#################################################
# ch6.3 Temporal Difference
# TD0 & constant step-size MC
#################################################
# TD0 model
TD0.ex1<-function(maxItr,alpha,gamma) {
V=c(0,rep(0.5,5),0)
V.hist=matrix(0,nrow=maxItr+1,ncol=5)
V.hist[1,]=V[2:6]
P.act=matrix(0.5,ncol=7,nrow=2)
for (i_rep in 1:maxItr) {
state=5
while (state!=1 & state!=7) {
action=sample(c(-1,1),1,prob=P.act[,state])
state.old=state
state=state+action
r=ifelse(state==7,1,0)
V[state.old]=V[state.old]+alpha*(r+gamma*V[state]-V[state.old])
}
V.hist[(i_rep+1),]=V[2:6]
}
return(V.hist)
}
# (re)creating Fig 6.6
true.V=1:5*(1/6)
res=TD0.ex1(1000,0.1,1)
plot(true.V,type='o',pch=15,ylim=c(0,1),ylab="Value",xaxt="n",
xlab="State",xlim=c(0.5,5.5),cex=2,lwd=2)
axis(1,at=1:5,labels=c("A","B","C","D","E"))
cols=c('red','blue','green','cyan','magenta')
ns=c(1,2,11,101,1001)
for (i_lines in 1:5) {
lines(res[ns[i_lines],],type='o',pch=15+i_lines,cex=2,lwd=2,col=cols[i_lines])
}
legend('topleft',c('True value','t=0','t=1','t=10','t=100','t=1000'),
col=c('black',cols),pch=15:20,lwd=1.5)
# constant step-size Monte Carlo
constMC.ex1<-function(maxItr,alpha) {
V=c(0,rep(0.5,5),0)
V.hist=matrix(0,nrow=maxItr+1,5)
V.hist[1,]=V[2:6]
P.act=matrix(0.5,ncol=7,nrow=2)
for (i_rep in 1:maxItr) {
state=5;
state.hist=state
while (state!=1 & state!=7) {
action=sample(c(-1,1),1,prob=P.act[,state])
state=state+action
state.hist=cbind(state.hist,state)
}
R=ifelse(state==7,1,0)
n.state=length(state.hist)
for (i_state in 1:(n.state-1)) {
V[state.hist[i_state]]=V[state.hist[i_state]]+
alpha*(R-V[state.hist[i_state]])
}
V.hist[(i_rep+1),]=V[2:6]
}
return(V.hist)
}
# (re)creating Fig 6.7
alphaTD=c(0.05,0.075,0.1,0.15)
alphaMC=c(0.01,0.02,0.03,0.04)
n.alphas=length(alphaTD)
pchs=0:(0+n.alphas)
true.V=1:5*(1/6)
n_rep=100
sqs=seq(1,101,2)
plot(0,0,type='n',xlim=c(0,100),ylim=c(0,0.25))
for (i_alpha in 1:n.alphas) {
rmsTD=matrix(0,101,n_rep)
rmsMC=matrix(0,101,n_rep)
for (i_rep in 1:n_rep) {
resTD=TD0.ex1(100,alphaTD[i_alpha],1)
resMC=constMC.ex1(100,alphaMC[i_alpha])
for (i_gen in 1:101) {
rmsTD[i_gen,i_rep]=sqrt(mean((resTD[i_gen,]-true.V)^2))
rmsMC[i_gen,i_rep]=sqrt(mean((resMC[i_gen,]-true.V)^2))
}
}
mTD=rowMeans(rmsTD)
mMC=rowMeans(rmsMC)
lines(mTD,col='red')
lines(mMC,col='blue')
lines(sqs,mTD[sqs],col='red',pch=pchs[i_alpha],type='p')
lines(sqs,mMC[sqs],col='blue',pch=pchs[i_alpha],type='p')
}
labs=c("MC, alpha=0.01",
"MC, alpha=0.02",
"MC, alpha=0.03",
"MC, alpha=0.04",
"TD, alpha=0.05",
"TD, alpha=0.075",
"TD, alpha=0.10",
"TD, alpha=0.15")
legend('topright',labs,col=c(rep('blue',4),rep('red',4)),pch=rep(0:3,2),lwd=1.5)
#################################################
# ch6.4 On-policy TD, Sarsa
#################################################
sarsa.ex6.5<-function(maxItr,alpha,gamma,epsilon) {
# field size: 7row x 10column
# horizontal move -> COLUMN
# vertical move -> ROW
# effect of wind -> ROW
# actions: 1-up, 2-right, 3-down, 4-left
act.V=matrix(c(1,0,0,1,-1,0,0,-1),nrow=4,byrow=T)
wind=matrix(c(0,0,0,0,0,0,1,0,1,0,1,0,2,0,2,0,1,0,0,0),byrow=T,nrow=10)
goal=c(4,8)
Qs=array(0,dim=c(7,10,4))
for (i_rep in 1:maxItr) {
state=c(4,1) # start
