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1506.02488#7	On the Fuzzy Stability of an Affine Functional Equation	The result of Rassias has inï¬ uenced the development of what is now called the Hyers- In 1994, a generalization of Ulam-Rassias stability theory for functional equations. Rassias theorem was obtained by Gavruta [11] by replacing the bound Îµ(kxkp + kykp) by a general control function Ï (x, y). During the last decades, the stability problems of several functional equations have been extensively investigated by a number of authors (c.f. [12], [13], [14], [17] and [20]â [26] etc.). In 1982-1989, J.M.Rassias [15, 16] replaced the sum appeared in right hand side of the equation (1.1) by the product of powers of norms. In fact, he proved the following theorem. Theorem 1.1 Let f :	1506.02488#6	1506.02488#8	1506.02488	[ "1506.02488" ]
1506.02488#8	On the Fuzzy Stability of an Affine Functional Equation	E1 â E2 be a mapping from a normed vector space E1 into Banach space E2 subject to the inequality k f (x + y) â f (x) â f (y) k6 Îµ(k x kpk y kp) (1.3) for all x, y â E1, where Îµ and p are constants with Îµ > 0 and 0 6 p < 1 limit f (2nx) 2n (1.4) L(x) = lim nâ	1506.02488#7	1506.02488#9	1506.02488	[ "1506.02488" ]
1506.02488#9	On the Fuzzy Stability of an Affine Functional Equation	â exists for all x â E1, and L : E1 â E2 is the unique additive mapping which satisï¬ es k f (x) â L(x) k6 Îµ 2 â 22p k x k2p (1.5) for all x â E1. If p > 1 2 the inequality (1.3) holds for x, y â E1 and the limit A(x) = lim nâ â 2nf x 2n (1.6) exists for all x â E1 and A : E1 â E2 is the unique additive mapping which satisï¬	1506.02488#8	1506.02488#10	1506.02488	[ "1506.02488" ]
1506.02488#10	On the Fuzzy Stability of an Affine Functional Equation	es k f (x) â A(x) k6 Îµ 22p â 2 k x k2p (â x â E1) (1.7) 2 (1.2) Recently, Cadariu et al [19] studied the generalized Hyers-Ulam stability by using the direct method as well as the ï¬ xed point method for the aï¬ ne type functional equation f (2x + y) + f (x + 2y) + f (x) + f (y) = 4f (x + y), for all x, y â G. (1.8) In the present paper, we obtain the general solution of the following functional equation f (3x + y + z) + f (x + 3y + z) + f (x + y + 3z) + f (x) + f (y) + f (z) = 6f (x + y + z). (1.9) where f : X â Y , X and Y are normed spaces.	1506.02488#9	1506.02488#11	1506.02488	[ "1506.02488" ]
1506.02488#11	On the Fuzzy Stability of an Affine Functional Equation	Then, we establish the fuzzy Hyers- Ulam-Rassias stability of the above functional equation. # 2. Preliminary Notes Before we proceed to the main results, we will introduce a deï¬ nition and some ex- amples to illustrate the idea of fuzzy norm. Deï¬ nition 2.1 Let X be a real linear space. A mapping N : X Ã R â [0, 1] (the so-called fuzzy subset) is said to be a f uzzy norm on X if for all x, y â X and all s, t â R, (N1) N(x, t) = 0 for t 6 0; (N2) x = 0 if and only if N(x, t) = 1 for all t > 0; (N3) N(cx, t) = N(x, t/ \| c \|) if c 6= 0; (N4) N(x + y, t + s) > min{N(x, t), N(y, s)}; (N5) N(x, .) is a non-decreasing function on R and lim tâ â (N6) for x 6= 0, N(x, .) is continuous on R. The pair (X, N) is called a f uzzy normed linear space. One may regard N(x, t) as the truth value of the statement that the norm of x is less than or equal to the real number t. Example 2.2 Let (X, k.k) be a normed linear space. One can be easily verify that for each p > 0, Np(x, t) = t t+pkxk 0 ( t > 0, x â X t 6 0, x â X is a fuzzy norm on X. Example 2.3 Let (X, k.k) be a normed linear space. The mapping N : X Ã R â [0, 1] by N(x, t) = t2â kxk2 t2+kxk2 0 ( t > kxk t 6 kxk	1506.02488#10	1506.02488#12	1506.02488	[ "1506.02488" ]
1506.02488#12	On the Fuzzy Stability of an Affine Functional Equation	3 is a fuzzy norm on X. Deï¬ nition 2.4 Let (X, N) be a fuzzy normed linear space. A sequence {xn} in X N(xn â x, t) = 1 for all is said to be convergent if there exists an x â X such that lim nâ â t > 0. In this case, x is called the limit of the sequence {xn} and we denote it by N â lim nâ â N(xn â x, t) = x.	1506.02488#11	1506.02488#13	1506.02488	[ "1506.02488" ]
1506.02488#13	On the Fuzzy Stability of an Affine Functional Equation	Deï¬ nition 2.5 Let (X, N) be a fuzzy normed linear space. A sequence {xn} in X is said to be Cauchy if for each Îµ > 0 and each Î´ > 0 there exists an n0 â N such that N(xm â xn, Î´) > 1 â Îµ (m, n > n0). It is well known that every convergent sequence in a fuzzy normed linear space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.	1506.02488#12	1506.02488#14	1506.02488	[ "1506.02488" ]
1506.02488#14	On the Fuzzy Stability of an Affine Functional Equation	The remaining part of the paper is organized as follows: We discuss the general solution of functional equation (1.9) in Section 3. Section 4 is devoted to investigate the non- uniform version of stability of functional equation (1.9) in fuzzy normed spaces and in section (5), we show under suitable conditions that in fuzzy normed spaces functional equation (1.9) is stable uniformly. Now we proceed to ï¬ nd the general solution of the functional equation (1.9) 3. Solution of the Functional Equation (1.9) Theorem 3.1 A mapping f :	1506.02488#13	1506.02488#15	1506.02488	[ "1506.02488" ]
1506.02488#15	On the Fuzzy Stability of an Affine Functional Equation	X â Y , X and Y are normed spaces, is a solution of the functional equation (1.9) if and only if it is an aï¬ ne mapping (i.e., it is the sum between a constant and an additive function). Proof. We can easily seen that any aï¬ ne function f is a solution of the equation (1.9). Conversely, we have two cases: Case 1 : f (0) = 0. If we take y = z = â x in (1.9), we obtain 2f (x) + 2f (â 3x) + 2f (â x) = 6f (â x), for all x â X. (3.1) Again replacing putting y = z = 0 in (1.9), we obtain f (3x) = 3f (x), for all x â X. (3.2) By (3.1) and (3.2), we have f (â x) = â f (x), for all x â X. It results that f is an odd mapping. Replace z by â y in (1.9), we get f (x + 2y) + f (x â 2y) = 2f (x) (3.3)	1506.02488#14	1506.02488#16	1506.02488	[ "1506.02488" ]
1506.02488#16	On the Fuzzy Stability of an Affine Functional Equation	4 If we replace x and y by u+v # 2 and uâ v 4 , respectively, in (3.3) and using (3.2), we have utvu 2 u-v 4 If we replace x and y by and , respectively, in (3.3) and using (3.2), we have f (u + v) = f (u) + f (v), for all u, v â X. So, f is an additive mapping. Case 2 : General case. Let us consider the function g(x) := f (x) â f (0). It is clear that g(0) = 0 and f (x) = g(x) + f (0). Replacing f by g in (1.9), it results g(3x + y + z) + g(x + 3y + z) + g(x + y + 3z) + g(x) + g(y) + g(z) = 6g(x + y + z). for all x, y, z â X. Taking in account that g(0) = 0, from Case 1, we obtain that g is an additive mapping, hence f (x) = g(x) + f (0) is an aï¬ ne function. This completes the proof. For a given mapping f :	1506.02488#15	1506.02488#17	1506.02488	[ "1506.02488" ]
1506.02488#17	On the Fuzzy Stability of an Affine Functional Equation	X â Y , let us denote Df (x, y, z) = f (3x + y + z) + f (x + 3y + z) + f (x + y + 3z) + f (x) + f (y) + f (z) â 6f (x + y + z) # 4. Fuzzy Hyers-Ulam-Rassias Stability: non-uniform version Theorem 4.1 Let X be a linear space and (Z, N â ²) a fuzzy normed space.	1506.02488#16	1506.02488#18	1506.02488	[ "1506.02488" ]
1506.02488#18	On the Fuzzy Stability of an Affine Functional Equation	Let Ï : X 3 â Z be a mapping such that for some Î± 6= 0 with 0 < Î± < 3 N â ²(Ï (3x, 0, 0), t) > N â ²(Î±Ï (x, 0, 0), t) (4.1) for all x â X, t > 0 and lim nâ â N â ²(Ï (3nx, 3ny, 3nz), 3nt) = 1, for all x, y, z â X and all t > 0. Suppose that (Y, N) be a fuzzy Banach space and an odd mapping f : X â Y satisï¬	1506.02488#17	1506.02488#19	1506.02488	[ "1506.02488" ]
1506.02488#19	On the Fuzzy Stability of an Affine Functional Equation	es the inequality N(Df (x, y, z), t) > N â ²(Ï (x, y, z), t) (4.2) for all x, y, z â X and all t > 0. Then the limit A(x) = N â lim nâ â f (3nx) 3n exists for all x â X and the mapping A : X â Y is the unique aï¬ ne mapping satisfying N(f (x) â A(x) â f (0), t) > N â ²(Ï (x, 0, 0), (3 â Î±)t) (4.3)	1506.02488#18	1506.02488#20	1506.02488	[ "1506.02488" ]
1506.02488#20	On the Fuzzy Stability of an Affine Functional Equation	5 for all x â X and all t > 0. Proof. Letting y = z = 0 in (4.2), we get N(f (3x) â 3f (x) + 2f (0), t) > N â ²(Ï (x, 0, 0), t) (4.4) for all x â X and all t > 0. If we deï¬ ne the mapping g : X â Y such that g(x) := f (x) â f (0) for all x â X. Indeed g(0) = 0. Then (4.4) implies N(g(3x) â 3g(x), t) > N â ²(Ï (x, 0, 0), t) Replacing x by 3nx in the last inequality, we obtain N(g(3"tta) â 3g(3"2), #) > Nâ (y(3"x, 0, 0), t) (â g(8"a ) t Jane (0.0.0).â ) 3ntl 3n > 9n4+1 g(3"tx) g(3"r) at n (2 SS) 2 NC. 0.0).1) (4.5) # for alla â ¬ X oore) nâ 1 g(3j+1x) 3j+1 â g(3j x) 3n â g(x) = 3j and (4.5) # j=0 P # that n-1 . n-1 . g(3"x) alt g(3!1x) â g(34x) alt v( ~ g(x), >) 3741) =N ae 3741 37Â° 37H) j=0 j-0 n-1 ; ; g(3ittxr) â g(3â x) alt > min Ute (4 arn 31 =} > N"(:p(@, 0,0), t). ce ceeeeeeeeeeseeseeseeeeeeeeeeeeeens (4.6) for all x â X and all t > 0. Replacing x by 3mx in (4.6), we get	1506.02488#19	1506.02488#21	1506.02488	[ "1506.02488" ]
1506.02488#21	On the Fuzzy Stability of an Affine Functional Equation	n-1 ; g Brtmy wero alt (> sta) > N' (ole, 0.0), 3rtm j=0 t qm # j=0 X and so : g(r) g(3â ¢ax) "LA* alt v( = a , Dd, azar) ) > N'(v(, 0,0), 1) j=m # j=m X > N â ² g(3"t"a g(3"x t (> â - d 2 N' (20,0), ae â (4.7) 3741 # j=m P 6 â ( Î± 3 )j < â , the Cauchy for all x â X, t > 0 and m, n > 0. Since 0 < Î± < 3 and # j=0 P criterion for convergence and (N5) imply that { g(3nx) 3n } is a Cauchy sequence in (Y, N). Since (Y, N) is a fuzzy Banach space, this sequence converges to some point A(x) â Y . f (3nx) Hence, we can deï¬ ne a mapping A : X â Y by A(x) = N â lim 3n nâ â g(3nx) 3n = N â lim nâ â for all x â X, namely. Since f is odd, A is odd. Letting m = 0 in (4.7), we get	1506.02488#20	1506.02488#22	1506.02488	[ "1506.02488" ]
1506.02488#22	On the Fuzzy Stability of an Affine Functional Equation	(oe _ ule).t)> Nâ (ote 0,0), a) # j=0 P Taking the limit as n â â and using (N6), we get N(A(a) â g(x), t) > W'(e(0.0,0), = t ) ad 3741 j=0 = N'(y(2, 0,0), (3 â a)t) N(f(2) ~~ A(z) ~~ f(0), t) 2 N'(y(@, 0, 0), (3 ~~ a)t) for all x â X and all t > 0. Now we claim that A is aï¬	1506.02488#21	1506.02488#23	1506.02488	[ "1506.02488" ]
