| 1: for curriculum stage c = 1 to 2 do |
| 2: | if c=1 then |
| 3: | Set number of agents N = 1 |
| 4: | Initialize Stage 1 main networks Qg := Q = Q1,π := π1 with parameters 0Q1,0π1 |
| 5: | Initialize target networks with 0'π1, 0Q1 |
| 6: | else if c= 2 then |
| 7: | Instantiate N >1 agents |
| 8: | Construct global Qg := Qπ(s,a) = {Q1,Q²},credit function Qc := Qπ(s,am) = |
| {Q1,Q²} and π := {π1,π²} using function augmentation with parameters 0Qg,0Q.,0π |
| 9: | Initialize target networks with 0Q,0Q,0 |
| 10: | Restore values of trained parameters 0Q1,0π1 into the respective subsets of 0Qg, 0Qc, 0π |
| 11: | end if |
| 12: | Set all target network weights to equal main networks weights |
| 13: | Initialize exploration parameter é =_ Estart and empty replay buffer B |
| 14: | for each training episode e = 1 to E do |
| 15: | Assign goal(s) g to agent(s) according to given distribution |
| 16: | Get initial state S1 and observation(s) 01 |
| 17: | fort=1 to T do// execute policies in environment |
| 18: | Sample action at ~ π(atlo;0π,∈) for each agent. |
| 19: | Execute action(s) at,receive {rn}n, St+1,and Ot+1 |
| 20: | Store(St,Ot,ge,at,{r}n,Rg,St+1,Ot+1) into B |
| 21: | St ← St+1,Ot ←Ot+1 |
| 22: 23: | end for |
| if e mod Etrain = O then |
| 24: | for epochs 1...K do// conduct training |
| 25: | Sample minibatch of S transitions (Si,Oi, gi, ai,{r²}n, Si+1,Oi+1) from B |
| 26: | Compute global target for all n: xn = r + γQ(Si+1, ai+1, g²;0Qg)lai+1~π' |
| 27: | |
| 28: | if c =1 then |
| 29: | A(Si,ai)=Q1(si,ai,gi;0Q1)-∑aπ(ailOi,gi)Q1(si,ai,gi;0Q1) |
| 30: | else if c=2 then |
| 31: | ∀m,n ∈ [1.N],compute target y² = r"+γQ(si+1,a+1,gn;0c)la1~π'm |
| 32: | |
| 33: 34: | Am(si,ai）:=Q(si,ai,g0Q）-∑amπ(am）Q(si,am,g;0Q） end if |
| 35: | V0J(π)=∑∑mn=1(Vθlg(on,g))An(si,a) |
| Update policy: 0π ←0π + βVθπJ(π) |
| 36: | |
| 37: | Update all target network parameters using: 0' ← Tθ + (1 - τ)0' |
| 38: | Reset buffer B |
| 39: | end for end if |
| 40: | |
| 41: | If ∈> ∈end,then ∈ ←∈- ∈step |
| 42: | end for |
| 43:end for | |
+
+Off-policy training with a large replay buffer allows RL algorithms to benefit from less correlated transitions (Silver et al., 2014; Lillicrap et al., 2016). The algorithmic modification for off-policy training is to maintain a circular replay buffer that does not reset (i.e. remove line 38), and conduct training (lines 24-41) while executing policies in the environment (lines 17-22). Despite introducing bias in MARL, we found that off-policy training benefited CM3 in SUMO and Checkers.
+
+# B DERIVATIONS
+
+# B.1 PROPOSITION 1
+
+By stationarity and relabeling $t$ , the credit function can be written:
+
+$$
+\partial _ { n } ^ { \pi } ( s , a ^ { m } ) : = \mathbb { E } _ { \pi } \left[ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } , \mathbf { a } _ { t } , g ^ { n } ) \ \middle | \ s _ { 0 } = s , a _ { 0 } ^ { m } = a ^ { m } \right] = \mathbb { E } _ { \pi } \left[ \sum _ { t = 1 } ^ { \infty } \gamma ^ { t - 1 } R ( s _ { t } , \mathbf { a } _ { t } , g ^ { n } ) \ \middle | \ s _ { 1 } = s , a _ { 1 } ^ { m } = s , a _ { 2 } ^ { m } = a ^ { m } \right]
+$$
+
+Using the law of iterated expectation, the credit function satisfies the Bellman expectation equation (2):
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \hat { \mathbf { x } } } ^ { ( 1 ) } ( \mathbf { x } , \mathbf { x } ^ { * } ) - \boldsymbol { \epsilon } _ { \hat { \mathbf { x } } } ^ { ( 2 ) } \frac { \mathcal { S } } { \epsilon _ { \hat { \mathbf { x } } } ^ { ( 3 ) } } \mathcal { S } ( \hat { \mathbf { x } } , \mathbf { x } , \theta ^ { * } ) ^ { 2 } \quad \forall \boldsymbol { \mathcal { S } } _ { \hat { \mathbf { x } } } - \boldsymbol { \mathcal { S } } _ { \hat { \mathbf { x } } } \boldsymbol { \mathcal { S } } _ { \hat { \mathbf { x } } } ^ { ( 5 ) } = \boldsymbol { \mathcal { M } } ^ { * } \bigg ] } \\ & { = - \mathbf { E } _ { \hat { \mathbf { x } } } \Bigg [ \prod ( \hat { \mathbf { x } } ) \alpha _ { \hat { \mathbf { x } } } \hat { \mathbf { x } } ^ { ( 3 ) } \boldsymbol { \mathcal { S } } \Bigg ] + \frac { \mathcal { S } } { \epsilon _ { \hat { \mathbf { x } } } ^ { ( 3 ) } } \frac { \mathcal { S } } { \epsilon _ { \hat { \mathbf { x } } } ^ { ( 4 ) } } \mathbb { E } ( \hat { \mathbf { x } } , \theta _ { 0 } , \boldsymbol { \mathcal { S } } _ { \hat { \mathbf { x } } } ^ { ( 5 ) } \Bigg ] \cdot \boldsymbol { \hat { \mathbf { x } } } _ { \hat { \mathbf { x } } } - \boldsymbol { \hat { \mathbf { x } } } _ { \hat { \mathbf { x } } } \hat { \mathbf { x } } _ { \hat { \mathbf { x } } } ^ { ( 5 ) } = \boldsymbol { \mathcal { B } } ^ { * } \Bigg ] } \\ & = \mathbf { E } _ { \hat { \mathbf { x } } , \hat { \mathbf { x } } ^ { * } } \Bigg [ \exp \Bigg [ \mathrm { i } \Bigg [ \prod ( \hat { \mathbf { x } } ) \alpha _ { \hat { \mathbf { x } } } \hat { \mathbf { x } } _ { \hat { \mathbf { x } } } ^ { ( 2 ) } \Bigg ] + \sum _ { \hat { \mathbf { x } } ^ { \prime } } ^ { \infty } \Bigg [ \prod ( \hat { \mathbf { x } } ) \alpha _ { \hat { \mathbf { x } } } \hat { \mathbf { x } } _ { \hat { \mathbf { x } } ^ { \prime } } \Bigg ] \cdot \boldsymbol { \hat { \mathbf { x } } } _ { \hat { \mathbf { x } } } ^ { ( 3 ) } \Bigg ] \cdot \boldsymbol { \hat { \mathbf { x } } } \\ & \end{array}
+$$
+
+$$
+\begin{array} { r l } & { \quad + \mathbb { E } _ { x } \cdot \frac { \sum _ { i = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } { \sum _ { j = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } \Bigg [ \ln _ { i } \left( \theta _ { i } - \theta _ { i } ^ { n } \right) - \exp ^ { - \alpha \epsilon } - \alpha ^ { \alpha \epsilon } - \alpha ^ { \alpha \epsilon } - \mathcal { E } _ { i } ^ { n } \Bigg ( \frac { \theta _ { i } ^ { n } } { \epsilon } - \alpha ^ { \alpha \epsilon } - \alpha ^ { \alpha \epsilon } \Bigg ) \Bigg ] \ , \ \kappa _ { 1 } = - \kappa _ { 2 } \times \beta - \alpha ^ { \alpha } } \\ & { = \frac { \sum _ { i = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } { \sum _ { j = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } \Bigg [ \exp ^ { - \alpha \epsilon } - \gamma _ { i } \Bigg ( \theta _ { i } ^ { n } \exp ^ { - \alpha \epsilon } - \alpha ^ { \alpha \epsilon } \Bigg ) \Bigg ] \ , \quad \kappa _ { 2 } = - \kappa _ { 2 } \times \beta - \kappa _ { 1 } = - \kappa _ { 2 } ^ { \alpha \epsilon } } \\ & { \quad + \mathbb { E } _ { x } \Bigg ( \alpha ^ { \alpha } - \alpha ^ { \alpha } - \kappa _ { 1 } \Bigg [ \exp ^ { - \alpha \epsilon } - \gamma _ { 1 } \Bigg ( \theta _ { i } ^ { n } \exp ^ { - \alpha \epsilon } - \alpha ^ { \alpha \epsilon } \Bigg ) \Bigg ] \ \Bigg ) \ , \ \kappa _ { 3 } = - \kappa _ { 2 } \times \alpha ^ { \alpha } - \kappa _ { 1 } - \beta \exp ^ { - \alpha \epsilon } - \alpha ^ { \alpha } \Bigg ] \ \ \sin ^ { \alpha - \alpha } } \\ & { = \frac { \sum _ { i = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } { \sum _ { j = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } \Bigg [ \exp ^ { - \alpha \epsilon } - \gamma _ { 1 } \Bigg ( \theta _ { i } ^ { \alpha } - \alpha ^ { \alpha } - \alpha ^ { \alpha } - \beta \exp ^ { - \alpha \epsilon } - \alpha ^ { \alpha } \Bigg ) \Bigg ] \ , \quad \kappa _ { 2 } = \kappa _ { 2 } \times \beta - \kappa _ { 1 } } \\ & \quad + \frac { \sum _ { i = 1 } ^ { n } \mathcal { E } _ { i } ^ { n } } \sum \end{array}
+$$
+
+The goal-specific joint value function is the marginal of the credit function:
+
+$$
+\begin{array} { c } { { V _ { n } ^ { \pi } ( s ) = \mathbb { E } _ { \pi } \left[ \displaystyle \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } , { \mathbf a } _ { t } , g ^ { n } ) \mid s _ { 0 } = s \right] } } \\ { { { } } } \\ { { = \mathbb { E } _ { a _ { 0 } ^ { m } \mid s _ { 0 } , \pi } \left[ \mathbb { E } _ { \pi } \left[ \displaystyle \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } , { \mathbf a } _ { t } , g ^ { n } ) \mid s _ { 0 } = s , a _ { 0 } ^ { m } = a ^ { m } \right] \Bigm | s _ { 0 } = s \right] } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { a ^ { m } } \pi ( a ^ { m } | o ^ { m } ( s ) , g ^ { m } ) Q _ { n } ^ { \pi } ( s , a ^ { m } ) } } \end{array}
+$$
+
+The credit function can be expressed in terms of the goal-specific action-value function:
+
+$$
+\begin{array} { c } { { { \displaystyle V _ { n } ^ { \pi } ( s ) = \sum _ { a ^ { \prime \prime } } \pi ( a ^ { \prime \prime } \vert o ^ { m } , g ^ { m } ) Q _ { n } ^ { \pi } ( s , a ^ { m } ) } } } \\ { { { \displaystyle V _ { n } ^ { \pi } ( s ) = \sum _ { \bf a } \pi ( { \bf a } \vert s , { \bf g } ) Q _ { n } ^ { \pi } ( s , { \bf a } ) } } } \\ { { { \displaystyle = \sum _ { a ^ { \prime \prime } } \sum _ { a ^ { - \prime \prime } } \pi ( a ^ { m } \vert o ^ { m } , g ^ { m } ) \pi ( a ^ { - m } \vert s , g ^ { - m } ) Q _ { n } ^ { \pi } ( s , ( a ^ { m } , a ^ { - m } ) ) } } } \\ { { { \displaystyle \Rightarrow Q _ { n } ^ { \pi } ( s , a ^ { m } ) = \sum _ { a ^ { - m } } \pi ( a ^ { - m } \vert s , g ^ { - m } ) Q _ { n } ^ { \pi } ( s , { \bf a } ) } } } \end{array}
+$$
+
+# B.2 PROPOSITION 2
+
+First we state some elementary relations between global functions $V _ { n } ^ { \pi } ( s )$ and $Q _ { n } ^ { \pi } ( s , \mathbf { a } )$ . These carry over directly from the case of an MDP, by treating the joint policy $\pi$ as as an effective “single-agent” policy and restricting attention to a single goal $g ^ { n }$ (standard derivations are included at the end of this section).
+
+$$
+\begin{array} { c } { { Q _ { n } ^ { \pi } ( s , \mathbf { a } ) = R ( s , \mathbf { a } , g ^ { n } ) + \gamma \displaystyle \sum _ { s ^ { \prime } } P ( s ^ { \prime } | s , \mathbf { a } ) V _ { n } ^ { \pi } ( s ^ { \prime } ) } } \\ { { { \cal V } _ { n } ^ { \pi } ( s ) = \displaystyle \sum _ { \mathbf { a } } \pi ( \mathbf { a } | s , \mathbf { g } ) Q _ { n } ^ { \pi } ( s , \mathbf { a } ) } } \end{array}
+$$
+
+We follow the proof of the policy gradient theorem (Sutton et al., 2000):
+
+$$
+\begin{array} { r l r } { \nabla _ { \phi } V _ { n } ^ { \prime \prime } ( s ) } & { = \nabla _ { s } \sum _ { \phi } \pi ( s | s , \phi , g | ^ { 2 } s , \theta ) , } \\ & { } & { = \displaystyle \sum _ { u = 1 } ^ { N } [ \nabla _ { \phi } \pi ( s , s , \phi ) | Q _ { u } ^ { \top \prime } ( s , a ) + \pi ( s , s , g ) \nabla _ { \phi } Q _ { u } ^ { \top \prime } ( s , a ) } \\ & { } & { \displaystyle - \sum _ { u = 1 } ^ { N } [ \nabla _ { \phi } \pi ( s , s , e ) | Q _ { u } ^ { \top \prime } ( s , a ) + \pi ( s , s , g ) \nabla _ { \phi } \pi ( B _ { u } ^ { \top } , a ) + \gamma \sum _ { u = 1 } ^ { N } P _ { u } ^ { \top \prime } ( s , a ) V _ { n } ^ { \top \prime } ( s ^ { \top } ) } \\ & { } & { \displaystyle - \sum _ { u = 1 } ^ { N } [ \nabla _ { \phi } \pi ( s , s , e ) | Q _ { u } ^ { \top \prime } ( s , a ) + \pi ( s , g ) \mathbb { E } _ { \phi } ^ { \top \prime } \pi } \\ & { } & { \displaystyle - \sum _ { u = 1 } ^ { N } [ \nabla _ { \phi } \pi ( s , s , e ) | Q _ { u } ^ { \top \prime } ( s , a ) + \pi ( s , g ) \mathbb { E } _ { \phi } ^ { \top \prime } \pi } \\ & { } & { \displaystyle - \sum _ { u = 1 } ^ { N } \frac { s } { s } ] \sum _ { u = 1 } ^ { N } \gamma _ { u } \mathcal { E } _ { u } ^ { \top \prime } ( s , a ) , \quad \mathrm { ~ o ~ y ~ } \theta ^ { \top } ( s , a ) \mathrm { ~ ( ~ s ~ p r o s s i v e ~ y r o ~ a n d i n g ) ~ } } \\ & { } & \displaystyle \sum _ { u = 1 } ^ { N } \frac { s } { s } \sum _ { u = 1 } ^ { N } \gamma _ { u } \mathcal { E } _ { u } ^ { \top } ( s , a ) \times ( \nabla _ { \phi } \pi ( s , e ) | Q _ { u } ^ { \top } ( s , a ) | Q _ { u } ^ \top \prime \end{array}
+$$
+
+We can replace $Q _ { n } ^ { \pi } ( s , \mathbf { a } )$ by the advantage function $A _ { n } ^ { \pi } ( s , \mathbf { a } ) : = Q _ { n } ^ { \pi } ( s , \mathbf { a } ) - V _ { n } ^ { \pi } ( s )$ , which does not change the expectation in Equation (9) because:
+
+$$
+\begin{array} { l } { { \displaystyle \mathbb { E } _ { \boldsymbol \pi } \left[ \nabla _ { \boldsymbol \theta } \log \boldsymbol \pi ( { \bf a } | s , { \bf g } ) V _ { n } ^ { \boldsymbol \pi } ( s ) \right] = \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) \sum _ { \bf a } \boldsymbol \pi ( { \bf a } | s , { \bf g } ) \nabla _ { \boldsymbol \theta } \log \boldsymbol \pi ( { \bf a } | s , { \bf g } ) V _ { n } ^ { \boldsymbol \pi } ( s ) } } \\ { { \displaystyle \qquad = \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) V _ { n } ^ { \boldsymbol \pi } ( s ) \nabla _ { \boldsymbol \theta } \sum _ { \bf a } \boldsymbol \pi ( { \bf a } | s , { \bf g } ) = 0 } } \end{array}
+$$
+
+So the gradient (9) can be written
+
+$$
+\nabla _ { \theta } J _ { n } ( \pi ) = \mathbb { E } _ { \pi } \Big [ \big ( \nabla _ { \theta } \sum _ { m = 1 } ^ { N } \log \pi ( a ^ { m } | o ^ { m } , g ^ { m } ) \big ) \left( Q _ { n } ^ { \pi } ( s , \mathbf { a } ) - V _ { n } ^ { \pi } ( s ) \right) \Big ]
+$$
+
+Recall that from (3), for any choice of agent label $k \in [ 1 . . N ]$ :
+
+$$
+V _ { n } ^ { \pi } ( s ) = \sum _ { a ^ { k } } \pi ( a ^ { k } | o ^ { k } , g ^ { k } ) Q _ { n } ^ { \pi } ( s , a ^ { k } )
+$$
+
+Then substituting (3) into (10):
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } J _ { n } ( \boldsymbol { \pi } ) = \mathbb { E } _ { \boldsymbol { \pi } } \left[ \left( \nabla _ { \theta } \displaystyle \sum _ { m = 1 } ^ { N } \log \pi ( a ^ { m } | o ^ { m } , g ^ { m } ) \right) A _ { n , k } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) \right] } \\ & { A _ { n , k } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) : = Q _ { n } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) - \displaystyle \sum _ { \hat { a } ^ { k } } \pi ( \hat { a } ^ { k } | o ^ { k } , g ^ { k } ) Q _ { n } ^ { \boldsymbol { \pi } } ( s , \hat { a } ^ { k } ) } \end{array}
+$$
+
+Now notice that the choice of $k$ in (13) is completely arbitrary, since (3) holds for any $k \in [ 1 . . N ]$ Therefore, it is valid to distribute $A _ { n , k } ^ { \pi } ( s , \mathbf { a } )$ into the summation in (12) using the summation index $m$ instead of $k$ . Further summing (12) over all $n$ , we arrive at the result of Proposition 2:
+
+$$
+\begin{array} { c l } { { } } & { { \displaystyle \nabla _ { \theta } J ( \boldsymbol { \pi } ) = \mathbb { E } _ { \boldsymbol { \pi } } \Big [ \sum _ { m = 1 } ^ { N } \sum _ { n = 1 } ^ { N } \Big ( \nabla _ { \theta } \log \pi ( a ^ { m } | o ^ { m } , g ^ { m } ) \Big ) A _ { n , m } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) \Big ] } } \\ { { } } & { { \displaystyle A _ { n , m } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) : = Q _ { n } ^ { \boldsymbol { \pi } } ( s , \mathbf { a } ) - \sum _ { \hat { a } ^ { m } } \pi ( \hat { a } ^ { m } | o ^ { m } , g ^ { m } ) Q _ { n } ^ { \boldsymbol { \pi } } ( s , \hat { a } ^ { m } ) } } \end{array}
+$$
+
+The relation between $V _ { n } ^ { \pi } ( s )$ and $Q _ { n } ^ { \pi } ( s , \mathbf { a } )$ in (7) and (8) are derived as follows:
+
+$$
+\begin{array} { r l } & { \frac { \gamma } { \mu _ { 1 } } ( x _ { 1 } ) = - \mathbb { E } _ { \mu _ { 1 } } \left[ \sum _ { j = 1 } ^ { n } y _ { 1 } ( x _ { 1 } , x _ { j + 1 } , y _ { 1 } ) \right] \left[ \mu _ { 1 } - \lambda _ { 1 } - x _ { 2 } - \lambda _ { 1 } \right] } \\ & { \quad - \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 1 } ( x _ { 1 } , x _ { 1 } ) \right] \left[ \sum _ { j = 1 } ^ { n } y _ { 1 } ( x _ { 1 } , x _ { j } ) \right] \left[ \lambda _ { 1 } - x _ { 2 } - x _ { 1 } \right] } \\ & { \quad - \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 1 } ( x _ { 1 } , x _ { 2 } ) \right] + \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] \left[ \lambda _ { 1 } - x _ { 2 } - x _ { 1 } \right] } \\ & { \quad - \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] + \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 1 } - x _ { 2 } \right] \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] \left[ \mu _ { 1 } - x _ { 2 } \right] \lambda _ { 2 } - \lambda _ { 1 } - x _ { 2 } \right] \left[ \mu _ { 1 } - \lambda _ { 2 } \lambda _ { 1 } \right] } \\ & { \quad - \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 1 } ( x _ { 1 } , x _ { 2 } ) \right] + \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] \mathbb { E } _ { \lambda } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] \left[ \lambda _ { 1 } - x _ { 2 } \right] } \\ & \quad - \mathbb { E } _ { \mu _ { 1 } } \lambda _ { 1 } - x _ { 2 } \lambda _ { 1 } ^ { \prime } + \mathbb { E } _ { \mu _ { 1 } } \left[ \lambda _ { 2 } ( x _ { 1 } , x _ { 2 } ) \right] \mathbb { E } _ { \lambda } \left[ \lambda _ { 1 } ( x _ { 1 } , \end{array}
+$$
+
+# C VARIANCE
+
+C.1 VARIANCE OF COMA GRADIENT.