if (runif(1) > epsilon) {
move=which.max(Qs[state[1],state[2],])
} else { move=sample(1:4,1)}
while (!all(state==goal)) {
st.old=state
mv.old=move
state=state+act.V[move,]+wind[state[2],]
if (state[1]<1) {state[1]=1}
if (state[1]>7) {state[1]=7}
if (state[2]<1) {state[2]=1}
if (state[2]>10) {state[2]=10}
if (runif(1) > epsilon) {
move=which.max(Qs[state[1],state[2],])
} else { move=sample(1:4,1)}
rew=ifelse(all(state==goal),0,-1)
Qs[st.old[1],st.old[2],mv.old]=Qs[st.old[1],st.old[2],mv.old]
+alpha*(rew+gamma* Qs[state[1],state[2],move]
-Qs[st.old[1],st.old[2],mv.old])
}
}
return(Qs)
}
# running example
Qs=sarsa.ex6.5(5e6,0.1,1,0.1)
# sim optimal actions
state=c(4,1);goal=c(4,8);
state.hist=state
while (!all(state==goal)) {
moveID=which.max(Qs[state[1],state[2],])
state=state+act.V[moveID,]+wind[state[2],]
if (state[1]<1) {state[1]=1}
if (state[1]>7) {state[1]=7}
if (state[2]<1) {state[2]=1}
if (state[2]>10) {state[2]=10}
state.hist=rbind(state.hist,state)
}
# plotting results
plot(0,0,type='n',xlim=c(0,11),ylim=c(0,8),xlab="",ylab="",
main="Learned policies -- Sarsa")
lines(1,4,type='p',pch=19,col='red',cex=2)
lines(8,4,type='p',pch=19,col='red',cex=2)
dirs=c("up","right","down","left" )
for (i_row in 1:7) {
for (i_col in 1:10) {
best.move=dirs[which.max(Qs[i_row,i_col,])]
text(i_col,i_row,best.move)
}
}
lines(state.hist[,2],state.hist[,1],col="red",lwd=2)
#################################################
# ch6.5 Off-policy TD, Q-learning
#################################################
Qlearn.ex6.5<-function(maxItr,alpha,gamma,epsilon) {
# field size: 7row x 10column
# horizontal move -> COLUMN
# vertical move -> ROW
# effect of wind -> ROW
# actions: 1-up, 2-right, 3-down, 4-left
act.V=matrix(c(1,0,0,1,-1,0,0,-1),nrow=4,byrow=T)
wind=matrix(c(0,0,0,0,0,0,1,0,1,0,1,0,2,0,2,0,1,0,0,0),byrow=T,nrow=10)
goal=c(4,8)
Qs=array(0,dim=c(7,10,4))
for (i_rep in 1:maxItr) {
state=c(4,1) # start
while (!all(state==goal)) {
if (runif(1) > epsilon) {
move=which.max(Qs[state[1],state[2],])
} else { move=sample(1:4,1)}
sIDX=state
state=state+act.V[move,]+wind[state[2],]
if (state[1]<1) {state[1]=1}
if (state[1]>7) {state[1]=7}
if (state[2]<1) {state[2]=1}
if (state[2]>10) {state[2]=10}
max.Q=max(Qs[state[1],state[2],])
rew=ifelse(all(state==goal),0,-1)
Qs[sIDX[1],sIDX[2],move]=Qs[sIDX[1],sIDX[2],move]
+alpha*(rew+gamma* max.Q-Qs[sIDX[1],sIDX[2],move])
}
}
return(Qs)
}
Qs=Qlearn.ex6.5(1e6,0.05,1,0.1)
# sim optimal actions
state=c(4,1);goal=c(4,8);
state.hist=state
while (!all(state==goal)) {
moveID=which.max(Qs[state[1],state[2],])
state=state+act.V[moveID,]+wind[state[2],]
if (state[1]<1) {state[1]=1}
if (state[1]>7) {state[1]=7}
if (state[2]<1) {state[2]=1}
if (state[2]>10) {state[2]=10}
state.hist=rbind(state.hist,state)
}
# plotting results
plot(0,0,type='n',xlim=c(0,11),ylim=c(0,8),xlab="",ylab="",
main="Learned policies -- Q-learning")
lines(1,4,type='p',pch=19,col='red',cex=2)
lines(8,4,type='p',pch=19,col='red',cex=2)
dirs=c("up","right","down","left" )
for (i_row in 1:7) {
for (i_col in 1:10) {
best.move=dirs[which.max(Qs[i_row,i_col,])]
text(i_col,i_row,best.move)
}
}
lines(state.hist[,2],state.hist[,1],col="red",lwd=2)