1506.02488#23	On the Fuzzy Stability of an Affine Functional Equation	ne. Replacing x, y, z by 3nx, 3ny, 3nz, respectively, in (4.2), we get 1 N (FOr 3â y, 3"), â ) > N'(y(3"2, 3â y, 3"z), 3"t) # for all x,y,z â ¬ X and allt > 0. Since lim nâ â N â ²(Ï (3nx, 3ny, 3nz), 3nt) = 1, A satisï¬ es the functional equation (1.9). Hence A is aï¬	1506.02488#22	1506.02488#24	1506.02488	[ "1506.02488" ]
1506.02488#24	On the Fuzzy Stability of an Affine Functional Equation	ne. To prove the uniqueness of A, let Aâ ² : X â Y be another aï¬ ne mapping satisfying (4.3). Fix x â X. Clearly A(3nx) = 3nA(x) and Aâ ²(3nx) = 3nAâ ²(x) for all x â X and all n â N. It follows from (4.3) that N(A(2) â Aâ (2),t) = (= AiS"a) â ) 3" 3â > mind N A(3"r) â g(3 t) t _N g(3"r) = AN(3 a) t 3â 3" 2 3â 3â 2 3"(3 â a)t > w'(o(3%2.0.0), ( ; a) ) 3"(3 â a)t > N'\| v(x, 0,0), â â â â (02.0.0), ar ) 7 3n(3â Î±) for all x â X and all t > 0.	1506.02488#23	1506.02488#25	1506.02488	[ "1506.02488" ]
1506.02488#25	On the Fuzzy Stability of an Affine Functional Equation	Since lim nâ â 2Î±n = â , we obtain lim nâ â N â ² Ï (x, 0, 0), 3n(3 â Î±)t 2Î±n = 1. Thus N(A(x) â Aâ ²(x), t) = 1 for all x â X and all t > 0, and so A(x) = Aâ ²(x). This completes the proof. 5. Fuzzy Hyers-Ulam-Rassias Stability: uniform version Theorem 5.1 Let X be a linear space and (Y, N) be a fuzzy Banach space.	1506.02488#24	1506.02488#26	1506.02488	[ "1506.02488" ]
1506.02488#26	On the Fuzzy Stability of an Affine Functional Equation	Let Ï : X 3 â [0, â ) be a function such that â Ë Ï (x, y, z) = 1 3n Ï (3nx, 3ny, 3nz) < â (5.1) # n=0 X for all x, y, z â X. Let f : X â Y be a uniformly approximately aï¬ ne mapping with respect to Ï in the sense that lim tâ â N(Df (x, y, z), tÏ (x, y, z)) = 1 (5.2) uniformly on X 3.	1506.02488#25	1506.02488#27	1506.02488	[ "1506.02488" ]
1506.02488#27	On the Fuzzy Stability of an Affine Functional Equation	Then A(x) := N â lim nâ â f (3nx) 3n for all x â X exists and deï¬ nes an aï¬ ne mapping A : X â Y such that if for some Î± > 0, Î´ > 0 N(Df (x, y, z), Î´Ï (x, y, z)) > Î± (5.3) for all x, y, z â X, then N(f (x) â A(x) â f (0), Î´ 3 Ë Ï (0, 0, , x)) > Î± for all x â X. Proof. Let Îµ > 0, by (5.2), we can ï¬ nd t0 > 0 such that N(Df (x, y, z), tÏ (x, y, z)) > 1 â Îµ (5.4) for all x, y, z â X and all t > t0.	1506.02488#26	1506.02488#28	1506.02488	[ "1506.02488" ]
1506.02488#28	On the Fuzzy Stability of an Affine Functional Equation	Deï¬ ne g : X â Y such that g(x) := f (x) â f (0). It is clear that g(0) = 0 and f (x) = g(x) + f (0). Now (5.4) implies that N(Dg(x, y, z), tÏ (x, y, z)) > 1 â Îµ (5.5) for all x, y, z â X and all t > t0. By induction on n, we will show that n-1 x(a" = 3"g(x),t 5Â° 3-1 0(0, 0, 3"0)) >l-eÂ« (5.6) m=0 # m=0 X	1506.02488#27	1506.02488#29	1506.02488	[ "1506.02488" ]
1506.02488#29	On the Fuzzy Stability of an Affine Functional Equation	8 for all x â X, all t > t0 and n â N. Putting x = y = 0 and z = x in (5.5), we get (5.6) for n = 1. Let (5.6) holds for some positive integers n. Then n N(g(3n+1x) â 3n+1g(x), t 3nâ mÏ (0, 0, 3mx)) # m=0 X > min{N(g(3n+1x) â 3g(3nx), tÏ (0, 0, 3nx)), n N(3g(3nx) â 3n+1g(x), t 3(nâ m)Ï (0, 0, 3mx))} m=0 X > min{1 â Îµ, 1 â Îµ} = 1 â Îµ.	1506.02488#28	1506.02488#30	1506.02488	[ "1506.02488" ]
1506.02488#30	On the Fuzzy Stability of an Affine Functional Equation	This completes the induction argument. Let t = t0 and put n = p. Then by replacing x with 3nx in (5.6), we obtain pâ 1 N(g(3n+px) â 3pg(3nx), t0 3pâ mâ 1Ï (0, 0, 3n+mx)) > 1 â Îµ # m=0 X g(3"*Pa) (n+m-+1) nm, (oo ee a Lise y(0,0,3"t"xr) )>1-â e (5.7) # m=0 X for all integers n > 0, p > 0. The convergence of (5.1) and the equation pâ 1 3â (n+m+1)Ï (0, 0, 3n+mx)) = 1 k n+pâ 1 3â mÏ (0, 0, 3mx) # m=0 X # m=n X guarantees that for given Î´ > 0, there exists n0 â N such that t0 3 n+pâ 1 3â mÏ (0, 0, 3mx) < Î´ # m=n X for all n > n0 and p > 0. It follows from (5.7) that n+p, Now n+p, x(& zt) 9(3 e) ns n(n a . to x3 (n+m-+1) y(0, 0, grtmy, )> lâ e 3n+p 3n 3n+p m=0 (5.8) m=0 (5.8) for each n > no and all p > 0.	1506.02488#29	1506.02488#31	1506.02488	[ "1506.02488" ]
1506.02488#31	On the Fuzzy Stability of an Affine Functional Equation	Hence {oe 3 Hy is a Cauchy sequence in Y. Since Y isa fuzzy Banach space, this sequence converges to ome A(x) â ¬ Y. Hence we can define a mapping A: X + Y by A(a) :-= N â jim 4 Â§ = 7 â Nâ lim ae for alla â ¬ X nâ 0o namely. For each t > 0 and x â ¬ X lim nâ â N A(x) â f (3nx) 3n , t = 1.	1506.02488#30	1506.02488#32	1506.02488	[ "1506.02488" ]
1506.02488#32	On the Fuzzy Stability of an Affine Functional Equation	9 Now, let x, y, z â X. Fix t > 0 and 0 < Îµ < 1. Since lim nâ â is some n1 > n0 such that 3n Ï (3nx, 3ny, 3nz) = 0, there 1 N(DA(x, y, z), t) > min N A(3x + y + z) â f (3n(3x + y + z)) 3n , t 8 N(DA(za,y, z),t) > min{ W (A(G +yt2z) :) ; 3â 8 n (Ae + 3y +2) [B"(e ate) ). x (A +y+3z) f3"w = + 32)), ), x (Ale) - pero. ), x (AW) - AY) 2) (4) - ame), n(Aety +2) fONe tye) 5): t N(DAG"e. 3"y,3"2), =) The ï¬	1506.02488#31	1506.02488#33	1506.02488	[ "1506.02488" ]
1506.02488#33	On the Fuzzy Stability of an Affine Functional Equation	rst 7 terms on the right hand side of the above inequality tend to 1 as n â â and the last term is greater than N(Df (3nx, 3ny, 3nz), t0Ï (3nx, 3ny, 3nz)), i.e., by (5.4), greater than or equal to 1â Îµ. Thus N(DA(x, y, z), t) > 1â Îµ for all t > 0 and 0 < Îµ < 1. It follows that N(DA(x, y, z), t) = 1 for all t > 0 and by (N2), we have DA(x, y, z) = 1, i.e., A(3x + y + z) + A(x + 3y + z) + A(x + y + 3z) + A(x) + A(y) + A(z) = 6A(x + y + z) To end the proof, let for some positive Î± and Î´, (5.3) holds.	1506.02488#32	1506.02488#34	1506.02488	[ "1506.02488" ]
1506.02488#34	On the Fuzzy Stability of an Affine Functional Equation	Let nâ 1 Ï n(x, y, z) := 3â (m+1)Ï (3mx, 3my, 3mz) # m=0 X for all x, y, z â X. Let x â X. By a similar discussion as in the begining of the proof, we can obtain from (5.3) nâ 1 N(g(3nx) â 3ng(x), Î´ 3(nâ mâ 1)Ï n(0, 0, 3mx)) > Î´ (5.9) # m=0 X for all n â N. Let s > 0. We have g(3"x) N(g(a)â A(2), 6Â¢n(0, 0, x) +s) > min{ Â¥ (g(a) 25 Bn (5.10) -5Â¢n(0,0,2)), (Sa(a, Combining (5.8), (5.9) and the fact that	1506.02488#33	1506.02488#35	1506.02488	[ "1506.02488" ]
1506.02488#35	On the Fuzzy Stability of an Affine Functional Equation	lim nâ â N g(3nx) 3n â A(x), s = lim nâ â N f (3nx) 3n â A(x), s = 1, 10 5) f we obtain that N(g(x) â A(x), Î´Ï n(0, 0, x) + s) > Î± for large enough n. By the (upper semi) continuity of real function N(g(x) â A(x), .), we obtain that N g(x) â A(x), Î´ 3 Ë Ï (0, 0, x) + s > Î±. Taking the limit as s â 0, we conclude that N g(x) â A(x), Î´ 3 Ë Ï (0, 0, x) > Î± N f (x) â A(x) â f (0), Î´ 3 Ë Ï (0, 0, x) > Î±. This completes the proof. Theorem 5.2 Let X be a linear space and (Y, N) be a fuzzy Banach space.	1506.02488#34	1506.02488#36	1506.02488	[ "1506.02488" ]
1506.02488#36	On the Fuzzy Stability of an Affine Functional Equation	Let Ï : X 3 â [0, â ) be a function satisfying (5.1). Let f : X â Y be a uniformly ap- proximately aï¬ ne mapping with respect to Ï . Then there is a unique aï¬ ne mapping A : X â Y such that lim tâ â N(f (x) â A(x) â f (0), t Ë Ï (0, 0, x)) = 1 (5.11) uniformly on X.	1506.02488#35	1506.02488#37	1506.02488	[ "1506.02488" ]
1506.02488#37	On the Fuzzy Stability of an Affine Functional Equation	Proof. The existence of uniform limit (5.11) immediately follows from Theorem 4.5. It remains to prove the uniqueness assertion. Let Aâ ² be another aï¬ ne mapping satisfying (5.11). Fix c > 0. Given Îµ > 0, by (5.11) for A and Aâ ², we can ï¬ nd some t0 > 0 such that N(g(x) â A(x), N(g(x) â Aâ ²(x), t 2 t 2 Ë Ï (0, 0, x)) > 1 â Îµ, Ë Ï (0, 0, x)) > 1 â Îµ for all x â X and all t > t0. Fix some x â X and ï¬ nd some integer n0 such that t0 # m=n P	1506.02488#36	1506.02488#38	1506.02488	[ "1506.02488" ]
1506.02488#38	On the Fuzzy Stability of an Affine Functional Equation	â â 1 3n 3â (mâ n)Ï (0, 0, 3mâ n(3nx)) 3â mÏ (0, 0, 3mx) = m=n X â m=n X 1 3n 1 3j Ï (0, 0, 3j(3nx)) = j=0 X 1 3n Ë Ï (0, 0, 3nx) =	1506.02488#37	1506.02488#39	1506.02488	[ "1506.02488" ]
1506.02488#39	On the Fuzzy Stability of an Affine Functional Equation	11 We have g(3nx) N(Aâ ²(x) â A(x), c) > min 3n â A(x), N c 2 , N Aâ ²(x) â g(3nx) 3n , c 2 â A(a),0) > min (4 vo ~ Ale), <), x(4@) _Â§ ee, ~ i ( age Aa 5), (pea) > ming Â¥ (93%) â A(3" x), 3"to > 3-0, 0, 0), m=n x(4'G%) â g(3"x), 3"to > k-â ¢ (0,0, sx) m=n = min Â¥ (43) â A(3â 2), toB(0, 0, 32), x (4G â g(3"x), top(0, 0, 3"x)) > 1 â Îµ. It follows that N(Aâ ²(x) â A(x), c) = 1, for all c > 0. Thus A(x) = Aâ ²(x) for all x â X.	1506.02488#38	1506.02488#40	1506.02488	[ "1506.02488" ]
1506.02488#40	On the Fuzzy Stability of an Affine Functional Equation	This completes the proof. Considering the control function Ï (x, y, z) = Îµ(kxkp + kykp + kzkp) for some Îµ > 0, we obtain the following: Corollary 5.3 Let X be a normed linear space, let (Y, N) be a fuzzy Banach space, let Îµ > 0, and let 0 6 p < 1. Suppose that f : X â Y is a function such that lim nâ â	1506.02488#39	1506.02488#41	1506.02488	[ "1506.02488" ]
1506.02488#41	On the Fuzzy Stability of an Affine Functional Equation	N(Df (x, y, z), tÎµ(kxkp + kykp + kzkp)) = 1 uniformly on X 3. Then there is a unique aï¬ ne mapping A : X â Y such that lim tâ â N f (x) â A(x) â f (0), Îµt31â pkxkp 31â p â 1 = 1 uniformly on X. 12 # References [1] L.A.	1506.02488#40	1506.02488#42	1506.02488	[ "1506.02488" ]
1506.02488#42	On the Fuzzy Stability of an Affine Functional Equation	Zadeh, Fuzzy sets, Information and Control 8 (1965) 338-353. [2] A.K. Katsaras, Fuzzy topological vector spaces II, Fuzzy Sets Syst., 12(1984), 143-154. [3] C. Wu and J. Fang, Fuzzy generalization of Kolomogoroï¬ s theorem, J.Harbin Inst. Technol., 1(1984), 1-7. [4] R. Biswas, Fuzzy inner product space and fuzzy norm functions, Inform. Sci., 53(1991), 185-190. [5] C. Felbin, Finite dimensional fuzzy normed space, Fuzzy Sets Syst., 48(1992), 239-248. [6] T. Bag and S.K.	1506.02488#41	1506.02488#43	1506.02488	[ "1506.02488" ]
1506.02488#43	On the Fuzzy Stability of an Affine Functional Equation	Samanta, Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math., 11:3(2003), 687-705. [7] S.C. Cheng and J.N. Mordeson, Fuzzy linear operator and fuzzy normed linear spaces, Bull. Calcuta Math. Soc., 86(1994), 429-436. [8] S.M. Ulam, Problems in Modern Mathematics, Science ed., John Wiley & Sons: New York; 1940 (Chapter VI, Some Questions in Analysis: Section 1, Stability). [9] D.H. Hyers, On the stability of the linear functional equation, Proc.	1506.02488#42	1506.02488#44	1506.02488	[ "1506.02488" ]