+
+Let $Q : = Q ^ { \pi } ( s , \mathbf { a } , \mathbf { g } )$ denote the centralized Q function, let $\pi ( a ^ { n } ) : = \pi ( a ^ { n } | o ^ { n } , g ^ { n } ) \quad$ denote a single agent’s policy, and let $\pi ( a ^ { - n } ) : = \pi ( a ^ { - n } | o ^ { - n } , g ^ { - n } ) $ denote the other agents’ joint policy.
+
+In cooperative multi-goal MARL, the direct application of COMA has the following gradient.
+
+$$
+\begin{array} { c } { { \displaystyle \nabla _ { \theta } J = \mathbb E \Big [ \sum _ { n } \nabla _ { \theta } \log \pi ( a ^ { n } | o ^ { n } , g ^ { n } ) \big ( Q - b _ { n } ( s , a ^ { - n } , \mathbf { g } ) \big ) \Big ] } } \\ { { \displaystyle b _ { n } \big ( s , a ^ { - n } , \mathbf { g } \big ) : = \sum _ { \hat { a } ^ { n } } \pi \big ( \hat { a } ^ { n } | o ^ { n } , g ^ { n } \big ) Q ^ { \pi } \big ( s , \hat { a } ^ { n } , a ^ { - n } , \mathbf { g } \big ) } } \end{array}
+$$
+
+Define the following:
+
+$$
+\begin{array} { r l } & { z _ { n } : = \nabla _ { \theta } \log \pi ( a ^ { n } | o ^ { n } , g ^ { n } ) } \\ & { f _ { n } : = \nabla _ { \theta } \log \pi ( a ^ { n } | o ^ { n } , g ^ { n } ) \big ( Q - b _ { n } ( s , a ^ { - n } ) \big ) = z _ { n } \big ( Q - b _ { n } ( s , a ^ { - n } , \mathbf { g } ) \big ) } \end{array}
+$$
+
+Define $\begin{array} { r } { M _ { n m } \ \mathrel { \mathop : } = \ \mathbb { E } _ { \pi } [ f _ { n } ] ^ { T } \mathbb { E } _ { \pi } [ f _ { m } ] } \end{array}$ and let $\begin{array} { r c l } { { M } } & { { : = } } & { { \sum _ { n , m } M _ { n m } } } \end{array}$ . Then we have $\begin{array} { r l } { M _ { n m } } & { { } = } \end{array}$ $\mathbb { E } _ { \pmb { \pi } } [ z _ { n } Q ] ^ { T } \mathbb { E } _ { \pmb { \pi } } [ z _ { m } Q ]$ since
+
+$$
+\begin{array} { l } { { \displaystyle { \mathbb E } _ { \boldsymbol \pi } [ z _ { n } b _ { n } ] = { \mathbb E } _ { \boldsymbol \pi } \Big [ \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) \sum _ { \mathbf { a } } \boldsymbol \pi ( \mathbf a | s , \mathbf g ) \nabla _ { \boldsymbol \theta } \log \boldsymbol \pi ( a ^ { n } | \boldsymbol \sigma ^ { n } , \boldsymbol g ^ { n } ) b _ { n } ( s , a ^ { - n } , \mathbf g ) \Big ] } } \\ { ~ = \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) \sum _ { a ^ { - n } } \boldsymbol \pi ^ { - n } ( a ^ { - n } | \boldsymbol \sigma ^ { - n } , \boldsymbol g ^ { - n } ) \sum _ { a ^ { n } } \boldsymbol \pi ( a ^ { n } | \boldsymbol \sigma ^ { n } , \boldsymbol g ^ { n } ) \nabla _ { \boldsymbol \theta } \log \boldsymbol \pi ( a ^ { n } | \boldsymbol \sigma ^ { n } , \boldsymbol g ^ { n } ) b _ { n } ( s , a ^ { - n } , \mathbf g ) } \\ { ~ = \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) \sum _ { a ^ { - n } } \boldsymbol \pi ^ { - n } ( a ^ { - n } | \boldsymbol \sigma ^ { - n } , \boldsymbol g ^ { - n } ) \sum _ { a ^ { n } } \nabla _ { \boldsymbol \theta } \boldsymbol \pi ( a ^ { n } | \boldsymbol \sigma ^ { n } , \boldsymbol g ^ { n } ) b _ { n } ( s , a ^ { - n } , \mathbf g ) } \\ { ~ = \sum _ { s } \rho ^ { \boldsymbol \pi } ( s ) \sum _ { a ^ { - n } } \boldsymbol \pi ^ { - n } ( a ^ { - n } | \boldsymbol \sigma ^ { - n } , \boldsymbol g ^ { - n } ) b _ { n } ( s , a ^ { - n } , \mathbf g ) \nabla _ { \boldsymbol \theta } \sum _ { a ^ { n } } \boldsymbol \pi ( a ^ { n } | \boldsymbol \sigma ^ { n } , \boldsymbol g ^ { n } ) = \boldsymbol 0 } \end{array}
+$$
+
+Since the COMA gradient is $\mathbb { E } _ { \pi } [ \sum _ { n = 1 } ^ { N } f _ { n } ]$ . its variance can be derived to be (Wu et al., 2018):
+
+$$
+\begin{array} { r l } { { \mathrm { V a r } ( \sum _ { n = 1 } ^ { N } f _ { n } ) = \sum _ { n } \mathbb { E } _ { \boldsymbol \pi } [ z _ { n } ^ { T } z _ { n } Q ^ { 2 } - 2 b _ { n } z _ { n } ^ { T } z _ { n } Q + b _ { n } ^ { 2 } z _ { n } ^ { T } z _ { n } ] } \quad } & { { } } \\ { + \sum _ { n } \sum _ { m \neq n } \mathbb { E } _ { \boldsymbol \pi } [ z _ { n } ^ { T } z _ { m } ( Q - b _ { n } ) ( Q - b _ { m } ) ] - M } \end{array}
+$$
+
+# C.2 VARIANCE OF THE CM3 GRADIENT
+
+For convenience, let $Q _ { n } : = Q _ { n } ^ { \pi } ( s , \mathbf { a } ) = Q ^ { \pi } ( s , \mathbf { a } , g ^ { n } )$ denote the global Q function for goal $g ^ { n }$ , and let $\pi ( a ^ { m } ) : = \pi ( a ^ { m } | o ^ { m } , g ^ { m } )$ . The CM3 gradient can be rewritten as
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } J ( \pmb { \pi } ) = \mathbb { E } _ { \pmb { \pi } } \Big [ \displaystyle \sum _ { n = 1 } ^ { N } \displaystyle \sum _ { m = 1 } ^ { N } \nabla _ { \theta } \log \pi ( a ^ { m } ) \big ( Q _ { n } - b _ { n m } ( s ) \big ) \Big ] } \\ & { b _ { n m } ( s ) : = \displaystyle \sum _ { \hat { a } ^ { m } } \pi ( \hat { a } ^ { m } ) Q _ { n } ^ { \pmb { \pi } } ( s , \hat { a } ^ { m } ) } \end{array}
+$$
+
+As before, $z _ { m } : = \nabla _ { \boldsymbol { \theta } } \log \pi ( a ^ { m } )$ . Define $h _ { n m } : = z _ { m } ( Q _ { n } - b _ { n m } ( s ) )$ and let $\begin{array} { r } { h _ { n } : = \sum _ { m } h _ { n m } } \end{array}$ . Then the variance is
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { V a r } ( \sum _ { n } h _ { n } ) = \sum _ { n } \mathrm { V a r } ( h _ { n } ) + \sum _ { n } \sum _ { m \neq n } \mathrm { C o v } ( h _ { n } , h _ { m } ) } \\ { \displaystyle = \sum _ { n } \biggl ( \sum _ { m } \mathrm { V a r } ( h _ { n m } ) + \sum _ { m } \sum _ { k \neq m } \mathrm { C o v } ( h _ { n m } , h _ { n k } ) \biggr ) + \sum _ { n } \sum _ { m \neq n } \mathrm { C o v } ( h _ { n } , h _ { m } ) } \end{array}
+$$
+
+# D EXAMPLE OF GREEDY INITIALIZATION FOR MARL EXPLORATION
+
+A greedy initialization can provide significant improvement in multi-agent exploration versus naïve random exploration, as shown by a simple thought experiment. Consider a two-player MG defined by a $4 \times 3$ gridworld with unit actions (up, down, left, right). Agent $A$ starts at (1,2) with goal (4,2), while agent $B$ starts at (4,2) with goal (1,2). The greedy policy for each agent in MG is to move horizontally toward its target, since this is optimal in the induced $\mathbf { M }$ (when the other agent is absent). Case 1: Suppose that for $\epsilon \in ( 0 , 1 )$ , $A$ and $B$ follow greedy policies with probability $1 - \epsilon$ , and take random actions $( p ( a ) = 1 / 4 )$ ) with probability $\epsilon$ . Then the probability of a symmetric optimal trajectory is $P ( \mathrm { c o o p e r a t e } ) = 2 \epsilon ^ { 2 } ( ( 1 - \epsilon ) + \epsilon / 4 ) ^ { 8 }$ . For $\epsilon = 0 . 5$ , $P$ (cooperate) $\approx 0 . 0 1$ . Case 2: If agents execute uniform random exploration, then $P ( { \mathrm { c o o p e r a t e } } ) = 3 . 0 5 { \mathrm { e } } - 5 \ll 0 . 0 1$ .
+
+# E GENERALIZATION
+
+Table 1: Test performance with heavy traffic on difficult initial and goal lanes configurations
+
+| Item | D(G(.)) | Utterance |
| MTC |
| Context_0 | NA | perhaps <person> had a word with the man upstairs. |
| Context_1 | NA | a word ?i’m sure by now he’s engineered a hostile takeover . |
| Response_0 0.996 | | <person>,i know what you’re saying， <person>,that’s not whati’m saying. |
| Response-1 0.991 | | <person>,i know .i was just about to help |
| Response_2 0.315 | | the guy. <person>,i'm sorry. |
| Response_3 | 0.203 | <person>,i’ma litle out . |
| Context_0 | NA | says he wanted food.<person>.he wanted the gold . |
| Context_1 | NA | how’s he going to want the gold ? he couldn ' t even know we had it. |
| Response_0 0.998 | | <person>,i know .but it’s not him,it’s the only way he’s got it all figured |
| Response_1 0.981 | | <person>,i know．but i have to tell you . these things are really stupid and you think i |
| Response_2 0.690 | | was wrong ? <person>,i'm sure he did. |
| Response-3 0.314 | | <person>,i'm not sure |
| UDC |
| Context_0 | NA | The netboot one is suppose to download pack- |
| Context_1 | NA | ages from the net. like the ones to be installed? or the installed to be run? |
| Response_O 0.993 | | you don ’t need to install the whole system , just install the ubuntu installer |
| Response-1 0.952 | | you can install the ubuntu installer from the |
| Response_2 0.749 | | ubuntu menu I'm not sure ,I don ’t know . |
| Response_3 | 0.184 | you can |
| Response_4 | 0.003 | youa. 1. |
| Context_0 | NA | DJones: update manager won't detect 12.04.1 as a new version if you are already running 12.04, because 12.04.1 = 12.04 + lots of pack- |
| Response_0 0.991 | | age updates did you try a clean install of the latest version ? |
| Response_1 0.981 | | try installing the latest_UNK and see if it works |
| Response_2 0.615 | | I’m not sure you have a problem.. |
| Response_3 0.191 | | try sudo apt-get remove the package that is not |
| Response_4 0.002 | | installed try the-UNK.1.1.1.1.1.-UNK.deb |
+
+Table 3: Sample responses of HRED, VHRED and hredGAN.
+
+| Movie | |
| Context_0 HRED VHRED | perhaps <person> had a word with the man upstairs . i'm sorry,<person>. <person<'tdo it. hredGAN_w<person>,i’m not gon na be okay.i just don ’t think it’s a |
| Context_1 HRED VHRED | good idea. hredGAN.u <person>,icause it was allthe way to you. a word ?i'm sure by now he’s engineered a hostile takeover . i'm sorry,<person>. <person>'tdo it. |
| hredGAN_w<person>,i know what you ’re saying，<person>,that’s not whati’m saying. hredGAN_u <person>,i have to tell youi’m not the same. |
| Context_0 HRED VHRED | <person>’s. i'm sorry,<person>. |
| <person>'tdo it . hredGAN_w <person>,i’m sure it’s the only thing i ever imagined . |
| |
| hredGAN_u <person>,i'l be right there . |
| |
| Context_1 | |
| can i ask you something about her ? |
| HRED | idon'tknow . |
| VHRED: | you’re not going to be here . |
| hredGAN_widon’tknow.i thinki’ve heard of it. |
| hredGAN-ui’m sorry,ican 't. |
| Ubuntu | |
| |
| Context_0 | this is Dacs i am in i used 'sudo statx and it let me in |
| HRED | what's the problem ? |
| VHRED | I don't know what you want to do .if you want to do it,you need to |
| do it to do it in the terminal ? |
| hredGAN-w you don’t have to do anything，just type in the terminal and it |
| should be to find the right device |
| hredGAN_u you can have a look at the output of the command . |
| Context.1 | good deal... cat /etc/X11/default-display-manager |
| HRED | what's the problem ? |
| VHRED | do you know what you want to do ? |
| hredGAN_w I’m trying to figure a command that I can find to find out the file |
| that Ican find in the file" |
| hredGAN-u Idon 't see the point,but I’m not sure how to do that . |
| Context_2 | /usr/sbin/lightdmhttp://paste.ubuntu.com/1286224/ |
| /etc/X1/xorg.conf |
| HRED | what 's the problem ? |
| VHRED | |
| is there a way to do that in the terminal ? |
| hredGAN_w did you just type\”sudo mount -a\”? |
| |
| |
| |
| hredGAN-u i have no idea,i just installed ubuntu and i have no idea how to do |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| that |
| |
| |
+
+Qualitative Generator Performance: The results of the human evaluation is reported in the last column of Table 1. The human evaluation agrees largely with the automatic evaluation. The hredGAN w performs best on both datasets although the gap is more on the MTC than on the UTC. This implies that the improvement of HRED with adversarial generation is better than with variational generation (VHRED). In addition, looking at the actual samples from the generator outputs in Table 6 shows that hredGAN especially hredGAN w performs better than (V)HRED. While other models produce short and generic utterances, hredGAN w mostly yields informative responses. For example, in the first dialogue in Table 6, when the speaker is sarcastic about ”the man upstairs”, hredGAN w responds with the most coherent utterance with respect to the dialogue history. We see similar behavior across other samples. We also note that although hredGAN u’s responses are the longest on Ubuntu (in line with the NASL score), the responses are less informative compared to hredGAN w resulting into a lower human evaluation score. We reckon this might be due to a mismatch between utterance-level noise and word-level discrimination or lack of capacity to capture the data distribution using single noise distribution. We hope to investigate this further in the future.
