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LETTER doi:10.1038/nature14236

Human-level control through deep reinforcement
learning
Volodymyr Mnih1*, Koray Kavukcuoglu1*, David Silver1*, Andrei A. Rusu1, Joel Veness1, Marc G. Bellemare1, Alex Graves1,
Martin Riedmiller1, Andreas K. Fidjeland1, Georg Ostrovski1, Stig Petersen1, Charles Beattie1, Amir Sadik1, Ioannis Antonoglou1,
Helen King1, Dharshan Kumaran1, Daan Wierstra1, Shane Legg1 & Demis Hassabis1

The theory of reinforcement learning provides a normative account1, agent is to select actions in a fashion that maximizes cumulative future
deeply rooted in psychological2 and neuroscientific3 perspectives on reward. More formally, we use a deep convolutional neural network to
animal behaviour, of how agents may optimize their control of an approximate the optimal action-value function
environment. To use reinforcement learning successfully in situations
Q ðs,aÞ~ max rt zcrtz1 zc2 rtz2 z . . . jst ~s, at ~a, p ,
approaching real-world complexity, however, agents are confronted p
with a difficult task: they must derive efficient representations of the which is the maximum sum of rewards rt discounted by c at each time-
environment from high-dimensional sensory inputs, and use these step t, achievable by a behaviour policy p 5 P(ajs), after making an
to generalize past experience to new situations. Remarkably, humans observation (s) and taking an action (a) (see Methods)19.
and other animals seem to solve this problem through a harmonious Reinforcement learning is known to be unstable or even to diverge
combination of reinforcement learning and hierarchical sensory pro- when a nonlinear function approximator such as a neural network is
cessing systems4,5, the former evidenced by a wealth of neural data used to represent the action-value (also known as Q) function20. This
revealing notable parallels between the phasic signals emitted by dopa- instability has several causes: the correlations present in the sequence
minergic neurons and temporal difference reinforcement learning of observations, the fact that small updates to Q may significantly change
algorithms3. While reinforcement learning agents have achieved some the policy and therefore change the data distribution, and the correlations
successes in a variety of domains6–8, their applicability has previously between the action-values (Q) and the target values rzc max Qðs0 , a0 Þ.
been limited to domains in which useful features can be handcrafted, a0
We address these instabilities with a novel variant of Q-learning, which
or to domains with fully observed, low-dimensional state spaces.
uses two key ideas. First, we used a biologically inspired mechanism
Here we use recent advances in training deep neural networks9–11 to
termed experience replay21–23 that randomizes over the data, thereby
develop a novel artificial agent, termed a deep Q-network, that can
removing correlations in the observation sequence and smoothing over
learn successful policies directly from high-dimensional sensory inputs
changes in the data distribution (see below for details). Second, we used
using end-to-end reinforcement learning. We tested this agent on
an iterative update that adjusts the action-values (Q) towards target
the challenging domain of classic Atari 2600 games12. We demon-
values that are only periodically updated, thereby reducing correlations
strate that the deep Q-network agent, receiving only the pixels and with the target.
the game score as inputs, was able to surpass the performance of all
While other stable methods exist for training neural networks in the
previous algorithms and achieve a level comparable to that of a pro-
reinforcement learning setting, such as neural fitted Q-iteration24, these
fessional human games tester across a set of 49 games, using the same
methods involve the repeated training of networks de novo on hundreds
algorithm, network architecture and hyperparameters. This work
of iterations. Consequently, these methods, unlike our algorithm, are
bridges the divide between high-dimensional sensory inputs and
too inefficient to be used successfully with large neural networks. We
actions, resulting in the first artificial agent that is capable of learn-
parameterize an approximate value function Q(s,a;hi) using the deep
ing to excel at a diverse array of challenging tasks.
convolutional neural network shown in Fig. 1, in which hi are the param-
We set out to create a single algorithm that would be able to develop eters (that is, weights) of the Q-network at iteration i. To perform
a wide range of competencies on a varied range of challenging tasks—a experience replay we store the agent’s experiences et 5 (st,at,rt,st 1 1)
central goal of general artificial intelligence13 that has eluded previous at each time-step t in a data set Dt 5 {e1,…,et}. During learning, we
efforts8,14,15. To achieve this, we developed a novel agent, a deep Q-network apply Q-learning updates, on samples (or minibatches) of experience
(DQN), which is able to combine reinforcement learning with a class (s,a,r,s9) , U(D), drawn uniformly at random from the pool of stored
of artificial neural network16 known as deep neural networks. Notably, samples. The Q-learning update at iteration i uses the following loss
recent advances in deep neural networks9–11, in which several layers of function:
nodes are used to build up progressively more abstract representations " # 2
of the data, have made it possible for artificial neural networks to learn
concepts such as object categories directly from raw sensory data. We Li ðhi Þ~ ðs,a,r,s0 Þ*UðDÞ rzc max
0
Q(s0 ,a0 ; h{
i ){Qðs,a; hi Þ
a
use one particularly successful architecture, the deep convolutional
network17, which uses hierarchical layers of tiled convolutional filters in which c is the discount factor determining the agent’s horizon, hi are
to mimic the effects of receptive fields—inspired by Hubel and Wiesel’s the parameters of the Q-network at iteration i and h{i are the network
seminal work on feedforward processing in early visual cortex18—thereby parameters used to compute the target at iteration i. The target net-
exploiting the local spatial correlations present in images, and building work parameters h{ i are only updated with the Q-network parameters
in robustness to natural transformations such as changes of viewpoint (hi) every C steps and are held fixed between individual updates (see
or scale. Methods).
We consider tasks in which the agent interacts with an environment To evaluate our DQN agent, we took advantage of the Atari 2600
through a sequence of observations, actions and rewards. The goal of the platform, which offers a diverse array of tasks (n 5 49) designed to be
1
Google DeepMind, 5 New Street Square, London EC4A 3TW, UK.
*These authors contributed equally to this work.

