function [] = opt_initial_values(nB,nA,nP,sigmaReward)
% 
% Demonstrate the technique of optimisitc initial values
% We compare two methods both epsilon=0.1 greedy
% 
% 1) evaluate the algorithm performance when the initial action values are taken = +5
% 2) evaluate the algorithm performance when the initial action values are taken = 0
%
% Inputs: 
%   nB: the number of bandits
%   nA: the number of arms
%   nP: the number of plays (times we will pull a arm)
%   sigmaReward: the standard deviation of the return from each of the arms
% 
% Written by:
% -- 
% John L. Weatherwax                2007-11-13
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

%close all; 
%clc; 
%clear; 

if( nargin<1 ) % the number of bandits: 
  nB = 2000;  
end
if( nargin<2 ) % the number of arms: 
  nA = 10; 
end
if( nargin<3 ) % the number of plays (times we will pull a arm):
  nP = 2000; 
end
if( nargin<4 ) % the standard deviation of the return from each of the arms: 
  sigmaReward = 1.0; 
end

%randn('seed',0); 

%-----
% Compare two methods: 
% M1) An epsion greedy (eps=0.1) algorithm with Q_0 = +0 (less exploration):
% M2) An epsion greedy (eps=0.1) algorithm with Q_0 = +5 (allows early exploration): 
%     both use the alpha=0.1 action value update algorithm
%-----
tEps  = 0.1;
alpha = 0.1; 

avgReward    = zeros(2,nP); 
perOptAction = zeros(2,nP); 

allRewards_M1      = zeros(nB,nP); 
pickedMaxAction_M1 = zeros(nB,nP); 
allRewards_M2      = zeros(nB,nP); 
pickedMaxAction_M2 = zeros(nB,nP); 
for bi=1:nB, % pick a bandit
  % generate the TRUE reward Q^{\star}: 
  qStarMeans = randn(1,nA); 
  qT_M1 = zeros(1,nA);  % <- initialize action value estimates to zero (no knowledge) 
  qT_M2 = 5*ones(1,nA); % <- initialize action value estimates to *5* allows exploration
  
  for pi=1:nP, % make a play ... one for each algorithm
    
    if( rand(1) <= tEps ) % pick a RANDOM arm ... explore: 
      [dum,arm_M1] = histc(rand(1),linspace(0,1+eps,nA+1));
      arm_M2 = arm_M1; 
    else                  % pick the GREEDY arm:
      [dum,arm_M1] = max( qT_M1 );
      [dum,arm_M2] = max( qT_M2 ); 
    end
    clear dum;
    
    % determine if the arm selected is the best possible: 
    [dum,bestArm] = max( qStarMeans ); 
    if( arm_M1==bestArm ) pickedMaxAction_M1(bi,pi) = 1; end
    if( arm_M2==bestArm ) pickedMaxAction_M2(bi,pi) = 1; end
    % get the reward from drawing on that arm: 
    reward_M1 = qStarMeans(arm_M1) + sigmaReward*randn(1); 
    reward_M2 = qStarMeans(arm_M2) + sigmaReward*randn(1); 
    allRewards_M1(bi,pi) = reward_M1; 
    allRewards_M2(bi,pi) = reward_M2; 
    % update qT using alpha incrementally: 
    qT_M1(arm_M1) = qT_M1(arm_M1) + alpha*( reward_M1-qT_M1(arm_M1) );
    qT_M2(arm_M2) = qT_M2(arm_M2) + alpha*( reward_M2-qT_M2(arm_M2) );
    
    [ qT_M1; qT_M2; qStarMeans ];
  end
end

avgRew          = mean(allRewards_M1,1);
avgReward(1,:) = avgRew(:).'; 
avgRew          = mean(allRewards_M2,1);
avgReward(2,:) = avgRew(:).'; 

percentOptAction  = mean(pickedMaxAction_M1,1);
perOptAction(1,:) = percentOptAction(:).';
percentOptAction  = mean(pickedMaxAction_M2,1);
perOptAction(2,:) = percentOptAction(:).';

% produce the average rewards plot: 
% 
figure; hold on; 
all_hnds = plot( 1:nP, avgReward );
legend( all_hnds, { 'Q_0=0', 'Q_0=5' }, 'Location', 'SouthEast' ); 
axis tight; grid on; 
xlabel( 'plays' ); ylabel( 'Average Reward' ); 

% produce the percent optimal action plot: 
% 
figure; hold on; 
all_hnds = plot( 1:nP, perOptAction );
legend( all_hnds, { 'Q_0=0', 'Q_0=5' }, 'Location', 'SouthEast' ); 
axis( [ 0, nP, 0, 1 ] ); axis tight; grid on; 
xlabel( 'plays' ); ylabel( '% Optimal Action' );