Maxmin Q-learning: Controlling the Estimation Bias of Q-learning

1 Introduction

Q-learning (Watkins, 1989)

is one of the most popular reinforcement learning algorithms. One of the reasons for this widespread adoption is the simplicity of the update. On each step, the agent updates its action value estimates towards the observed reward and the estimated value of the maximal action in the next state. This target represents the highest value the agent thinks it could obtain from the current state and action, given the observed reward.

Unfortunately, this simple update rule has been shown to suffer from overestimation bias (Thrun and Schwartz, 1993; van Hasselt, 2010)

. The agent updates with the maximum over action values might be large because an action’s value actually is high, or it can be misleadingly high simply because of the stochasticity or errors in the estimator. With many actions, there is a higher probability that one of the estimates is large simply due to stochasticity and the agent will overestimate the value. This issue is particularly problematic under function approximation, and can significant impede the quality of the learned policy

(Thrun and Schwartz, 1993; Szita and Lőrincz, 2008; Strehl et al., 2009) or even lead to failures of Q-learning (Thrun and Schwartz, 1993). More recently, experiments across several domains suggest that this overestimation problem is common (Hado van Hasselt et al., 2016).

Double Q-learning (van Hasselt, 2010) is introduced to instead ensure underestimation bias. The idea is to maintain two unbiased independent estimators of the action values. The expected action value of estimator one is selected for the maximal action from estimator two, which is guaranteed not to overestimate the true maximum action value. Double DQN (Hado van Hasselt et al., 2016)

, the extension of this idea to Q-learning with neural networks, has been shown to significantly improve performance over Q-learning. However, this is not a complete answer to this problem, because trading overestimation bias for underestimation bias is not always desirable, as we show in our experiments.

Several other methods have been introduced to reduce overestimation bias, without fully moving towards underestimation. Weighted Double Q-learning (Zhang et al., 2017) uses a weighted combination of the Double Q-learning estimate, which likely has underestimation bias, and the Q-learning estimate, which likely has overestimation bias. Bias-corrected Q-Learning (Lee et al., 2013) reduces the overestimation bias through a bias correction term. Ensemble Q-learning and Averaged Q-learning (Anschel et al., 2017) take averages of multiple action values, to both reduce the overestimation bias and the estimation variance. However, with a finite number of action-value functions, the average operation in these two algorithms will never completely remove the overestimation bias, as the average of several overestimation biases is always positive. Further, these strategies do not guide how strongly we should correct for overestimation bias, nor how to determine—or control—the level of bias.

The overestimation bias also appears in the actor-critic setting (Fujimoto et al., 2018; Haarnoja et al., 2018). For example, Fujimoto et al. (2018) propose the Twin Delayed Deep Deterministic policy gradient algorithm (TD3) which reduces the overestimation bias by taking the minimum value between two critics. However, they do not provide a rigorous theoretical analysis for the effect of applying the minimum operator. There is also no theoretical guide for choosing the number of estimators such that the overestimation bias can be reduced to 0.

In this paper, we study the effects of overestimation and underestimation bias on learning performance, and use them to motivate a generalization of Q-learning called Maxmin Q-learning. Maxmin Q-learning directly mitigates the overestimation bias by using a minimization over multiple action-value estimates. Moreover, it is able to control the estimation bias varying from positive to negative which helps improve learning efficiency as we will show in next sections. We prove that, theoretically, with an appropriate number of action-value estimators, we are able to acquire an unbiased estimator with a lower approximation variance than Q-learning. We empirically verify our claims on several benchmarks. We study the convergence properties of our algorithm within a novel Generalized Q-learning framework, which is suitable for studying several of the recently proposed Q-learning variants. We also combine deep neural networks with Maxmin Q-learning (Maxmin DQN) and demonstrate its effectiveness in several benchmark domains.

2 Problem Setting

We formalize the problem as a Markov Decision Process (MDP),

(S, A, P, r, γ)

, where

S

is the state space,

A

is the action space,

P : S \times A \times S \to [0, 1]

is the transition probabilities,

r : S \times A \times S \to R

is the reward mapping, and

γ \in [0, 1]

is the discount factor. At each time step

t

, the agent observes a state

S_{t} \in S

and takes an action

A_{t} \in A

and then transitions to a new state

S_{t + 1} \in S

according to the transition probabilities

P

and receives a scalar reward

R_{t + 1} = r (S_{t}, A_{t}, S_{t + 1}) \in R

. The goal of the agent is to find a policy

π : S \times A \to [0, 1]

that maximizes the expected return starting from some initial state.

Q-learning is an off-policy algorithm which attempts to learn the state-action values $Q : S \times A \to R$ for the optimal policy. It tries to solve for

Q^{*} (s, a) = E [R_{t + 1} + max a^{'} \in A Q^{*} (S_{t + 1}, a^{'}) ∣ ∣ S_{t} = s, A_{t} = a]

The optimal policy is to act greedily with respect to these action values: from each $s$ select $a$ from ${a r g m a x}_{a \in A} Q^{*} (s, a)$ . The update rule for an approximation $Q$ for a sampled transition $s_{t}, a_{t}, r_{t + 1}, s_{t + 1}$ is:

Q (s_{t}, a_{t})

\leftarrow Q (s_{t}, a_{t}) + α (Y_{t}^{Q} - Q (s_{t}, a_{t}))

for YQt\makebox[0.0pt]\tiny def% =rt+1+γmaxa′∈AQ(st+1,a′)

(1)

where $α$ is the step-size. The transition can be generated off-policy, from any behaviour that sufficiently covers the state space. This algorithm is known to converge in the tabular setting (Tsitsiklis, 1994), with some limited results for the function approximation setting (Melo and Ribeiro, 2007).

3 Understanding when Overestimation Bias Helps and Hurts

In this section, we briefly discuss the estimation bias issue, and empirically show that both overestimation and underestimation bias may improve learning performance, depending on the environment. This motivates our Maxmin Q-learning algorithm described in the next section, which allows us to flexibly control the estimation bias and reduce the estimation variance.