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RESEARCH LETTER

Convolution Convolution Fully connected Fully connected

No input

Figure 1 | Schematic illustration of the convolutional neural network. The symbolizes sliding of each filter across input image) and two fully connected
details of the architecture are explained in the Methods. The input to the neural layers with a single output for each valid action. Each hidden layer is followed
network consists of an 84 3 84 3 4 image produced by the preprocessing by a rectifier nonlinearity (that is, maxð0,xÞ).
map w, followed by three convolutional layers (note: snaking blue line

difficult and engaging for human players. We used the same network We compared DQN with the best performing methods from the
architecture, hyperparameter values (see Extended Data Table 1) and reinforcement learning literature on the 49 games where results were
learning procedure throughout—taking high-dimensional data (210|160 available12,15. In addition to the learned agents, we also report scores for
colour video at 60 Hz) as input—to demonstrate that our approach a professional human games tester playing under controlled conditions
robustly learns successful policies over a variety of games based solely and a policy that selects actions uniformly at random (Extended Data
on sensory inputs with only very minimal prior knowledge (that is, merely Table 2 and Fig. 3, denoted by 100% (human) and 0% (random) on y
the input data were visual images, and the number of actions available axis; see Methods). Our DQN method outperforms the best existing
in each game, but not their correspondences; see Methods). Notably, reinforcement learning methods on 43 of the games without incorpo-
our method was able to train large neural networks using a reinforce- rating any of the additional prior knowledge about Atari 2600 games
ment learning signal and stochastic gradient descent in a stable manner— used by other approaches (for example, refs 12, 15). Furthermore, our
illustrated by the temporal evolution of two indices of learning (the DQN agent performed at a level that was comparable to that of a pro-
agent’s average score-per-episode and average predicted Q-values; see fessional human games tester across the set of 49 games, achieving more
Fig. 2 and Supplementary Discussion for details). than 75% of the human score on more than half of the games (29 games;
a 2,200 b 6,000
2,000

Average score per episode Average score per episode
1,800 5,000
1,600
4,000
1,400
1,200
3,000
1,000
800 2,000
600
400 1,000
200
0 0
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
Training epochs Training epochs

c 10 d 11
9 10

Average action value (Q) Average action value (Q)
8 9
7 8
7
6
6
5
5
4
4
3 3
2 2
1 1
0 0
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
Training epochs Training epochs

Figure 2 | Training curves tracking the agent’s average score and average on the curve is the average of the action-value Q computed over the held-out
predicted action-value. a, Each point is the average score achieved per episode set of states. Note that Q-values are scaled due to clipping of rewards (see
after the agent is run with e-greedy policy (e 5 0.05) for 520 k frames on Space Methods). d, Average predicted action-value on Seaquest. See Supplementary
Invaders. b, Average score achieved per episode for Seaquest. c, Average Discussion for details.
predicted action-value on a held-out set of states on Space Invaders. Each point