+
+Discriminator Performance: Although only hredGAN uses a discriminator, the observed discriminator behavior is interesting. We observe that the discriminator score is generally reasonable with longer, more informative and more persona-related responses receiving higher scores as shown in Table 2. It worth to note that this behavior, although similar to the behavior of a human judge is learned without supervision. Moreover, the discriminator seems to have learned to assign average score to more frequent or generic responses such as “I don’t know”, “I’m not sure” and so on, and high score to rearer answers. That’s why we sample a modified noise distribution during inference so that the generator can produce rearer utterances that will be scored high by the discriminator.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we have introduced an adversarial learning approach that addresses response diversity and control of generator outputs, using an HRED-derived generator and discriminator. The proposed system outperforms existing state-of-the-art (V)HRED models for generating responses in multiturn dialogue with respect to automatic and human evaluations. The superiority of the adversarial generation (hredGAN) over the variational generation (VHRED) is in line with other generative models employing these approaches. Our analysis also concludes that the word-level noise injection seems to perform better in general.
+
+While this is a good starting point, we recognize the need to explore further improvements to the proposed adversarial framework: In the future, we hope to: explore which noise level works with which discrimination level; consider a multi-resolution discriminator with combined word- and utterancelevel discriminations; and explore further tuning of the generator and discriminator models.
+
+# REFERENCES
+
+D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of International Conference of Learning Representation (ICLR 2015), 2015.
+R. E. Banchs. Movie-dic: A movie dialogue corpus for research and development. In Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics, pp. 203–207, 2012.
+E. Bruni and R. Fernndez. Adversarial evaluation for open-domain dialogue generation. In Proceedings of the 18th Annual SIGdial Meeting, 2018.
+T. Che, Y. Li, R. Zhang, R. D. Hjelm, W. Li, Y. Song, and Y. Bengio. Maximum-likelihood augmented discrete generative adversarial networks. In arXiv preprint arXiv:1702.07983, 2017.
+K. Cho, B. Merrienboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. In Proceedings of International Conference of Learning Representation (ICLR 2015), pp. 1724– 1734, 2014.
+X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In International conference on artificial intelligence and statistics, 2010.
+I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In Proceedings of Advances in Neural Information Processing Systems (NIPS 2014), 2014.
+P. Isola, J. Y. Zhu, T. Zhou, and A. A. Efros. Image-to-image translation with conditional adversarial networks. In Conference on Computer Vision and Pattern Recognition (CVPR, 2017), 2017.
+S. Jean, K. Cho, R. Memisevic, and Y. Bengio. On using very large target vocabulary for neural machine translation. In arXiv preprint arXiv:1412.2007, 2015.
+A. Kannan and O. Vinyals. Adversarial evaluation of dialogue models. In arXiv preprint arXiv:1701.08198v1, 2017.
+A. Lamb, A. Goyah, Y. Zhang, S. Zhang, A. Courville, and Y. Bengio. Professor forcing: A new algorithm for training recurrent networks. In Proceedings of Advances in Neural Information Processing Systems (NIPS 2016), 2016.
+J. Li, M. Galley, C. Brockett, J. Gao, and B. Dolan. A diversity-promoting objective function for neural conversation models. In Proceedings of NAACL-HLT, 2016a.
+J. Li, W. Monroe, A. Ritter, M. Galley, J. Gao, and D. Jurafsky. Deep reinforcement learning for dialogue generation. In arXiv preprint arXiv:arXiv:arXiv:1606.01541v4, 2016b.
+J. Li, W. Monroe, T. Shi, A. Ritter, and D. Jurafsky. Adversarial learning for neural dialogue generation. In arXiv preprint arXiv:1701.06547, 2017.
+C. Y. Lin. Rouge: a package for automatic evaluation of summaries. In Proceedings of the Workshop on Text Summarization Branches Out, 2014.
+C. Liu, R. Lowe, I. V. Serban, M. Noseworthy, L. Charlin, and J. Pineau. How not to evaluate your dialogue system: An empirical study of unsupervised evaluation metrics for dialogue response generation. In Proceedings of EMNLP, pp. 2122–2132, 2016.
+M. T. Luong, I. Sutskever, Q. V. Le, O. Vinyals, and W. Zaremba. Addressing the rare word problem in neural machine translation. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics, 2015.
+K. Papineni, S. Roukos, T. Ward, and W. Zhu. Bleu: A method for automatic evalution of machine translation. In Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics, pp. 311–318, 2002.
+I. Serban, A. Sordoni, Y. Bengio, A. Courville, and J. Pineau. Building end-to-end dialogue systems using generative hierarchical neural network models. In Proceedings of The Thirtieth AAAI Conference on Artificial Intelligence (AAAI 2016), pp. 3776–3784, 2016.
+I. V. Serban, T. Klinger, G. Tesauro, K. Talamadupula, B. Zhou, Y. Bengio, and A. Courville. Multiresolution recurrent neural networks: An application to dialogue response generation. In Proceedings of The Thirty-first AAAI Conference on Artificial Intelligence (AAAI 2017), 2017a.
+I. V. Serban, A. Sordoni, R. Lowe, L. Charlin, J. Pineau, A. Courville, and Y. Bengio. A hierarchical latent variable encoder-decoder model for generating dialogue. In Proceedings of The Thirty-first AAAI Conference on Artificial Intelligence (AAAI 2017), 2017b.
+I. Sutskever, O. Vinyals, and Q. Le. Sequence to sequence learning with neural networks. In Proceedings of Advances in Neural Information Processing Systems (NIPS), pp. 3104–3112, 2014.
+O. Vinyals and Q. Le. A neural conversational model. In Proceedings of ICML Deep Learning Workshop, 2015.
+R. J. Williams and D. Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural computation, 1(2):270–280, 1989.
+C. Xing, W. Wu, Y. Wu, M. Zhou, Y. Huang, and W. Ma. Hierarchical recurrent attention network for response generation. In arXiv preprint arXiv:1701.07149, 2017.
+Z. Xu, B. Liu, B. Wang, S. Chengjie, X. Wang, Z. Wang, and C. Qi. Neural response generation via gan with an approximate embedding layer. In EMNLP, 2017.
+L. Yu, W. Zhang, J. Wang, and Y. Yu. Seqgan: sequence generative adversarial nets with policy gradient. In Proceedings of The Thirty-first AAAI Conference on Artificial Intelligence (AAAI 2017), 2017.
+Y. Zhang, Z. Gan, K. Fan, Z. Chen, R. Henao, D. Shen, and L. Carin. Adversarial feature matching for text generation. In arXiv preprint arXiv:1706.03850, 2017.
+Y. Zhang, M. Galley, J. Gao, Z. Gan, X. Li, C. Brockett, and B. Dolan. Generating informative and diverse conversational responses via adversarial information maximization. In arXiv preprint arXiv:arXiv:1809.05972v5, 2018.
+
+# APPENDIX
+
+# 6 RELATED WORK
+
+Our work is related to end-to-end neural network–based open domain dialogue models. Most neural dialogue models use transduction frameworks adapted from neural machine translations (Sutskever et al., 2014; Bahdanau et al., 2015). These $\mathtt { S e q 2 S e q }$ networks are trained end-to-end with MLE criteria using large corpora of human-to-human conversation data. Others use GAN’s discriminator as a reward function in a reinforcement learning framework (Yu et al., 2017) and in conjunction with MLE (Li et al., 2017; Che et al., 2017). Zhang et al. (2017) explored the idea of GAN with a feature matching criterion. Xu et al. (2017) and Zhang et al. (2018) employed GAN with an approximate embedding layer as well as with adversarial information maximization respectively to improve Seq2Seq’s diversity performance.
+
+Still, $\mathtt { S e q 2 S e q }$ models are limited in their ability to capture long temporal dependencies in multiturn conversation. Although, Li et al. (2016b) attempted to optimize a pair Seq2Seq models for multi-turn dialogue, the multi-turn objective is only applied at inference and not used for actual model training. Hence, the introduction of HRED models (Serban et al., 2016; 2017a;b; Xing et al., 2017) for modeling dialogue response in multi-turn conversations. However, these HRED models suffer from lack of diversity since they are trained with only MLE criteria. On other hand, adversarial system has been used for evaluating open domain dialogue models (Bruni & Fernndez, 2018; Kannan & Vinyals, 2017). Our work, hredGAN is closest to the combination of HRED generation models (Serban et al., 2016) and adversarial evaluation (Kannan & Vinyals, 2017).
+
+Table 4: Generator Performance: HRED vs. HRED $^ +$ Attn
+
+| Movie | |
| Context.0 HRED | <person>i've raised my rate.imight even cut my sessions to forty minutes |
| VHRED | i’m sorry,<person>. <person>,<person>,<person>,<person> |
| hredGAN_w | <person>,i’m sorry,buti’m not sure if you’re really a very good guy |
| hredGAN-u | i' ll never see them again,<person>. |
| i'm going to handle this exactly the way your father would have handled this .i’ve got it solved . |
| Context_0 Context.1 | you’ve got it solved ?where is she ？ |
| HRED | idon't know. |
| VHRED | <person>,,i’m sorry. |
| hredGAN_w | to the end of äabeautiful woman who willbe very happy. |
| hredGAN_u | i'm sorry,ijust didn ’t know- |
| Context_0 | ofcoursenot！jperson’slikeadaughtertome.herfatheranditaughtatuniversitytogetheragoodman,butheneverlistened. |
| HRED | i'm sorry,<person>. |
| VHRED hredGAN-w | <person>,mr.<person>. |
| hredGAN-u | <person>,i know it’s beenallsix years sinceitold himabout you. i'm sure you didn’t,buti would have donealitle more. |
| |
| Context_0 Context_1 | ofcourse not！iperson’slikeadaughtertome.erfatheranditaughtatuniversitytogether.agoodman,buthe neverlistened. |
| HRED | neither did i. i'm sorry，,<person>. |
| VHRED | <person>,you'rea<unk>. |
| hredGAN-w | <person>,i know you’re think about it. think it'I be time,right ? |
| hredGAN-u | i'Ilbe thefirstone.ifitdidyou,you'lleit. |
| Context_0 | come on,<person>,we could have a great week up in vermont . |
| Context.1 | i see what you mean.but wednesday we’ve got to be back in new york . |
| HRED VHRED | <person>. <person>,idon't know what you’re going to do. |
| hredGAN_w | <person>,iknow .i'm kind of busy here. |
| hredGAN_u | i'm sorry,ican 't. |
| Context_0 | isabel ？ |
| Context_1 | hi. |
| HRED | that’s whati’m talking about . |
| VHRED: | you're not going to be a<unk>,<person>. |
| hredGAN_w | <person>？ |
| hredGAN_u | what do you want to know ? |
| Context.0 | you understand mr.<person> is constantly barraged with nuisance litigation and irs audits |
| Context_1 | of course . |
| HRED | that’s right,<person>. |
| VHRED: | <person>,you’re going to bea<unk>of the<unk>. |
| hredGAN_w | <person>,i’msure,but you’re probably stillin to the new<person>’s house,huh ? |
| hredGAN_u | you're gon nabe alitle boy,andi’m not gon na get out ！ |
| Context_0 | <person>'t go into my room,she’s asleep. |
| Context_1 | <person>,but the nanny,mrs.<person>,she wants to see her. |
| HRED | i'm sorry,<person>. |
| VHRED: | <person>,mr.<person>. |
| hredGAN_w | <person>,i know .but she’s never seen me anyway. |
| hredGAN-u | i'm sorry ididn't know what to say . |
| |
| Context_0 | what was that for ? |
| Context_1 | i'd forgotten how exciting it is working together . |
| HRED | i'm sorry,<person>. |
| VHRED: | <person>,<person>.idon'tknow. |
| hredGAN_w | i'm sorry,<person>.so many people are so interested,but you’re a very nice guy . |
| hredGAN_u | i've always been so sure to<person> |
| |
| Context_0 | a teacher’s. |
| Context_1 | oh my god. i'm sorry,<person> |
| HRED VHRED: | <person>,<person>'t worry about it.idon’t know what you’re doing. |
| hredGAN_w | i'msorry.ididn’tmean to.ididn't meanto hurt you.ijust wanted you |
| hredGAN-u | <person>,i don 't want to be your friend . |
| |
| Context_0 HRED | this is all we have . <person>,i’m sorry. |
| VHRED: | <person>,idon't know what you're doing .idon't know whati’m going to do. |
| hredGAN_w | i'm sorry,sir,buti’ma very good man. |
| hredGAN-u | but you’ve been in the apartment . |
| |
| Context_0 | i took the initiative . |
| Context_1 | you haven'tbeen given an assignment in this case. |
| HRED | <person>. |
| VHRED: | |
| <person>,<person>,<person>,<person>.i’m not going to get out of the <unk>. |
| hredGAN_w | i'm not sure.<continued_utterance>i’m not gon na do you this shit anyway,<person> |
| hredGAN-u | |
| <person>,i guess i didn 't. |
| |
| Context_0 | <person>,i was just gon na leave and seeificould pick up their trail . |
| |
| Context_1 | no,you wait right there for me.<person> beon the piedmontfight tomorrow at seven.meet me at the airport. |
| |
| |
| HRED | i'm sorry，,<person> |
| |
| |
| |
| |
| VHRED: | |
| |
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| hredGAN_w | |
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| what do youthinki’msaying?idon’tknow whati’m going todo.idon’tknow whati’mgoing todo. |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| hredGAN-u | |
| |
| |
| |
| |
| |
| |
| i'm not sure i don 't.<continued_utterance>do you want to go over to the station ? <person>,i don'tknowifi'm not. |
|
+
+Table 6: Additional sample responses of HRED, VHRED and hredGAN on Ubuntu.
+
+| Ubuntu | |
| Context.0 | what's the command you are running to chroot and the error message you get in return? |
| Context.1 | i mountded the device sudo mount/dev/sdal/mnt-that works-then sudo chroot/mnt-eroris no such file or command |
| HRED | what 's the problem ? |
| VHRED | is there a way to do that ? what do you mean ? |
| hredGAN_w | I have no idea,I just did a sudo fdisk and it said it was mounted |
| hredGAN_u | I'm trying to get the latest version of ubuntu,butI’m getting a lot of problems . |
| Context_0 | Hmm,have you checked that ICMP is being allowedallthe way throughand thatthe serverresponds tootherICMPrequests? |
| Context_1 | nowicanpingit(iadtenabletlocompressooncienticanoteachthmotesubetaddingaruleihefrewallfoardthelan |
| to vpn |
| HRED VHRED | what 's the problem ? I don't know what you want to do |
| hredGAN_w | I have no idea,I’m just trying to get the ip address from the router to the router" |
| hredGAN_u | I'm not sure if that’sthe case,butI’m not sure if it’sa good idea to have a look at the output . |
| |
| Context_0 | The netboot one is suppose to download packages from the net. |
| HRED | I don't know what you want to do . |
| VHRED | how do I do that ? I'm not sure of a command to do so Ican just install the ubuntu .iso and burn a new iso |
| hredGAN_w hredGAN_u | you can 't have a look at the man page ... |
| |
| Context_0 | The netboot one is suppose to download packages from the net. |
| Context_1 | like the ones to be installed? or the installed to be run? |
| HRED | I don't know what the problem is . |
| VHRED | is there a way to install the package manager ? |
| hredGAN_w hredGAN_u | you don ’t need to install the whole system, just install the ubuntu installer I'm trying to installa -UNK version of ubuntu,butI'm trying to install it on my laptop |
| |
| Context_0 | You trying to mount it open with the password? |
| HRED | Ihave to do that,butI have a problem with the same problem. is there a way to do ? |
| VHRED | yes,I'm not sure if it’s mounted or not |
| hredGAN_w | I'm trying to install the latest version of Ubuntu,butI'm running on a -UNK |
| hredGAN-u | |
| Context_0 | are you using lvm for your root partition? |
| HRED | yes,but I have a problem with a new user,butI don't want to do that. |
| VHRED | what is the best way to do that ?what is that ? |
| hredGAN_w | yes,I'm on alive system,I’m trying to use the partition to boot from windows |
| hredGAN_u | I'm not sure if it’s a good idea to have a backup of the system.. |
| Context.0 | are you using lvm for your root partition? |
| Context_1 | ihavethesameofgorkigoa04box.Itppearsin2.04lvisonustartandutipathisnotidouselforyootderifty is thereanywaytworkarounditaveyuhardofayonepostingaugabouthistohaveitookedtdar.welliopethycanfiis |
| eventuallyiopetheykeepitrdroundandmoveeverytingtoustartUpstartisoteasytosetupforprogamstatdon’tcomepackaged |
| for upstart. I don't know what the problem is,but I don't know what the problem is .? |
| HRED VHRED | is there a way to do that ?or **unknown** ？ |
| hredGAN_w | you can also try the live cd and see if you can get it back up the live" |
| hredGAN-u | I'mtryintotywirelessorkingomyapto,ut’mgeingtoheintatcaeabletousteirelescardtogethieless |
| working. |
| Context_0 | are you using lvm for your root partition? ihavethesameofgorkingoa0.04box.Itppearsin2.04lisonupstartandultipathisot.idouselforyotfolderifty |
| Context.1 | istheranyaytoworkaroundihaveyouhardofanyonepostingabugabouttis tohaveitokedatdar.welliopethyanfiis eventuallyiopetheyepiitrdoudandmoveeverytngtoupstartUpstartisoteasytosetupforprogamstatdon'tcoepackaged |
| for upstart. |
| Context_2 HRED | yes,becauseicant supply itatboottime,iwantthe truecrypt drive to come up by itself without manual intervention what is the problem ? |
| VHRED | what do you mean ? |
| hredGAN_w | you can also mount a new one and put the mount command to the mount point” |
| hredGAN_u | I'm trying to get my sound working,butI'm trying to get my sound working. |
| are you using lvm for your root partition? |
| Context_0 | |
| Context_1 | ihavethesameconfgorkingona4box.Itappearsin04lisonupstartandutipathisnotidouselforootfolderifty isthereanyytworkroudiaveyouheardofanyonepostingugaboutistoveitlokedatdar.welliopeteyafiis |
| eventuallyiopetheykeepitrdroundandmoveeverytingtoustartUpstartisoteasytosetupforprogamstatdon’tcomepackaged |
| for upstart. |
| Context_2 | yes,becauseicantsupply itatboottime,i wantthe truecrypt drive to come up by itself without manual intervention |
| Context_3 | Kinda defeats the use of it anyone could get in don't you think? |
| HRED | what is the problem ? |
| VHRED | is there a way to mount the file ?if you want to do it ? |
| hredGAN_w | I have no idea,I just want to get the data from the other computer |
| I'm trying to getthe latest driver from the nvidia driver,butI’m trying to get the nvidia driver working |
| hredGAN_u | |
| |
\ No newline at end of file
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+# Knowledge Matters: Importance of Prior Information for Optimization
+
+C¸ a˘glar G¨ul¸cehre D´epartement d’informatique et de recherche op´erationnelle Universit´e de Montr´eal, Montr´eal, QC, Canada
+
+gulcehrc@iro.umontreal.ca
+
+Yoshua Bengio D´epartement d’informatique et de recherche op´erationnelle Universit´e de Montr´eal, Montr´eal, QC, Canada
+
+bengioy@iro.umontreal.ca
+
+Editor: Not Assigned
+
+# Abstract
+
+We explore the effect of introducing prior information into the intermediate level of deep supervised neural networks for a learning task on which all the black-box state-of-the-art machine learning algorithms tested have failed to learn. We motivate our work from the hypothesis that there is an optimization obstacle involved in the nature of such tasks, and that humans learn useful intermediate concepts from other individuals via a form of supervision or guidance using a curriculum. The experiments we have conducted provide positive evidence in favor of this hypothesis. In our experiments, a two-tiered MLP architecture is trained on a dataset for which each image input contains three sprites, and the binary target class is 1 if all three have the same shape. Black-box machine learning algorithms only got chance on this task. Standard deep supervised neural networks also failed. However, using a particular structure and guiding the learner by providing intermediate targets in the form of intermediate concepts (the presence of each object) allows to nail the task. Much better than chance but imperfect results are also obtained by exploring architecture and optimization variants, pointing towards a difficult optimization task. We hypothesize that the learning difficulty is due to the composition of two highly non-linear tasks. Our findings are also consistent with hypotheses on cultural learning inspired by the observations of effective local minima (possibly due to ill-conditioning and the training procedure not being able to escape what appears like a local minimum).