1506.02488#44	On the Fuzzy Stability of an Affine Functional Equation	Natl. Acad. Sci., 27(1941) 222â 224. [10] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300. [11] P. Gavruta, A generalization of the HyersUlamRassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994) 431436. [12] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientiï¬	1506.02488#43	1506.02488#45	1506.02488	[ "1506.02488" ]
1506.02488#45	On the Fuzzy Stability of an Affine Functional Equation	c Publishing Co., Inc., River Edge, NJ, 2002. [13] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in Sev- eral Variables, BirkhÂ¨auser, Basel; 1998. [14] P. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009. [15] J.M. Rassias, On approximation of approximately linear mappings by linear map- ping, J.Funct. Anal., 46:1(1982), 126-130. [16] J.M. Rassias, On approximation of approximately linear mappings by linear map- pings, Bull.Sci. Math. (2), 108:4(1984), 445-446.	1506.02488#44	1506.02488#46	1506.02488	[ "1506.02488" ]
1506.02488#46	On the Fuzzy Stability of an Affine Functional Equation	13 [17] M. Mursaleen, Khursheed J. Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci. 7, No. 5, 1685-1692 (2013). [18] S. Javadi, J. M. Rassias, Stability of General Cubic Mapping in Fuzzy Normed Spaces, An. SÂ¸t. Univ.	1506.02488#45	1506.02488#47	1506.02488	[ "1506.02488" ]
1506.02488#47	On the Fuzzy Stability of an Affine Functional Equation	Ovidius ConstantÂ¸a, Vol. 20(1), 2012, 129-150. [19] L.Cadariu, L. Gavruta, P. Gavruta, On the stability of an aï¬ ne functional equa- tion, J. Nonlinear Sci. Appl, 6(2013) 60-67. [20] S. A. Mohiuddine, Stability of Jensen functional equation in intuitionistic fuzzy normed space, Chaos, Solitons & Fract., 42 (2009) 2989â 2996. [21] S. A.	1506.02488#46	1506.02488#48	1506.02488	[ "1506.02488" ]
1506.02488#48	On the Fuzzy Stability of an Affine Functional Equation	Mohiuddine and M.A. Alghamdi, Stability of functional equation obtained through a ï¬ xed-point alternative in intuitionistic fuzzy normed spaces, Adv. Dif- ference Equ. 2012, 2012:141. [22] S. A. Mohiuddine and H. SÂ¸evli, Stability of Pexiderized quadratic functional equa- tion in intuitionistic fuzzy normed space, J. Comput. Appl. Math., 235 (2011) 2137â 2146. [23] M. Mursaleen and K. J.	1506.02488#47	1506.02488#49	1506.02488	[ "1506.02488" ]
1506.02488#49	On the Fuzzy Stability of an Affine Functional Equation	Ansari, Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation, Appl. Math. Inf. Sci., 7(5) (2013) 1685â 1692. [24] M. Mursaleen and S. A. Mohiuddine, On stability of a cubic functional equation in intuitionistic fuzzy normed spaces, Chaos, Solitons Fract. 42 (2009) 2997â 3005. [25] S. A. Mohiuddine and A. Alotaibi, Fuzzy stability of a cubic functional equation via ï¬	1506.02488#48	1506.02488#50	1506.02488	[ "1506.02488" ]
1506.02488#50	On the Fuzzy Stability of an Affine Functional Equation	xed point technique, Adv. Diï¬ erence Equ. 2012, 2012:48 [26] S. A. Mohiuddine and M. Cancan, H. SÂ¸evli, Intuitionistic fuzzy stability of a Jensen functional equation via ï¬ xed point technique, Math. Comput. Modelling, 54 (2011) 2403â 2409. 14	1506.02488#49		1506.02488	[ "1506.02488" ]
1505.00853#0	Empirical Evaluation of Rectified Activations in Convolutional Network	5 1 0 2 v o N 7 2 ] G L . s c [ 2 v 3 5 8 0 0 . 5 0 5 1 : v i X r a # Empirical Evaluation of Rectiï¬ ed Activations in Convolution Network # Bing Xu University of Alberta [email protected] # Naiyan Wang Hong Kong University of Science and Technology [email protected] # Tianqi Chen University of Washington [email protected] # Mu Li Carnegie Mellon University [email protected] # Abstract In this paper we investigate the performance of diï¬ erent types of rectiï¬ ed activation func- tions in convolutional neural network: stan- dard rectiï¬ ed linear unit (ReLU), leaky rec- tiï¬ ed linear unit (Leaky ReLU), parametric rectiï¬ ed linear unit (PReLU) and a new ran- domized leaky rectiï¬ ed linear units (RReLU). We evaluate these activation function on standard image classiï¬ cation task. Our ex- periments suggest that incorporating a non- zero slope for negative part in rectiï¬ ed acti- vation units could consistently improve the results. Thus our ï¬ ndings are negative on the common belief that sparsity is the key of good performance in ReLU. Moreover, on small scale dataset, using deterministic neg- ative slope or learning it are both prone to overï¬ tting. They are not as eï¬ ective as us- ing their randomized counterpart. By us- ing RReLU, we achieved 75.68% accuracy on CIFAR-100 test set without multiple test or ensemble. et al., 2014), object detection(Girshick et al., 2014) and tracking(Wang et al., 2015). Despite its depth, one of the key characteristics of modern deep learn- ing system is to use non-saturated activation function (e.g. ReLU) to replace its saturated counterpart (e.g. sigmoid, tanh). The advantage of using non-saturated activation function lies in two aspects: The ï¬ rst is to solve the so called â exploding/vanishing gradientâ . The second is to accelerate the convergence speed. In all of these non-saturated activation functions, the most notable one is rectiï¬		1505.00853#1	1505.00853	[ "1502.03167" ]
1505.00853#1	Empirical Evaluation of Rectified Activations in Convolutional Network	ed linear unit (ReLU) (Nair & Hinton, 2010; Sun et al., 2014). Brieï¬ y speaking, it is a piecewise linear function which prunes the nega- tive part to zero, and retains the positive part. It has a desirable property that the activations are sparse af- ter passing ReLU. It is commonly believed that the superior performance of ReLU comes from the spar- In this sity (Glorot et al., 2011; Sun et al., 2014). paper, we want to ask two questions: First, is spar- sity the most important factor for a good performance? Second, can we design better non-saturated activation functions that could beat ReLU? # 1. Introduction	1505.00853#0	1505.00853#2	1505.00853	[ "1502.03167" ]
1505.00853#2	Empirical Evaluation of Rectified Activations in Convolutional Network	Convolutional neural network (CNN) has made great success in various computer vision tasks, such as im- age classiï¬ cation (Krizhevsky et al., 2012; Szegedy We consider a broader class of activation functions, namely the rectiï¬ ed unit family. In particular, we are interested in the leaky ReLU and its variants. In con- trast to ReLU, in which the negative part is totally dropped, leaky ReLU assigns a noon-zero slope to it.	1505.00853#1	1505.00853#3	1505.00853	[ "1502.03167" ]
1505.00853#3	Empirical Evaluation of Rectified Activations in Convolutional Network	The ï¬ rst variant is called parametric rectiï¬ ed linear unit (PReLU) (He et al., 2015). In PReLU, the slopes of negative part are learned form data rather than pre- deï¬ ned. The authors claimed that PReLU is the key factor of surpassing human-level performance on Im- ageNet classiï¬ cation (Russakovsky et al., 2015) task. Empirical Evaluation of Rectiï¬ ed Activations in Convolutional Network The second variant is called randomized rectiï¬ ed lin- ear unit (RReLU). In RReLU, the slopes of negative parts are randomized in a given range in the training, and then ï¬ xed in the testing. In a recent Kaggle Na- tional Data Science Bowl (NDSB) competition1, it is reported that RReLU could reduce overï¬ tting due to its randomized nature. In this paper, we empirically evaluate these four kinds of activation functions. Based on our experiment, we conclude on small dataset, Leaky ReLU and its vari- ants are consistently better than ReLU in convolu- tional neural networks. RReLU is favorable due to its randomness in training which reduces the risk of overï¬	1505.00853#2	1505.00853#4	1505.00853	[ "1502.03167" ]
1505.00853#4	Empirical Evaluation of Rectified Activations in Convolutional Network	tting. While in case of large dataset, more inves- tigation should be done in future. # 2. Rectiï¬ ed Units In this section, we introduce the four kinds of rectiï¬ ed units: rectiï¬ ed linear (ReLU), leaky rectiï¬ ed linear (Leaky ReLU), parametric rectiï¬ ed linear (PReLU) and randomized rectiï¬ ed linear (RReLU). We illus- trate them in Fig.1 for comparisons. In the sequel, we use xji to denote the input of ith channel in jth example , and yji to denote the corresponding output after passing the activation function. In the following subsections, we introduce each rectiï¬ ed unit formally. # 2.2. Leaky Rectiï¬ ed Linear Unit Leaky Rectiï¬ ed Linear activation is ï¬ rst introduced in acoustic model(Maas et al., 2013). Mathematically, we have # Li if xi â ¥ 0 if xi < 0, yi = (2) xi ai where ai is a ï¬ xed parameter in range (1, +â ). In original paper, the authors suggest to set ai to a large number like 100. In additional to this setting, we also experiment smaller ai = 5.5 in our paper. # 2.3. Parametric Rectiï¬ ed Linear Unit Parametric rectiï¬ ed linear is proposed by (He et al., 2015). The authors reported its performance is much better than ReLU in large scale image classiï¬ cation task. It is the same as leaky ReLU (Eqn.2) with the exception that ai is learned in the training via back propagation. # 2.4. Randomized Leaky Rectiï¬ ed Linear Unit Randomized Leaky Rectiï¬ ed Linear is the randomized version of leaky ReLU. It is ï¬ rst proposed and used in Kaggle NDSB Competition. The highlight of RReLU is that in training process, aji is a random number sampled from a uniform distribution U (l, u).	1505.00853#3	1505.00853#5	1505.00853	[ "1502.03167" ]
1505.00853#5	Empirical Evaluation of Rectified Activations in Convolutional Network	Formally, we have: t \ { \ \ i Leaky ReLU/PReLU Randomized Leaky ReLU Figure 1: ReLU, Leaky ReLU, PReLU and RReLU. For PReLU, ai is learned and for Leaky ReLU ai is ï¬ xed. For RReLU, aji is a random variable keeps sam- pling in a given range, and remains ï¬ xed in testing. Seyi ifr, >0 . Yai = (a, if aj; <0, (3) where aji â ¼ U (l, u), l < u and l, u â [0, 1) (4) In the test phase, we take average of all the aji in training as in the method of dropout (Srivastava et al., 2014) , and thus set aji to l+u to get a deterministic 2 result. Suggested by the NDSB competition winner, aji is sampled from U (3, 8). We use the same conï¬ gu- ration in this paper. In test time, we use: # 2.1. Rectiï¬ ed Linear Unit yji = xji l+u 2 (5) Rectiï¬ ed Linear is ï¬ rst used in Restricted Boltzmann Machines(Nair & Hinton, 2010). Formally, rectiï¬ ed linear activation is deï¬ ned as: Xi w= {5 ifa; >0 ifa; <0. (1) # 3. Experiment Settings We evaluate classiï¬ cation performance on same con- volutional network structure with diï¬ erent activa- tion functions. Due to the large parameter search- ing space, we use two state-of-art convolutional net- work structure and same hyper parameters for diï¬ er- ent activation setting. All models are trained by using CXXNET2.	1505.00853#4	1505.00853#6	1505.00853	[ "1502.03167" ]