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LETTER RESEARCH

Video Pinball
Boxing
Breakout
Star Gunner
Robotank
Atlantis
Crazy Climber
Gopher
Demon Attack
Name This Game
Krull
Assault
Road Runner
Kangaroo
James Bond
Tennis
Pong
Space Invaders
Beam Rider
Tutankham
Kung-Fu Master
Freeway
Time Pilot
Enduro
Fishing Derby
Up and Down
Ice Hockey
Q*bert
H.E.R.O. At human-level or above
Asterix Below human-level
Battle Zone
Wizard of Wor
Chopper Command
Centipede
Bank Heist
River Raid
Zaxxon
Amidar
Alien
Venture
Seaquest
Double Dunk
Bowling
Ms. Pac-Man
Asteroids
Frostbite
Gravitar DQN
Private Eye
Montezuma's Revenge Best linear learner

0 100 200 300 400 500 600 1,000 4,500%

Figure 3 | Comparison of the DQN agent with the best reinforcement outperforms competing methods (also see Extended Data Table 2) in almost all
learning methods15 in the literature. The performance of DQN is normalized the games, and performs at a level that is broadly comparable with or superior
with respect to a professional human games tester (that is, 100% level) and to a professional human games tester (that is, operationalized as a level of
random play (that is, 0% level). Note that the normalized performance of DQN, 75% or above) in the majority of games. Audio output was disabled for both
expressed as a percentage, is calculated as: 100 3 (DQN score 2 random play human players and agents. Error bars indicate s.d. across the 30 evaluation
score)/(human score 2 random play score). It can be seen that DQN episodes, starting with different initial conditions.

see Fig. 3, Supplementary Discussion and Extended Data Table 2). In perceptually dissimilar (Fig. 4, bottom right, top left and middle), con-
additional simulations (see Supplementary Discussion and Extended sistent with the notion that the network is able to learn representations
Data Tables 3 and 4), we demonstrate the importance of the individual that support adaptive behaviour from high-dimensional sensory inputs.
core components of the DQN agent—the replay memory, separate target Furthermore, we also show that the representations learned by DQN
Q-network and deep convolutional network architecture—by disabling are able to generalize to data generated from policies other than its
them and demonstrating the detrimental effects on performance. own—in simulations where we presented as input to the network game
We next examined the representations learned by DQN that under- states experienced during human and agent play, recorded the repre-
pinned the successful performance of the agent in the context of the game sentations of the last hidden layer, and visualized the embeddings gen-
Space Invaders (see Supplementary Video 1 for a demonstration of the erated by the t-SNE algorithm (Extended Data Fig. 1 and Supplementary
performance of DQN), by using a technique developed for the visual- Discussion). Extended Data Fig. 2 provides an additional illustration of
ization of high-dimensional data called ‘t-SNE’25 (Fig. 4). As expected, how the representations learned by DQN allow it to accurately predict
the t-SNE algorithm tends to map the DQN representation of percep- state and action values.
tually similar states to nearby points. Interestingly, we also found instances It is worth noting that the games in which DQN excels are extremely
in which the t-SNE algorithm generated similar embeddings for DQN varied in their nature, from side-scrolling shooters (River Raid) to box-
representations of states that are close in terms of expected reward but ing games (Boxing) and three-dimensional car-racing games (Enduro).
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RESEARCH LETTER

Figure 4 | Two-dimensional t-SNE embedding of the representations in the predicts high state values for both full (top right screenshots) and nearly
last hidden layer assigned by DQN to game states experienced while playing complete screens (bottom left screenshots) because it has learned that
Space Invaders. The plot was generated by letting the DQN agent play for completing a screen leads to a new screen full of enemy ships. Partially
2 h of real game time and running the t-SNE algorithm25 on the last hidden layer completed screens (bottom screenshots) are assigned lower state values because
representations assigned by DQN to each experienced game state. The less immediate reward is available. The screens shown on the bottom right
points are coloured according to the state values (V, maximum expected reward and top left and middle are less perceptually similar than the other examples but
of a state) predicted by DQN for the corresponding game states (ranging are still mapped to nearby representations and similar values because the
from dark red (highest V) to dark blue (lowest V)). The screenshots orange bunkers do not carry great significance near the end of a level. With
corresponding to a selected number of points are shown. The DQN agent permission from Square Enix Limited.