+
+Keywords: Deep Learning, Neural Networks, Optimization, Evolution of Culture, Curriculum Learning, Training with Hints
+
+# 1. Introduction
+
+There is a recent emerging interest in different fields of science for cultural learning (Henrich and McElreath, 2003) and how groups of individuals exchanging information can learn in ways superior to individual learning. This is also witnessed by the emergence of new research fields such as ”Social Neuroscience”. Learning from other agents in an environment by the means of cultural transmission of knowledge with a peer-to-peer communication is an efficient and natural way of acquiring or propagating common knowledge. The most popular belief on how the information is transmitted between individuals is that bits of information are transmitted by small units, called memes, which share some characteristics of genes, such as self-replication, mutation and response to selective pressures (Dawkins, 1976).
+
+This paper is based on the hypothesis (which is further elaborated in Bengio (2013a)) that human culture and the evolution of ideas have been crucial to counter an optimization issue: this difficulty would otherwise make it difficult for human brains to capture high level knowledge of the world without the help of other educated humans. In this paper machine learning experiments are used to investigate some elements of this hypothesis by seeking answers for the following questions: are there machine learning tasks which are intrinsically hard for a lone learning agent but that may become very easy when intermediate concepts are provided by another agent as additional intermediate learning cues, in the spirit of Curriculum Learning (Bengio et al., 2009b)? What makes such learning tasks more difficult? Can specific initial values of the neural network parameters yield success when random initialization yield complete failure? Is it possible to verify that the problem being faced is an optimization problem or with a regularization problem? These are the questions discussed (if not completely addressed) here, which relate to the following broader question: how can humans (and potentially one day, machines) learn complex concepts?
+
+In this paper, results of different machine learning algorithms on an artificial learning task involving binary 64 $\times$ 64 images are presented. In that task, each image in the dataset contains 3 Pentomino tetris sprites (simple shapes). The task is to figure out if all the sprites in the image are the same or if there are different sprite shapes in the image. Several state-of-the-art machine learning algorithms have been tested and none of them could perform better than a random predictor on the test set. Nevertheless by providing hints about the intermediate concepts (the presence and location of particular sprite classes), the problem can easily be solved where the same-architecture neural network without the intermediate concepts guidance fails. Surprisingly, our attempts at solving this problem with unsupervised pre-training algorithms failed solve this problem. However, with specific variations in the network architecture or training procedure, it is found that one can make a big dent in the problem. For showing the impact of intermediate level guidance, we experimented with a two-tiered neural network, with supervised pre-training of the first part to recognize the category of sprites independently of their orientation and scale, at different locations, while the second part learns from the output of the first part and predicts the binary task of interest.
+
+The objective of this paper is not to propose a novel learning algorithm or architecture, but rather to refine our understanding of the learning difficulties involved with composed tasks (here a logical formula composed with the detection of object classes), in particular the training difficulties involved for deep neural networks. The results also bring empirical evidence in favor of some of the hypotheses from Bengio (2013a), discussed below, as well as introducing a particular form of curriculum learning (Bengio et al., 2009b).
+
+Building difficult AI problems has a long history in computer science. Specifically hard AI problems have been studied to create CAPTCHA’s that are easy to solve for humans, but hard to solve for machines (Von Ahn et al., 2003). In this paper we are investigating a difficult problem for the off-the-shelf black-box machine learning algorithms.1
+
+# 1.1 Curriculum Learning and Cultural Evolution Against Effective Local Minima
+
+What Bengio (2013a) calls an effective local minimum is a point where iterative training stalls, either because of an actual local minimum or because the optimization algorithm is unable (in reasonable time) to find a descent path (e.g., because of serious ill-conditioning). In this paper, it is hypothesized that some more abstract learning tasks such as those obtained by composing simpler tasks are more likely to yield effective local minima for neural networks, and are generally hard for general-purpose machine learning algorithms.
+
+The idea that learning can be enhanced by guiding the learner through intermediate easier tasks is old, starting with animal training by shaping (Skinner, 1958; Peterson, 2004; Krueger and Dayan, 2009). Bengio et al. (2009b) introduce a computational hypothesis related to a presumed issue with effective local minima when directly learning the target task: the good solutions correspond to hard-to-find-by-chance effective local minima, and intermediate tasks prepare the learner’s internal configuration (parameters) in a way similar to continuation methods in global optimization (which go through a sequence of intermediate optimization problems, starting with a convex one where local minima are no issue, and gradually morphing into the target task of interest).
+
+In a related vein, Bengio (2013a) makes the following inferences based on experimental observations of deep learning and neural network learning:
+
+Point 1: Training deep architectures is easier when some hints are given about the function that the intermediate levels should compute (Hinton et al., 2006; Weston et al., 2008; Salakhutdinov and Hinton, 2009; Bengio, 2009). The experiments performed here expand in particular on this point.
+
+Point 2: It is much easier to train a neural network with supervision (where examples ar provided to it of when a concept is present and when it is not present in a variety of examples) than to expect unsupervised learning to discover the concept (which may also happen but usually leads to poorer renditions of the concept). The poor results obtained with unsupervised pre-training reinforce that hypothesis.
+
+Point 3: Directly training all the layers of a deep network together not only makes it difficult to exploit all the extra modeling power of a deeper architecture but in many cases it actually yields worse results as the number of required layers is increased (Larochelle et al., 2009; Erhan et al., 2010). The experiments performed here also reinforce that hypothesis.
+
+Point 4: Erhan et al. (2010) observed that no two training trajectories ended up in the same effective local minimum, out of hundreds of runs, even when comparing solutions as functions from input to output, rather than in parameter space (thus eliminating from the picture the presence of symmetries and multiple local minima due to relabeling and other reparametrizations). This suggests that the number of different effective local minima (even when considering them only in function space) must be huge.
+
+Point 5: Unsupervised pre-training, which changes the initial conditions of the descent procedure, sometimes allows to reach substantially better effective local minima (in terms of generalization error!), and these better local minima do not appear to be reachable by chance alone (Erhan et al., 2010). The experiments performed here provide another piece of evidence in favor of the hypothesis that where random initialization can yield rather poor results, specifically targeted initialization can have a drastic impact, i.e., that effective local minima are not just numerous but that some small subset of them are much better and hard to reach by chance.2
+
+Based on the above points, Bengio (2013a) then proposed the following hypotheses regarding learning of high-level abstractions.
+
+• Optimization Hypothesis: When it learns, a biological agent performs an approximate optimization with respect to some implicit objective function.
+
+• Deep Abstractions Hypothesis: Higher level abstractions represented in brains require deeper computations (involving the composition of more non-linearities).
+
+• Local Descent Hypothesis: The brain of a biological agent relies on approximate local descent and gradually improves itself while learning.
+
+• Effective Local Minima Hypothesis: The learning process of a single human learner (not helped by others) is limited by effective local minima.
+
+• Deeper Harder Hypothesis: Effective local minima are more likely to hamper learning as the required depth of the architecture increases.
+
+• Abstractions Harder Hypothesis: High-level abstractions are unlikely to be discovered by a single human learner by chance, because these abstractions are represented by a deep subnetwork of the brain, which learns by local descent.
+
+• Guided Learning Hypothesis: A human brain can learn high level abstractions if guided by the signals produced by other agents that act as hints or indirect supervision for these high-level abstractions.
+
+• Memes Divide-and-Conquer Hypothesis: Linguistic exchange, individual learning and the recombination of memes constitute an efficient evolutionary recombination operator in the meme-space. This helps human learners to collectively build better internal representations of their environment, including fairly high-level abstractions.
+
+This paper is focused on “Point 1 ” and testing the “Guided Learning Hypothesis”, using machine learning algorithms to provide experimental evidence. The experiments performed also provide evidence in favor of the “Deeper Harder Hypothesis” and associated “Abstractions Harder Hypothesis”. Machine Learning is still far beyond the current capabilities of humans, and it is important to tackle the remaining obstacles to approach AI. For this purpose, the question to be answered is why tasks that humans learn effortlessly from very few examples, while machine learning algorithms fail miserably?
+
+# 2. Culture and Optimization Difficulty
+
+As hypothesized in the “Local Descent Hypothesis”, human brains would rely on a local approximate descent, just like a Multi-Layer Perceptron trained by a gradient-based iterative optimization. The main argument in favor of this hypothesis relies on the biologically-grounded assumption that although firing patterns in the brain change rapidly, synaptic strengths underlying these neural activities change only gradually, making sure that behaviors are generally consistent across time. If a learning algorithm is based on a form of local (e.g. gradient-based) descent, it can be sensitive to effective local minima (Bengio, 2013a).
+
+When one trains a neural network, at some point in the training phase the evaluation of error seems to saturate, even if new examples are introduced. In particular Erhan et al. (2010) find that early examples have a much larger weight in the final solution. It looks like the learner is stuck in or near a local minimum. But since it is difficult to verify if this is near a true local minimum or simply an effect of strong ill-conditioning, we call such a “stuck” configuration an effective local minimum, whose definition depends not just on the optimization objective but also on the limitations of the optimization algorithm.
+
+Erhan et al. (2010) highlighted both the issue of effective local minima and a regularization effect when initializing a deep network with unsupervised pre-training. Interestingly, as the network gets deeper the difficulty due to effective local minima seems to be get more pronounced. That might be because of the number of effective local minima increases (more like an actual local minima issue), or maybe because the good ones are harder to reach (more like an ill-conditioning issue) and more work will be needed to clarify this question.
+
+As a result of Point 4 we hypothesize that it is very difficult for an individual’s brain to discover some higher level abstractions by chance only. As mentioned in the “Guided Learning Hypothesis” humans get hints from other humans and learn high-level concepts with the guidance of other humans3. Curriculum learning (Bengio et al., 2009a) and incremental learning (Solomonoff, 1989), are examples of this. This is done by properly choosing the sequence of examples seen by the learner, where simpler examples are introduced first and more complex examples shown when the learner is ready for them. One of the hypothesis on why curriculum works states that curriculum learning acts as a continuation method that allows one to discover a good minimum, by first finding a good minimum of a smoother error function. Recent experiments on human subjects also indicates that humans teach by using a curriculum strategy (Khan et al., 2011).
+
+Some parts of the human brain are known to have a hierarchical organization (i.e. visual cortex) consistent with the deep architecture studied in machine learning papers. As we go from the sensory level to higher levels of the visual cortex, we find higher level areas corresponding to more abstract concepts. This is consistent with the Deep Abstractions Hypothesis.
+
+Training neural networks and machine learning algorithms by decomposing the learning task into sub-tasks and exploiting prior information about the task is well-established and in fact constitutes the main approach to solving industrial problems with machine learning. The contribution of this paper is rather on rendering explicit the effective local minima issue and providing evidence on the type of problems for which this difficulty arises. This prior information and hints given to the learner can be viewed as inductive bias for a particular task, an important ingredient to obtain a good generalization error (Mitchell, 1980). An interesting earlier finding in that line of research was done with Explanation Based Neural Networks (EBNN) in which a neural network transfers knowledge across multiple learning tasks. An EBNN uses previously learned domain knowledge as an initialization or search bias (i.e. to constrain the learner in the parameter space) (O’Sullivan, 1996; Mitchell and Thrun, 1993).
+
+Another related work in machine learning is mainly focused on reinforcement learning algorithms, based on incorporating prior knowledge in terms of logical rules to the learning algorithm as a prior knowledge to speed up and bias learning (Kunapuli et al., 2010; Towell and Shavlik, 1994).
+
+As discussed in “Memes Divide and Conquer Hypothesis“ societies can be viewed as a distributed computational processing systems. In civilized societies knowledge is distributed across different individuals, this yields a space efficiency. Moreover computation, i.e. each individual can specialize on a particular task/topic, is also divided across the individuals in the society and hence this will yield a computational efficiency. Considering the limitations of the human brain, the whole processing can not be done just by a single agent in an efficient manner. A recent study in paleoantropology states that there is a substantial decline in endocranial volume of the brain in the last 30000 years Henneberg (1988). The volume of the brain shrunk to 1241 ml from 1502 ml (Henneberg and Steyn, 1993). One of the hypothesis on the reduction of the volume of skull claims that, decline in the volume of the brain might be related to the functional changes in brain that arose as a result of cultural development and emergence of societies given that this time period overlaps with the transition from hunter-gatherer lifestyle to agricultural societies.
+
+# 3. Experimental Setup
+
+Some tasks, which seem reasonably easy for humans to learn4, are nonetheless appearing almost impossible to learn for current generic state-of-art machine learning algorithms.
+
+Here we study more closely such a task, which becomes learnable if one provides hints to the learner about appropriate intermediate concepts. Interestingly, the task we used in our experiments is not only hard for deep neural networks but also for non-parametric machine learning algorithms such as SVM’s, boosting and decision trees.
+
+The result of the experiments for varying size of dataset with several off-the-shelf black box machine learning algorithms and some popular deep learning algorithms are provided in Table 1. The detailed explanations about the algorithms and the hyperparameters used for those algorithms are given in the Appendix Section 5.2. We also provide some explanations about the methodologies conducted for the experiments at Section 3.2.
+
+# 3.1 Pentomino Dataset
+
+In order to test our hypothesis, an artificial dataset for object recognition using 64 $\times$ 64 binary images is designed5. If the task is two tiered (i.e., with guidance provided), the task in the first part is to recognize and locate each Pentomino object class $_ 6$ in the image. The second part/final binary classification task is to figure out if all the Pentominos in the image are of the same shape class or not. If a neural network learned to detect the categories of each object at each location in an image, the remaining task becomes an XOR-like operation between the detected object categories. The types of Pentomino objects that is used for generating the dataset are as follows:
+
+
+Figure 1: Left (a): An example image from the dataset which has a different sprite type in it. Right (b): An example image from the dataset that has only one type of Pentomino object in it, but with different orientations and scales.
+
+Pentomino sprites N, P, F, Y, J, and Q, along with the Pentomino N2 sprite (mirror of “Pentomino N” sprite), the Pentomino F2 sprite (mirror of “Pentomino F” sprite), and the Pentomino Y2 sprite (mirror of “Pentomino Y” sprite).
+
+
+Figure 2: Different classes of Pentomino shapes used in our dataset.
+
+As shown in Figures 1(a) and 1(b), the synthesized images are fairly simple and do not have any texture. Foreground pixels are “1” and background pixels are “0”. Images of the training and test sets are generated iid. For notational convenience, assume that the domain of raw input images is $X$ , the set of sprites is $S$ , the set of intermediate object categories is $Y$ for each possible location in the image and the set of final binary task outcomes is $Z = \{ 0 , 1 \}$ . Two different types of rigid body transformation is performed: sprite rotation $r o t ( X , \gamma )$ where $\Gamma = \{ \gamma \colon ( \gamma = 9 0 \times \phi ) \wedge [ ( \phi \in \mathbb { N } ) , ( 0 \leq \phi \leq 3 ) ] \}$ and scaling $s c a l e ( X , \alpha )$ where $\alpha \in \{ 1 , 2 \}$ is the scaling factor. The data generating procedure is summarized below.