The overestimation bias occurs since the target ${max}_{a^{'} \in A} Q (s_{t + 1}, a^{'})$ is used in the Q-learning update. Because $Q$

is an approximation, it is probable that the approximation is higher than the true value for one or more of the actions. The maximum over these estimators, then, is likely to be skewed towards an overestimate. For example, even unbiased estimates

Q (s_{t + 1}, a^{'})

for all

a^{'}

, will vary due to stochasticity.

Q (s_{t + 1}, a^{'}) = Q^{*} (s_{t + 1}, a^{'}) + e_{a^{'}}

, and for some actions,

e_{a^{'}}

will be positive. As a result,

E [{max}_{a^{'} \in A} Q (s_{t + 1}, a^{'})] \geq {max}_{a^{'} \in A} E [Q (s_{t + 1}, a^{'})] = {max}_{a^{'} \in A} Q^{*} (s_{t + 1}, a^{'})

This overestimation bias, however, may not always be detrimental. And, further, in some cases, erring towards an underestimation bias can be harmful. Overestimation bias can help encourage exploration for overestimated actions, whereas underestimation bias might discourage exploration. In particular, we expect more overestimation bias in highly stochastic areas of the world; if those highly stochastic areas correspond to high-value regions, then encouraging exploration there might be beneficial. An underestimation bias might actually prevent an agent from learning that a region is high-value. Alternatively, if highly stochastic areas also have low values, overestimation bias might cause an agent to over-explore a low-value region.

We show this effect in the simple MDP, shown in Figure 1. The MDP for state $A$ has only two actions: Left and Right. It has a deterministic neutral reward for both the Left action and the Right action. The Left action transitions to state $B$ where there are eight actions transitions to a terminate state with a highly stochastic reward. The mean of this stochastic reward is $μ$ . By selecting $μ > 0$ , the stochastic region becomes high-value, and we expect overestimation bias to help and underestimation bias to hurt. By selecting $μ < 0$ , the stochastic region becomes low-value, and we expect overestimation bias to hurt and underestimation bias to help.

Figure 1: A simple episodic MDP, adapted from Figure

6.5

in Sutton and Barto (2018) which is used to highlight the difference between Double Q-learning and Q-learning. This MDP has two non-terminal states

A

and

B

. Every episode starts from

A

which has two actions: Left and Right. The Right action transitions to a terminal state with reward

0

. The Left action transitions to state

B

with reward

0

. From state

B

, there are 8 actions that all transition to a terminal state with a reward

μ + ξ

, where

ξ

is drawn from a uniform distribution

U (- 1, 1)

. When

μ > 0

, the optimal action in state

A

is Left; when

μ < 0

, it is Right.

We test Q-learning, Double Q-learning and our new algorithm Maxmin Q-learning in this environment. Maxmin Q-learning (described fully in the next section) uses $N$ estimates of the action values in the targets. For $N = 1$ , it corresponds to Q-learning; otherwise, it progresses from overestimation bias at $N = 1$ towards underestimation bias with increasing $N$ . In the experiment, we used a discount factor $γ = 1$ ; a replay buffer with size $100$ ; an $ϵ$ -greedy behaviour with $ϵ = 0.1$

; tabular action-values, initialized with a Gaussian distribution

N (0, 0.01)

; and a step-size of

0.01

for all algorithms.

The results in Figure 2 verify our hypotheses for when overestimation and underestimation bias help and hurt. Double Q-learning underestimates too much for $μ = + 1$ , and converges to a suboptimal policy. Q-learning learns the optimal policy the fastest, though for all values of $N = 2, 4, 6, 8$ , Maxmin Q-learning does progress towards the optimal policy. All methods get to the optimal policy for $μ = - 1$ , but now Double Q-learning reaches the optimal policy the fastest, and followed by Maxmin Q-learning with larger $N$ .

4 Maxmin Q-learning

In this section, we develop Maxmin Q-learning, a simple generalization of Q-learning designed to control the estimation bias, as well as reduce the estimation variance of action values. The idea is to maintain $N$ estimates of the action values, $Q^{i}$ , and use the minimum of these estimates in the Q-learning target: ${max}_{a^{'}} {min}_{i \in {1, \dots, N}} Q^{i} (s^{'}, a^{'})$ . For $N = 1$ , the update is simply Q-learning, and so likely has overestimation bias. As $N$ increase, the overestimation decreases; for some $N > 1$ , this maxmin estimator switches from an overestimate, in expectation, to an underestimate. We characterize the relationship between $N$ and the expected estimation bias below in Theorem 1. Note that Maxmin Q-learning uses a different mechanism to reduce overestimation bias than Double Q-learning; Maxmin Q-learning with $N = 2$ is not Double Q-learning.

The full algorithm is summarized in Algorithm 1, and is a simple modification of Q-learning with experience replay. We use random subsamples of the observed data for each of the $N$ estimators, to make them nearly independent. To do this training online, we keep a replay buffer. On each step, a random estimator $i$ is chosen and updated using a mini-batch from the buffer. Multiple such updates can be performed on each step, just like in experience replay, meaning multiple estimators can be updated per step using different random mini-batches. In our experiments, to better match DQN, we simply do one update per step. Finally, it is also straightforward to incorporate target networks to get Maxmin DQN, by maintaining a target network for each estimator.

Input: step-size

α

, exploration parameter

ϵ > 0

, number of action-value functions

N

Initialize

N

action-value functions

{Q^{1}, \dots, Q^{N}}

randomly

Initialize empty replay buffer

D

Observe initial state

s

while Agent is interacting with the Environment do

Q^{m i n} (s, a) \leftarrow {min}_{k \in {1, \dots, N}} Q^{k} (s, a)

\forall a \in A

Choose action

a

ϵ

-greedy based on

Q^{m i n}

Take action

a

, observe

r

s^{'}

Store transition

(s, a, r, s^{'})

D

Select a subset

S

from

{1, \dots, N}

(e.g., randomly select one

i

to update)

for $i \in S$ do

Sample random mini-batch of transitions

(s_{D}, a_{D}, r_{D}, s_{D}^{'})

from

D

Get update target:

Y^{M Q} \leftarrow r_{D} + γ {max}_{a^{'} \in A} Q^{m i n} (s_{D}^{'}, a^{'})

Update action-value

Q^{i}

Q^{i} (s_{D}, a_{D}) \leftarrow Q^{i} (s_{D}, a_{D}) + α [Y^{M Q} - Q^{i} (s_{D}, a_{D})]

end for

s \leftarrow s^{'}

end while

Algorithm 1 Maxmin Q-learning

We now characterize the relation between the number of action-value functions used in Maxmin Q-learning and the estimation bias of action values. For compactness, we write $Q_{s a}^{i}$ instead of $Q^{i} (s, a)$ . Each $Q_{s a}^{i}$ has random approximation error $e_{s a}^{i}$

Q_{s a}^{i} = Q_{s a}^{*} + e_{s a}^{i} .

