Applying Q-Learning to Flappy Bird

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1 Applying Q-Lerning to Flppy Bird Moritz Ebeling-Rump, Mnfred Ko, Zchry Hervieux-Moore Abstrct The field of mchine lerning is n interesting nd reltively new re of reserch in rtificil intelligence. In this pper, specil type of reinforcement lerning, Q-Lerning, ws pplied to the populr mobile gme Flppy Bird. The Q- Lerning lgorithm ws tested on two different environments. The originl version nd simplified version. The mximum score chieved on the originl version nd simplified version were 169 nd 28,851, respectively. The trde-off between runtime nd ccurcy ws investigted. Using pproprite settings, the Q-Lerning lgorithm ws proven to be successful with reltively quick convergence time. I. INTRODUCTION Q-Lerning is form reinforcement lerning tht does not require the gents to hve prior knowledge of the environment dynmics 1. The gents do not hve ccess to the cost function or the trnsition probbilities. Reinforcement lerning is type of lerning strtegy where the gents re not told which ctions to tke. Insted, the gents discovers which ction yields the highest rewrd from experimenttion 2. Q-Lerning ws first introduced by Wtkins in nd it ws not until 1992 tht the convergence ws shown by Wtkins nd Dyn 1. bird is two-dimensionl side-scrolling gme, illustrted in Figure 1, feturing retro style grphics. The gol of the gme is to direct the bird through series of pipes. If the bird touches the floor or pipe, then the bird will die nd the gme restrts. The only ction tht plyers re ble to perform is to mke the bird jump. Otherwise, the bird will fll due to grvity. Ech pipe the bird successfully clers, the score is incremented. In the originl gme, the gp between the pipes re held constnt but the height of the gp is uniformly distributed. In the simplified version tht ws considered, the height of the gp is held constnt. The gme is ment to test plyers urnce s there is no to the gme. II. THEORY Wtkins nd Dyn first showed convergence of the Q- Lerning lgorithm in 1. However, Tsitsiklis gives n lternte proof of convergence with stronger results in 6. It is Tsitsiklis s results tht will be shown here. To begin, define the stndrd form of stochstic pproximtion lgorithms. x i := x i + α i (F i (x) x i + ω i ) (1) Where x = (x 1, x 2,..., x n ) R n, F 1, F 2,..., F n re mppings from R n R, ω i is noise term, nd α i is the step size. Assumptions on equtions of this form re now listed. Note, in Tsitsiklis s pper, he llows for outdted informtion. For simplicity, it is ssumed ll informtion vilble to the controller is up to dte which will simplify ths list. This is justified s the ppliction considered hs controller tht never uses outdted informtion. Assumption 1: () x(0) is F (0)-mesurble. (b) i nd t, w i is F (t + 1)-mesurble. (c) i nd t, α i (t) is F (t)-mesurble. (d) i nd t, we hve Eω i (t) F (t) = 0. (e) constnts A nd B such tht Fig. 1. The gmeply of Flppy Bird Flppy Bird is mobile gme developed in 2013 by Dong Nyugen 4 nd published by dotgers, smll indepent gme developer compny bsed in Vietnm 5. Flppy Eωi 2 (t) F (t) A + B mx mx x j(τ) 2, i, t j τ t Assumption 1 is the typicl restrictions imposed on stochstic systems. First, the present cn be determined entirely by previous informtion. Second, the noise term hs men 0 nd bounded vrince. Assumption 2: () i, α i (t) =, w.p. 1