1505.00853#6	Empirical Evaluation of Rectified Activations in Convolutional Network	1Kaggle National Data Science Bowl Competition: https://www.kaggle.com/c/datasciencebowl 2CXXNET: https://github.com/dmlc/cxxnet Empirical Evaluation of Rectiï¬ ed Activations in Convolutional Network # 3.1. CIFAR-10 and CIFAR-100 The CIFAR-10 and CIFAR-100 dataset (Krizhevsky & Hinton, 2009) are tiny nature image dataset. CIFAR- 10 datasets contains 10 diï¬ erent classes images and CIFAR-100 datasets contains 100 diï¬ erent classes. Each image is an RGB image in size 32x32. There are 50,000 training images and 10,000 test images. We use raw images directly without any pre-processing and augmentation. The result is from on single view test without any ensemble.	1505.00853#5	1505.00853#7	1505.00853	[ "1502.03167" ]
1505.00853#7	Empirical Evaluation of Rectified Activations in Convolutional Network	The network structure is shown in Table 1. It is taken from Network in Network(NIN)(Lin et al., 2013). Input Size NIN 32 Ã 32 32 Ã 32 32 Ã 32 32 Ã 32 16 Ã 16 16 Ã 16 16 Ã 16 16 Ã 16 16 Ã 16 8 Ã 8 8 Ã 8 8 Ã 8 8 Ã 8 8 Ã 8 10 or 100 5x5, 192 1x1, 160 1x1, 96 3x3 max pooling, /2 dropout, 0.5 5x5, 192 1x1, 192 1x1, 192 3x3,avg pooling, /2 dropout, 0.5 3x3, 192 1x1, 192 1x1, 10 8x8, avg pooling, /1 softmax We refer the network and augmentation setting from team AuroraXie4, one of competition winners. The network structure is shown in Table 5. We only use single view test in our experiment, which is diï¬ erent to original multi-view, multi-scale test. Input Size NDSB Net 70 Ã 70 70 Ã 70 70 Ã 70 35 Ã 35 35 Ã 35 35 Ã 35 35 Ã 35 17 Ã 17 17 Ã 17 17 Ã 17 17 Ã 17 17 Ã 17 17 Ã 17 17 Ã 17 8 Ã 8 8 Ã 8 8 Ã 8 8 Ã 8 8 Ã 8 8 Ã 8 12544 Ã 1 1024 Ã 1 1024 Ã 1 121 3x3, 32 3x3, 32 3x3, max pooling, /2 3x3, 64 3x3, 64 3x3, 64 3x3, max pooling, /2 split: branch1 â branch 2 3x3, 96 â 3x3, 96 3x3, 96 â 3x3, 96 3x3, 96 â	1505.00853#6	1505.00853#8	1505.00853	[ "1502.03167" ]
1505.00853#8	Empirical Evaluation of Rectified Activations in Convolutional Network	3x3, 96 3x3, 96 channel concat, 192 3x3, max pooling, /2 3x3, 256 3x3, 256 3x3, 256 3x3, 256 3x3, 256 SPP (He et al., 2014) {1, 2, 4} ï¬ atten fc1 fc2 softmax Table 1. CIFAR-10/CIFAR-100 network structure. Each layer is a convolutional layer if not otherwise speciï¬ ed. Ac- tivation function is followed by each convolutional layer. In CIFAR-100 experiment, we also tested RReLU on Batch Norm Inception Network (Ioï¬ e & Szegedy, 2015). We use a subset of Inception Network which is started from inception-3a module. This network achieved 75.68% test accuracy without any ensemble or multiple view test 3. # 3.2. National Data Science Bowl Competition The task for National Data Science Bowl competition is to classify plankton animals from image with award of $170k. There are 30,336 labeled gray scale images in 121 classes and there are 130,400 test data. Since the test set is private, we divide training set into two parts: 25,000 images for training and 5,336 images for validation. The competition uses multi-class log-loss to evaluate classiï¬ cation performance. 3CIFAR-100 Reproduce code: https://github. com/dmlc/mxnet/blob/master/example/notebooks/ cifar-100.ipynb Table 2. National Data Science Bowl Competition Net- work.	1505.00853#7	1505.00853#9	1505.00853	[ "1502.03167" ]
1505.00853#9	Empirical Evaluation of Rectified Activations in Convolutional Network	All layers are convolutional layers if not otherwise speciï¬ ed. Activation function is followed by each convolu- tional layer. # 4. Result and Discussion Table 3 and 4 show the results of CIFAR-10/CIFAR- 100 dataset, respectively. Table 5 shows the NDSB result. We use ReLU network as baseline, and com- pare the convergence curve with other three activa- tions pairwisely in Fig. 2, 3 and 4, respectively. All these three leaky ReLU variants are better than base- line on test set. We have the following observations based on our experiment: 1. Not surprisingly, we ï¬ nd the performance of nor- mal leaky ReLU (a = 100) is similar to that of ReLU, but very leaky ReLU with larger a = 5.5 is much better. 4Winning Doc of AuroraXie: https://github.com/ auroraxie/Kaggle-NDSB Empirical Evaluation of Rectiï¬ ed Activations in Convolutional Network 2. On training set, the error of PReLU is always the lowest, and the error of Leaky ReLU and RReLU are higher than ReLU. It indicates that PReLU may suï¬ er from severe overï¬ tting issue in small scale dataset. reasons of their superior performances still lack rigor- ous justiï¬ cation from theoretic aspect. Also, how the activations perform on large scale data is still need to be investigated. This is an open question worth pur- suing in the future. 3. The superiority of RReLU is more signiï¬ cant than that on CIFAR-10/CIFAR-100. We conjec- ture that it is because the in the NDSB dataset, the training set is smaller than that of CIFAR- 10/CIFAR-100, but the network we use is even bigger. This validates the eï¬ ectiveness of RReLU when combating with overï¬ tting. # Acknowledgement We would like to thank Jason Rolfe from D-Wave sys- tem for helpful discussion on test network for random- ized leaky ReLU. # References 4. For RReLU, we still need to investigate how the randomness inï¬ uences the network training and testing process.	1505.00853#8	1505.00853#10	1505.00853	[ "1502.03167" ]
1505.00853#10	Empirical Evaluation of Rectified Activations in Convolutional Network	Girshick, Ross, Donahue, Jeï¬ , Darrell, Trevor, and Malik, Jitendra. Rich feature hierarchies for accu- rate object detection and semantic segmentation. In CVPR, pp. 580â 587, 2014. Activation ReLU Leaky ReLU, a = 100 Leaky ReLU, a = 5.5 PReLU RReLU (yji = xji/ l+u 2 ) Training Error Test Error 0.00318 0.0031 0.00362 0.00178 0.00550 0.1245 0.1266 0.1120 0.1179 0.1119 Glorot, Xavier, Bordes, Antoine, and Bengio, Yoshua. In Proceedings of Deep sparse rectiï¬ er networks. the 14th International Conference on Artiï¬ cial In- telligence and Statistics. JMLR W&CP Volume, vol- ume 15, pp. 315â 323, 2011. Table 3.	1505.00853#9	1505.00853#11	1505.00853	[ "1502.03167" ]
1505.00853#11	Empirical Evaluation of Rectified Activations in Convolutional Network	Error rate of CIFAR-10 Network in Network with diï¬ erent activation function Activation ReLU Leaky ReLU, a = 100 Leaky ReLU, a = 5.5 PReLU RReLU (yji = xji/ l+u 2 ) Training Error Test Error 0.1356 0.11552 0.08536 0.0633 0.1141 0.429 0.4205 0.4042 0.4163 0.4025 He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Spatial pyramid pooling in deep convo- lutional networks for visual recognition. In ECCV, pp. 346â 361, 2014. He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Delving deep into rectiï¬ ers: Surpassing human-level performance on imagenet classiï¬ cation. arXiv preprint arXiv:1502.01852, 2015. Table 4. Error rate of CIFAR-100 Network in Network with diï¬ erent activation function Activation ReLU Leaky ReLU, a = 100 Leaky ReLU, a = 5.5 PReLU RReLU (yji = xji/ l+u 2 ) Train Log-Loss Val Log-Loss 0.8092 0.7846 0.7831 0.7187 0.8090 0.7727 0.7601 0.7391 0.7454 0.7292 Ioï¬ e, Sergey and Szegedy, Christian.	1505.00853#10	1505.00853#12	1505.00853	[ "1502.03167" ]
1505.00853#12	Empirical Evaluation of Rectified Activations in Convolutional Network	Batch nor- malization: Accelerating deep network training by arXiv preprint reducing internal covariate shift. arXiv:1502.03167, 2015. Krizhevsky, Alex and Hinton, Geoï¬ rey. Learning mul- tiple layers of features from tiny images. Computer Science Department, University of Toronto, Tech. Rep, 1(4):7, 2009. Table 5. Multi-classes Log-Loss of NDSB Network with dif- ferent activation function # 5. Conclusion	1505.00853#11	1505.00853#13	1505.00853	[ "1502.03167" ]
1505.00853#13	Empirical Evaluation of Rectified Activations in Convolutional Network	Krizhevsky, Alex, Sutskever, Ilya, and Hinton, Geof- frey E. Imagenet classiï¬ cation with deep convolu- In NIPS, pp. 1097â 1105, tional neural networks. 2012. In this paper, we analyzed four rectiï¬ ed activation functions using various network architectures on three datasets. Our ï¬ ndings strongly suggest that the most popular activation function ReLU is not the end of story: Three types of (modiï¬ ed) leaky ReLU all con- sistently outperform the original ReLU. However, the Lin, Min, Chen, Qiang, and Yan, Shuicheng. Network in network. arXiv preprint arXiv:1312.4400, 2013. Maas, Andrew L, Hannun, Awni Y, and Ng, An- drew Y. Rectiï¬ er nonlinearities improve neural net- work acoustic models. In ICML, volume 30, 2013. Empirical Evaluation of Rectiï¬ ed Activations in Convolutional Network ReLU Train ReLU Val Leaky ReLU,a=100 Train ReLU Train ReLU Val Leaky ReLU,a=100 Train ReLU Train ReLU Val Leaky ReLU,a=5.5 Train ReLU Train ReLU Val PReLU Train PReLU Val ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val 100 200 30 00-150 r 0 Epoch Epoch 100 30200250 r Epoch ReLU Train ReLU Val Leaky ReLU,a=5.5 Train 100 30 r Epoch ReLU Train ReLU Val PReLU Train PReLU Val 200 00-150 0 Epoch ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val 100 30200250 r Epoch Figure 2: Convergence curves for training and test sets of diï¬ erent activations on CIFAR-10 Network in Network. ReLU Train ReLU Val Error ReLU Train ReLU Val PReLU Train PReLU Val 10 30 200 250 ot Epoch	1505.00853#12	1505.00853#14	1505.00853	[ "1502.03167" ]
1505.00853#14	Empirical Evaluation of Rectified Activations in Convolutional Network	10 o.s\| o.8\| 07] 5 0.6\| 5 os\| o.a\| 0.3\| o2\| oul ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val 0 100 50 Epoch 10 ReLU Train ReLU Val ReLU Train ReLU Val PReLU Train PReLU Val o.s\| o.8\| 07] Error 5 0.6\| 5 os\| o.a\| 0.3\| o2\| oul ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val 10 30 200 250 0 ot Epoch 100 50 Epoch Figure 3: Convergence curves for training and test sets of diï¬ erent activations on CIFAR-100 Network in Network. ReLU Train ReLU Val Leaky ReLU, Leaky ReLU, 100 150 Epoch 200 Bo 300 ReLU Train ReLU Val Leaky ReLU, Leaky ReLU, â 030100 150 Epoch 200-250-300 â 030100380 Epoch 200-230-300 ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val â 030100 150 Epoch 200-250-300 ReLU Train ReLU Val Leaky ReLU, Leaky ReLU, ReLU Train ReLU Val Leaky ReLU, Leaky ReLU, ReLU Train ReLU Val RReLU,[3,8] Train RReLU,[3.8] Val 100 150 Epoch 200 Bo 300 â 030100 150 Epoch 200-250-300 â 030100380 Epoch 200-230-300 â 030100 150 Epoch 200-250-300 Figure 4: Convergence curves for training and test sets of diï¬ erent activations on NDSB Net. Nair, Vinod and Hinton, Geoï¬ rey E. Rectiï¬	1505.00853#13	1505.00853#15	1505.00853	[ "1502.03167" ]