Indeed, in certain games DQN is able to discover a relatively long-term realization of such a process in the mammalian brain, with the time-
strategy (for example, Breakout: the agent learns the optimal strategy, compressed reactivation of recently experienced trajectories during
which is to first dig a tunnel around the side of the wall allowing the ball offline periods21,22 (for example, waking rest) providing a putative mech-
to be sent around the back to destroy a large number of blocks; see Sup- anism by which value functions may be efficiently updated through
plementary Video 2 for illustration of development of DQN’s perfor- interactions with the basal ganglia22. In the future, it will be important
mance over the course of training). Nevertheless, games demanding more to explore the potential use of biasing the content of experience replay
temporally extended planning strategies still constitute a major chal- towards salient events, a phenomenon that characterizes empirically
lenge for all existing agents including DQN (for example, Montezuma’s observed hippocampal replay29, and relates to the notion of ‘prioritized
Revenge). sweeping’30 in reinforcement learning. Taken together, our work illus-
In this work, we demonstrate that a single architecture can success- trates the power of harnessing state-of-the-art machine learning tech-
fully learn control policies in a range of different environments with only niques with biologically inspired mechanisms to create agents that are
very minimal prior knowledge, receiving only the pixels and the game capable of learning to master a diverse array of challenging tasks.
score as inputs, and using the same algorithm, network architecture and Online Content Methods, along with any additional Extended Data display items
hyperparameters on each game, privy only to the inputs a human player and Source Data, are available in the online version of the paper; references unique
would have. In contrast to previous work24,26, our approach incorpo- to these sections appear only in the online paper.
rates ‘end-to-end’ reinforcement learning that uses reward to continu-
Received 10 July 2014; accepted 16 January 2015.
ously shape representations within the convolutional network towards
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LETTER RESEARCH

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18. Hubel, D. H. & Wiesel, T. N. Shape and arrangement of columns in cat’s striate Author Contributions V.M., K.K., D.S., J.V., M.G.B., M.R., A.G., D.W., S.L. and D.H.
cortex. J. Physiol. 165, 559–568 (1963). conceptualized the problem and the technical framework. V.M., K.K., A.A.R. and D.S.
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20. Tsitsiklis, J. & Roy, B. V. An analysis of temporal-difference learning with function A.S. created the testing platform. K.K., H.K., S.L. and D.H. managed the project. K.K., D.K.,
approximation. IEEE Trans. Automat. Contr. 42, 674–690 (1997). D.H., V.M., D.S., A.G., A.A.R., J.V. and M.G.B. wrote the paper.
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learning systems in the hippocampus and neocortex: insights from the successes Author Information Reprints and permissions information is available at
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419–457 (1995). Readers are welcome to comment on the online version of the paper. Correspondence
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RESEARCH LETTER