2 (b) C such tht i, α i (t) 2 C, w.p. 1 Assumption 2 re restrictions on the step size to ensure tht it pproches 0 t slow enough speed to ensure convergence of the lgorithm. The following conventions re now used. If x y then x i y i i. The norm v is defined s: x v = mx i x i v i, x R n Assumption 3: () The mpping F is monotone. Tht is, if x y then F (x) F (y) (b) The mpping F is continuous. (c) The mpping F hs unique fixed point x. (d) If e R n is the vector whose components re ll equl to 1, nd r > 0, then, F (X) re F (x re) F (x + re) F (X) + re It is not hrd to imgine how Assumption 3 might be used to generte inequlities tht re essentil in proof. Assumption 4: vector x R n, positive vector v, nd β 0, 1) such tht, F (x) x v β x x v, x R n It cn be seen tht the previous ssumption provides the frmework for contrction rgument (since β 0, 1)). This cn be used to prove convergence to sttionry point. Assumption 5: positive vector v, nd β 0, 1) nd D such tht, F (x) v β x v + D, x R n Tsitsiklis goes on the prove the following three theorems. Theorem 1: Let Assumptions 1,2, nd 5 hold. Then, then sequence x(t) in the stochstic pproximtion is bounded with probbility 1. Theorem 2: Let Assumptions 1,2, nd 3 hold. Then, then sequence x(t) in the stochstic pproximtion is bounded with probbility 1. Theorem 3: Let Assumptions 1,2, nd 4 hold. Then, then sequence x(t) in the stochstic pproximtion converges to x with probbility 1. From these theorems, it is cler tht Theorem 3 is the strongest in the sense tht we need the fewest ssumptions to gurntee convergence. Also, Theorem 2 requires boundedness which is the result of Theorem 1. Thus, one cn think of these s: to gurntee convergence, one needs Assumptions 1, 2, 4, nd 5 or 1, 2, nd 3. With regrds to Q-Lerning, the wy these theorems re importnt deps on writing F (x) s follows: F iu (Q) = Ec iu + β min Q s(i,u),v v U(s(t,u)) when F (x) is written in this form, it cn be shown tht Assumptions 1 nd 4 re met. Hence, by Theorem 3, we only need Assumption 2 for convergence to be gurnteed. This is n ssumption on the step size α so it is very esy for the lgorithm user to impose this restriction in prctice. For our ppliction, we will rewrite this cost minimiztion s rewrd mximiztion nd use different nottion s shorthnd: F (Q t (s t, t )) = E R t+1 + γe mx Q t (s t+1, ) We now strt with the Q-Lerning lgorithm nd work our wy bckwrds to the stochstic pproximtion lgorithm in eqution 1. Q t+1 (s t, t ) = Q t (s t, t ) + α t (s t, t ) { } R t+1 + γ mx Q t (s t+1, ) Q t (s t, t ) = Q t (s t, t ) + α t (s t, t ) {R t+1 + E R t+1 E R t+1 +γe mx Q t (s t+1, ) E mx Q t (s t+1, ) I t } + mx Q t (s t+1, ) Q t (s t, t ), where I t denotes the informtion vilble to the controller t time t. nd w t (s t, t ) =R t+1 E R t+1 + mx Q t (s t+1, ) E mx Q t (s t+1, ) I t. Rewriting the expression for Q we get Q t+1 (s t, t ) = Q t (s t, t ) + α t (s t, t ) (F (Q t (s t, t )) Q t (s t, t ) + w t (s t, t )) Q is now in the desired form! Thus, we only need to impose the restictions on α(t) to gurntee convergence of Q. The lst cse we wish to consider is when the problem is undiscounted (β = 1). For this, one further ssumption is needed. Assumption 6: () t lest one proper sttionry policy (b) Every improper sttionry policy yields infinite expected cost for t lest one initil stte Where proper policy is defined to be sttionry policy tht the probbility of going to n bsorbing stte with 0 rewrd converges to 1. Otherwise, it is improper. With this, we get the lst theorem: Theorem 4: The Q-Lerning lgorithm converges with probbility 1 if we impose Assumption 2 nd: () β < 1, (b) β = 1, Q iu (0) = 0, nd ll policies re proper, or (c) β = 1, Q iu (0) = 0, Assumption 6 holds, nd Q(t) is gurnteed to be bounded with probbility 1.