+
+Sprite transformations: Before placing the sprites in an empty image, for each image $x \in X$ , a value for $z \in Z$ is randomly sampled which is to have (or not) the same three sprite shapes in the image. Conditioned on the constraint given by $z$ , three sprites are randomly selected $s _ { i j }$ from $S$ without replacement. Using a uniform probability distribution over all possible scales, a scale is chosen and accordingly each sprite image is scaled. Then rotate each sprite is randomly rotated by a multiple of 90 degrees.
+
+Sprite placement: Upon completion of sprite transformations, a 64 $\times$ 64 uniform grid is generated which is divided into 8 $\times$ 8 blocks, each block being of size 8 $\times$ 8 pixels, and randomly select three different blocks from the 64=8 $\times$ 8 on the grid and place the transformed objects into different blocks (so they cannot overlap, by construction).
+
+Each sprite is centered in the block in which it is located. Thus there is no object translation inside the blocks. The only translation invariance is due to the location of the block inside the image.
+
+A Pentomino sprite is guaranteed to not overflow the block in which it is located, and there are no collisions or overlaps between sprites, making the task simpler. The largest possible Pentomino sprite can be fit into an 8 $\times$ 4 mask.
+
+# 3.2 Learning Algorithms Evaluated
+
+Initially the models are cross-validated by using 5-fold cross-validation. With 40,000 examples, this gives 32,000 examples for training and 8,000 examples for testing. For neural network algorithms, stochastic gradient descent (SGD) is used for training. The following standard learning algorithms were first evaluated: decision trees, SVMs with Gaussian kernel, ordinary fully-connected Multi-Layer Perceptrons, Random Forests, k-Nearest Neighbors, Convolutional Neural Networks, and Stacked Denoising Auto-Encoders with supervised fine-tuning. More details of the configurations and hyper-parameters for each of them are given in Appendix Section 5.2. The only better than chance results were obtained with variations of the Structured Multi-Layer Perceptron described below.
+
+# 3.2.1 Structured Multi-Layer Perceptron (SMLP)
+
+The neural network architecture that is used to solve this task is called the SMLP (Structured Multi-Layer Perceptron), a deep neural network with two parts as illustrated in Figure 5 and 7:
+
+The lower part, P1NN (Part 1 Neural Network, as it is called in the rest of the paper), has shared weights and local connectivity, with one identical MLP instance of the P1NN for each patch of the image, and typically an 11-element output vector per patch (unless otherwise noted). The idea is that these 11 outputs per patch could represent the detection of the sprite shape category (or the absence of sprite in the patch). The upper part, P2NN (Part 2 Neural Network) is a fully connected one hidden layer MLP that takes the concatenation of the outputs of all patch-wise P1NNs as input. Note that the first layer of P1NN is similar to a convolutional layer but where the stride equals the kernel size, so that windows do not overlap, i.e., P1NN can be decomposed into separate networks sharing the same parameters but applied on different patches of the input image, so that each network can actually be trained patch-wise in the case where a target is provided for the P1NN outputs. The P1NN output for patch $\mathbf { p _ { i } }$ which is extracted from the image $\mathbf { x }$ is computed as follows:
+
+$$
+f _ { \theta } ( \mathbf { p _ { i } } ) = g _ { 2 } ( V g _ { 1 } ( U \mathbf { p _ { i } } + \mathbf { b } ) + \mathbf { c } )
+$$
+
+where $\mathbf { p _ { i } } \in R ^ { d }$ is the input patch/receptive field extracted from location $i$ of a single image. $U \in R ^ { d _ { h } \times d }$ is the weight matrix for the first layer of P1NN and $ { \mathbf { b } } \in R _ { h } ^ { d }$ is the vector of biases for the first layer of P1NN. $g _ { 1 } ( \cdot )$ is the activation function of the first layer and $g _ { 2 } ( \cdot )$ is the activation function of the second layer. In many of the experiments, best results were obtained with $g _ { 1 } ( \cdot )$ a rectifying non-linearity (a.k.a. as RELU), which is $m a x ( 0 , \ X )$ (Jarrett et al., 2009b; Nair and Hinton, 2010; Glorot et al., 2011a; Krizhevsky et al., 2012). $V \in R ^ { d _ { h } \times d _ { o } }$ is the second layer’s weights matrix, such that and $\mathbf { c } \in R _ { d _ { o } }$ are the biases of the second layer of the P1NN, with $d _ { o }$ expected to be smaller than $d _ { h }$ .
+
+In this way, $g _ { 1 } ( U { \bf p } _ { \bf i } + { \bf b } )$ is an overcomplete representation of the input patch that can potentially represent all the possible Pentomino shapes for all factors of variations in the patch (rotation, scaling and Pentomino shape type). On the other hand, when trained with hints, $f _ { \boldsymbol \theta } ( \mathbf { p _ { i } } )$ is expected to be the lower dimensional representation of a Pentomino shape category invariant to scaling and rotation in the given patch.
+
+In the experiments with SMLP trained with hints (targets at the output of P1NN), the P1NN is expected to perform classification of each 8 $\times$ 8 non-overlapping patches of the original 64 $\times$ 64 input image without having any prior knowledge of whether that specific patch contains a Pentomino shape or not. P1NN in SMLP without hints just outputs the local activations for each patch, and gradients on $f _ { \boldsymbol \theta } ( \mathbf { p } _ { i } )$ are backpropagated from the upper layers. In both cases P1NN produces the input representation for the Part 2 Neural Net (P2NN). Thus the input representation of P2NN is the concatenated output of P1NN across all the 64 patch locations:
+
+$\mathbf { h _ { o } } = [ f _ { \theta } ( \mathbf { p _ { 0 } } ) , . . . , f _ { \theta } ( \mathbf { p _ { i } } ) , . . . , f _ { \theta } ( \mathbf { p _ { N } } ) ) ]$ where $N$ is the number of patches and the $h _ { o } \in R ^ { d _ { i } } , d _ { i } =$ $d _ { o } \times N$ . $\mathbf { h _ { 0 } }$ is the concatenated output of the P1NN at each patch.
+
+There is a standardization layer on top of the output of P1NN that centers the activations and performs divisive normalization by dividing by the standard deviation over a minibatch of the activations of that layer. We denote the standardization function $z ( \cdot )$ . Standardization makes use of the mean and standard deviation computed for each hidden unit such that each hidden unit of $\mathbf { h _ { 0 } }$ will have 0 activation and unit standard deviation on average over the minibatch. $X$ is the set of pentomino images in the minibatch, where $X \in R ^ { d _ { i n } \times N }$ is a matrix with $N$ images. $h _ { o } ^ { ( i ) } ( \mathbf { x _ { j } } )$ is the vector of activations of the $i$ -th hidden unit of hidden layer $h _ { o } ( \mathbf { x _ { j } } )$ for the $j$ -th example, with $x _ { j } \in X$ .
+
+$$
+\mu _ { h _ { o } ^ { ( i ) } } = \frac { 1 } { N } \sum _ { \mathbf { x _ { j } } \in X } h _ { o } ^ { ( i ) } ( \mathbf { x _ { j } } )
+$$
+
+$$
+\sigma _ { h _ { o } ^ { ( i ) } } = \sqrt { \frac { \sum _ { j } ^ { N } ( h _ { o } ^ { ( i ) } ( \mathbf { x _ { j } } ) - \mu _ { h _ { o } ^ { ( i ) } } ) ^ { 2 } } { N } + \epsilon }
+$$
+
+$$
+z ( h _ { o } ^ { ( i ) } ( \mathbf { x _ { j } } ) ) = \frac { h _ { o } ^ { ( i ) } ( \mathbf { x _ { j } } ) - \mu _ { h _ { o } ^ { ( i ) } } } { \operatorname* { m a x } ( \boldsymbol { \sigma } _ { h _ { o } ^ { ( i ) } } , \epsilon ) }
+$$
+
+where $\epsilon$ is a very small constant, that is used to prevent numerical underflows in the standard deviation. P1NN is trained on each 8 $\times$ 8 patches extracted from the image.
+
+$\mathbf { h _ { o } }$ is standardized for each training and test sample separately. Different values of $\epsilon$ were used for SMLP-hints and SMLP-nohints.
+
+The concatenated output of P1NN is fed as an input to the P2NN. P2NN is a feedforward MLP with a sigmoid output layer using a single RELU hidden layer. The task of P2NN is to perform a nonlinear logical operation on the representation provided at the output of P1NN.
+
+The SMLP-hints architecture exploits a hint about the presence and category of Pentomino objects, specifying a semantics for the P1NN outputs. P1NN is trained with the intermediate target $Y$ , specifying the type of Pentomino sprite shape present (if any) at each of the 64 patches (8 $\times$ 8 non-overlapping blocks) of the image. Because a possible answer at a given location can be “none of the object types” i.e., an empty patch, $y _ { p }$ (for patch $p$ ) can take one of the 11 possible values, 1 for rejection and the rest is for the Pentomino shape classes, illustrated in Figure 2:
+
+$$
+y _ { p } = { \left\{ \begin{array} { l l } { 0 } & { { \mathrm { i f ~ p a t c h ~ } } p { \mathrm { ~ i s ~ e m p t y } } } \\ { s \in S } & { { \mathrm { i f ~ t h e ~ p a t c h ~ } } p { \mathrm { ~ c o n t a i n s ~ a ~ P e n t o m i n o ~ s p r i t e ~ } } . } \end{array} \right. }
+$$
+
+A similar task has been studied by Fleuret et al. (2011) (at SI appendix Problem 17), who compared the performance of humans vs computers.
+
+The SMLP-hints architecture takes advantage of dividing the task into two subtasks during training with prior information about intermediate-level relevant factors. Because the sum of the training losses decomposes into the loss on each patch, the P1NN can be pre-trained patchwise. Each patch-specific component of the P1NN is a fully connected MLP with 8 $\times$ 8 inputs and 11 outputs with a softmax output layer. SMLP-hints uses the the standardization given in Equation 3 but with $\epsilon = 0$ .
+
+The standardization is a crucial step for training the SMLP on the Pentomino dataset, and yields much sparser outputs, as seen on Figures 3 and 4. If the standardization is not used, even SMLP-hints could not solve the Pentomino task. In general, the standardization step dampens the small activations and augments larger ones(reducing the noise). Centering the activations of each feature detector in a neural network has been studied in (Raiko et al., 2012) and (Vatanen et al., 2013). They proposed that transforming the outputs of each hidden neuron in a multi-layer perceptron network to have zero output and zero slope on average makes first order optimization methods closer to the second order techniques.
+
+By default, the SMLP uses rectifier hidden units as activation function, we found a significant boost by using rectification compared to hyperbolic tangent and sigmoid activation functions. The P1NN has a highly overcomplete architecture with 1024 hidden units per patch, and L1 and L2 weight decay regularization coefficients on the weights (not the biases) are respectively 1e-6 and 1e-5. The learning rate for the P1NN is 0.75. 1 training epoch was enough for the P1NN to learn the features of Pentomino shapes perfectly on the 40000 training examples. The P2NN has 2048 hidden units. L1 and L2 penalty coefficients for the P2NN are 1e-6, and the learning rate is 0.1. These were selected by trial and error based on validation set error. Both P1NN (for each patch) and P2NN are fully-connected neural networks, even though P1NN globally is a special kind of convolutional neural network.
+
+Filters of the first layer of SMLP are shown in Figure 6. These are the examples of the filters obtained with the SLMP-hints trained with 40k examples, whose results are given in Table 1. Those filters look very noisy but they work perfectly on the Pentomino task.
+
+# 3.2.3 Deep and Structured Supervised MLP without Hints (SMLP-nohints)
+
+SMLP-nohints uses the same connectivity pattern (and deep architecture) that is also used in the SMLP-hints architecture, but without using the intermediate targets ( $Y$ ). It directly predicts the final outcome of the task ( $Z$ ), using the same number of hidden units, the same connectivity and the same activation function for the hidden units as SMLP-hints. 120 hyperparameter values have been evaluated by randomly selecting the number of hidden units from [64, 128, 256, 512, 1024, 1200, 2048] and randomly sampling 20 learning rates uniformly in the log-domain within the interval of [0.008, 0.8]. Two fully connected hidden layers with 1024 hidden units (same as P1NN) per patch is used and 2048 (same as P2NN) for the last hidden layer, with twenty training epochs. For this network the best results are obtained with a learning rate of 0.05.7
+
+
+Figure 3: Bar chart of concatenated softmax output activations $\mathbf { h _ { o } }$ of P1NN ( $1 1 \times 6 4 = 7 0 4$ outputs) in SMLP-hints before standardization, for a selected example. There are very large spikes at each location for one of the possible 11 outcome (1 of K representation).
+
+
+Figure 4: Softmax output activations $\mathbf { h _ { 0 } }$ of P1NN at SMLP-hints before standardization. There are positive spiked outputs at the locations where there is a Pentomino shape. Positive and negative spikes arise because most of the outputs are near an average value. Activations are higher at the locations where there is a pentomino shape.
+
+
+Figure 5: Structured MLP architecture, used with hints (trained in two phases, first P1NN, bottom two layers, then P2NN, top two layers). In SMLP-hints, P1NN is trained on each 8x8 patch extracted from the image and the softmax output probabilities of all 64 patches are concatenated into a 64 $\times$ 11 vector that forms the input of P2NN. Only $U$ and $V$ are learned in the P1NN and its output on each patch is fed into P2NN. The first level and the second level neural networks are trained separately, not jointly.
+
+
+Figure 6: Filters of Structured MLP architecture, trained with hints on 40k examples.
+
+
+Figure 7: Structured MLP architecture, used without hints (SMLP-nohints). It is the same architecture as SMLP-hints (Figure 5) but with both parts (P1NN and P2NN) trained jointly with respect to the final binary classification task.
+
+We chose to experiment with various SMLP-nohint architectures and optimization procedures, trying unsuccessfully to achieve as good results with SMLP-nohint as with SMLP-hints.
+
+Rectifier Non-Linearity A rectifier nonlinearity is used for the activations of MLP hidden layers. We observed that using piecewise linear nonlinearity activation function such as the rectifier can make the optimization more tractable.
+
+
+Figure 8: First layer filters learned by the Structured MLP architecture, trained without using hints on 447600 examples with online SGD and a sigmoid intermediate layer activation.
+
+Intermediate Layer The output of the P1NN is considered as an intermediate layer of the SMLP. For the SMLP-hints, only softmax output activations have been tried at the intermediate layer, and that sufficed to learn the task. Since things did not work nearly as well with the SMLP-nohints, several different activation functions have been tried: softmax $( \cdot )$ , $\operatorname { t a n h } ( { \cdot } )$ , sigmoid $( \cdot )$ and linear activation functions.
+
+Standardization Layer Normalization at the last layer of the convolutional neural networks has been used occasionaly to encourage the competition between the hidden units. (Jarrett et al., 2009a) used a local contrast normalization layer in their architecture which performs subtractive and divisive normalization. A local contrast normalization layer enforces a local competition between adjacent features in the feature map and between features at the same spatial location in different feature maps. Similarly (Krizhevsky et al., 2012) observed that using a local response layer that enjoys the benefit of using local normalization scheme aids generalization.
+
+Standardization has been observed to be crucial for both SMLP trained with or without hints. In both SMLP-hints and SMLP-nohints experiments, the neural network was not able to generalize or even learn the training set without using standardization in the SMLP intermediate layer, doing just chance performance. More specifically, in the SMLP-nohints architecture, standardization is part of the computational graph, hence the gradients are being backpropagated through it. The mean and the standard deviation is computed for each hidden unit separately at the intermediate layer as in Equation 4. But in order to prevent numerical underflows or overflows during the backpropagation we have used $\epsilon = 1 e - 8$ (Equation 3).
+
+The benefit of having sparse activations may be specifically important for the ill-conditioned problems, for the following reasons. When a hidden unit is “off”, its gradient (the derivative of the loss with respect to its output) is usually close to 0 as well, as seen here. That means that all off-diagonal second derivatives involving that hidden unit (e.g. its input weights) are also near 0. This is basically like removing some columns and rows from the Hessian matrix associated with a particular example. It has been observed that the condition number of the Hessian matrix (specifically, its largest eigenvalue) increases as the size of the network increases (Dauphin and Bengio, 2013), making training considerably slower and inefficient (Dauphin and Bengio, 2013). Hence one would expect that as sparsity of the gradients (obtained because of sparsity of the activations) increases, training would become more efficient, as if we were training a smaller sub-network for each example, with shared weights across examples, as in dropouts (Hinton et al., 2012).
+
+In Figure 9, the activation of each hidden unit in a bar chart is shown: the effect of standardization is significant, making the activations sparser.
+
+
+Figure 9: Activations of the intermediate-level hidden units of an SLMP-nohints for a particular examples (x-axis: hidden unit number, y-axis: activation value). Left (a): before standardization. Right (b): after standardization.
+
+In Figure 10, one can see the activation histogram of the SMLP-nohints intermediate layer, showing the distribution of activation values, before and after standardization. Again the sparsifying effect of standardization is very apparent.
+
+
+Figure 10: Distribution histogram of activation values of SMLP-nohints intermediate layer. Left (a): before standardization. Right (b): after standardization.
+
+In Figures 10 and 9, the intermediate level activations of SMLP-nohints are shown before and after standardization. These are for the same SMLP-nohints architecture whose results are presented on Table 1. For that same SMLP, the Adadelta (Zeiler, 2012) adaptive learning rate scheme has been used, with 512 hidden units for the hidden layer of P1NN and rectifier activation function. For the output of the P1NN, 11 sigmoidal units have been used while P2NN had 1200 hidden units with rectifier activation function. The output nonlinearity of the P2NN is a sigmoid and the training objective is the binary crossentropy.
+
+Adaptive Learning Rates We have experimented with several different adaptive learning rate algorithms. We tried rmsprop 8, Adadelta (Zeiler, 2012), Adagrad (Duchi et al., 2010) and a linearly $\left( 1 / \mathrm { t } \right)$ decaying learning rate (Bengio, 2013b). For the SMLP-nohints with sigmoid activation function we have found Adadelta(Zeiler, 2012) converging faster to an effective local minima and usually yielding better generalization error compared to the others.