We assume that $e_{s a}^{i}$

is a uniform random variable

U (- τ, τ)

for some

τ > 0

. The uniform random assumption was used by Thrun and Schwartz (1993) to demonstrate bias in Q-learning, and reflects that non-negligible positive and negative

e_{s a}^{i}

are possible. Notice that for

N

estimators with

n_{s a}

samples, the

τ

will be proportional to some function of

n_{s a} / N

, because the data will be shared amongst the

N

estimators. For the general theorem, we use a generic

τ

, and in the following corollary provide a specific form for

τ

in terms of

N

and

n_{s a}

Recall that $M$ is the number of actions applicable at state $s^{'}$ . Define the estimation bias $Z_{M N}$ for transition $s, a, r, s^{'}$ to be

	$Z_{M N}$	$\makebox[0.0pt]\tiny def=(r+γmaxa′Qmins′a′)−(r+γmaxa′Q∗s′a′)$
		$= γ (max a^{'} Q_{s^{'} a^{'}}^{m i n} - max a^{'} Q_{s^{'} a^{'}}^{*})$

where

Qminsa\makebox[0.0pt]\tiny def=mini∈{1,…,N}Qisa=Q∗sa+mini∈{1,…,N}eisa

We now show how the expected estimation bias $E [Z_{M N}]$ and the variance of $Q_{s a}^{m i n}$ are related to the number of action-value functions $N$ in Maxmin Q-learning.

Theorem 1

Under the conditions stated above,

[label=(),leftmargin=0cm,itemindent=.5cm,itemsep=0em]
the expected estimation bias is

$E [Z_{M N}] = γ τ [1 - 2 t_{M N}]$ $where t_{M N} = \frac{M (M - 1) \dots 1}{(M + \frac{1}{N}) (M - 1 + \frac{1}{N}) \dots (1 + \frac{1}{N})} .$

$E [Z_{M N}]$ decreases as $N$ increases: $E [Z_{M, N = 1}] = γ τ \frac{M - 1}{M + 1}$ and $E [Z_{M, N \to \infty}] = - γ τ$ .
$V a r [Q_{s a}^{m i n}] = \frac{4 N τ^{2}}{(N + 1)^{2} (N + 2)} .$

$V a r [Q_{s a}^{m i n}]$ decreases as $N$ increases: $V a r [Q_{s a}^{m i n}] = \frac{τ^{2}}{3}$ for N=1 and $V a r [Q_{s a}^{m i n}] = 0$ for $N \to \infty$ .

Theorem 1 is a generalization of the first lemma in Thrun and Schwartz (1993); we provide the proof in Appendix A as well as a visualization of the expected bias for varying $M$ and $N$ . This theorem shows that the average estimation bias $E [Z_{M N}]$ , decreases as $N$ increases. Thus, we can control the bias by changing the number of estimators in Maxmin Q-learning. Specifically, the average estimation bias can be reduced from positive to negative as $N$ increases. Notice that $E [Z_{M N}] = 0$ when $t_{M N} = \frac{1}{2}$ . This suggests that by choosing $N$ such that $t_{M N} \approx \frac{1}{2}$ , we can reduce the bias to near $0$ .

Furthermore, $V a r [Q_{s a}^{m i n}]$ decreases as $N$ increases. This indicates that we can control the estimation variance of target action value through $N$ . We show just this in the following Corollary. The subtlety is that with increasing $N$ , each estimator will receive less data. The fair comparison is to compare the variance of a single estimator that uses all of the data, as compared to the maxmin estimator which shares the samples across $N$ estimators. We show that there is an $N$ such that the variance is lower, which arises largely due to the fact that the variance of each estimator decreases linearly in $n$ , but the $τ$ parameter for each estimator only decreases at a square root rate in the number of samples.

Corollary 1

Assuming the $n_{s a}$ samples are evenly allocated amongst the $N$ estimators, then $τ = \sqrt{3 σ^{2} N / n_{s a}}$ where $σ^{2}$ is the variance of samples for $(s, a)$ and, for $Q_{s a}$ the estimator that uses all $n_{s a}$ samples for a single estimate,

V a r [Q_{s a}^{m i n}] = \frac{12 N^{2}}{(N + 1)^{2} (N + 2)} V a r [Q_{s a}] .

Under this uniform random noise assumption, for $N \geq 8$ , $V a r [Q_{s a}^{m i n}] < V a r [Q_{s a}]$ .

5 Experiments

In this section, we first investigate robustness to reward variance, in a simple environment (Mountain Car) in which we can perform more exhaustive experiments. Then, we investigate performance in seven benchmark environments.

Robustness under increasing reward variance in Mountain Car

Mountain Car (Sutton and Barto, 2018) is a classic testbed in Reinforcement Learning, where the agent receives a reward of $- 1$ per step with $γ = 1$ , until the car reaches the goal position and the episode ends. In our experiment, we modify the rewards to be stochastic with the same mean value: the reward signal is sampled from a Gaussian distribution $N (- 1, σ^{2})$ on each time step. An agent should learn to reach the goal position in as few steps as possible.

The experimental setup is as follows. We trained each algorithm with $1, 000$ episodes. The number of steps to reach the goal position in the last training episode was used as the performance measure. The fewer steps, the better performance. All experimental results were averaged over $100$ runs. The key algorithm settings included the function approximator, step-sizes, exploration parameter and replay buffer size. All algorithm used $ϵ$ -greedy with $ϵ = 0.1$ and a buffer size of $100$ . For each algorithm, the best step-size was chosen from ${0.005, 0.01, 0.02, 0.04, 0.08}$ , separately for each reward setting. Tile-coding was used to approximate the action-value function, where we used $8$ tilings with each tile covering $1 / 8$ th of the bounded distance in each dimension. For Maxmin Q-learning, we randomly chose one action-value function to update at each step.