3 III. IMPLEMENTATION The key to successful implementtion of Q-Lerning is modeling the problem efficiently. As long s the stte spce is finite, Q-Lerning will converge. In prcticl pplictions, we re looking to obtin results in limited time frme. If the stte spce is very lrge, the lgorithm will tke too long to converge. On the other hnd, if the stte spce is very smll, ccurcy will be lost. Thus, there is trde-off between run-time nd ccurcy. The dimension of the stte spce will be kept to minimum by only considering three ttributes: X distnce from the next pipe Y distnce from the next pipe Life of the bird A visul representtion of the first two ttributes re shown in Figure 2 exct vlues of the rewrd function re not s importnt s long s reinforcement is positive nd punishment is negtive. We weigh the benefit of survivl lower thn the punishment of deth, s the gol is to void deth. { 15, if bird live R t+1 = 1000, if bird ded The lerning rte determines to wht extent new informtion overrides old informtion. The following lerning ws chosen: 1 α t (s t, t ) = 1 + t k=0 1 {s k =s t, k = t} The discount fctor determines the importnce of future rewrds. This vlue is given below. γ = 1. These re the essentil definitions tht re needed to implement the Q-Lerning lgorithm. Now the Q-Lerning pseudocode tht we used in our implementtion is shown below. 1) Initilize Q mtrix 2) Repet ) Determine the ction,, in stte, s, bsed on Q mtrix b) Tke the ction,, nd observe the outcome stte, s, nd rewrd, r c) Updte Q mtrix bsed on Eqution 2 Fig. 2. The first two ttributes: X distnce & Y distnce Only integers re considered to ensure finite stte spce. The distnce in X direction is bounded by 0 from below nd by 300 from bove. The Y distnce rnges from 200 to 200. The stte of the bird is chrcterized by ded s 0 nd the live s 1. Thus the stte spce is defined s: X = 0, , 200 {0, 1}. At every point in time, there re two possible ctions - jump or don t jump. Thus the ction spce is defines s: U = {0, 1}. After the previous ction hs been pplied, the resulting stte is nlyzed. A rewrd function is invoked, which reinforces or punishes the previous ction. If the bird survived, the previous ction is seen s positive nd is being reinforced. However, if the bird died, we pply the punishment. The Q t+1 (s t, t ) = Q t (s t, t ) + α t (s t, t ) ( ) R t+1 + γ mx Q t (s t+1, ) Q t (s t, t ) IV. DISCUSSION When strting the Q-Lerning lgorithm, the Q mtrix is initilized to zero. The Q mtrix cn be seen s the brin of the rtificil intelligence. At the beginning, it does not hve ny knowledge bout its environment. When pplying Eqution 2 for the first time step, the mximum is tken between 0 nd 0. In other words: fced with the decision of jumping or not jumping, both ctions hve the sme Q vlues - tie occurs. We decided to choose the ction don t jump s tie breker becuse it is the more humn-like option. The bird ws expected to fll to the ground in the first couple of trils, but lern to void the ground firly quickly since flling to the ground is not good option. The Q mtrix described before is of size 300 x 400 x 2. Even though tht doesn t seem tht big, it is still totl of 240,000 different sttes. Tht mens tht every time the bird is positioned slightly different reltive to the pipe, it does not hve ny knowledge. However, the lgorithm still converges. It just tkes while to fill the Q mtrix. Tht is why the stte spce ws discretized further for sttes where the bird is fr wy from the initil pipe. The first intelligent behvior is the bird lerns tht flling to the ground is bd ide. Note tht the bird doesn t hve (2)

4 informtion bout wht ground is or even where it is locted. The only vilble informtion is the distnce to the next pipe nd the current life sttus. The next stge in the bird s lerning process is tht it will crsh into the lower pipe until it lerns tht this leds to negtive rewrd. With the tie breker being don t jump, it will fll whenever it encounters new stte. After getting feedbck, it will relize tht flling ws not good move in the prior sitution. If the bird is locted in the sme position reltive to pipe gin it will use this informtion nd jump insted. This propgtion continues leding to the bird eventully clering the first pipe. Another problem rouse: even