+
+# 3.2.4 Deep and Structured MLP with Unsupervised Pre-Training
+
+Several experiments have been conducted using an architecture similar to the SMLP-nohints, but by using unsupervised pre-training of P1NN, with Denoising Auto-Encoder (DAE) and/or Contractive Auto-Encoders (CAE). Supervised fine-tuning proceeds as in the deep and structured MLP without hints. Because an unsupervised learner may not focus the representation just on the shapes, a larger number of intermediate-level units at the output of P1NN has been explored: previous work on unsupervised pre-training generally found that larger hidden layers were optimal when using unsupervised pre-training, because not all unsupervised features will be relevant to the task at hand. Instead of limiting to 11 units per patch, we experimented with networks with up to 20 hidden (i.e., code) units per patch in the second-layer patch-wise auto-encoder.
+
+In Appendix 5.1 we also provided the result of some experiments with binary-binary RBMs trained on 8 $\times$ 8 patches from the 40k training dataset.
+
+In unsupervised pretraining experiments in this paper, both contractive auto-encoder (CAE) with sigmoid nonlinearity and binary cross entropy cost function and denoising auto-encoder (DAE) have been used. In the second layer, experiments were performed with a DAE with rectifier hidden units utilizing L1 sparsity and weight decay on the weights of the auto-encoder. Greedy layerwise unsupervised training procedure is used to train the deep auto-encoder architecture (Bengio et al., 2007). In unsupervised pretraining experiments, tied weights have been used. Different combinations of CAE and DAE for unsupervised pretraining have been tested, but none of the configurations tested managed to learn the Pentomino task, as shown in Table 1.
+
+# 3.3 Experiments with 1 of $\mathbf { K }$ representation
+
+To explore the effect of changing the complexity of the input representation on the difficulty of the task, a set of experiments have been designed with symbolic representations of the information in each patch. In all cases an empty patch is represented with a 0 vector. These representation can be seen as an alternative input for a P2NN-like network, i.e., they were fed as input to an MLP or another black-box classifier.
+
+The following four experiments have been conducted, each one using one using a different input representation for each patch:
+
+| Method | Dependence | Transformationspace | Stochasticity(0 sampling) | Phase |
| Gaussian noise on input(Bishop,1995a; An,1996) | p(） | Input | Random | Training |
| Gaussian noise on hidden units(DeVries and Taylor, 2017) | p(0) | Hidden features | Random | Training |
| Dropout (Hinton et al., 2012; Srivastavaet al., 2014) | p(） | Input andhidden features | Random | Training |
| Random dropout probability(Bouthillier et al., 2015, Sec. 4) | p() | Input andhidden features | Random | Training |
| Curriculum dropout(Morerio et al., 2017) | p(0|time) | Input andhidden features | Random | Training |
| Bayesian dropout(Maeda,2014) | p(0|π) | Input andhidden features | Random | Training |
| Standout (adaptive dropout)(Ba and Frey, 2013) | p(0|x,π) | Input andhidden features | Random | Training |
| "Projection” of dropout noise into inputspace (Bouthillier et al., 2015, Sec. 3) | p(0lx,f,w) | InputUses auxiliary Tin hidden-featurespace. | Random | Training |
| Approximation of Gaussian process bytest-time dropout(Gal and Ghahramani, 2016) | p(0) | Input andhidden features | Random | Test |
| Stochastic depth (Huang et al., 2016b) | p（） | Hidden features | Random | Training |
| Noisy activation functions(Nair and Hinton, 2010; Xu et al., 2015;Gülcehre et al., 2016a) | p(0x） | Hidden features | Random | Training |
| Training with adversarial examples(Szegedy et al., 2014) | p(0|x,t') | Input | AdaptiveConstrained | Training |
| Network fooling (adversarial examples)(Szegedy et al., 2014)(Not for regularization) | p(0lx,t） | Input | AdaptiveConstrained | Test |
| Synthetic minority oversampling inhidden-feature space (Wong et al., 2016) | p(0x,t,D) | Hidden features | Random | Training |
| Inter-and extrapolation in hidden-featurespace (DeVries and Taylor, 2017) | p（0|x,t,D） | Hidden features | Random | Training |
| Batch normalization (Ioffe and Szegedy,2015),Ghost batch normalization (Hofferet al., 2017) | p(0|x) | Hidden features | Deterministic | Trainingand test |
| Layer normalization(Ba et al., 2016) | p(0|x) | Hidden features | Deterministic | Trainingand test |
| Annealed noise on targets(Wang and Principe, 1999) | p(0|time) | Target | Random | Training |
| Label smoothing (Szegedy et al., 2016,Sec. 7; Goodfellow et al.,2016, Chap.7) | p() | Target | Deterministic | Training |
| Model compression (mimic models,distilled models) (Bucila et al., 2006; Baand Caruana,2014; Hinton et al., 2015) | p（0|x,D) | Target | Deterministic | Training |
+
+Table 2: Existing domain-specific data-based methods classified according to our taxonomy. Table columns are described in Section 3. Note that these methods are never applied on the hidden features, because domain knowledge cannot be applied on them.
+
+| Method | Dependence | Transformationspace | Stochasticity(0 sampling) | Phase |
| Rigid and elastic image transformation(Baird,1990; Yaegger et al., 1996; Simardet al.,2003; Ciresan et al., 2010) | p(0） | Input | Random | Training |
| Test-time image transformations(Simonyan and Zisserman, 2015;Dielemanet al., 2015) | p(0) | Input | Random | Test |
| Sound transformations(Salamon and Bello, 2017) | p(） | Input | Random | Training |
| Error-maximizing rigid imagetransformations(Loosli et al., 2007; Fawzi et al., 2016) | p(） | Input | Adaptivestochastic&constrained,respectively | Training |
| Learning class-specific elasticimage-deformation fields(Hauberg et al., 2016) | p(0|t,D) | Input | Random | Training |
| Any handcrafted data preprocessing, forexample scale-invariant feature transform(SIFT) for images (Lowe,1999) | p(） | Input | Deterministic | Trainingand test |
| Overfeat (Sermanet et al., 2013) | p(0) | Input | Deterministic | Trainingand test |
+
+If the transformation is additionally representation-preserving, then the distribution $Q$ created by the transformation $\tau _ { \theta }$ attempts to mimic the ground truth data distribution $P$ . Otherwise, the notion of a “ground truth data distribution” in the modified representation may be vague. We provide more details about the transition from $\mathcal { D }$ to $Q$ in Appendix B.
+
+Summary of data-based methods Data-based regularization is a popular and very useful way to improve the results of deep learning. In this section we formalized this group of methods and showed that seemingly unrelated techniques such as Target-preserving data augmentation, Dropout, or Batch normalization are methodologically surprisingly close to each other. In Section 8 we discuss future directions that we find promising.
+
+# 4 Regularization via the network architecture
+
+A network architecture $f$ can be selected to have certain properties or match certain assumptions in order to have a regularizing effect.1
+
+Table 3: Methods based on network architecture, and rough description of assumptions that they encode. There are partial overlaps between some listed methods. For example, Residual learning uses Skip-connections. Many noise-based methods also fit Table 1 (cf. Appendix A).
+
+| Method | Method class | Assumptions about an appropriate learnable input-output mapping |
| Any chosen (not overlycomplex） architecture | * | Mapping can be well approximated by functions from the chosen familywhich are easily accessible by optimization. |
| Small network | * | Mapping is simple (complexity of the mapping depends on the number ofnetwork units and layers). |
| Deep network | * | The mapping is complex,but can be decomposed into a composition (orgenerally into a directed acyclic graph) of simple nonlinear transformations,e.g.affine transformation followed by simple nonlinearity(fully-connectedlayer),“multi-channel convolution”followed by simple nonlinearity (convo-lutional layer),etc. |
| Hard bottleneck (layer withfew neurons)；soft bottleneck(e.g.Jacobian penalty (Rifaiet al.,201lc),see Section 6) | Layer operation | Data concentrates around a lower-dimensional manifold; has few factors ofvariation. |
| Convolutional networks(Fukushima and Miyake,1982;Rumelhart et al.,1986,Pp.348-352; LeCun et al.,1989;Simard et al., 2003) | Layer operation | Spatially local and shift-equivariant feature extraction is all we need. |
| Dilated convolutions(Yu and Koltun,2015) | Layer operation | Like convolutional networks.Additionally:Sparse sampling of wide localneighborhoods provides relevant information,and better preserves rele-vant high-resolution information than architectures with downscaling andupsampling. |
| Strided convolutions (seeDumoulin and Visin,2016) | Layer operation | The mapping is reliable at reacting to features that do not vary tooabruptly in space,i.e.which are present in several neighboring pixels andcan be detected even if the filter center skips some of the pixels.The out-put is robust towards slight changes of the location of features,and changesof strength/presence of spatially strongly varying features. |
| Pooling | Layer operation | The output is invariant to slight spatial distortions of the input (slightchanges of the location of (deep) features).Features that are sensitive tosuch distortions can be discarded. |
| Stochastic pooling(Zeiler and Fergus,2013) | Layer operation | The output is robust towards slight changes of the location (like pooling)but also of the strength/presence of (deep) features. |
| Training with different kindsof noise (including Dropout;see Section 3) | Noise | The mapping is robust to noise:the given class of perturbations of theinput or deep features should not affect the output too much. |
| Dropout (Hinton et al., 2012;Srivastava etal.,2014),DropConnect (Wan et al.,2013),and related methods | Noise | Extracting complementary (non-coadapted） features is helpful.Non-coadapted features are more informative,better disentangle factors of vari-ation.(We want to disentangle factors of variation because they are en-tangled in different ways in inputs vs.in outputs.)When interpreted as ensemble learning:usual assumptions of ensemblelearning (predictions of weak learners have complementary info and can becombined to strong prediction). |
| Maxout units(Goodfellow et al., 2013) | Layer operation | Assumptions similar to Dropout,with more accurate approximation ofmodel averaging (when interpreted as ensemble learning) |
| Skip-connections (Long et al.,2015;Huang et al.,2016a) | Connections be-tween layers | Certain lower-level features can directly be reused in a meaningful way at(several) higher levels of abstraction |
| Linearly augmentedfeed-forward network (van derSmagt and Hirzinger,1998) | Connections be-tween layers | Skip-connections that share weights with the non-skip-connections.Helpsagainst vanishing gradients. Rather changes the learning algorithm thanthe network mapping. |
| Residual learning(He et al., 2016) | Connections be-tween layers | Learning additive difference of a mapping f (or its compositional parts)from the identity mapping is easier than learning f itself. Meaningful deepfeatures can be composed asa sum of lower-level and intermediate-levelfeatures. |
| Stochastic depth(Huang et al.,2016b),DropIn(Smith et al., 2015) | Connectionsbetween layers;noise | Similar to Dropout:extracting complementary (non-coadapted) featuresacross different levels of abstraction is helpful; implicit model ensem-ble.Similar to Residual learning: meaningful deep features can be com-posed as a sum of lower-level and intermediate-level features,with theintermediate-level ones being optional,and leaving them out beingmeaningful data augmentation. Similar to Mollifying networks: simpli-fying random parts of the mapping improves training. |
| Mollifying networks(Gülcehre et al., 2016b) | Connectionsbetweenlayers;noise | The mapping can be easier approximated by estimating its decreasinglylinear simplified version |
| Network information criterion(Murata et al.,1994),Networkgrowing and network pruning(see Bishop,1995a, Sec. 9.5) | Model selection | Optimal generalization is reached by a network that has the right numberof units (not too few,not too many) |
| Multi-task learning (seeCaruana,1998;Ruder,2017) | * | Several tasks can help each other to learn mutually useful feature extrac-tors,as long as the tasks do not compete for resources (network capacity) |
+
+Assumptions about the mapping An input-output mapping $f _ { w }$ must have certain properties in order to fit the data $P$ well. Although it may be intractable to enforce the precise properties of an ideal mapping, it may be possible to approximate them by simplified assumptions about the mapping. These properties and assumptions can then be imposed upon model fitting in a hard or soft manner. This limits the search space of models and allows finding better solutions. An example is the decision about the number of layers and units, which allows the mapping to be neither too simple nor too complex (thus avoiding underfitting and overfitting). Another example are certain invariances of the mapping, such as locality and shift-equivariance of feature extraction hardwired in convolutional layers. Overall, the approach of imposing assumptions about the input-output mapping discussed in this section is the selection of the network architecture $f$ . The choice of architecture $f$ on the one hand hardwires certain properties of the mapping; additionally, in an interplay between $f$ and the optimization algorithm (Section 7), certain weight configurations are more likely accessible by optimization than others, further limiting the likely search space in a soft way. A complementary way of imposing certain assumptions about the mapping are regularization terms (Section 6), as well as invariances present in the (augmented) data set (Section 3).
+
+Assumptions can be hardwired into the definition of the operation performed by certain layers, and/or into the connections between layers. This distinction is made in Table 3, where these and other methods are listed.
+
+In Section 3 about data, we mentioned regularization methods that transform data in the hidden-feature space. They can be considered part of the architecture. In other words, they fit both Sections 3 (data) and 4 (architecture). These methods are listed in Table 1 with hidden features as their transformation space.
+
+Weight sharing Reusing a certain trainable parameter in several parts of the network is referred to as weight sharing. This usually makes the model less complex than using separately trainable parameters. An example are convolutional networks (LeCun et al., 1989). Here the weight sharing does not merely reduce the number of weights that need to be learned; it also encodes the prior knowledge about the shift-equivariance and locality of feature extraction. Another example is weight sharing in autoencoders.
+
+Activation functions Choosing the right activation function is quite important; for example, using Rectified linear units (ReLUs) improved the performance of many deep architectures both in the sense of training times and accuracy as well as overcoming the need for greedy layer-wise pre-training (Hahnloser et al., 2000; Jarrett et al., 2009; Nair and Hinton, 2010; Glorot et al., 2011). The success of ReLUs can be partially attributed to the fact that they provide more expressive families of mappings compared to sigmoid activations (in the sense that the classical sigmoid nonlinearity can be approximated very well $^ 2$ with only two ReLUs, but it takes an infinite number of sigmoid units to approximate a ReLU) and their affine extrapolation to unknown regions of data space seems to provide better generalization in practice than the “stagnating” extrapolation of sigmoid units. However, their hard negative cut-off and unbounded positive part are not always desired properties. Some activation functions were designed explicitly for regularization. For Dropout, Maxout units (Goodfellow et al., 2013) allow a more precise approximation of the geometric mean of the model ensemble predictions at test time. Stochastic pooling (Zeiler and Fergus, 2013), on the other hand, is a noisy version of max-pooling. The authors claim that this allows modelling distributions of activations instead of taking just the maximum.
+
+Noisy models Stochastic pooling was one example of a stochastic generalization of a deterministic model. Some models are stochastic by injecting random noise into various parts of the model. The most frequently used noisy model is Dropout (Hinton et al., 2012; Srivastava et al., 2014).
+
+Multi-task learning A special type of regularization is multi-task learning (see Caruana, 1998; Ruder, 2017), where the network is modified to predict targets for several tasks at once. It can be combined with semi-supervised learning to utilize unlabeled data on an auxiliary task (Rasmus et al., 2015). A similar concept of sharing knowledge between tasks is also utilized in meta-learning, where multiple tasks from the same domain are learned sequentially, using previously gained knowledge as bias for new tasks (Baxter, 2000); and transfer learning, where knowledge from one domain is transferred into another domain (Pan and Yang, 2010). These approaches differ from other methods in the sense that they require some additional target data, which are not always available.
+
+Model selection The best among several trained models (e.g. with different architectures) can be selected by evaluating the predictions on a validation set. It should be noted that this holds for selecting the best combination of all techniques (Sections 3–7), not just architecture; and that the validation set used for model selection in the “outer loop” should be different from the validation set used e.g. for Early stopping (Section 7), and different from the test set (Cawley and Talbot, 2010). However, there are also model selection methods that specifically target the selection of the number of units in a specific network architecture, e.g. using network growing and network pruning (see Bishop, 1995a, Sec. 9.5), or additionally do not require a validation set, e.g. the Network information criterion to compare models based on the training error and second derivatives of the loss function (Murata et al., 1994).
+
+# 5 Regularization via the error function
+
+Ideally, the error function $E$ reflects an appropriate notion of quality, and in some cases some assumptions about the data distribution. Typical examples are mean squared error or cross-entropy. The error function $E$ can also have a regularizing effect. An example is Dice coefficient optimization (Milletari et al., 2016) which is robust to class imbalance. Moreover, the overall form of the loss function can be different than Eq. (3). For example, in certain loss functions that are robust to class imbalance, the sum is taken over pairwise combinations $\mathcal { D } \times \mathcal { D }$ of training samples (Yan et al., 2003), rather than over training samples. But such alternatives to Eq. (3) are rather rare, and similar principles apply. If additional tasks are added for a regularizing effect (multi-task learning (see Caruana, 1998; Ruder, 2017)), then targets $t$ are modified to consist of several tasks, the mapping $f _ { w }$ is modified to produce an according output $y$ , and $E$ is modified to account for the modified $t$ and $y$ . Besides, there are regularization terms that depend on $\partial E / \partial x$ . They depend on $t$ and thus in our definition are considered part of $E$ rather than of $R$ , but they are listed in Section 6 among $R$ (rather than here) for a better overview.
+
+# 6 Regularization via the regularization term
+
+Regularization can be achieved by adding a regularizer $R$ into the loss function. Unlike the error function $E$ (which expresses consistency of outputs with targets), the regularization term is independent of the targets. Instead, it is used to encode other properties of the desired model, to provide inductive bias (i.e. assumptions about the mapping other than consistency of outputs with targets). The value of $R$ can thus be computed for an unlabeled test sample, whereas the value of $E$ cannot.
+
+The independence of $R$ from $t$ has an important implication: it allows additionally using unlabeled samples (semi-supervised learning) to improve the learned model based on its compliance with some desired properties (Sajjadi et al., 2016). For example, semi-supervised learning with ladder networks (Rasmus et al., 2015) combines a supervised task with an unsupervised auxiliary denoising task in a “multi-task” learning fashion. (For alternative interpretations, see Appendix A.) Unlabeled samples are extremely useful when labeled samples are scarce. A Bayesian perspective on the combination of labeled and unlabeled data in a semi-supervised manner is offered by Lasserre et al. (2006).