As shown in Figure 3, when the reward variance is small, the performance of Q-learning, Double Q-learning, Averaged Q-learning, and Maxmin Q-learning are comparable. However, as the variance increases, Q-learning, Double Q-learning, and Averaged Q-learning became much less stable than Maxmin Q-learning. In fact, when the variance was very high ( $σ = 50$ , see Appendix C.2), Q-learning and Averaged Q-learning failed to reach the goal position in $5, 000$ steps, and Double Q-learning produced runs $> 400$ steps, even after many episodes.

(a) Robustness under increasing reward variance

Results on Benchmark Environments

To evaluate Maxmin DQN, we choose seven games from Gym (Brockman et al., 2016), PyGame Learning Environment (PLE) (Tasfi, 2016), and MinAtar (Young and Tian, 2019): Lunarlander, Catcher, Pixelcopter, Asterix, Seaquest, Breakout, and Space Invaders. For games in MinAtar (i.e. Asterix, Seaquest, Breakout, and Space Invaders), we reused the hyper-parameters and settings of neural networks in (Young and Tian, 2019). And the step-size was chosen from $[3 * 10^{- 3}, 10^{- 3}, 3 * 10^{- 4}, 10^{- 4}, 3 * 10^{- 5}]$

. For Lunarlander, Catcher, and Pixelcopter, the neural network was a multi-layer perceptron with hidden layers fixed to

[64, 64]

. The discount factor was

0.99

. The size of the replay buffer was

10, 000

. The weights of neural networks were optimized by RMSprop with gradient clip

5

. The batch size was

32

. The target network was updated every

200

frames.

ϵ

-greedy was applied as the exploration strategy with

ϵ

decreasing linearly from

1.0

0.01

1, 000

steps. After

1, 000

steps,

ϵ

was fixed to

0.01

. For Lunarlander, the best step-size was chosen from

[3 * 10^{- 3}, 10^{- 3}, 3 * 10^{- 4}, 10^{- 4}, 3 * 10^{- 5}]

. For Catcher and Pixelcopter, the best step-size was chosen from

[10^{- 3}, 3 * 10^{- 4}, 10^{- 4}, 3 * 10^{- 5}, 10^{- 5}]

For both Maxmin DQN and Averaged DQN, the number of target networks $N$ was chosen from $[2, 3, 4, 5, 6, 7, 8, 9]$ . And we randomly chose one action-value function to update at each step. We first trained each algorithm in a game for certain number of steps. After that, each algorithm was tested by running $100$ test episodes with $ϵ$ -greedy where $ϵ = 0.01$ . Results were averaged over $20$ runs for each algorithm, with learning curves shown for the best hyper-parameter setting (see Appendix C.3 for the parameter sensitivity curves).

We see from Figure 4 that Maxmin DQN performs as well as or better than other algorithms. In environments where final performance is noticeably better—-Pixelcopter, Lunarlander and Asterix—the initial learning is slower. A possible explanation for this is that the Maxmin agent more extensively explored early on, promoting better final performance. We additionally show on Pixelcopter and Asterix that for smaller $N$ , Maxmin DQN learns faster but reaches suboptimal performance—behaving more like Q-learning—and for larger $N$ learns more slowly but reaches better final performance.

6 Convergence Analysis of Maxmin Q-learning

In this section, we show Maxmin Q-learning is convergent in the tabular setting. We do so by providing a more general result for what we call Generalized Q-learning: Q-learning where the bootstrap target uses a function $G$ of $N$ action values. The main condition on $G$ is that it maintains relative maximum values, as stated in Assumption 1. We use this more general result to prove Maxmin Q-learning is convergent, and then discuss how it provides convergence results for Q-learning, Ensemble Q-learning, Averaged Q-learning and Historical Best Q-learning as special cases.

Many variants of Q-learning have been proposed, including Double Q-learning (van Hasselt, 2010), Weighted Double Q-learning (Zhang et al., 2017), Ensemble Q-learning (Anschel et al., 2017), Averaged Q-learning (Anschel et al., 2017), and Historical Best Q-learning (Yu et al., 2018). These algorithms differ in their estimate of the one-step bootstrap target. To encompass all variants, the target action-value of Generalized Q-learning $Y^{G Q}$ is defined based on action-value estimates from both dimensions:

Y^{G Q} = r + γ Q_{s^{'}}^{G Q} (t - 1)

(2)

where $t$ is the current time step and the action-value function $Q_{s}^{G Q} (t)$ is a function of $Q_{s}^{1} (t - K), \dots, Q_{s}^{1} (t - 1), \dots, Q_{s}^{N} (t - K), \dots, Q_{s}^{N} (t - 1)$ :

Q_{s}^{G Q} (t) = G ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} Q_{s}^{1} (t - K) & \dots & Q_{s}^{1} (t - 1) Q_{s}^{2} (t - K) & \dots & Q_{s}^{2} (t - 1) ⋮ & ⋱ & ⋮ Q_{s}^{N} (t - K) & \dots & Q_{s}^{N} (t - 1) \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3)

For simplicity, the vector

(Q_{s a}^{G Q} (t))_{a \in A}

is denoted as

Q_{s}^{G Q} (t)

, same for

Q_{s}^{i} (t)

. The corresponding update rule is

Q_{s a}^{i} (t) \leftarrow Q_{s a}^{i} (t - 1) + α_{s a}^{i} (t - 1) (Y^{G Q} - Q_{s a}^{i} (t - 1))

(4)

For different $G$ functions, Generalized Q-learning reduces to different variants of Q-learning, including Q-learning itself. For example, Generalized Q-learning can be reduced to Q-learning simply by setting $K = 1$ , $N = 1$ with $G (Q_{s}) = {max}_{a \in A} Q_{s a}$ . Double Q-learning can be specified with $K = 1$ , $N = 2$ , and $G (Q_{s}^{1}, Q_{s}^{2}) = Q_{s, arg {max}_{a^{'} \in A} Q_{s a^{'}}^{1}}^{2}$ .