1505.00853#15	Empirical Evaluation of Rectified Activations in Convolutional Network	ed linear units improve restricted Boltzmann machines. In ICML, pp. 807â 814, 2010. Erhan, Dumitru, Vanhoucke, Vincent, and Rabi- novich, Andrew. Going deeper with convolutions. arXiv preprint arXiv:1409.4842, 2014. Russakovsky, Olga, Deng, Jia, Su, Hao, Krause, Jonathan, Satheesh, Sanjeev, Ma, Sean, Huang, Zhi- heng, Karpathy, Andrej, Khosla, Aditya, Bernstein, Michael, Berg, Alexander C., and Fei-Fei, Li. Im- ageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 2015. doi: 10.1007/s11263-015-0816-y. Wang, Naiyan, Li, Siyi, Gupta, Abhinav, and Ye- ung, Dit-Yan. Transferring rich feature hierar- arXiv preprint chies for robust visual tracking. arXiv:1501.04587, 2015. Srivastava, Nitish, Hinton, Geoï¬ rey, Krizhevsky, Alex, Sutskever, Ilya, and Salakhutdinov, Ruslan. Dropout: A simple way to prevent neural networks from overï¬ tting. The Journal of Machine Learning Research, 15(1):1929â 1958, 2014.	1505.00853#14	1505.00853#16	1505.00853	[ "1502.03167" ]
1505.00853#16	Empirical Evaluation of Rectified Activations in Convolutional Network	Sun, Yi, Wang, Xiaogang, and Tang, Xiaoou. Deeply learned face representations are sparse, selective, and robust. arXiv preprint arXiv:1412.1265, 2014. Szegedy, Christian, Liu, Wei, Jia, Yangqing, Ser- manet, Pierre, Reed, Scott, Anguelov, Dragomir,	1505.00853#15		1505.00853	[ "1502.03167" ]
1505.00521#0	Reinforcement Learning Neural Turing Machines - Revised	6 1 0 2 n a J 2 1 ] G L . s c [ 3 v 1 2 5 0 0 . 5 0 5 1 : v i X r a Under review as a conference paper at ICLR 2016 # REINFORCEMENT LEARNING NEURAL TURING MACHINES - REVISED Wojciech Zaremba1,2 New York University Facebook AI Research [email protected] Ilya Sutskever2 Google Brain [email protected] # ABSTRACT The Neural Turing Machine (NTM) is more expressive than all previously considered models because of its external memory. It can be viewed as a broader effort to use abstract external Interfaces and to learn a parametric model that interacts with them. The capabilities of a model can be extended by providing it with proper Interfaces that interact with the world. These external Interfaces include memory, a database, a search engine, or a piece of software such as a theorem veriï¬		1505.00521#1	1505.00521	[ "1503.01007" ]
1505.00521#1	Reinforcement Learning Neural Turing Machines - Revised	er. Some of these Interfaces are provided by the developers of the model. However, many important existing Interfaces, such as databases and search engines, are discrete. We examine feasibility of learning models to interact with discrete Interfaces. We investigate the following discrete Interfaces: a memory Tape, an input Tape, and an output Tape. We use a Reinforcement Learning algorithm to train a neural network that interacts with such Interfaces to solve simple algorithmic tasks. Our Interfaces are expressive enough to make our model Turing complete. # INTRODUCTION Graves et al. (2014b)â s Neural Turing Machine (NTM) is model that learns to interact with an external memory that is differentiable and continuous. An external memory extends the capabilities of the NTM, allowing it to solve tasks that were previously unsolvable by conventional machine learning methods. In general, it appears that ML models become This is the source of the NTMâ	1505.00521#0	1505.00521#2	1505.00521	[ "1503.01007" ]
1505.00521#2	Reinforcement Learning Neural Turing Machines - Revised	s expressive power. signiï¬ cantly more powerful if they are able to learn to interact with external interfaces. There exist a vast number of Interfaces that could be used with our models. For example, the Google search engine is an example of such Interface. The search engine consumes queries (which are actions), and outputs search results. However, the search engine is not differentiable, and the model interacts with the Interface using discrete actions. This work examines the feasibility of learning to interact with discrete Interfaces using the reinforce algorithm. Discrete Interfaces cannot be trained directly with standard backpropagation because they are not dif- ferentiable. It is most natural to learn to interact with discrete Interfaces using Reinforcement Learning methods. In this work, we consider an Input Tape and a Memory Tape interface with discrete access. Our concrete proposal is to use the Reinforce algorithm to learn where to access the discrete interfaces, and to use the backpropagation algorithm to determine what to write to the memory and to the output. We call this model the RLâ	1505.00521#1	1505.00521#3	1505.00521	[ "1503.01007" ]
1505.00521#3	Reinforcement Learning Neural Turing Machines - Revised	NTM. Discrete Interfaces are computationally attractive because the cost of accessing a discrete Interface is often independent of its size. It is not the case for the continuous Interfaces, where the cost of access scales linearly with size. It is a signiï¬ cant disadvantage since slow models cannot scale to large difï¬ cult In addition, an output Interface that lets problems that require intensive training on large datasets. the model decide when it wants to make a prediction allows the modelâ s runtime to be in principle unbounded. If the model has an output interface of this kind together with an interface to an unbounded memory, the model becomes Turing complete. We evaluate the RL-NTM on a number of simple algorithmic tasks. The RL-NTM succeeds on problems such as copying an input several times to the output tape (the â repeat copyâ task from Graves et al. (2014b)), reversing a sequence, and a few more tasks of comparable difï¬ culty. However, its success is highly dependent on the architecture of the â controllerâ .	1505.00521#2	1505.00521#4	1505.00521	[ "1503.01007" ]
1505.00521#4	Reinforcement Learning Neural Turing Machines - Revised	We discuss this in more details in Section 8. 1Work done while the author was at Google. 2Both authors contributed equally to this work. 1 # Under review as a conference paper at ICLR 2016 Finally, we found it non-trivial to correctly implement the RL-NTM due its large number of interacting components. We developed a simple procedure to numerically check the gradients of the Reinforce algorithm (Section 5). The procedure can be applied to problems unrelated to NTMs, and is of the independent interest. The code for this work can be found at https://github.com/ilyasu123/rlntm. # 2 THE MODEL	1505.00521#3	1505.00521#5	1505.00521	[ "1503.01007" ]
1505.00521#5	Reinforcement Learning Neural Turing Machines - Revised	Many difï¬ cult tasks require a prolonged, multi-step interaction with an external environment. Examples of such environments include computer games (Mnih et al., 2013), the stock market, an advertisement system, or the physical world (Levine et al., 2015). A model can observe a partial state from the environment, and inï¬ uence the environment through its actions. This is seen as a general reinforcement leaning problem. However, our setting departs from the classical RL, i.e. we have a freedom to design tools available to solve a given problem. Tools might cooperate with the model (i.e. backpropagation through memory), and the tools specify the actions over the environment. We formalize this concept under the name Interfaceâ Controller interaction. The external environment is exposed to the model through a number of Interfaces, each with its own API. For instance, a human perceives the world through its senses, which include the vision Interface and the touch Interface. The touch Interface provides methods for contracting the various muscles, and methods for sensing the current state of the muscles, pain level, temperature and a few others. In this work, we explore a number of simple Interfaces that allow the controller to access an input tape, a memory tape, and an output tape. The part of the model that communicates with Interfaces is called the Controller, which is the only part of the system which learns. The Controller can have prior knowledge about behavior of its Interfaces, but it is not the case in our experiments. The Controller learns to interact with Interfaces in a way that allows it to solve a given task. Fig. 1 illustrates the complete Interfacesâ Controller abstraction. Input Interface Output Interface Memory Interface input position increment -1 0 1 Target prediction to output symbol or not? 0 1 memory address increment -1 0 1 new memory value vector Controller Output Controller Output Past State Controller Future State Past State LSTM Future State Controller Input Controller Input Input Interface Output Interface Memory Interface Current Input Current Memory An abstract Interfaceâ Controller model Our model as an Interfaceâ Controller Figure 1: (Left) The Interfaceâ Controller abstraction, (Right) an instantiation of our model as an Interfaceâ Controller. The bottom boxes are the read methods, and the top are the write methods. The RLâ	1505.00521#4	1505.00521#6	1505.00521	[ "1503.01007" ]
1505.00521#6	Reinforcement Learning Neural Turing Machines - Revised	NTM makes discrete decisions regarding the move over the input tape, the memory tape, and whether to make a prediction at a given timestep. During training, the modelâ s prediction is compared with the desired output, and is used to train the model when the RL-NTM chooses to advance its position on the output tape; otherwise it is ignored. The memory value vector is a vector of content that is stored in the memory cell. We now describe the RLâ NTM. As a controller, it uses either LSTM, direct access, or LSTM (see sec. 8.1 for a deï¬ nition). It has a one-dimensional input tape, a one-dimensional memory, and a one- dimensional output tape as Interfaces. Both the input tape and the memory tape have a head that reads the Tapeâ s content at the current location. The head of the input tape and the memory tape can move in any direction. However, the output tape is a write-only tape, and its head can either stay at the current position or move forward.	1505.00521#5	1505.00521#7	1505.00521	[ "1503.01007" ]
