t , A t ) ⏟ current value + α ⏟ learning rate ⋅ ( R t + 1 ⏟ reward + γ ⏟ discount factor ⋅ max a Q ( S t + 1 , a ) ⏟ estimate of optimal future value ⏟ new value (temporal difference target) ) {\displaystyle Q^{new}(S_{t},A_{t})\leftarrow (1-\underbrace {\alpha } _{\text{learning rate}})\cdot \underbrace {Q(S_{t},A_{t})} _{\text{current value}}+\underbrace {\alpha } _{\text{learning rate}}\cdot {\bigg (}\underbrace {\underbrace {R_{t+1}} _{\text{reward}}+\underbrace {\gamma } _{\text{discount factor}}\cdot \underbrace {\max _{a}Q(S_{t+1},a)} _{\text{estimate of optimal future value}}} _{\text{new value (temporal difference target)}}{\bigg )}} 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})} is the sum of three terms: ( 1 − α ) Q ( S t , A t ) {\displaystyle (1-\alpha )Q(S_{t},A_{t})} : the current value (weighted by one minus the learning rate) α R t + 1 {\displaystyle \alpha \,R_{t+1}} : the reward R t + 1 {\displaystyle R_{t+1}} to obtain if action A t {\displaystyle A_{t}} is taken when in state S t {\displaystyle S_{t}} (weighted by learning rate) α γ max a Q ( S t + 1 , a ) {\displaystyle \alpha \gamma \max _{a}Q(S_{t+1},a)} : the maximum reward that can be obtained from state S t + 1 {\displaystyle S_{t+1}} (weighted by learning rate and discount factor) 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 or paths. For all final states s f {\displaystyle s_{f}} , Q ( s f , 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. == Influence of variables == === Learning rate === The learning rate 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 α t = 1 {\displaystyle \alpha _{t}=1} 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 α t = 0.1 {\displaystyle \alpha _{t}=0.1} for all t {\displaystyle t} . === Discount factor === The discount factor γ {\displaystyle \gamma } 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. 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 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 a lower discount factor and increasing it towards its final value accelerates learning. === Initial conditions (Q0) === 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 the initial conditions. According to this idea, the first time an action is taken the reward is used to set the value
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