We first introduce Assumption 1 for function $G$ in Generalized Q-learning, and then state the theorem. The proof can be found in Appendix B.

Assumption 1 (Conditions on $G$ )

Let $G : R^{n N K} \mapsto R$ and $G (Q) = q$ where $Q = (Q_{a}^{i j}) \in R^{n N K}$ , $a \in A$ and $| A | = n$ , $i \in {1, \dots, N}, j \in {0, \dots, K - 1}$ and $q \in R$ .

[label=()]
If $Q_{a}^{i j} = Q_{a}^{k l}$ , $\forall i, k$ , $\forall j, l$ , and $\forall a$ , then $q = {max}_{a} Q_{a}^{i j}$ .
$\forall Q, Q^{'} \in R^{n N K}$ , $∣ G (Q) - G (Q^{'}) ∣\leq {max}_{a, i, j} ∣ Q_{a}^{i j} - {Q^{'}}_{a}^{i j} ∣$ .

We can verify that Assumption 1 holds for Maxmin Q-learning. Set $K = 1$ and set $N$ to be a positive integer. Let $Q_{s} = (Q_{s}^{1}, \dots, Q_{s}^{N})$ and define $G^{M Q} (Q_{s}) = {max}_{a \in A} {min}_{i \in {1, \dots, N}} Q_{s a}^{i}$ . It is easy to check that part (i) of Assumption 1 is satisfied. Part (ii) is also satisfied because

∣ G (Q_{s}) - G (Q_{s}^{'}) ∣\leq∣ max a min i Q_{s a}^{i} - max a^{'} min i^{'} Q_{s a^{'}}^{' i^{'}} ∣\leq max a, i ∣ Q_{s a}^{i} - Q_{s a}^{' i} ∣ .

Assumption 2 (Conditions on the step-sizes)

There exists some (deterministic) constant $C$ such that for every $(s, a) \in S \times A, i \in {1, \dots, N}$ , $0 \leq α_{s a}^{i} (t) \leq 1$ , and with probability $1$ ,

\infty \sum t = 0 (α_{s a}^{i} (t))^{2} \leq C, \infty \sum t = 0 α_{s a}^{i} (t) = \infty

Theorem 2

Assume a finite MDP $(S, A, P, R)$ and that Assumption 1 and 2 hold. Then the action-value functions in Generalized Q-learning, using the tabular update in Equation (3), will converge to the optimal action-value function with probability $1$ , in either of the following cases: (i) $γ < 1$ , or (ii) $γ = 1$ , $\forall a \in A, Q_{s_{1} a}^{i} (t = 0) = 0$ where $s_{1}$ is an absorbing state and all policies are proper.

As shown above, because the function $G$ for Maxmin Q-learning satisfies Assumption 1, then by Theorem 2 it converges. Next, we apply Theorem 2 to Q-learning and its variants, proving the convergence of these algorithms in the tabular case. For Q-learning, set $K = 1$ and $N = 1$ . Let $G^{Q} (Q_{s}) = {max}_{a \in A} Q_{s a}$ . It is straightforward to check that Assumption 1 holds for function $G^{Q}$ . For Ensemble Q-learning, set $K = 1$ and set $N$ to be a positive integer. Let $G^{E Q} ((Q_{s}^{1}, \dots, Q_{s}^{N})) = {max}_{a \in A} \frac{1}{N} \sum_{i = 1}^{N} Q_{s a}^{i}$ . Easy to check that Assumption 1 is satisfied. For Averaged Q-learning, the proof is similar to Ensemble Q-learning except that $N = 1$ and $K$ is a positive integer. For Historical Best Q-learning, set $N = 1$ and $K$ to be a positive integer. We assume that all auxiliary action-value functions are selected from action-value functions at most $K$ updates ago. Define $G^{H B Q}$ to be the largest action-value among $Q_{s a} (t - 1), \dots, Q_{s a} (t - K)$ for state $s$ . Assumption 1 is satisfied and the convergence is guaranteed.

7 Conclusion

Overestimation bias is a byproduct of Q-learning, stemming from the selection of a maximal value to estimate the expected maximal value. In practice, overestimation bias leads to poor performance in a variety of settings. Though multiple Q-learning variants have been proposed, Maxmin Q-learning is the first solution that allows for a flexible control of bias, allowing for overestimation or underestimation determined by the choice of

N

and the environment. We showed theoretically that we can decrease the estimation bias and the estimation variance by choosing an appropriate number

N

of action-value functions. We empirically showed that advantages of Maxmin Q-learning, both on toy problems where we investigated the effect of reward noise and on several benchmark environments. Finally, we introduced a new Generalized Q-learning framework which we used to prove the convergence of Maxmin Q-learning as well as several other Q-learning variants that use

N

action-value estimates.

Acknowledgments

We would like to thank Huizhen Yu and Yi Wan for their valuable feedback and helpful discussion.