1505.00521#7	Reinforcement Learning Neural Turing Machines - Revised	Fig. 2 shows an example execution trace for the entire RLâ NTM on the reverse task (sec. 6). At the core of the RLâ NTM is an LSTM controller which receives multiple inputs and has to generate multiple outputs at each timestep. Table 1 summarizes the controllerâ s inputs and outputs, and the way in which the RLâ NTM is trained to produce them. The objective function of the RLâ NTM is the expected log probability of the desired outputs, where the expectation is taken over all possible sequences of actions, weighted with probability of taking these actions. Both backpropagation and Reinforce maximize this objective. Backpropagation maximizes the log probabilities of the modelâ s predictions, while the reinforce algorithm inï¬ uences the probabilities of action sequences.	1505.00521#6	1505.00521#8	1505.00521	[ "1503.01007" ]
1505.00521#8	Reinforcement Learning Neural Turing Machines - Revised	2 # Under review as a conference paper at ICLR 2016 Output @ \|Nenory wh) Output @ \|Nenoryatr\| (Output @ \|Nemory +4) [output > \|Menory s+] utout 9 \|Menory wi) . u raul) ia emety -Â§ Goh g ath. SS ah 6 vor Â« bbe tL itr tai het Lonel tetell TL fae t \|\| n ah tL hE fie Memoryi) \|nput Wor] [Memory *l] \| Input 4Â°] [Memory Â«\|\| Input 490] Memory Â«BK ] Input 49H] Memory WeÂ«\|[ Input 490] Emty nidsan 2 . Z tb t ts ty ts Time cap Fre Raden sete	1505.00521#7	1505.00521#9	1505.00521	[ "1503.01007" ]
1505.00521#9	Reinforcement Learning Neural Turing Machines - Revised	Figure 2: Execution of RLâ NTM on the ForwardReverse task. At each timestep, the RL-NTM con- sumes the value of the current input tape, the value of the current memory cell, and a representation of all the actions that have been taken in the previous timestep (not marked on the ï¬ gures). The RL- NTM then outputs a new value for the current memory cell (marked with a star), a prediction for the next target symbol, and discrete decisions for changing the positions of the heads on the various tapes. The RL-NTM learns to make discrete decisions using the Reinforce algorithm, and learns to produce continuous outputs using backpropagation.	1505.00521#8	1505.00521#10	1505.00521	[ "1503.01007" ]
1505.00521#10	Reinforcement Learning Neural Turing Machines - Revised	The global objective can be written formally as: n Ss Dreinforce (1, @2, --.,@n\|) p> log (pop (Yili... +, 2s, 01,-.- i, 8)\| [a1,42,...,anJEAt i=l Aâ represents the space of sequences of actions that lead to the end of episode. The probabilities in the above equation are parametrized with a neural network (the Controller). We have marked with preinforce the part of the equation which is learned with Reinforce. pbp indicates the part of the equation optimized with the classical backpropagation. Interface Read Write Training Type Input Tape Output Tape Head Head Content window of values surrounding the current position â	1505.00521#9	1505.00521#11	1505.00521	[ "1503.01007" ]
1505.00521#11	Reinforcement Learning Neural Turing Machines - Revised	â distribution over [â 1, 0, 1] distribution over [0, 1] distribution over output vocabulary Reinforce Reinforce Backpropagation Memory Tape Miscellaneous Head Content window of memory values surrounding the current address all actions taken in the previous time step distribution over [â 1, 0, 1] vector of real values to store â Reinforce Backpropagation â Table 1: Table summarizes what the Controller reads at every time step, and what it has to produce.	1505.00521#10	1505.00521#12	1505.00521	[ "1503.01007" ]
1505.00521#12	Reinforcement Learning Neural Turing Machines - Revised	The â trainingâ column indicates how the given part of the model is trained. The RLâ NTM receives a direct learning signal only when it decides to make a prediction. If it chooses to not make a prediction at a given timestep, then it will not receive a direct learning signal. Theoretically, we can allow the RLâ NTM to run for an arbitrary number of steps without making any prediction, hoping that after sufï¬ ciently many steps, it would decide to make a prediction. Doing so will also provide the RLâ NTM with arbitrary computational capability. However, this strategy is both unstable and computationally infeasible. Thus, we resort to limiting the total number of computational steps to a ï¬ xed upper bound, and force the RLâ NTM to predict the next desired output whenever the number of remaining desired outputs is equal to the number of remaining computational steps. # 3 RELATED WORK This work is the most similar to the Neural Turing Machine Graves et al. (2014b). The NTM is an ambitious, computationally universal model that can be trained (or â automatically programmedâ ) with the backpropagation algorithm using only input-output examples. Following the introduction NTM, several other memory-based models have been introduced.	1505.00521#11	1505.00521#13	1505.00521	[ "1503.01007" ]
1505.00521#13	Reinforcement Learning Neural Turing Machines - Revised	All of them can be seen as part of a larger community effort. These models are constructed according to the Interfaceâ Controller abstraction (Section 2). Neural Turing Machine (NTM) (Graves et al., 2014a) has a modiï¬ ed LSTM as the Controller, and the following three Interfaces: a sequential input, a delayed Output, and a differentiable Memory. 3 # Under review as a conference paper at ICLR 2016 Weakly supervised Memory Network (Sukhbaatar et al., 2015) uses a feed forward network as the Controller, and has a differentiable soft-attention Input, and Delayed Output as Interfaces. Stack RNN (Joulin & Mikolov, 2015) has a RNN as the Controller, and the sequential input, a differen- tiable memory stack, and sequential output as Interfaces.	1505.00521#12	1505.00521#14	1505.00521	[ "1503.01007" ]
1505.00521#14	Reinforcement Learning Neural Turing Machines - Revised	Also uses search to improve its performance. Neural DeQue (Grefenstette et al., 2015) has a LSTM as the Controller, and a Sequential Input, a differentiable Memory Queue, and the Sequential Output as Interfaces. Our model ï¬ ts into the Interfacesâ Controller abstraction. It has a direct access LSTM as the Controller (or LSTM or feed forward network), and its three interfaces are the Input Tape, the Memory Tape, and the Output Tape. All three Interfaces of the RLâ NTM are discrete and cannot be trained only with backpropagation. This prior work investigates continuous and differentiable Interfaces, while we consider discrete In- terfaces. Discrete Interfaces are more challenging to train because backpropagation cannot be used. However, many external Interfaces are inherently discrete, even though humans can easily use them (apparently without using continuous backpropagation). For instance, one interacts with the Google search engine with discrete actions. This work examines the possibility of learning models that interact with discrete Interfaces with the Reinforce algorithm. The Reinforce algorithm (Williams, 1992) is a classical RL algorithm, which has been applied to the broad spectrum of planning problems (Peters & Schaal, 2006; Kohl & Stone, 2004; Aberdeen & Baxter, 2002). In addition, it has been applied in object recognition to implement visual attention (Mnih et al., 2014; Ba et al., 2014). This work uses Reinforce to train an attention mechanism: we use it to train how to access the various tapes provided to the model.	1505.00521#13	1505.00521#15	1505.00521	[ "1503.01007" ]
1505.00521#15	Reinforcement Learning Neural Turing Machines - Revised	The RLâ NTM can postpone prediction for an arbitrary number of timesteps, and in principle has access to the unbounded memory. As a result, the RL-NTM is Turing complete in principle. There have been very few prior models that are Turing complete Schmidhuber (2012; 2004). Although our model is Turing complete, it is not very powerful because it is very difï¬ cult to train, and our model can solve only relatively simple problems. Moreover, the RLâ NTM does not exploit Turing completeness, as none of tasks that it solves require superlinear runtime to be solved. # 4 THE REINFORCE ALGORITHM Notation Let A be a space of actions, and At be a space of all sequences of actions that cause an episode to end (so At c A*). Anaction at time-step Â¢ is denoted by a;. We denote time at the end of episode by T (this is not completely formal as some episodes can vary in time). Let a1,, stand for a sequence of actions [a1,@2,..., a4]. Let r(a1,,) denote the reward achieved at time t, having executed the sequence of ac- tions a.,, and R(aj,r) is the cumulative reward, namely R(ax:7) = an r(a1:t)- Let po (ar\|1.(1-1)) be a parametric conditional probability of an action a; given all previous actions @1.;_1). Finally, po is a policy parametrized by 6. This work relies on learning discrete actions with the Reinforce algorithm (Williams, 1992). We now describe this algorithm in detail. Moreover, the supplementary materials include descriptions of tech- niques for reducing variance of the gradient estimators.	1505.00521#14	1505.00521#16	1505.00521	[ "1503.01007" ]
1505.00521#16	Reinforcement Learning Neural Turing Machines - Revised	The goal of reinforcement learning is to maximize the sum of future rewards. The Reinforce algorithm (Williams, 1992) does so directly by optimizing the parameters of the policy pÎ¸(at\|a1:(tâ 1)). Reinforce follows the gradient of the sum of the future rewards. The objective function for episodic reinforce can be expressed as the sum over all sequences of valid actions that cause the episode to end: J(Î¸) = pÎ¸(a1, a2, . . . , aT )R(a1, a2, . . . , aT ) = pÎ¸(a1:T )R(a1:T ) [a1,a2,...,aT ]â Aâ a1:T â Aâ	1505.00521#15	1505.00521#17	1505.00521	[ "1503.01007" ]
1505.00521#17	Reinforcement Learning Neural Turing Machines - Revised	This sum iterates over sequences of all possible actions. This set is usually exponential or even inï¬ nite, so it cannot be computed exactly and cheaply for most of problems. However, it can be written as 4 # Under review as a conference paper at ICLR 2016 expectation, which can be approximated with an unbiased estimator. We have that: = Ss po(ar.r)R(ar.7) = ay:7â ¬At Eur~pe > r(a12) = t=1 T Eay~po(a:)Eas~po(aslar) +++ Ear~polar\|arcrâ 1y) >)" (@1:t) t=1 J(Î¸) = The last expression suggests a procedure to estimate J(Î¸): simply sequentially sample each at from the model distribution pÎ¸(at\|a1:(tâ 1)) for t from 1 to T . The unbiased estimator of J(Î¸) is the sum of r(a1:t). This gives us an algorithm to estimate J(Î¸). However, the main interest is in training a model to maximize this quantity. The reinforce algorithm maximizes J(0) by following the gradient of it: AJ(0)= S> [Aopo(ar-r)] R(a-r) â Î¸J(Î¸) = a1:T â Aâ However, the above expression is a sum over the set of the possible action sequences, so it cannot be computed directly for most At. Once again, the Reinforce algorithm rewrites this sum as an expectation that is approximated with sampling. It relies on the equation: 0 f (0) = f(@) â ee = f(0)Oo[log f (0)]. This identity is valid as long as f(x) 4 0. As typical neural network parametrizations of distributions assign non-zero probability to every action, this condition holds for f = pg.	1505.00521#16	1505.00521#18	1505.00521	[ "1503.01007" ]