fter flying through the gp the bird would eventully die. The bird lerned tht flying through the pipe will led to deth. Due to the bck propgtion, the Q mtrix hd negtive entries for sttes close to the gp. Following the lgorithm, the sttes bove the gp with Q vlues of zero would get explored. For ll the AI knows, there could be better option thn flying through the gp. It continues to go through ll other sttes trying to find n lterntive before relizing tht flying through the gp ws the best wy. It ws observed tht the speed of the bird mkes big difference. The gme engine works with grvity. The longer the bird is flling, the fster it gets. This influences the choice of the optiml policy. We could include the speed of the bird in the stte spce, but tht would dd nother dimension to the stte spce, which would slow down the lerning process tremously. Tht s why different pproch ws tken. The lerning rte ws lowered with regrd to how often certin stte ws visited before. The dynmic lerning rte tkes cre of both problems described bove. The AI lerns tht flying closely bove the pipe leds to positive benefits. Even if it does eventully crsh, it doesn t disregrd this informtion. The cse of the bird crshing becuse of missing informtion regrding speed is rther seldom. The djusted lerning rte mkes sure tht these improbble events do not hve too big of n impct on the Q mtrix. The bird successfully lerned how to ply the gme using Q- Lerning without ny prior knowledge of the gme dynmics. A high score of 169 nd 28,851 ws chieved on the originl nd simplified gme, respectively. Figure 3 shows the lerning rte of the bird by compring the number of trils nd their respective scores. Fig. 3. The score plotted with the number of trils of the AI The convergence of the lgorithm is ensured due to the choice of suitble α(t) nd ppropriteness of the stte spce since the discounted fctor γ = 1 ws chosen. We know tht Assumption 2 on α is stisfied due to: α t (s t, t ) = Likewise, = n= t k=0 1 {s k =s t, k = t} 1 n = α t (s t, t ) 2 1 = n 2 = 2 n=1 Since γ = 1 is used, it needs to stisfy the conditions of Theorem 4, in prticulr, prt b or c. We initilize the Q mtrix to 0, which is requirement for both. Next, we impose dditionl structure on the Q. We bound Q from bove by 10,000. Tht is, rewrd becomes 0 once the Q mtrix entry becomes sufficiently lrge. Also, it is known priori tht the gme is solvble. Tht is, there is n optiml policy tht will gurntee the bird s survivl for eternity. This is known by the construction of the gme. The code does not llow for sequence of pipes tht will ensure deth. Hence, we hve two bsorbing sttes. One being the deth of the bird, s the bird cnnot revive. The other is the collection of sttes tht mke the bird live forever. Since we bound the Q mtrix, we hve proper sttionry policy tht hs 0 rewrd nd ll other improper policies (those leding to deth) hve negtive infinite expected rewrd. Thus, we stisfy the conditions for prt c of Theorem 4 nd, by consequence, hve convergence of the Q mtrix. Although convergence is gurnteed, in prctice, one does not hve the luxury of running the lgorithm forever. Thus, the gretest critique of this lgorithm is tht convergence my tke long time, which is undesirble. For instnce, if the controller lerns good behvior, but sub-optiml, it will

5 continue to perform this ction even though better ctions exist until those ctions re explored. In this ppliction, the mjority of the sttes in the stte spce re undesirble. If the controller misses the smll window of desirble sttes, it might hve to explore the entire stte spce before returning to tht smll window which will gretly increse the time of convergence. V. POSSIBLE IMPROVEMENTS If given more time, mny ides could hve been implemented to improve the Q-Lerning lgorithm. Some of these ides include techniques for fster convergence nd methods to strengthen the survivbility rte of the bird. In the project, the grphics ws displying throughout the lerning process of the bird. This significntly slows down the lgorithm. If the grphics were to be removed from the code, the optiml policy cn be chieved in significntly reduced time frme. Only one bird ws used to find the optiml policy throughout the project. If multiple birds were simultneously lerning