References

O. Anschel, N. Baram, and N. Shimkin (2017) Averaged-DQN: Variance Reduction and Stabilization for Deep Reinforcement Learning. In
International Conference on Machine Learning
, pp. 176–185. Cited by: §1, §6.
D. P. Bertsekas and J. N. Tsitsiklis (1989) Parallel and distributed computation: numerical methods. Vol. 23, Prentice hall Englewood Cliffs, NJ. Cited by: §B.3, §B.3, §B.3.
D. P. Bertsekas and J. N. Tsitsiklis (1996) Neuro-dynamic programming. Vol. 5, Athena Scientific Belmont, MA. Cited by: §B.2.
G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, and W. Zaremba (2016) OpenAI Gym. arXiv preprint arXiv:1606.01540. Cited by: §5.
H. A. David and H. N. Nagaraja (2004) Order Statistics. Encyclopedia of Statistical Sciences. Cited by: item 1, Appendix A.
S. Fujimoto, H. Hoof, and D. Meger (2018) Addressing function approximation error in actor-critic methods. In International Conference on Machine Learning, pp. 1587–1596. Cited by: §1.
T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine (2018) Soft actor-critic: off-policy maximum entropy deep reinforcement learning with a stochastic actor. In International Conference on Machine Learning, pp. 1861–1870. Cited by: §1.
H. Hado van Hasselt, A. Guez, and D. Silver (2016) Deep Reinforcement Learning with Double Q-learning. In
AAAI Conference on Artificial Intelligence
, Cited by: §1, §1.
D. Lee, B. Defourny, and W. B. Powell (2013) Bias-corrected Q-learning to Control Max-operator Bias in Q-learning. In IEEE Symposium on Adaptive Dynamic Programming and Reinforcement Learning, pp. 93–99. Cited by: §1.
F. S. Melo and M. I. Ribeiro (2007) Q-learning with Linear Function Approximation. In
International Conference on Computational Learning Theory
, pp. 308–322. Cited by: §2.
A. L. Strehl, L. Li, and M. L. Littman (2009) Reinforcement Learning in Finite MDPs: PAC Analysis. Journal of Machine Learning Research 10 (Nov), pp. 2413–2444. Cited by: §1.
R. S. Sutton and A. G. Barto (2018) Reinforcement Learning: An Introduction. Second edition, MIT Press. Cited by: Figure 1, §5.
I. Szita and A. Lőrincz (2008) The Many Faces of Optimism: A Unifying Approach. In International Conference on Machine learning, pp. 1048–1055. Cited by: §1.
N. Tasfi (2016) PyGame learning environment. GitHub. Note: https://github.com/ntasfi/PyGame-Learning-Environment Cited by: §5.
S. Thrun and A. Schwartz (1993) Issues in Using Function Approximation for Reinforcement Learning. In Fourth Connectionist Models Summer School, Cited by: §1, §4, §4.
J. N. Tsitsiklis (1994) Asynchronous Stochastic Approximation and Q-learning. Machine learning. Cited by: Appendix B, §2.
H. van Hasselt (2010) Double Q-learning. In Advances in Neural Information Processing Systems, pp. 2613–2621. Cited by: §1, §1, §6.
C. Watkins (1989) Learning from delayed rewards. Ph.D. Thesis, King’s College, Cambridge. Cited by: §1.
K. Young and T. Tian (2019) MinAtar: An Atari-inspired Testbed for More Efficient Reinforcement Learning Experiments. arXiv preprint arXiv:1903.03176. Cited by: Figure 4, §5.
W. Yu, R. Wang, R. Li, J. Gao, and X. Hu (2018) Historical Best Q-Networks for Deep Reinforcement Learning. In International Conference on Tools with Artificial Intelligence, pp. 6–11. Cited by: §6.
Z. Zhang, Z. Pan, and M. J. Kochenderfer (2017) Weighted Double Q-learning. In International Joint Conference on Artificial Intelligence, pp. 3455–3461. Cited by: §1, §6.

Appendix A The Proof of Theorem 1

We first present Lemma 1 here as a tool to prove Theorem 1. Note that the first three properties in this lemma are well-known results of order statistics (David and Nagaraja, 2004).

Lemma 1

Let $X_{1}, \dots, X_{N}$ be $N$

i.i.d. random variables from an absolutely continuous distribution with probability density function(PDF)

f (x)

and cumulative distribution function (CDF)

F (x)

. Denote

μ\makebox[0.0pt]\tiny def=E[Xi]

and

σ2\makebox[0.0pt]\tiny def=Var[Xi]<+∞

. Set

X1:N\makebox[0.0pt]\tiny def=mini∈{1,…,N}Xi

and

XN:N\makebox[0.0pt]\tiny def=maxi∈{1,…,N}Xi

. Denote the PDF and CDF of

X_{1 : N}

f_{1 : N} (x)

and

F_{1 : N} (x)

, respectively. Similarly, denote the PDF and CDF of

X_{N : N}

f_{N : N} (x)

and

F_{N : N} (x)

, respectively. We then have

[label=()]
$μ - \frac{(N - 1) σ}{\sqrt{2 n - 1}} \leq E [X_{1 : N}] \leq μ$ and $E [X_{1 : N + 1}] \leq E [X_{1 : N}]$ .
$F_{1 : N} (x) = 1 - (1 - F (x))^{N}$ . $f_{1 : N} (x) = N f (x) (1 - F (x))^{N - 1}$ .
$F_{N : N} (x) = (F (x))^{N}$ . $f_{N : N} (x) = N f (x) (F (x))^{N - 1}$ .
If $X_{1}, \dots, X_{N} \sim U (- τ, τ)$ , we have $V a r (X_{1 : N}) = \frac{4 N τ^{2}}{(N + 1)^{2} (N + 2)}$ and $V a r (X_{1 : N + 1}) < V a r (X_{1 : N}) \leq V a r (X_{1 : 1}) = σ^{2}$ for any positive integer $N$ .

Proof.

[label=()]
By the definition of $X_{1 : N}$ , we have $X_{1 : N + 1} \leq X_{1 : N}$ . Thus $E [X_{1 : N + 1}] \leq E [X_{1 : N}]$ . Since $E [X_{1 : 1}] = E [X_{1}] = μ$ , $E [X_{1 : N}] \leq E [X_{1 : 1}] = μ$ . The proof of $μ - \frac{(N - 1) σ}{\sqrt{2 N - 1}} \leq E [X_{1 : N}]$ can be found in (David and Nagaraja, 2004, Chapter 4 Section 4.2).
We first consider the cdf of $X_{1 : N}$ . $F_{1 : N} (x) := P (X_{1 : N} \leq x) = 1 - P (X_{1 : N} > x) = 1 - P (X_{1} > x, \dots, X_{M} > x) = 1 - P (X_{1} > x) \dots P (X_{N} > x) = 1 - (1 - F (x))^{N}$ . Then the pdf of $X_{1 : N}$ is $f_{1 : N} (x) := \frac{d F_{1 : N}}{d x} = N f (x) (1 - F (x))^{N - 1}$ .
Similar to (ii), we first consider cdf of $X_{N : N}$ . $F_{N : N} (x) := P (X_{N : N} \leq x) = P (X_{1} \leq x, \dots, X_{N} \leq x) = P (X_{1} \leq x) \dots P (X_{M} \leq x) = (F (x))^{N}$ . Then the pdf of $X_{N : N}$ is $f_{N : N} (x) := \frac{d F_{N : N}}{d x} = N f (x) (F (x))^{N - 1}$ .
Since $X_{1}, \dots, X_{N} \sim U n i f o r m (- τ, τ)$ , we have $F (x) = \frac{1}{2} + \frac{x}{2 τ}$ and $f (x) = \frac{1}{2 τ}$ . $V a r (X_{1 : N}) = E [{X_{1 : N}}^{2}] - {E [X_{1 : N}]}^{2} = 4 τ^{2} (\frac{2}{(N + 1) (N + 2)} - \frac{1}{(N + 1)^{2}}) = \frac{4 n τ^{2}}{(N + 1)^{2} (N + 2)}$ . It is easy to check that $V a r (X_{1 : N + 1}) < V a r (X_{1 : N}) \leq V a r (X_{1 : 1}) = σ^{2}$ for any positive integer $N$ .