METHODS Our experimental setup amounts to using the following minimal prior know-
ledge: that the input data consisted of visual images (motivating our use of a con-
Preprocessing. Working directly with raw Atari 2600 frames, which are 210 3 160
volutional deep network), the game-specific score (with no modification), number
pixel images with a 128-colour palette, can be demanding in terms of computation
of actions, although not their correspondences (for example, specification of the
and memory requirements. We apply a basic preprocessing step aimed at reducing
up ‘button’) and the life count.
the input dimensionality and dealing with some artefacts of the Atari 2600 emu-
Evaluation procedure. The trained agents were evaluated by playing each game
lator. First, to encode a single frame we take the maximum value for each pixel colour
30 times for up to 5 min each time with different initial random conditions (‘no-
value over the frame being encoded and the previous frame. This was necessary to
op’; see Extended Data Table 1) and an e-greedy policy with e 5 0.05. This pro-
remove flickering that is present in games where some objects appear only in even
cedure is adopted to minimize the possibility of overfitting during evaluation. The
frames while other objects appear only in odd frames, an artefact caused by the
random agent served as a baseline comparison and chose a random action at 10 Hz
limited number of sprites Atari 2600 can display at once. Second, we then extract
which is every sixth frame, repeating its last action on intervening frames. 10 Hz is
the Y channel, also known as luminance, from the RGB frame and rescale it to
about the fastest that a human player can select the ‘fire’ button, and setting the
84 3 84. The function w from algorithm 1 described below applies this preprocess-
random agent to this frequency avoids spurious baseline scores in a handful of the
ing to the m most recent frames and stacks them to produce the input to the
games. We did also assess the performance of a random agent that selected an action
Q-function, in which m 5 4, although the algorithm is robust to different values of
at 60 Hz (that is, every frame). This had a minimal effect: changing the normalized
m (for example, 3 or 5).
DQN performance by more than 5% in only six games (Boxing, Breakout, Crazy
Code availability. The source code can be accessed at https://sites.google.com/a/
Climber, Demon Attack, Krull and Robotank), and in all these games DQN out-
deepmind.com/dqn for non-commercial uses only.
performed the expert human by a considerable margin.
Model architecture. There are several possible ways of parameterizing Q using a
The professional human tester used the same emulator engine as the agents, and
neural network. Because Q maps history–action pairs to scalar estimates of their
played under controlled conditions. The human tester was not allowed to pause,
Q-value, the history and the action have been used as inputs to the neural network
save or reload games. As in the original Atari 2600 environment, the emulator was
by some previous approaches24,26. The main drawback of this type of architecture
run at 60 Hz and the audio output was disabled: as such, the sensory input was
is that a separate forward pass is required to compute the Q-value of each action,
equated between human player and agents. The human performance is the average
resulting in a cost that scales linearly with the number of actions. We instead use an
reward achieved from around 20 episodes of each game lasting a maximum of 5 min
architecture in which there is a separate output unit for each possible action, and
each, following around 2 h of practice playing each game.
only the state representation is an input to the neural network. The outputs cor-
Algorithm. We consider tasks in which an agent interacts with an environment,
respond to the predicted Q-values of the individual actions for the input state. The
in this case the Atari emulator, in a sequence of actions, observations and rewards.
main advantage of this type of architecture is the ability to compute Q-values for all
At each time-step the agent selects an action at from the set of legal game actions,
possible actions in a given state with only a single forward pass through the network.
A~f1, . . . ,K g. The action is passed to the emulator and modifies its internal state
The exact architecture, shown schematically in Fig. 1, is as follows. The input to
and the game score. In general the environment may be stochastic. The emulator’s
the neural network consists of an 84 3 84 3 4 image produced by the preprocess-
internal state is not observed by the agent; instead the agent observes an image
ing map w. The first hidden layer convolves 32 filters of 8 3 8 with stride 4 with the
xt [Rd from the emulator, which is a vector of pixel values representing the current
input image and applies a rectifier nonlinearity31,32. The second hidden layer con-
screen. In addition it receives a reward rt representing the change in game score.
volves 64 filters of 4 3 4 with stride 2, again followed by a rectifier nonlinearity.
Note that in general the game score may depend on the whole previous sequence of
This is followed by a third convolutional layer that convolves 64 filters of 3 3 3 with
actions and observations; feedback about an action may only be received after many
stride 1 followed by a rectifier. The final hidden layer is fully-connected and con-
thousands of time-steps have elapsed.
sists of 512 rectifier units. The output layer is a fully-connected linear layer with a
Because the agent only observes the current screen, the task is partially observed33
single output for each valid action. The number of valid actions varied between 4
and many emulator states are perceptually aliased (that is, it is impossible to fully
and 18 on the games we considered.
understand the current situation from only the current screen xt ). Therefore,
Training details. We performed experiments on 49 Atari 2600 games where results
sequences of actions and observations, st ~x1 ,a1 ,x2 ,:::,at{1 ,xt , are input to the
were available for all other comparable methods12,15. A different network was trained
algorithm, which then learns game strategies depending upon these sequences. All
on each game: the same network architecture, learning algorithm and hyperpara-
sequences in the emulator are assumed to terminate in a finite number of time-
meter settings (see Extended Data Table 1) were used across all games, showing that steps. This formalism gives rise to a large but finite Markov decision process (MDP)
our approach is robust enough to work on a variety of games while incorporating in which each sequence is a distinct state. As a result, we can apply standard rein-
only minimal prior knowledge (see below). While we evaluated our agents on unmodi- forcement learning methods for MDPs, simply by using the complete sequence st
fied games, we made one change to the reward structure of the games during training as the state representation at time t.
only. As the scale of scores varies greatly from game to game, we clipped all posi-
The goal of the agent is to interact with the emulator by selecting actions in a way
tive rewards at 1 and all negative rewards at 21, leaving 0 rewards unchanged.
that maximizes future rewards. We make the standard assumption that future rewards
Clipping the rewards in this manner limits the scale of the error derivatives and
are discounted by a factor of c per time-step (c was set to 0.99 throughout), and
makes it easier to use the same learning rate across multiple games. At the same time, XT
0
it could affect the performance of our agent since it cannot differentiate between define the future discounted return at time t as Rt ~ ct {t rt 0 , in which T is the
rewards of different magnitude. For games where there is a life counter, the Atari t 0 ~t
time-step at which the game terminates. We define the optimal action-value
2600 emulator also sends the number of lives left in the game, which is then used to
function Q ðs,aÞ as the maximum expected return achievable by following any
mark the end of an episode during training.
policy, after seeing some sequence s and then taking some action a, Q ðs,aÞ~
In these experiments, we used the RMSProp (see http://www.cs.toronto.edu/
maxp ½Rt Dst ~s,at ~a,p in which p is a policy mapping sequences to actions (or
,tijmen/csc321/slides/lecture_slides_lec6.pdf ) algorithm with minibatches of size
distributions over actions).
32. The behaviour policy during training was e-greedy with e annealed linearly
The optimal action-value function obeys an important identity known as the
from 1.0 to 0.1 over the first million frames, and fixed at 0.1 thereafter. We trained
Bellman equation. This is based on the following intuition: if the optimal value
for a total of 50 million frames (that is, around 38 days of game experience in total)
Q ðs0 ,a0 Þ of the sequence s9 at the next time-step was known for all possible actions
and used a replay memory of 1 million most recent frames.
a9, then the optimal strategy is to select the action a9 maximizing the expected value
Following previous approaches to playing Atari 2600 games, we also use a simple of rzcQ ðs0 ,a0 Þ:
frame-skipping technique15. More precisely, the agent sees and selects actions on
every kth frame instead of every frame, and its last action is repeated on skipped
Q ðs,aÞ ~ s0 rzc max Q 0 0
ðs ,a ÞDs,a
frames. Because running the emulator forward for one step requires much less a 0