nd they collectively updted the mtrix Q, then this will gretly reduce the time needed to fill the Q mtrix. This will result in the birds converging to the optiml policy much quicker. When the bird dies, the gme restrts with the bird t the sme position every time. If rndom strting position were to be implemented, then the bird will be ble to explore the stte spce more efficiently. This my reduce the time it tkes for the bird to converge to the optiml policy. To improve the bird s survivbility, possible solution is to dd nother dimension to the stte spce. The bird s stte is highly depent on the velocity of the bird. If velocity is included in the stte spce, then the trnsition probbility would be deterministic. This would llow the bird to lern precisely wht the next stte is going to be given its current stte nd ction. After the bird hs converged to the optiml policy, it ws noticed tht there ws only one cse tht led to the deth of the bird. This cse ws when the height of the gp of the current pipe ws low to the ground while the height of the gp of the next pipe ws ner the ceiling. This cse did not give the bird enough room to mke it to the next pipe. A solution to this would be to include the position of the next pipe in the stte spce. This would llow the bird to pln hed which in turn would increse its survivbility rte. VI. CONCLUSION The big dvntge of the Q-Lerning lgorithm is tht it converges without knowledge of the system. However, the dt given to the lgorithm highly influences the run-time. The key to receiving meningful results in limited timefrme is the choice of n efficient stte spce. Following this observtion, pproprite spces nd prmeters were chosen. Even with short lerning time, the rtificil intelligence exceeded expecttions by performing better thn beginner humn plyer. The Q-Lerning lgorithm cn be improved upon by implementing the improvements outlined in Section V. APPENDIX The Mtlb code for the Q-L erning lgorithm is shown below. Note tht only the relevnt prt of the code is included. x m e t r i c = round ( Tubes. ScreenX... ( Tubes. FrontP ) Bird. ScreenPos ( 1 ) ); y m e t r i c = round (177 ( Tubes. VOffset... ( Tubes. FrontP ) 1) Bird. ScreenPos ( 2 ) ); %I f t h e x m e t r i c i s n e g t i v e %( pssed t h e p i p e ) l o o k t t h e n e x t p i p e i f x m e t r i c < 0 x m e t r i c = round ( Tubes. ScreenX ( mod... ( Tubes. FrontP, 3 ) + 1 )... Bird. ScreenPos ( 1 ) ); y m e t r i c = round ( ( Tubes. VOffset ( mod... ( Tubes. FrontP, 3 ) + 1 )... 1) Bird. ScreenPos (2)+5 10); %Find i n d i c e s f o r k =1: l e n g t h ( x ) i f x m e t r i c <x ( k ) x i n d =k ; brek ; f o r k =1: l e n g t h ( y ) i f y m e t r i c <y ( k ) y i n d =k ; brek ; x i n d o =1; f o r k =1: l e n g t h ( x ) i f x m e t r i c o l d <x ( k ) x i n d o =k ; brek ; y i n d o =1; f o r k =1: l e n g t h ( y ) i f y m e t r i c o l d <y ( k ) y i n d o =k ; brek ;

6 %C l c u l t e lph s t t e C o u n t ( x i n d o, y i n d o ) =... s t t e C o u n t ( x i n d o, y i n d o ) + 1 ; l p h = 1 / ( 1 + s t t e C o u n t ) ; c t i o n i n d e x = c t i o n +1; i f d e t h F l g == t r u e R = 1000; e l s e R = 1 5 ; %Q l e r n i n g l g o r i t h m Q temp =... Q( x i n d o, y i n d o, c t i o n i n d e x ) +... l p h (R mx(q( x ind, y ind, : ) )... Q( x i n d o, y i n d o, c t i o n i n d e x ) ) ; i f Q temp <= Q( x i n d o, y i n d o, c t i o n i n d e x )... = Q temp ; d e t h F l g = f l s e ; %Tke c t i o n bsed on Q m t r i x i f Q( x ind, y ind,1) <Q( x ind, y ind, 2 ) c t i o n =1; F l y K e y S t t u s = t r u e ; e l s e c t i o n =0; x m e t r i c o l d = x m e t r i c ; y m e t r i c o l d = y m e t r i c ; REFERENCES 1 C. J. Wtkins nd P. Dyn, Q-lerning, Mchine lerning, vol. 8, no. 3-4, pp , R. S. Sutton nd A. G. Brto, Reinforcement lerning: An introduction. MIT press, C. J. C. H. Wtkins, Lerning from delyed rewrds. PhD thesis, University of Cmbridge Englnd, R. Willims, Wht is flppy bird? the gme tking the pp store by storm, jn M. Berth, Everything you need to know bout your new fvorite cell phone gme, flppy bird, jn J. N. Tsitsiklis, Asynchronous stochstic pproximtion nd q- lerning, Mchine Lerning, vol. 16, no. 3, pp , 1994.

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