+
+A classical regularizer is weight decay (see Plaut et al., 1986; Lang and Hinton, 1990; Goodfellow et al., 2016, Chap. 7):
+
+$$
+R ( w ) = \lambda \frac 1 2 \| w \| _ { 2 } ^ { 2 } ,
+$$
+
+where $\lambda$ is a weighting term controlling the importance of the regularization over the consistency. From the Bayesian perspective, weight decay corresponds to using a symmetric multivariate normal distribution as prior for the weights: $p ( w ) = \mathcal { N } ( w | \mathbf { 0 } , \lambda ^ { - 1 } \mathbf { I } )$ (Nowlan and Hinton, 1992). Indeed, $\begin{array} { r } { - \log \mathcal { N } ( w | \mathbf { 0 } , \lambda _ { \cdot } ^ { - 1 } \mathbf { I } ) \propto - \log \exp \left( - \frac { \lambda } { 2 } \| w \| _ { 2 } ^ { 2 } \right) = \frac { \lambda } { 2 } \| w \| _ { 2 } ^ { 2 } = R ( w ) } \end{array}$ . Weight decay has gained big popularity, and it is being successfully used; Krizhevsky et al. (2012) even observe reduction of the error on the training set.
+
+Another common prior assumption that can be expressed via the regularization term is “smoothness” of the learned mapping (see Bengio et al., 2013, Section 3.2): if $x _ { 1 } \approx x _ { 2 }$ , then $f _ { w } ( x _ { 1 } ) \approx f _ { w } ( x _ { 2 } )$ . It can be expressed by the following loss term:
+
+$$
+R ( f _ { w } , x ) = \left. J _ { f _ { w } } ( x ) \right. _ { F } ^ { 2 } ,
+$$
+
+where $\left\| \cdot \right\| _ { F }$ denotes the Frobenius norm, and $J _ { f _ { w } } ( x )$ is the Jacobian of the neural network input-to-output mapping $f _ { w }$ for some fixed network weights $w$ . This term penalizes mappings with large derivatives, and is used in contractive autoencoders (Rifai et al., 2011c).
+
+The domain of loss regularizers is very heterogeneous. We propose a natural way to categorize them by their dependence. We saw in Eq. (5) that weight decay depends on $w$ only, whereas the Jacobian penalty in Eq. (6) depends on $w$ , $f$ , and $x$ . More precisely, the Jacobian penalty uses the derivative $\partial y / \partial x$ of output $y = f _ { w } ( x )$ w.r.t. input $x$ . (We use vector-by-vector derivative notation from matrix calculus, i.e. $\partial y / \partial x = \partial f _ { w } ( x ) / \partial x = J _ { f _ { w } }$ is the Jacobian of $f _ { w }$ with fixed weights $w$ .) We identify the following dependencies of $R$ :
+
+∙ Dependence on the weights $w$
+∙ Dependence on the network output $y = f _ { w } ( x )$
+∙ Dependence on the derivative $\partial y / \partial w$ of the output $y = f _ { w } ( x )$ w.r.t. the weights $w$
+∙ Dependence on the derivative $\partial y / \partial x$ of the output $y = f _ { w } ( x )$ w.r.t. the input $x$
+∙ Dependence on the derivative $\partial E / \partial x$ of the error term $E$ w.r.t. the input $x$ ( $E$ depends on $t$ , and according to our definition such methods belong to Section 5, but they are listed here for overview)
+
+A review of existing methods can be found in Table 4. Weight decay seems to be still the most popular of the regularization terms. Some of the methods are equivalent or nearly equivalent to other methods from different taxonomy branches. For example, Tangent prop simulates minimal data augmentation (Simard et al., 1992); Injection of small-variance Gaussian noise (Bishop, 1995b; An, 1996) is an approximation of Jacobian penalty (Rifai et al., 2011c); and Fast dropout (Wang and Manning, 2013) is (in shallow networks) a deterministic approximation of Dropout. This is indicated in the Equivalence column in Table 4.
+
+# 7 Regularization via optimization
+
+The last class of the regularization methods according to our taxonomy is the regularization through optimization. While this may sound unusual, optimization and regularization cannot be clearly separated in the context of deep learning where it is not so crucial what the optimum of the empirical risk is (because it cannot be found exactly, and the ultimate goal is minimizing the expected risk anyway). Instead, the shape of the loss function and the optimization procedure play together to dictate how the training proceeds in the weight space and where it ends up. To demonstrate the overlap of regularization and optimization, we show in Figure 1 how one of the most prominent regularization methods, Dropout, can be seen as a modification of the optimization procedure.
+
+Stochastic gradient descent (SGD) (see Bottou, 1998) (along with its derivations) is the most frequently used optimization algorithm in the context of deep neural networks and is the center of our attention. We also list some alternative methods below.
+
+| Method | Description | Dependency | Equivalence |
| w | y | | | |
| Weight decay (see Plaut et al.,1986;Lang and Hinton,1990;Goodfellow et al.,2016,Chap.7) | L² norm on network weights (notbiases). Favors smaller weights,thus for usual architectures tendsto make the mapping less“extreme",more robust to noise in the input. | × | | | | | Early stopping (seeCollobert and Bengio,2004;Goodfellow et al., 2016,Chap.7) |
| Weight smoothing(Lang and Hinton,1990) | Penalizes L²norm of gradientsof learned filters,making themsmooth. Not beneficial in practice. | × | | | | | |
| Weight elimination(Weigend et al.,1991) | Similar to weight decay but favorsfew stronger connections over manyweak ones. | × | | | | | Goal similar to Narrow andbroad Gaussians |
| Soft weight-sharing(Nowlan and Hinton,1992) | Mixture-of-Gaussians prior onweights.Generalization of weightdecay.Weights are pushed to forma predefined number of groups withsimilar values. | X | | | | | |
| Narrow and broad Gaussians(Nowlan and Hinton,1992;Blundell et al., 2015) | Weights come from two Gaussians,a narrow and a broad one.Specialcase of Soft weight-sharing. | X | | | | | Goal similar to Weightelimination |
| Fast dropout approximation(Wang and Manning,2013) | Approximates the loss that dropoutminimizes. Weighted L2weightpenalty. Only for shallow networks. | X | x | | | | Dropout |
| Mutual exclusivity(Sajjadi et al., 2016) | Unlabeled samples push decisionboundaries to low-density regions ininput space,promoting sharp (con-fident） predictions. | | X | | | | |
| Segmentation with binarypotentials (BenTaieb andHamarneh,2016) | Penalty on anatomically implausi-ble image segmentations. | | X | | | | |
| Flat minima search(Hochreiter and Schmidhuber,1995) | Penalty for sharp minima,i.e. forweight configurations where smallweight perturbation leads to higherror increase.Flat minima havelow Minimum description length(i.e.exhibit ideal balance betweentraining error and model complex-ity)and thus should generalize bet-ter (Rissanen,1986). | × | | X | | | |
| Tangent prop(Simard et al.,1992) | L² penalty on directional derivativeof mapping in the predefined tan-gent directions that correspond toknown input-space transformations. | | | | X | | Simple data augmentation |
| Jacobian penalty(Rifai et al., 2011c) | L2penalty on the Jacobian of(partsof） the network mapping-smoothness prior. | | | | X | | Noise on inputs injection(not exact (see An,1996)) |
| Manifold tangent classifier(Rifai et al., 2011a) | Like tangent prop,but the input“tangent” directions are extractedfrom manifold learned by a stack ofcontractive autoencoders and thenperforming SVD of the Jacobian ateach input sample. | | | | × | | |
| Hessian penalty(Rifai et al.,2011b) | Fast waytoapproximateL²penalty of thepenalizingJacobianwithnoisyinput. | | | | X | | |
| Tikhonov regularizers(Bishop,1995b) | L² penalty on (up to) n-th deriva-tive of the learned mapping w.r.t.input. | | | | × | | Forpenalty on firstderivative: noise on inputsinjection (not exact (see An,1996)） |
| Loss-invariant backpropagation(Demyanov et al., 2015, Sec. 3.1;Lyu et al., 2015) | (L²)norm of gradient of loss w.r.t.input.Changes the mapping suchthat the loss becomes rather invari-ant to changes of the input. | | | | | × | Adversarial training |
| Prediction-invariantbackpropagation(Demyanov et al.,2015, Sec. 3.2) | (L2） norm of directional derivativeof mapping w.r.t.input in the di-rection of x causing the largest in-crease in loss. | | | | × | × | Adversarial training |
+
+Table 4: Regularization terms, with dependencies marked by $\pmb { * }$ . Methods that depend on $\partial E / \partial x$ implicitly depend on targets $t$ and thus can be considered part of the error function (Section 5) rather than regularization term (Section 6).
+
+Stochastic gradient descent is an iterative optimization algorithm using the following update rule:
+
+$$
+w _ { t + 1 } = w _ { t } - \eta _ { t } \nabla _ { w } \mathcal { L } ( w _ { t } , d _ { t } ) ,
+$$
+
+where $\nabla \mathcal { L } ( w _ { t } , d _ { t } )$ is the gradient of the loss $\mathcal { L }$ evaluated on a mini-batch $d _ { t }$ from the training set $\mathcal { D }$ . It is frequently used in combination with momentum and other tweaks improving the convergence speed (see Wilson et al., 2017). Moreover, the noise induced by the varying mini-batches helps the algorithm escape saddle points (Ge et al., 2015); this can be further reinforced by adding supplementary gradient noise (Neelakantan et al., 2015; Chaudhari and Soatto, 2015).
+
+If the algorithm reaches a low training error in a reasonable time (linear in the size of the training set, allowing multiple passes through $\mathcal { D }$ ), the solution generalizes well under certain mild assumptions; in that sense SGD works as an implicit regularizer : a short training time prevents overfitting even without any additional regularizer used (Hardt et al., 2016). This is in line with (Zhang et al., 2017) who find in a series of experiments that regularization (such as Dropout, data augmentation, and weight decay) is by itself neither necessary nor sufficient for good generalization.
+
+We divide the methods into three groups: initialization/warm-start methods, update methods, and termination methods, discussed in the following.
+
+Initialization and warm-start methods These methods affect the initial selection of the model weights. Currently the most frequently used method is sampling the initial weights from a carefully tuned distribution. There are multiple strategies based on the architecture choice, aiming at keeping the variance of activations in all layers around 1, thus preventing vanishing or exploding activations (and gradients) in deeper layers (Glorot and Bengio, 2010, Sec. 4.2; He et al., 2015).
+
+Another (complementary) option is pre-training on different data, or with a different objective, or with partially different architecture. This can prime the learning algorithm towards a good solution before the fine-tuning on the actual objective starts. Pre-training the model on a different task in the same domain may lead to learning useful features, making the primary task easier. However, pre-trained models are also often misused as a lazy approach to problems where training from scratch or using thorough domain adaptation, transfer learning, or multi-task learning methods would be worth trying. On the other hand, pre-training or similar techniques may be a useful part of such methods.
+
+Finally, with some methods such as Curriculum learning (Bengio et al., 2009), the transition between pre-training and fine-tuning is smooth. We refer to them as warm-start methods.
+
+∙ Initialization without pre-training
+
+– Random weight initialization (Rumelhart et al., 1986, p. 330; Glorot and Bengio, 2010; He et al., 2015; Hendrycks and Gimpel, 2016)
+– Orthogonal weight matrices (Saxe et al., 2013)
+– Data-dependent weight initialization (Krähenbühl et al., 2015)
+
+∙ Initialization with pre-training
+
+– Greedy layer-wise pre-training (Hinton et al., 2006; Bengio et al., 2007; Erhan et al., 2010) (has become less important due to advances (e.g. ReLUs) in effective end-to-end training that optimizes all parameters simultaneously)
+– Curriculum learning (Bengio et al., 2009)
+– Spatial contrasting (Hoffer et al., 2016)
+– Subtask splitting (Gülçehre and Bengio, 2016)
+
+Update methods This class of methods affects individual weight updates. There are two complementary subgroups: Update rules modify the form of the update formula; Weight and gradient filters are methods that affect the value of the gradient or weights, which are used in the update formula, e.g. by injecting noise into the gradient (Neelakantan et al., 2015).
+
+
+Figure 1: Effect of Dropout on weight optimization. Starting from the current weight configuration (red dot), all weights of certain neurons are set to zero (black arrow), descent step is performed in that subspace (teal arrow), and then the discarded weight-space coordinates are restored (blue arrow).
+
+Again, it is not entirely clear which of the methods only speed up the optimization and which actually help the generalization. Wilson et al. (2017) show that some of the methods such as AdaGrad or Adam even lose the regularization abilities of SGD.
+
+∙ Update rules
+
+– Momentum, Nesterov’s accelerated gradient method, AdaGrad, AdaDelta, RMSProp, Adam—overview in (Wilson et al., 2017) Learning rate schedules (Girosi et al., 1995; Hoffer et al., 2017)
+Online batch selection (Loshchilov and Hutter, 2015) SGD alternatives: L-BFGS (Liu and Nocedal, 1989; Le et al., 2011), Hessianfree methods (Martens, 2010), Sum-of-functions optimizer (Sohl-Dickstein et al., 2014), ProxProp (Frerix et al., 2017)
+
+∙ Gradient and weight filters
+
+– Annealed Langevin noise (Neelakantan et al., 2015)
+– AnnealSGD (Chaudhari and Soatto, 2015) Dropout (Hinton et al., 2012; Srivastava et al., 2014) corresponds to optimization steps in subspaces of weight space, see Figure 1
+Annealed noise on targets (Wang and Principe, 1999) (works as noise on gradient, but belongs rather to data-based methods, Section 3)
+
+Termination methods There are numerous possible stopping criteria and selecting the right moment to stop the optimization procedure may improve the generalization by reducing the error caused by the discrepancy between the minimizers of expected and empirical risk: The network first learns general concepts that work for all samples from the ground truth distribution $P$ before fitting the specific sample $\mathcal { D }$ and its noise (Krueger et al., 2017).
+
+The most successful and popular termination methods put a portion of the labeled data aside as a validation set and use it to evaluate performance (validation error ). The most prominent example is Early stopping (see Prechelt, 1998). Collobert and Bengio (2004) show that Early stopping has the same effect as Weight decay regularization penalty term in multi-layered perceptrons with linear output units; however, its hyperparameters are easier to tune.
+
+In scenarios where the training data are scarce it is possible to resort to termination methods that do not use a validation set. The simplest case is fixing the number of passes through the training set.
+
+∙ Termination using a validation set – Early stopping (see Morgan and Bourlard, 1990; Prechelt, 1998) – Choice of validation set size based on test set size (Amari et al., 1997)
+∙ Termination without using a validation set – Fixed number of iterations – Optimized approximation algorithm (Liu et al., 2008)
+
+# 8 Recommendations, discussion, conclusions
+
+We see the main benefits of our taxonomy to be two-fold: Firstly, it provides an overview of the existing techniques to the users of regularization methods and gives them a better idea of how to choose the ideal combination of regularization techniques for their problem. Secondly, it is useful for development of new methods, as it gives a comprehensive overview of the main principles that can be exploited to regularize the models. We summarize our recommendations $^ 3$ in the following paragraphs:
+
+Recommendations for users of existing regularization methods Overall, using the information contained in data as well as prior knowledge as much as possible, and primarily starting with popular methods, the following procedure can be helpful:
+
+∙ Common recommendations for the first steps:
+
+– Deep learning is about disentangling the factors of variation. An appropriate data representation should be chosen; known meaningful data transformations should not be outsourced to the learning. Redundantly providing the same information in several representations is okay. Output nonlinearity and error function should reflect the learning goals.
+– A good starting point are techniques that usually work well (e.g. ReLU, successful architectures). Hyperparameters (and architecture) can be tuned jointly, but “lazily” (interpolating/extrapolating from experience instead of trying too many combinations). Often it is helpful to start with a simplified dataset (e.g. fewer and/or easier samples) and a simple network, and after obtaining promising results gradually increasing the complexity of both data and network while tuning hyperparameters and trying regularization methods.
+
+∙ Regularization via data:
+
+– When not working with nearly infinite/abundant data: $^ *$ Gathering more real data (and using methods that take its properties into account) is advisable if possible: · Labeled samples are best, but unlabeled ones can also be helpful (compatible with semi-supervised learning). Samples from the same domain are best, but samples from similar domains can also be helpful (compatible with domain adaptation and transfer learning). · Reliable high-quality samples are best, but lower-quality ones can also be helpful (their confidence/importance can be adjusted accordingly). · Labels for an additional task can be helpful (compatible with multi-task learning). · Additional input features (from additional information sources) and/or data preprocessing (i.e. domain-specific data transformations) can be helpful (the network architecture needs to be adjusted accordingly).
+
+$^ *$ Data augmentation (e.g. target-preserving handcrafted domain-specific transformations) can well compensate for limited data. If natural ways to augment data (to mimic natural transformations sufficiently well) are known, they can be tried (and combined). \* If natural ways to augment data are unknown or turn out to be insufficient, it may be possible to infer the transformation from data (e.g. learning imagedeformation fields) if a sufficient amount of data is available for that.
+
+– Popular generic methods (e.g. advanced variants of Dropout) often also help.
+
+∙ Architecture and regularization terms:
+
+– Knowledge about possible meaningful properties of the mapping can be used to e.g. hardwire invariances (to certain transformations) into the architecture, or be formulated as regularization terms.
+
+– Popular methods may help as well (see Tables 3–4), but should be chosen to match the assumptions about the mapping (e.g. convolutional layers are fully appropriate only if local and shift-equivariant feature extraction on regular-grid data is desired).
+
+∙ Optimization:
+
+Initialization: Even though pre-trained ready-made models greatly speed up prototyping, training from a good random initialization should also be considered. – Optimizers: Trying a few different ones, including advanced ones (e.g. Nesterov momentum, Adam, ProxProp), may lead to improved results. Correctly chosen parameters, such as learning rate, usually make a big difference.