computation than having the agent select an action, this technique allows the agent
to play roughly k times more games without significantly increasing the runtime. The basic idea behind many reinforcement learning algorithms is to estimate
We use k 5 4 for all games. the action-value function by using the Bellman equation as an iterative update,
The values of all the hyperparameters and optimization parameters were selected Qiz1 ðs,aÞ~ s0 ½rzc maxa0 Qi ðs0 ,a0 ÞDs,a. Such value iteration algorithms converge
by performing an informal search on the games Pong, Breakout, Seaquest, Space to the optimal action-value function, Qi ?Q as i??. In practice, this basic approach
Invaders and Beam Rider. We did not perform a systematic grid search owing to is impractical, because the action-value function is estimated separately for each
the high computational cost. These parameters were then held fixed across all other sequence, without any generalization. Instead, it is common to use a function approx-
games. The values and descriptions of all hyperparameters are provided in Extended imator to estimate the action-value function, Qðs,a; hÞ<Q ðs,aÞ. In the reinforce-
Data Table 1. ment learning community this is typically a linear function approximator, but

sometimes a nonlinear function approximator is used instead, such as a neural replay the behaviour distribution is averaged over many of its previous states,
network. We refer to a neural network function approximator with weights h as a smoothing out learning and avoiding oscillations or divergence in the parameters.
Q-network. A Q-network can be trained by adjusting the parameters hi at iteration Note that when learning by experience replay, it is necessary to learn off-policy
i to reduce the mean-squared error in the Bellman equation, where the optimal (because our current parameters are different to those used to generate the sam-
target values rzc maxa0 Q ðs0 ,a0 Þ are substituted with approximate target values ple), which motivates the choice of Q-learning.
y~rzc maxa0 Q s0 ,a0 ; h{ {
i , using parameters hi from some previous iteration. In practice, our algorithm only stores the last N experience tuples in the replay
This leads to a sequence of loss functions Li(hi) that changes at each iteration i, memory, and samples uniformly at random from D when performing updates. This
approach is in some respects limited because the memory buffer does not differ-
Li ðhi Þ~ s,a,r ðEs0 ½yDs,a{Qðs,a; hi ÞÞ2
entiate important transitions and always overwrites with recent transitions owing