1505.00521#18	Reinforcement Learning Neural Turing Machines - Revised	We have that: S> [Aope(arr)] Rar) = [ar.rJeAt = Ss po(a1:r) [Oo log po(ai:r)\| (arr) a1.r â ¬At n = Ss polar) \|) dp log po(aslar.Â¢â 1))] R(a-r) aypeAt t=1 T = Eay~po(ar)Ea2~po(aslar) +++ Ear~po(arlarr1)\| >, 00 log po(ailar-eâ 1)] [ 9) r(ar)] t=1 t=1 â Î¸J(Î¸) = The last expression gives us an algorithm for estimating 0gJ(@). We have sketched it at the left side of the Figure\|3} Itâ s easiest to describe it with respect to computational graph behind a neural network. Reinforce can be implemented as follows. A neural network outputs: 1; = log pg (az \|a1.(Â¢1)). Sequen- tially sample action a, from the distribution eâ *, and execute the sampled action a,. Simultaneously, experience a reward r(aj,,). Backpropagate the sum of the rewards al r(a1:4) to the every node 0p log po (ar 1:(-1))- We have derived an unbiased estimator for the sum of future rewards, and the unbiased estimator of its gradient. However, the derived gradient estimator has high variance, which makes learning difï¬	1505.00521#17	1505.00521#19	1505.00521	[ "1503.01007" ]
1505.00521#19	Reinforcement Learning Neural Turing Machines - Revised	cult. RLâ NTM employs several techniques to reduce gradient estimator variance: (1) future rewards back- propagation, (2) online baseline prediction, and (3) ofï¬ ine baseline prediction. All these techniques are crucial to solve our tasks. We provide detailed description of techniques in the Supplementary material. Finally, we needed a way of verifying the correctness of our implementation. We discovered a technique that makes it possible to easily implement a gradient checker for nearly any model that uses Reinforce. Following Section 5 describes this technique. # 5 GRADIENT CHECKING The RLâ NTM is complex, so we needed to ï¬ nd an automated way of verifying the correctness of our implementation. We discovered a technique that makes it possible to easily implement a gradient checker for nearly any model that uses Reinforce. This discovery is an independent contribution of this	1505.00521#18	1505.00521#20	1505.00521	[ "1503.01007" ]
1505.00521#20	Reinforcement Learning Neural Turing Machines - Revised	5 # Under review as a conference paper at ICLR 2016 Reinforce Gradient Checking of Reinforce sample(t) , def sample(time=t): {For rowsiin the minibatch 1 (a.@2,...ar] = Al j tetum a 1 'Loop until the end of the episode Execute in the environment â Accumulate reward T \ [Deer (41:2) \|p0(ar-r) Backpropagate 06 log pe(at\|41:(Â¢-1)) samplett) def sample(time=t) sample from Po(at\|ai-(t-1)) Execute in the environment !Loop until the end of the episode Y â Accumulate reward we (ais) Backpropagate Oo log po(aela1:(4â -1)) Figure 3: Figure sketches algorithms: (Left) the reinforce algorithm, (Right) gradient checking for the reinforce algorithm. The red color indicates necessary steps to override the reinforce to become the gradient checker for the reinforce.	1505.00521#19	1505.00521#21	1505.00521	[ "1503.01007" ]
1505.00521#21	Reinforcement Learning Neural Turing Machines - Revised	work. This Section describes the gradient checking for any implementation of the reinforce algorithm that uses a general function for sampling from multinomial distribution. The reinforce gradient veriï¬ cation should ensure that expected gradient over all sequences of actions matches the numerical derivative of the expected objective. However, even for a tiny problem, we would need to draw billions of samples to achieve estimates accurate enough to state if there is match or mis- match. Instead, we developed a technique which avoids sampling, and allows for gradient veriï¬	1505.00521#20	1505.00521#22	1505.00521	[ "1503.01007" ]
1505.00521#22	Reinforcement Learning Neural Turing Machines - Revised	cation of reinforce within seconds on a laptop. First, we have to reduce the size of our a task to make sure that the number of possible actions is manageable (e.g., < 104). This is similar to conventional gradient checkers, which can only be applied to small models. Next, we enumerate all possible sequences of actions that terminate the episode. By deï¬ nition, these are precisely all the elements of Aâ . The key idea is the following: we override the sampling function which turns a multinomial distribu- tion into a random sample with a deterministic function that deterministically chooses actions from an appropriate action sequence from At, while accumulating their probabilities. By calling the modified sampler, it will produce every possible action sequence from Aâ	1505.00521#21	1505.00521#23	1505.00521	[ "1503.01007" ]
1505.00521#23	Reinforcement Learning Neural Turing Machines - Revised	exactly once. For efficiency, it is desirable to use a single minibatch whose size is ##A'. The sampling function needs to be adapted in such a way, so that it incrementally outputs the appropriate sequence from At as we repeatedly call the sampling function. At the end of the minibatch, the sampling function will have access to the total probability of each action sequence ([], 79 (a+\|a1:e-1)), which in turn can be used to exactly compute J(6) and its derivative. To compute the derivative, the reinforce gradient produced by each sequence a1. â ¬ At should be weighted by its probability pg (a1.r). We summarize this procedure on Figure[3] The gradient checking is critical for ensuring the correctness of our implementation. While the basic reinforce algorithm is conceptually simple, the RLâ NTM is fairly complicated, as reinforce is used to train several Interfaces of our model. Moreover, the RLâ NTM uses three separate techniques for reducing the variance of the gradient estimators. The modelâ	1505.00521#22	1505.00521#24	1505.00521	[ "1503.01007" ]
1505.00521#24	Reinforcement Learning Neural Turing Machines - Revised	s high complexity greatly increases the probability of a code error. In particular, our early implementations were incorrect, and we were able to ï¬ x them only after implementing gradient checking. # 6 TASKS This section deï¬ nes tasks used in the experiments. Figure 4 shows exemplary instantiations of our tasks. Table 2 summarizes the Interfaces that are available for each task. 6 # Under review as a conference paper at ICLR 2016 Task Interface \| tyoutTape Memory Tape Copy v x DuplicatedInput v x Reverse v x RepeatCopy v x ForwardReverse x v	1505.00521#23	1505.00521#25	1505.00521	[ "1503.01007" ]
1505.00521#25	Reinforcement Learning Neural Turing Machines - Revised	Table 2: This table marks the available Interfaces for each task. The difï¬ culty of a task is dependent on the type of Interfaces available to the model. Copy DuplicatedInput Reverse RepeatCopy ForwardReverse Figure 4: This Figure presents the initial state for every task. The yellow box indicates the starting position of the reading head over the Input Interface. The gray characters on the Output Tape represent the target symbols. Our tasks involve reordering symbols, and and the symbols xi have been picked uniformly from the set of size 30. Copy. A generic input is x1x2x3 . . . xCâ and the desired output is x1x2 . . . xCâ . Thus the goal is to repeat the input. The length of the input sequence is variable and is allowed to change. The input sequence and the desired output both terminate with a special end-of-sequence symbol â	1505.00521#24	1505.00521#26	1505.00521	[ "1503.01007" ]
1505.00521#26	Reinforcement Learning Neural Turing Machines - Revised	. DuplicatedInput. A generic input has the form x1x1x1x2x2x2x3 . . . xCâ 1xCxCxCâ while the desired output is x1x2x3 . . . xCâ . Thus each input symbol is replicated three times, so the RL-NTM must emit every third input symbol. Reverse. A generic input is x1x2 . . . xCâ 1xCâ and the desired output is xCxCâ 1 . . . x2x1â . RepeatCopy. is x1x2 . . . xCx1 . . . xCx1 . . . xCâ , where the number of copies is given by m. Thus the goal is to copy the input m times, where m can be only 2 or 3.	1505.00521#25	1505.00521#27	1505.00521	[ "1503.01007" ]
1505.00521#27	Reinforcement Learning Neural Turing Machines - Revised	ForwardReverse. The task is identical to Reverse, but the RL-NTM is only allowed to move its input tape pointer forward. It means that a perfect solution must use the NTMâ s external memory. is mx1x2x3 . . . xCâ and # 7 CURRICULUM LEARNING Humans and animals learn much better when the examples are not randomly presented but organized in a meaningful order which illustrates gradually more concepts, and gradually more complex ones. . . . and call them â curriculum learningâ . Bengio et al. (2009) We were unable to solve tasks with RLâ NTM by training it on the difï¬ cult instances of the problems (where difï¬ cult usually means long). To succeed, we had to create a curriculum of tasks of increasing complexity. We veriï¬ ed that our tasks were completely unsolvable (in an all-or-nothing sense) for all but the shortest sequences when we did not use a curriculum. In our experiments, we measure the complexity c of a problem instance by the maximal length of the desired output to typical inputs. During training, we maintain a distribution over the task complexity. We shift the distribution over the task complexities whenever the performance of the RLâ NTM exceeds a threshold. Then, our model focuses on more difï¬ cult problem instances as its performance improves. Probability 10% 25% 65% Procedure to pick complexity d uniformly at random from the possible task complexities. uniformly from [1, C + e] d = D + e. Table 3: The curriculum learning distribution, indexed by C. Here e is a sample from a geometric 2 , i.e., p(e = k) = 1 distribution whose success probability is 1 2k .	1505.00521#26	1505.00521#28	1505.00521	[ "1503.01007" ]
