Dueling Network Architectures for DeepReinforcement Learning (ICML 2016)

Yoonho Lee

Department of Computer Science and EngineeringPohang University of Science and Technology

October 11, 2016

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Definition of RLgeneral setting of RL

Definition of RLatari setting

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Mathematical formulationsobjective of RL

DefinitionReturn Gt is the cumulative discounted reward from time t

Gt = rt+1 + γrt+2 + γ2rt+3 + . . .

DefinitionA policy π is a function that selects actions given states

π(s) = a

I The goal of RL is to find π that maximizes G0

Mathematical formulationsobjective of RL

DefinitionReturn Gt is the cumulative discounted reward from time t

Gt = rt+1 + γrt+2 + γ2rt+3 + . . .

DefinitionA policy π is a function that selects actions given states

π(s) = a

I The goal of RL is to find π that maximizes G0

Mathematical formulationsQ-Value

Gt =∞∑i=0

γk rt+i+1

DefinitionThe action-value (Q-value) function Qπ(s, a) is the expectation ofGt under taking action a, and then following policy π afterwards

Qπ(s, a) = Eπ[Gt |St = s,At = a,At+i = π(St+i )∀i ∈ N]

I ”How good is action a in state s?”

Mathematical formulationsOptimal Q-value

DefinitionThe optimal Q-value function Q∗(s, a) is the maximum Q-valueover all policies

Q∗(s, a) = maxπ

Qπ(s, a)

TheoremThere exists a policy π∗ such that Qπ∗(s, a) = Q∗(s, a) for all s, a

I Thus, it suffices to find Q∗

Mathematical formulationsOptimal Q-value

DefinitionThe optimal Q-value function Q∗(s, a) is the maximum Q-valueover all policies

Q∗(s, a) = maxπ

Qπ(s, a)

TheoremThere exists a policy π∗ such that Qπ∗(s, a) = Q∗(s, a) for all s, a

I Thus, it suffices to find Q∗

Mathematical formulationsQ-Learning

Bellman Optimality Equation

Q∗(s, a) satisfies the following equation:

Q∗(s, a) = R(s, a) + γ∑s′

P(s ′|s, a)maxa′

Q∗(s′, a′)

Q-Learning

Let a be ε-greedy w.r.t. Q, and a′ be optimal w.r.t. Q. Qconverges to Q∗ if we iteratively apply the following update:

Q(s, a)← α(R(s, a) + γQ(s ′, a′)) + (1− α)Q(s, a)

Mathematical formulationsQ-Learning

Bellman Optimality Equation

Q∗(s, a) satisfies the following equation:

Q∗(s, a) = R(s, a) + γ∑s′

P(s ′|s, a)maxa′

Q∗(s′, a′)

Q-Learning

Let a be ε-greedy w.r.t. Q, and a′ be optimal w.r.t. Q. Qconverges to Q∗ if we iteratively apply the following update:

Q(s, a)← α(R(s, a) + γQ(s ′, a′)) + (1− α)Q(s, a)

Mathematical formulationsOther approcahes to RL

Value-Based RL

I Estimate Q∗(s, a)

I Deep Q Network

Policy-Based RL

I Search directly for optimal policy π∗I DDPG, TRPO. . .

Model-Based RL

I Use the (learned or given) transition model of environment

I Tree Search, DYNA . . .

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Neural Fitted Q Iteration

I Supervised learning on (s,a,r,s’)

Neural Fitted Q Iterationinput

Neural Fitted Q Iterationtarget equation

Neural Fitted Q Iteration

Shortcomings

I Exploration is independent of experience

I Exploitation does not occur at all

I Policy evaluation does not occur at all

I Even in exact(table lookup) case, not guaranteed to convergeto Q∗

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Deep Q Networkaction policy

I Choose an ε-greedy policy w.r.t. Q

Deep Q Networknetwork freezing

I Get a copy of the network every C steps for stability

Deep Q Network

Deep Q Network

Deep Q Networkperformance

Deep Q Networkoveroptimism

I What happens when we overestimate Q?

Deep Q Network

Problems

I Overestimation of the Q function at any s spills over toactions that lead to s → Double DQN

I Sampling transitions uniformly from D is inefficient →Prioritized Experience Replay

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Double Deep Q Networktarget

We can write the DQN target as:

Y DQNt = Rt+1 + γQ(St+1, argmax

aQ(St+1, a; θ−t ); θ−t )

Double DQN’s target is:

Y DDQNt = Rt+1 + γQ(St+1, argmax

aQ(St+1, a; θt); θ−t )

This has the effect of decoupling action selection and actionevaluation

Double Deep Q Networkperformance

I Much stabler with very little change in code

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Prioritized Experience Replay

I Update ’more surprising’ experiences more frequently

Outline

Reinforcement LearningDefinition of RLMathematical formulations

AlgorithmsNeural Fitted Q IterationDeep Q NetworkDouble Deep Q NetworkPrioritized Experience ReplayDueling Network

Dueling Network

I The scalar approximates V , and the vector approximates A

Dueling Networkforward pass equation

The exact forward pass equation is:

Q(s, a; θ, α, β) = V (s; θ, β) + (A(s, a; θ, α)−maxa′

A(s, a′; θ, α))

The following module was found to be more stable without losingmuch performance:

Q(s, a; θ, α, β) = V (s; θ, β) + (A(s, a; θ, α)− 1

|A|∑a′

A(s, a′; θ, α))

Dueling Networkattention

I Advantage network only pays attention when current actioncrucial

Dueling Networkperformance

I Achieves state of the art in the Atari domain among DQNalgorithms

Dueling Network

Summary

I Since this is an improvement only in network architecture,methods that improve DQN(e.g. Double DQN) are allapplicable here as well

I Solves problem of V and A typically being of different scale

I Updates Q values more frequently than a single-stream DQN,where only a single Q value is updated for each observation

I Implicitly splits the credit assignment problem into a recursivebinary problem of “now or later”

Dueling Network

Shortcomings

I Only works for |A| <∞I Still not able to solve tasks involving long-term planning

I Better than DQN, but sample complexity is still high

I ε-greedy exploration is essentially random guessing