~ s,a,r,s0 ðy{Qðs,a; hi ÞÞ2 zEs,a,r ½Vs0 ½ y: to the finite memory size N. Similarly, the uniform sampling gives equal impor-
tance to all transitions in the replay memory. A more sophisticated sampling strat-
Note that the targets depend on the network weights; this is in contrast with the egy might emphasize transitions from which we can learn the most, similar to
targets used for supervised learning, which are fixed before learning begins. At prioritized sweeping30.
each stage of optimization, we hold the parameters from the previous iteration hi2 The second modification to online Q-learning aimed at further improving the
fixed when optimizing the ith loss function Li(hi), resulting in a sequence of well- stability of our method with neural networks is to use a separate network for gen-
defined optimization problems. The final term is the variance of the targets, which erating the targets yj in the Q-learning update. More precisely, every C updates we
does not depend on the parameters hi that we are currently optimizing, and may clone the network Q to obtain a target network ^ Q and use ^ Q for generating the
therefore be ignored. Differentiating the loss function with respect to the weights Q-learning targets yj for the following C updates to Q. This modification makes the
we arrive at the following gradient: algorithm more stable compared to standard online Q-learning, where an update
that increases Q(st,at) often also increases Q(st 1 1,a) for all a and hence also increases
the target yj, possibly leading to oscillations or divergence of the policy. Generating
+hi Lðhi Þ ~ s,a,r,s0 rzc max 0
Q s0 ,a0 ; h{
i {Qðs,a; hi Þ +hi Qðs,a; hi Þ :
a the targets using an older set of parameters adds a delay between the time an update
to Q is made and the time the update affects the targets yj, making divergence or
Rather than computing the full expectations in the above gradient, it is often oscillations much more unlikely.
computationally expedient to optimize the loss function by stochastic gradient also found it helpful to clip the error term from the update rzc maxa0 Q
0We
descent. The familiar Q-learning algorithm19 can be recovered in this framework s ,a0 ; h{
i {Qðs,a; hi Þ to be between 21 and 1. Because the absolute value loss
by updating the weights after every time step, replacing the expectations using function jxj has a derivative of 21 for all negative values of x and a derivative of 1
single samples, and setting h{ i ~hi{1 . for all positive values of x, clipping the squared error to be between 21 and 1 cor-
Note that this algorithm is model-free: it solves the reinforcement learning task responds to using an absolute value loss function for errors outside of the (21,1)
directly using samples from the emulator, without explicitly estimating the reward interval. This form of error clipping further improved the stability of the algorithm.
and transition dynamics Pðr,s0 Ds,aÞ. It is also off-policy: it learns about the greedy Algorithm 1: deep Q-learning with experience replay.
policy a~argmaxa0 Qðs,a0 ; hÞ, while following a behaviour distribution that ensures Initialize replay memory D to capacity N
adequate exploration of the state space. In practice, the behaviour distribution is Initialize action-value function Q with random weights h
often selected by an e-greedy policy that follows the greedy policy with probability Initialize target action-value function ^ Q with weights h2 5 h
1 2 e and selects a random action with probability e. For episode 5 1, M do
Training algorithm for deep Q-networks. The full algorithm for training deep
Initialize sequence s1 ~fx1 g and preprocessed sequence w1 ~wðs1 Þ
Q-networks is presented in Algorithm 1. The agent selects and executes actions
For t 5 1,T do
according to an e-greedy policy based on Q. Because using histories of arbitrary
With probability e select a random action at
length as inputs to a neural network can be difficult, our Q-function instead works
on a fixed length representation of histories produced by the function w described otherwise select at ~argmaxa Qðwðst Þ,a; hÞ
above. The algorithm modifies standard online Q-learning in two ways to make it Execute action at in emulator and observe reward rt and image xt 1 1
suitable for training large neural networks without diverging. Set stz1 ~st ,at ,xtz1 and preprocess wtz1 ~wðstz1 Þ
First, we use a technique known as experience replay23 in which we store the Store transition wt ,at ,rt ,wtz1 in D
agent’s experiences at each time-step, et 5 (st, at, rt, st 1 1), in a data set Dt 5 {e1,…,et}, Sample random minibatch of transitions wj ,aj ,rj ,wjz1 from D
(
pooled over many episodes (where the end of an episode occurs when a termi- rj if episode terminates at step jz1
nal state is reached) into a replay memory. During the inner loop of the algorithm, Set yj ~ ^ wjz1 ,a0 ; h{
rj zc maxa0 Q otherwise
we apply Q-learning updates, or minibatch updates, to samples of experience, 2
(s, a, r, s9) , U(D), drawn at random from the pool of stored samples. This approach Perform a gradient descent step on yj {Q wj ,aj ; h with respect to the
has several advantages over standard online Q-learning. First, each step of experience network parameters h
is potentially used in many weight updates, which allows for greater data efficiency. Every C steps reset ^
Q~Q
Second, learning directly from consecutive samples is inefficient, owing to the strong End For
correlations between the samples; randomizing the samples breaks these correla- End For
tions and therefore reduces the variance of the updates. Third, when learning on-
policy the current parameters determine the next data sample that the parameters 31. Jarrett, K., Kavukcuoglu, K., Ranzato, M. A. & LeCun, Y. What is the best multi-stage
are trained on. For example, if the maximizing action is to move left then the train- architecture for object recognition? Proc. IEEE. Int. Conf. Comput. Vis. 2146–2153
(2009).
ing samples will be dominated by samples from the left-hand side; if the maximiz-
32. Nair, V. & Hinton, G. E. Rectified linear units improve restricted Boltzmann
ing action then switches to the right then the training distribution will also switch. machines. Proc. Int. Conf. Mach. Learn. 807–814 (2010).
It is easy to see how unwanted feedback loops may arise and the parameters could get 33. Kaelbling, L. P., Littman, M. L. & Cassandra, A. R. Planning and acting in partially
stuck in a poor local minimum, or even diverge catastrophically20. By using experience observable stochastic domains. Artificial Intelligence 101, 99–134 (1994).