1505.00521#28	Reinforcement Learning Neural Turing Machines - Revised	7 # Under review as a conference paper at ICLR 2016 The distribution over task complexities is indexed with an integer c, and is deï¬ ned in Table 3. While we have not tuned the coefï¬ cients in the curriculum learning setup, we experimentally veriï¬ ed that it is critical to always maintain non-negligible mass over the hardest difï¬ culty levels (Zaremba & Sutskever, 2014). Removing it makes the curriculum much less effective. Whenever the average zero-one-loss (normalized by the length of the target sequence) of our RLâ NTM decreases below 0.2, we increase c by 1. We kept doing so until c reaches its maximal allowable value. Finally, we enforced a refractory period to ensure that successive increments of C are separated by at least 100 parameter updates, since we encountered situations where C increased in rapid succession which consistently caused learning to fail.	1505.00521#27	1505.00521#29	1505.00521	[ "1503.01007" ]
1505.00521#29	Reinforcement Learning Neural Turing Machines - Revised	# 8 CONTROLLERS The success of reinforcement learning training highly depends on the complexity of the controller, and its ease of training. Itâ s common to either limit number of parameters of the network, or to constraint it by initialization from pretrained model on some other task (for instance, object recognition network for robotics). Ideally, models should be generic enough to not need such â tricksâ . However, still some tasks require building task speciï¬ c architectures. Figure 6: The direct access controller. Figure 5: LSTM as a controller. This work considers two controllers.	1505.00521#28	1505.00521#30	1505.00521	[ "1503.01007" ]
1505.00521#30	Reinforcement Learning Neural Turing Machines - Revised	The ï¬ rst is a LSTM (Fig. 5), and the second is a direct access controller (Fig. 6). LSTM is a generic controller, that in principle should be powerful enough to solve any of the considered tasks. However, it has trouble solving many of them. Direct access controller, is a much better ï¬ t for symbol rearrangement tasks, however itâ s not a generic solution. 8.1 DIRECT ACCESS CONTROLLER All the tasks that we consider involve rearranging the input symbols in some way. For example, a typical task is to reverse a sequence (section 6 lists the tasks). For such tasks, the controller would beneï¬ t from a built-in mechanism for directly copying an appropriate input to memory and to the output. Such a mechanism would free the LSTM controller from remembering the input symbol in its control variables (â	1505.00521#29	1505.00521#31	1505.00521	[ "1503.01007" ]
1505.00521#31	Reinforcement Learning Neural Turing Machines - Revised	registersâ ), and would shorten the backpropagation paths and therefore make learning easier. We implemented this mechanism by adding the input to the memory and the output, and also adding the memory to the output and to the adjacent memories (ï¬ gure 6), while modulating these additive contribution by a dynamic scalar (sigmoid) which is computed from the controllerâ s state. This way, the controller can decide to effectively not add the current input to the output at a given timestep. Unfortunately the necessity of this architectural modiï¬ cation is a drawback of our implementation, since it is not domain independent and would therefore not improve the performance of the RLâ NTM on many tasks of interest. Controller \| TstM Direct Access Task Copy v v Duplicatedinput Reverse ForwardReverse RepeatCopy xxx KAKA Table 4: Success of training on various task for a given controller.	1505.00521#30	1505.00521#32	1505.00521	[ "1503.01007" ]
1505.00521#32	Reinforcement Learning Neural Turing Machines - Revised	8 # Under review as a conference paper at ICLR 2016 # 9 EXPERIMENTS We presents results of training RLâ NTM on all aforementioned tasks. The main drawback of our experiments is in the lack of comparison to the other models. However, the tasks that we consider have to be considered in conjunction with available Interfaces, and other models havenâ t been considered with the same set of interfaces. The statement, â this model solves additionâ is difï¬ cult to assess, as the way that digits are delivered deï¬ nes task difï¬ culty. The closest model to ours is NTM, and the shared task that they consider is copying. We are able to generalize with copying to an arbitrary length. However, our Interfaces make this task very simple.	1505.00521#31	1505.00521#33	1505.00521	[ "1503.01007" ]
1505.00521#33	Reinforcement Learning Neural Turing Machines - Revised	Table 4 summarizes results. We trained our model using SGD with a ï¬ xed learning rate of 0.05 and a ï¬ xed momentum of 0.9. We used a batch of size 200, which we found to work better than smaller batch sizes (such as 50 or 20). We normalized the gradient by batch size but not by sequence length. We independently clip the norm of the gradients w.r.t. the RL-NTM parameters to 5, and the gradient w.r.t. the baseline network to 2. We initialize the RLâ NTM controller and the baseline model using a Gaussian with standard deviation 0.1. We used an inverse temperature of 0.01 for the different action distributions. Doing so reduced the effective learning rate of the Reinforce derivatives. The memory consists of 35 real values through which we backpropagate. The initial memory state and the controllerâ s initial hidden states were set to the zero vector. Input Tape \| Memory Output Tape Input Tape Output Tape ERFCS7R3BGA AGB3R75CFREO WWW6667778S88SSRRRWWWYYY \| W67TE8SRWYO e Â» \|e W â R . * Ww Ww F \| W * Â¢ : 6 s 5 . : 6 : 7 * + 6 6 R + * 7 * 8 : A 1 ; 8 â : 7 : 5 : : 8 : _ : 3 : A * ic 8 8 A . 8 8 * A * 3 8 8 A : R 8 : A â 7 Â® : A * 5 R R A : c Â® . A Ã© â : A * * ; : A * t i : A . R ; . A : : y : A â 3 7 : Input Tape Output Tape WWW6667778S88SSRRRWWWYYY \| W67TE8SRWYO W â Ww Ww W * 6 s 6 : 6 6 7 * 1 ; 7 : 8 : 3 : 8 8 8 * 8 8 8 : Â® : R R Â® . â : ; : i : ; . y : 7 :	1505.00521#32	1505.00521#34	1505.00521	[ "1503.01007" ]
1505.00521#34	Reinforcement Learning Neural Turing Machines - Revised	Input Tape \| Memory Output Tape ERFCS7R3BGA AGB3R75CFREO e Â» \|e R . * F \| Â¢ : 5 . : 7 * + R + * 8 : A 8 â : 5 : : _ : A * ic A . 8 A * 3 A : R A â 7 A * 5 A : c A Ã© A * * A * t A . R A : : A â 3 # : Time Figure 7: (Left) Trace of ForwardReverse solution, (Right) trace of RepeatInput. The vertical depicts execution time. The rows show the input pointer, output pointer, and memory pointer (with the â symbol) at each step of the RL-NTMâ s execution. Note that we represent the set {1, . . . , 30} with 30 distinct symbols, and lack of prediction with #. The ForwardReverse task is particularly interesting. In order to solve the problem, the RLâ NTM has to move to the end of the sequence without making any predictions. While doing so, it has to store the input sequence into its memory (encoded in real values), and use its memory when reversing the sequence (Fig. 7).	1505.00521#33	1505.00521#35	1505.00521	[ "1503.01007" ]
1505.00521#35	Reinforcement Learning Neural Turing Machines - Revised	We have also experimented with a number of additional tasks but with less empirical success. Tasks we found to be too difï¬ cult include sorting and long integer addition (in base 3 for simplicity), and Repeat- Copy when the input tape is forced to only move forward. While we were able to achieve reasonable performance on the sorting task, the RLâ NTM learned an ad-hoc algorithm and made excessive use of its controller memory in order to sort the sequence. Empirically, we found all the components of the RL-NTM essential to successfully solving these prob- lems. All our tasks are either solvable in under 20,000 parameter updates or fail in arbitrary number of updates. We were completely unable to solve RepeatCopy, Reverse, and Forward reverse with the LSTM controller, but with direct access controller we succeeded. Moreover, we were also unable to solve any of these problems at all without a curriculum (except for short sequences of length 5). We present more traces for our tasks in the supplementary material (together with failure traces).	1505.00521#34	1505.00521#36	1505.00521	[ "1503.01007" ]
1505.00521#36	Reinforcement Learning Neural Turing Machines - Revised	9 # Under review as a conference paper at ICLR 2016 # 10 CONCLUSIONS We have shown that the Reinforce algorithm is capable of training an NTM-style model to solve very simple algorithmic problems. While the Reinforce algorithm is very general and is easily applicable to a wide range of problems, it seems that learning memory access patterns with Reinforce is difï¬ cult. Our gradient checking procedure for Reinforce can be applied to a wide variety of implementations. We also found it extremely useful: without it, we had no way of being sure that our gradient was correct, which made debugging and tuning much more difï¬	1505.00521#35	1505.00521#37	1505.00521	[ "1503.01007" ]
1505.00521#37	Reinforcement Learning Neural Turing Machines - Revised	cult. # 11 ACKNOWLEDGMENTS We thank Christopher Olah for the LSTM ï¬ gure that have been used in the paper, and to Tencia Lee for revising the paper. # REFERENCES Aberdeen, Douglas and Baxter, Jonathan. Scaling internal-state policy-gradient methods for pomdps. In MACHINE LEARNING-INTERNATIONAL WORKSHOP THEN CONFERENCE-, pp. 3â 10, 2002. Ba, Jimmy, Mnih, Volodymyr, and Kavukcuoglu, Koray. Multiple object recognition with visual attention. arXiv preprint arXiv:1412.7755, 2014. Bengio, Yoshua, Louradour, JÂ´erË ome, Collobert, Ronan, and Weston, Jason.	1505.00521#36	1505.00521#38	1505.00521	[ "1503.01007" ]
1505.00521#38	Reinforcement Learning Neural Turing Machines - Revised	Curriculum learning. In Proceedings of the 26th annual international conference on machine learning, pp. 41â 48. ACM, 2009. Graves, Alex, Wayne, Greg, and Danihelka, Ivo. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014a. Graves, Alex, Wayne, Greg, and Danihelka, Ivo. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014b. Grefenstette, Edward, Hermann, Karl Moritz, Suleyman, Mustafa, and Blunsom, Phil. Learning to transduce with unbounded memory. arXiv preprint arXiv:1506.02516, 2015. Joulin, Armand and Mikolov, Tomas. Inferring algorithmic patterns with stack-augmented recurrent nets. arXiv preprint arXiv:1503.01007, 2015. Kohl, Nate and Stone, Peter. Policy gradient reinforcement learning for fast quadrupedal locomotion. In Robotics and Automation, 2004.	1505.00521#37	1505.00521#39	1505.00521	[ "1503.01007" ]

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