Next, we prove Theorem 1.

Proof. Let $f (x)$ and $F (x)$ be the cdf and pdf of $e_{s a}$ , respectively. Similarly, Let $f_{N} (x)$ and $F_{N} (x)$ be the cdf and pdf of ${min}_{i \in {1, \dots, N}} e_{s a}^{i}$ . Since $e_{s a}$ is sampled from $U n i f o r m (- τ, τ)$ , it is easy to get $f (x) = \frac{1}{2 τ}$ and $F (x) = \frac{1}{2} + \frac{x}{2 τ}$ . By Lemma 1, we have $f_{N} (x) = N f (x) [1 - F (x)]^{N - 1} = \frac{N}{2 τ} (\frac{1}{2} - \frac{x}{2 τ})^{N - 1}$ and $F_{N} (x) = 1 - (1 - F (x))^{N} = 1 - (\frac{1}{2} - \frac{x}{2 τ})^{N}$ . The expectation of $Z_{M N}$ is

	$E [Z_{M N}]$	$= γ E [(max a^{'} Q_{s^{'} a^{'}}^{m i n} - max a^{'} Q_{s^{'} a^{'}}^{*})]$
		$= γ E [max a^{'} min i \in {1, \dots, N} e_{s a^{'}}^{i}]$
		$= γ \int_{- τ}^{τ} M x f_{N} (x) {F_{N} (x)}^{M - 1} d x$
		$= γ \int_{- τ}^{τ} M N \frac{x}{2 τ} (\frac{1}{2} - \frac{x}{2 τ})^{N - 1} [1 - (\frac{1}{2} - \frac{x}{2 τ})^{N}]^{M - 1} d x$
		$= γ \int_{- τ}^{τ} x d [1 - (\frac{1}{2} - \frac{x}{2 τ})^{N}]^{M}$
		$= γ τ - γ \int_{- τ}^{τ} [1 - (\frac{1}{2} - \frac{x}{2 τ})^{N}]^{M} d x$
		$=γτ[1−2∫10(1−yN)Mdy](y\makebox[0.0pt]\tiny def=12−x2τ)$

Let $t_{M N} = \int_{0}^{1} (1 - y^{N})^{M} d y$ , so that $E [Z_{M N}] = γ τ [1 - 2 t_{M N}]$ . Substitute $y$ by $t$ where $t = y^{N}$ , then

	$t_{M N}$	$= \frac{1}{N} \int_{0}^{1} t^{\frac{1}{N} - 1} (1 - t)^{M} d t$
		$= \frac{1}{N} B (\frac{1}{N}, M + 1)$
		$= \frac{1}{N} \frac{Γ (M + 1) Γ (\frac{1}{N})}{Γ (M + \frac{1}{N} + 1)}$
		$= \frac{Γ (M + 1) Γ (1 + \frac{1}{N})}{Γ (M + \frac{1}{N} + 1)}$
		$= \frac{M (M - 1) \dots 1}{(M + \frac{1}{N}) (M - 1 + \frac{1}{N}) \dots (1 + \frac{1}{N})}$

Each term in the denominator decreases as $N$ increases, because $1 / N$ gets smaller. Therefore, $t_{M, N = 1} = \frac{1}{M + 1}$ and $t_{M, N \to \infty} = 1$ . Using this, we conclude that $E [Z_{M N}]$ decreases as $N$ increases and $E [Z_{M, N = 1}] = γ τ \frac{M - 1}{M + 1}$ and $E [Z_{M, N \to \infty}] = - γ τ$ .

By Lemma 1, the variance of $Q_{s a}^{m i n}$ is

V a r [Q_{s a}^{m i n}] = \frac{4 N τ^{2}}{(N + 1)^{2} (N + 2)}

$V a r [Q_{s a}^{m i n}]$ decreases as $N$ increases. In particular, $V a r [Q_{s a}^{m i n}] = \frac{τ^{2}}{3}$ for $N = 1$ and $V a r [Q_{s a}^{m i n}] = 0$ for $N \to \infty$ .

The bias-variance trade-off of Maxmin Q-learning is illustrated by the empirical results in Figure 5, which support Theorem 1. For each $M$ , $N$ can be selected such that the absolute value of the expected estimation bias is close to $0$ according to Theorem 1. As $M$ increases, we can adjust $N$ to reduce both the estimation variance and the estimation bias.

Finally, we prove the result of the Corollary.

Corollary 1 Assuming the $n_{s a}$ samples are evenly allocated amongst the $N$ estimators, then $τ = \sqrt{3 σ^{2} N / n_{s a}}$ where $σ^{2}$ is the variance of samples for $(s, a)$ and, for $Q_{s a}$ the estimator that uses all $n_{s a}$ samples for a single estimate,

V a r [Q_{s a}^{m i n}] = \frac{12 N^{2}}{(N + 1)^{2} (N + 2)} V a r [Q_{s a}] .

Under this uniform random noise assumption, for $N \geq 8$ , $V a r [Q_{s a}^{m i n}] < V a r [Q_{s a}]$ .