Extended Data Figure 1 | Two-dimensional t-SNE embedding of the points) and DQN play (blue points) suggests that the representations learned
representations in the last hidden layer assigned by DQN to game states by DQN do indeed generalize to data generated from policies other than its
experienced during a combination of human and agent play in Space own. The presence in the t-SNE embedding of overlapping clusters of points
Invaders. The plot was generated by running the t-SNE algorithm25 on the last corresponding to the network representation of states experienced during
hidden layer representation assigned by DQN to game states experienced human and agent play shows that the DQN agent also follows sequences of
during a combination of human (30 min) and agent (2 h) play. The fact that states similar to those found in human play. Screenshots corresponding to
there is similar structure in the two-dimensional embeddings corresponding to selected states are shown (human: orange border; DQN: blue border).
the DQN representation of states experienced during human play (orange

Extended Data Figure 2 | Visualization of learned value functions on two all actions are around 0.7, reflecting the expected value of this state based on
games, Breakout and Pong. a, A visualization of the learned value function on previous experience. At time point 2, the agent starts moving the paddle
the game Breakout. At time points 1 and 2, the state value is predicted to be ,17 towards the ball and the value of the ‘up’ action stays high while the value of the
and the agent is clearing the bricks at the lowest level. Each of the peaks in ‘down’ action falls to 20.9. This reflects the fact that pressing ‘down’ would lead
the value function curve corresponds to a reward obtained by clearing a brick. to the agent losing the ball and incurring a reward of 21. At time point 3,
At time point 3, the agent is about to break through to the top level of bricks and the agent hits the ball by pressing ‘up’ and the expected reward keeps increasing
the value increases to ,21 in anticipation of breaking out and clearing a until time point 4, when the ball reaches the left edge of the screen and the value
large set of bricks. At point 4, the value is above 23 and the agent has broken of all actions reflects that the agent is about to receive a reward of 1. Note,
through. After this point, the ball will bounce at the upper part of the bricks the dashed line shows the past trajectory of the ball purely for illustrative
clearing many of them by itself. b, A visualization of the learned action-value purposes (that is, not shown during the game). With permission from Atari
function on the game Pong. At time point 1, the ball is moving towards the Interactive, Inc.
paddle controlled by the agent on the right side of the screen and the values of

Extended Data Table 1 | List of hyperparameters and their values

The values of all the hyperparameters were selected by performing an informal search on the games Pong, Breakout, Seaquest, Space Invaders and Beam Rider. We did not perform a systematic grid search owing
to the high computational cost, although it is conceivable that even better results could be obtained by systematically tuning the hyperparameter values.

Extended Data Table 2 | Comparison of games scores obtained by DQN agents with methods from the literature12,15 and a professional
human games tester

Best Linear Learner is the best result obtained by a linear function approximator on different types of hand designed features12. Contingency (SARSA) agent figures are the results obtained in ref. 15. Note the
figures in the last column indicate the performance of DQN relative to the human games tester, expressed as a percentage, that is, 100 3 (DQN score 2 random play score)/(human score 2 random play score).

Extended Data Table 3 | The effects of replay and separating the target Q-network

DQN agents were trained for 10 million frames using standard hyperparameters for all possible combinations of turning replay on or off, using or not using a separate target Q-network, and three different learning
rates. Each agent was evaluated every 250,000 training frames for 135,000 validation frames and the highest average episode score is reported. Note that these evaluation episodes were not truncated at 5 min
leading to higher scores on Enduro than the ones reported in Extended Data Table 2. Note also that the number of training frames was shorter (10 million frames) as compared to the main results presented in
Extended Data Table 2 (50 million frames).

Extended Data Table 4 | Comparison of DQN performance with lin-
ear function approximator

The performance of the DQN agent is compared with the performance of a linear function approximator
on the 5 validation games (that is, where a single linear layer was used instead of the convolutional
network, in combination with replay and separate target network). Agents were trained for 10 million
frames using standard hyperparameters, and three different learning rates. Each agent was evaluated
every 250,000 training frames for 135,000 validation frames and the highest average episode score is
reported. Note that these evaluation episodes were not truncated at 5 min leading to higher scores on
Enduro than the ones reported in Extended Data Table 2. Note also that the number of training frames
was shorter (10 million frames) as compared to the main results presented in Extended Data Table 2
(50 million frames).