Proof. Because $Q_{s a}^{i}$ is a sample mean, its variance is $σ^{2} N / n_{s a}$ where $σ^{2}$ is the variance of samples for $(s, a)$ and its mean is $Q_{s a}^{*}$ (because it is an unbiased sample average). Consequently, $e_{s a}$ has mean zero and variance $σ^{2} N / n_{s a}$ . Because $e_{s a}$ is a uniform random variable which has variance $\frac{1}{3} τ^{2}$ , we know that $τ = \sqrt{3 σ^{2} N / n_{s a}}$ . Plugging this value into the variance formula in Theorem 1, we get that

	$V a r [Q_{s a}^{m i n}]$	$= \frac{4 N τ^{2}}{(N + 1)^{2} (N + 2)}$
		$= \frac{12 N^{2} σ^{2} / n_{s a}}{(N + 1)^{2} (N + 2)}$
		$= \frac{12 N^{2}}{(N + 1)^{2} (N + 2)} V a r [Q_{s a}]$

because $V a r [Q_{s a}] = σ^{2} / n_{s a}$ for the sample average $Q_{s a}$ that uses all the samples for one estimator. Easy to verify that for $N \geq 8$ , $V a r [Q_{s a}^{m i n}] < V a r [Q_{s a}]$ .

Appendix B The Convergence Proof of Generalized Q-learning

The convergence proof of Generalized Q-learning is based on Tsitsiklis (1994). The key steps to use this result for Generalized Q-learning include showing that the operator is a contraction and verifying the noise conditions. We first show these two steps in Lemma 2 and Lemma 3. We then use these lemmas to make the standard argument for convergence.

b.1 Problem Setting for Generalized Q-learning

Consider a Markov decision problem defined on a finite state space $S$ . For every state $s \in S$ , there is a finite set $A$ of possible actions for state $s$ and a set of non-negative scalars $p_{s s^{'}} (a)$ , $a \in A$ , $s^{'} \in S$ , such that $\sum_{j \in S} p_{s s^{'}} (a) = 1$ for all $a \in A$ . The scalar $p_{s s^{'}} (a)$ is interpreted as the probability of a transition to $s^{'}$ , given that the current state is $s$ and action $a$ is applied. Furthermore, for every state $s$ and action $a$ , there is a random variable $r_{s a}$ which represents the reward if action $a$ is applied at state $s$ . We assume that the variance of $r_{s a}$ is finite for every $s$ and $a \in A$ .

A stationary policy is a function $π$ defined on $S$ such that $π (s) \in A$ for all $s \in S$

. Given a stationary policy, we obtain a discrete-time Markov chain

f^{π} (t)

with transition probabilities

Pr (f^{π} (t + 1) = s^{'} | f^{π} (t) = s) = p_{s s^{'}} (π (s))

(5)

Let $γ \in [0, 1]$ be a discount factor. For any stationary policy $π$ and initial state $s$ , the state value $V_{s}^{π}$ is defined by

V_{s}^{π} = lim T \to \infty E [T \sum t = 0 γ^{t} r_{f^{π} (t), π (f^{π} (t))} | f^{π} (0) = s]

(6)

The optimal state value function $V^{*}$ is defined by

V_{s}^{*} = sup π V_{s}^{π}, s \in S

(7)

The Markov decision problem is to evaluate the function $V^{*}$ . Once this is done, an optimal policy is easily determined.

Markov decision problems are easiest when the discount $γ$ is strictly smaller than $1$ . For the undiscounted case ( $γ = 1$ ), we will assume throughout that there is a reward-free state, say state $1$ , which is absorbing; that is, $p_{11} (a) = 1$ and $r_{1 u} = 0$ for all $a \in A$ . The objective is then to reach that state at maximum expected reward. We say that a stationary policy is proper if the probability of being at the absorbing state converges to $1$ as time converges to infinity; otherwise, we say that the policy is improper.

We define the dynamic programming operator $T : R^{| S |} \mapsto R^{|}$

Maxmin Q-learning: Controlling the Estimation Bias of Q-learning

Authors

Estimating the Maximum Expected Value: An Analysis of (Nested) Cross Validation and the Maximum Sample Average

Generalized Emphatic Temporal Difference Learning: Bias-Variance Analysis

A Novel Algorithm for Unbiased Learning to Rank

Decorrelated Double Q-learning

Unbiased Estimation Equation under f-Separable Bregman Distortion Measures

National Bias of International Gymnastics Judges during the 2013-2016 Olympic Cycle

Cross Learning in Deep Q-Networks

1 Introduction

2 Problem Setting

3 Understanding when Overestimation Bias Helps and Hurts

4 Maxmin Q-learning

Theorem 1

Corollary 1

5 Experiments

Robustness under increasing reward variance in Mountain Car

Results on Benchmark Environments

6 Convergence Analysis of Maxmin Q-learning

Assumption 1 (Conditions on $G$ )

Assumption 2 (Conditions on the step-sizes)

Theorem 2

7 Conclusion

Acknowledgments

References

Appendix A The Proof of Theorem 1

Lemma 1

Appendix B The Convergence Proof of Generalized Q-learning

b.1 Problem Setting for Generalized Q-learning

Maxmin Q-learning: Controlling the Estimation Bias of Q-learning

Authors

Estimating the Maximum Expected Value: An Analysis of (Nested) Cross Validation and the Maximum Sample Average

Generalized Emphatic Temporal Difference Learning: Bias-Variance Analysis

A Novel Algorithm for Unbiased Learning to Rank

Decorrelated Double Q-learning

Unbiased Estimation Equation under f-Separable Bregman Distortion Measures

National Bias of International Gymnastics Judges during the 2013-2016 Olympic Cycle

Cross Learning in Deep Q-Networks

This week in AI

1 Introduction

2 Problem Setting

3 Understanding when Overestimation Bias Helps and Hurts

4 Maxmin Q-learning

Theorem 1

Corollary 1

5 Experiments

Robustness under increasing reward variance in Mountain Car

Results on Benchmark Environments

6 Convergence Analysis of Maxmin Q-learning

Assumption 1 (Conditions on G)

Assumption 2 (Conditions on the step-sizes)

Theorem 2

7 Conclusion

Acknowledgments

References

Appendix A The Proof of Theorem 1

Lemma 1

Appendix B The Convergence Proof of Generalized Q-learning

b.1 Problem Setting for Generalized Q-learning

Assumption 1 (Conditions on $G$ )