+
+Recommendations for developers of novel regularization methods Getting an overview and understanding the reasons for the success of the best methods is a great foundation. Promising empty niches (certain combinations of taxonomy properties) exist that can be addressed. The assumptions to be imposed upon the model can have a strong impact on most elements of the taxonomy. Data augmentation is more expressive than loss terms (loss terms enforce properties only in infinitesimally small neighborhood of the training samples; data augmentation can use rich transformation parameter distributions). Data and loss terms impose assumptions and invariances in a rather soft manner, and their influence can be tuned, whereas hardwiring the network architecture is a harsher way to impose assumptions. Different assumptions and options to impose them have different advantages and disadvantages.
+
+Future directions for data-based methods There are several promising directions that in our opinion require more investigation: Adaptive sampling of $\theta$ might lead to lower errors and shorter training times (Fawzi et al., 2016) (in turn, shorter training times may additionally work as implicit regularization (Hardt et al., 2016), see also Section 7). Secondly, learning class-dependent transformations (i.e. $p ( \theta | t )$ ) in our opinion might lead to more plausible samples. Furthermore, the field of adversarial examples (and network robustness to them) is gaining increased attention after the recently sparked discussion on real-world adversarial examples and their robustness/invariance to transformations such as the change of camera position (Lu et al., 2017; Athalye and Sutskever, 2017). Countering strong adversarial examples may require better regularization techniques.
+
+Summary In this work we proposed a broad definition of regularization for deep learning, identified five main elements of neural network training (data, architecture, error term, regularization term, optimization procedure), described regularization via each of them, including a further, finer taxonomy for each, and presented example methods from these subcategories. Instead of attempting to explain referenced works in detail, we merely pinpointed their properties relevant to our categorization. Our work demonstrates some links between existing methods. Moreover, our systematic approach enables the discovery of new, improved regularization methods by combining the best properties of the existing ones.
+
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+
+# A Ambiguities in the taxonomy
+
+Although our proposed taxonomy seems intuitive, there are some ambiguities: Certain methods have multiple interpretations matching various categories. Viewed from the exterior, a neural network maps inputs $x$ to outputs $y$ . We formulate this as $y = f _ { w } ( \tau _ { \theta } ( x ) )$ for transformations $\tau _ { \theta }$ in input space (and similarly for hidden-feature space, where $^ Ḋ \prime \theta Ḍ$ is applied in between layers of the network $f _ { w }$ ). However, how to split this $x$ -to- $y$ mapping into “the $\tau _ { \theta }$ part” and “the $f _ { w }$ part”, and thus into Section 3 vs. Section 4, is ambiguous and up to one’s taste and goals. In our choices (marked with “ ” below), we attempt to use common notions and Occam’s razor.
+
+Ambiguity of attributing noise to $f$ , or to $w$ , or to data transformations $\tau _ { \theta }$ : – Stochastic methods such as Stochastic depth (Huang et al., 2016b) can have several interpretations if stochastic transformations are allowed for $f$ or $w$ : Stochastic transformation of the architecture $f$ (randomly dropping some connections), Table 3 $\sqcup$ Stochastic transformation of the weights $w$ (setting some weights to $0$ in a certain random pattern) $\sqcup$ Stochastic transformation $\tau _ { \theta }$ of data in hidden-feature space; dependence is $p ( \theta )$ , described in Table 1 for completeness
+
+∙ Ambiguity of splitting $^ Ḋ \prime \theta Ḍ$ into $\tau$ and $\theta$ :
+
+– Dropout:
+
+Parameters $\theta$ are the dropout mask; dependence is $p ( \theta )$ ; transformation $\tau$ applies the dropout mask to the hidden features
+
+$\sqcup$ Parameters $\theta$ are the seed state of a pseudorandom number generator; dependence is $p ( \theta )$ ; transformation $\tau$ internally generates the random dropout mask from the random seed and applies it to the hidden features
+
+– Projecting dropout noise into input space (Bouthillier et al., 2015, Sec. 3) can fit our taxonomy in different ways by defining $\tau$ and $\theta$ accordingly. It can have similar interpretations as Dropout above (if $\tau$ is generalized to allow for dependence on $x , f , w )$ , but we prefer the third interpretation without such generalizations:
+
+$\sqcup$ Parameters $\theta$ are the dropout mask (to be applied in a hidden layer); dependence is $p ( \theta )$ ; transformation $\gamma$ transforms the input to mimic the effect of the mask
+$\sqcup$ Parameters $\theta$ are the seed state of a pseudorandom number generator; dependence is $p ( \theta )$ ; transformation $\tau$ internally generates the random dropout mask from the random seed and transforms the input to mimic the effect of the mask
+V Parameters $\theta$ describe the transformation of the input in any formulation; dependence is $p ( \theta | x , f , w )$ ; transformation $\tau$ merely applies the transformation in input space
+
+∙ Ambiguity of splitting the network operation $f _ { w }$ into layers: There are several possibilities to represent a function (neural network) as a composition (or directed acyclic graph) of functions (layers).
+
+∙ Many of the input and hidden-feature transformations (Section 3) can be considered layers of the network (Section 4). In fact, the term “layer” is not uncommon for Dropout or Batch normalization.
+
+∙ The usage of a trainable parameter in several parts of the network is called weight sharing. However, some mappings can be expressed with two equivalent formulas such that a parameter appears only once in one formulation, and several times in the other.
+
+∙ Ambiguity of $E$ vs. $R$ : Auxiliary denoising task in ladder networks (Rasmus et al., 2015) and similar autoencoder-style loss terms can be interpreted in different ways:
+
+V Regularization term $R$ without given auxiliary targets $t$
+$\sqcup$ The ideal reconstructions can be considered as targets $t$ (if the definition of “targets” is slightly modified) and thus the denoising task becomes part of the error term $E$
+
+# B Data-augmented loss function
+
+To understand the success of target-preserving data augmentation methods, we consider the data-augmented loss function, which we obtain by replacing the training samples $( x _ { i } , t _ { i } ) \in \mathcal { D }$ in the empirical risk loss function (Eq. (3)) by augmented training samples $( \tau _ { \theta } ( x _ { i } ) , t _ { i } )$ :
+
+$$
+\begin{array} { r l } & { \hat { \mathcal { L } } _ { A } = \displaystyle \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \mathbb { E } _ { \theta } \Big [ \ell \big ( \tau _ { \theta } ( x _ { i } ) , t _ { i } \big ) \Big ] } \\ & { \quad \quad = \displaystyle \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \int \Big ( \ell \big ( \tau _ { \theta } ( x _ { i } ) , t _ { i } \big ) \Big ) p ( \theta ) \mathrm { d } \theta , } \end{array}
+$$
+
+where we have replaced the inner part ( $E$ and $R$ ) of the loss function by $\ell$ to simplify the notation. Moreover, $\hat { \mathcal { L } } _ { A }$ can be rewritten as
+
+$$
+\begin{array} { r l } & { \hat { \mathcal { L } } _ { A } = \displaystyle \iint \frac { 1 } { X , T } \displaystyle \sum _ { ( x , t _ { i } ) \in \mathcal { D } _ { \Theta } } \int \ell ( x , t ) \ p ( \theta ) \ \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \ \delta ( t - t _ { i } ) \ \mathrm { d } \theta \mathrm { d } t \mathrm { d } x } \\ & { \quad = \displaystyle \iint \ell ( x , t ) \Bigg [ \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } _ { \Theta } } \int \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \ \delta ( t - t _ { i } ) \ p ( \theta ) \mathrm { d } \theta \Bigg ] \mathrm { d } t \mathrm { d } x } \\ & { \quad = \displaystyle \iint \ell ( x , t ) \ q ( x , t ) \ \mathrm { d } t \mathrm { d } x , } \end{array}
+$$
+
+where $\delta ( x )$ is the Dirac delta function: $\delta ( x ) = 0 \forall x \neq 0$ and $\textstyle \int \delta ( x ) \mathrm { d } x = 1$ ; and $\boldsymbol { q } ( \boldsymbol { x } , t )$ is defined as
+
+$$
+q ( x , t ) = \frac { 1 } { | \mathcal { D } | } \sum _ { ( x _ { i } , t _ { i } ) \in \mathcal { D } } \int _ { \Theta } \delta \big ( x - \tau _ { \theta } ( x _ { i } ) \big ) \delta ( t - t _ { i } ) p ( \theta ) \mathrm { d } \theta .
+$$
+
+Since $q$ is non-negative and $\int _ { \ a } ^ { \cdot } \int q ( x , t ) \mathrm { d } x \mathrm { d } t = 1$ , it is a valid probability density function inducing the distribution $Q$ of augmented data. Therefore,
+
+$$
+\begin{array} { r } { \hat { \mathcal { L } } _ { A } = \mathbb { E } _ { ( x , t ) \sim Q } \big [ \ell ( x , t ) \big ] . } \end{array}
+$$
+
+When $Q = P$ , Eq. (11) becomes the expected risk (2). We can show how this is related to importance sampling:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } = \mathbb { E } _ { ( x , t ) \sim P } \big [ \ell ( x , t ) \big ] } \\ { \displaystyle = \int _ { X , T } \ell ( x , t ) p ( x , t ) \mathrm { d } t \mathrm { d } x } \\ { \displaystyle \quad \times \int _ { X , T } \ell ( x , \ell ) \frac { p ( x , t ) } { q ( x , t ) } q ( x , \ell ) \mathrm { d } t \mathrm { d } x } \\ { \displaystyle \quad \times _ { X , T } } \\ { \displaystyle = \mathbb { E } _ { ( x , t ) \sim Q } \Big [ \ell ( x , t ) \frac { p ( x , t ) } { q ( x , t ) } \Big ] } \\ { \displaystyle \quad \neq \mathbb { E } _ { ( x , t ) \sim Q } \big [ \ell ( x , t ) \big ] } \\ { \displaystyle \quad = \hat { \ell } _ { A + } } \end{array}
+$$
+
+The difference between $\mathcal { L }$ and $\hat { \mathcal { L } } _ { A }$ is the re-weighting term $p ( x , t ) / q ( x , t )$ identical to the one known from importance sampling (see Bishop, 1995a). The more similar $Q$ is to $P$ (i.e. the closer $Q$ models the ground truth distribution $P$ ), the more similar the augmented-data loss $\hat { \mathcal { L } } _ { A }$ is to the expected loss $\mathcal { L }$ . We see that data augmentation tries to simulate the real distribution $P$ by creating new samples from the training set $\mathcal { D }$ , bridging the gap between the expected and the empirical risk.
\ No newline at end of file
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+# MODEL-FREE CONTROL OF NONLINEAR STOCHASTIC SYSTEMS WITH STABILITY GUARANTEE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Reinforcement learning (RL) offers a principled way to achieve the optimal cumulative performance index in discrete-time nonlinear stochastic systems, which are modeled as Markov decision processes. Its integration with deep learning techniques has promoted the field of deep RL with an impressive performance in complicated continuous control tasks. However, from a control-theoretic perspective, the first and most important property of a system to be guaranteed is stability. Unfortunately, stability is rarely assured in RL and remains an open question. In this paper, we propose a stability guaranteed RL framework which simultaneously learns a Lyapunov function along with the controller or policy, both of which are parameterized by deep neural networks, by borrowing the concept of Lyapunov function from control theory. Our framework can not only offer comparable or superior control performance over the state-of-the-art RL algorithms, but also construct a Lyapunov function to validate the closed-loop stability. In the simulated experiments, our approach is evaluated on several well-known examples including classic CartPole balancing, 3-dimensional robot control and control of synthetic biology gene regulatory networks. Compared with RL algorithms without stability guarantee, our approach can enable the system to recover to the operating point when interfered by uncertainties such as unseen disturbances and system parametric variations to a certain extent. (Anonymous code is available to reproduce the experimental results1.)
+
+# 1 INTRODUCTION
+
+Control of discrete-time nonlinear stochastic systems is an important topic in both control theory and reinforcement learning. In the past decades, the advancement of nonlinear control theory in the control community has been successfully applied in aircraft, automobiles, advanced robots and space systems (Slotine et al., 1991; Isidori, 1995). Concurrently, reinforcement learning was developed in the machine learning community to address similar nonlinear control problems (Sutton et al., 1992; Tesauro, 1995; Bertsekas & Tsitsiklis, 1996). Until recently, significant progress has been made by combining advances in deep learning (LeCun et al., 2015) with reinforcement learning. Impressive results are obtained in a series of high-dimensional continuous nonlinear control problems (Duan et al., 2016; Zhang et al., 2016; Zhu et al., 2017; Gu et al., 2017) in which control-theoretic approach is typically difficult to apply.
+
+Given a control system, regardless of which controller design method is used, control theory or reinforcement learning, the first and most important property of a system needs to be guaranteed is stability, because an unstable control system is typically useless and potentially dangerous (Slotine et al., 1991). Qualitatively, a system is described as stable if starting the system in the neighborhood of its desired operating point implies that it will stay around the point ever after. For aircraft control systems, a typical stability problem is intuitively related to the following question: will a trajectory perturbation caused by a gust result in a significant deviation in the later flight trajectory? Here, the desired operating point of the system is the flight trajectory in the absence of disturbance. Every control system, whether linear or nonlinear, involves a stability problem which should be carefully studied.
+
+The most useful and general approach for studying the stability of control systems is Lyapunov method (Lyapunov, 1892), which is dominant in control engineering (Astr ˚ om & Wittenmark, 1989; ¨ Mayne et al., 2000). In Lyapunov method, a scalar “energy-like” function called Lyapunov function is constructed to analyze the stability of the system. For a linear dynamical system, a quadratic function is typically chosen as Lyapunov function in some classic controller design method, such as linear quadratic regulator (LQR) and model predictive control (MPC). Unfortunately, there is no universal method for constructing Lyapunov functions.
+
+In this paper, we propose a stability guaranteed reinforcement learning framework to jointly learn the controller or policy2 and a Lyapunov function both of which are parameterized by deep neural networks, with a focus on stabilization and tracking problems in discrete-time nonlinear stochastic systems modeled by Markov decision process. The contribution of our paper can be summarized as follows: 1) a novel data-based approach for analyzing the stability of the closed-loop system is proposed by constructing a Lyapunov function parameterized by deep neural network; 2) a practical learning algorithm is designed to search the stability guaranteed controller; 3) the learned controller is able to stabilize the system when interfered by uncertainties such as unseen disturbance and system parameters variations of certain extent. In our experiment, we can show that the stability guaranteed controller is more capable of handling uncertainties compared to those without such guarantees in nonlinear control problems including classic CartPole stabilization tasks, control of 3D legged robots and manipulator and reference tracking tasks for synthetic biology gene regulatory networks.
+
+# 1.1 RELATED WORKS
+
+In model-free reinforcement learning (RL), stability is rarely addressed due to the formidable challenge of analyzing and designing the closed-loop system dynamics in a model-free manner (Bus¸oniu et al., 2018), and the associated stability theory in model-free RL remains as an open problem (Bus¸oniu et al., 2018; Gorges, 2017).
+
+Recently, Lyapunov analysis is used in model-free RL to solve control problems with safety constraints (Chow et al., 2018; 2019). In Chow et al. (2018), Lyapunov-based approach for solving constrained Markov decision process is proposed with a novel way of constructing the Lyapunov function through linear programming. In Chow et al. (2019), the above results were further generalized to continuous control tasks. Even though Lyapunov-based methods were adopted in these results, neither of them addressed the stability of the system.
+
+As a basic tool in control theory, the construction/learning of Lyapunov function is not a trivial issue and many works are devoted to this problem. In Perkins & Barto (2002), the RL agent controls the switch between controllers designed using Lyapunov domain knowledge, so that any policy is safe and reliable. Petridis & Petridis (2006) proposes a straightforward approach for constructing Lyapunov function for nonlinear systems using neural networks. Richards et al. (2018) proposes a learning-based approach for constructing Lyapunov neural networks with maximized region of attraction. Results on learning and construction of Lyapunov functions are referred to Noroozi et al. (2008); Prokhorov (1994); Serpen (2005); Prokhorov & Feldkamp (1999).
+
+Other interesting results on the stability of learning-based control systems are reported in recent years. In Postoyan et al. (2017), an initial result is proposed for the stability analysis of deterministic nonlinear systems with optimal controller for infinite-horizon discounted cost, based on the assumption that discount is sufficiently close to 1. In Berkenkamp et al. (2017), a learning model-based safe RL approach with safety guarantee during exploration is introduced but limited to Lipschitz continuous nonlinear systems such as Gaussian process model. In addition, the verification of stability condition requires the discretization of state space, which limits its application to tasks with low-dimensional finite state space.
+
+# 2 PROBLEM STATEMENT
+
+We consider discrete-time nonlinear stochastic systems modeled by the Markov decision process (MDP). A MDP is defined by the tuple $( S , { \mathcal { A } } , c , P , \rho )$ , where $S \subseteq \mathbb { R } ^ { n }$ is the set of states, $\mathcal { A } \subseteq \mathbb { R } ^ { m }$ is the set of actions, $c ( s , a ) : S \times \mathcal { A } \to \mathbb { R } _ { + }$ is the cost function, $P ( s ^ { \prime } | s , a )$ is the transition probability function, and $\rho ( s )$ is the starting state distribution.
+
+In this paper, we focus on the stabilization and tracking problems for discrete-time nonlinear stochastic systems modeled by MDP. For both problems, the goal is to find a policy $\pi$ which can bring the cost $c$ to zero. In stabilization problems, the cost function is defined as the norm of states $\| s \|$ where $\| \cdot \|$ denotes the Euclidean norm. In tracking problems, we divide the state $s$ into two vectors, $s ^ { 1 }$ and $s ^ { 2 }$ , where $s ^ { 1 }$ is composed of elements of $s$ that are aimed at tracking the reference signal $r$ while $s ^ { 2 }$ contains the rest. For tracking, $\| s ^ { 1 } - r \|$ is chosen to be the cost function.
+
+From a control theoretic perspective, the task of stabilization and tracking could be addressed as ensuring the closed-loop system or error system to be asymptotically stable, i.e., starting from an initial point, the trajectories of state always converge to a single point or the reference trajectory. Let $c _ { \pi } ( s ) \triangleq \mathbb { E } _ { a \sim \pi } c ( s , a )$ denote the cost function under policy $\pi$ , the definition of stability studied in this paper is given as follows.
+
+Definition 1 The stochastic system is said to be stable in mean cost if $\begin{array} { r } { \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { s _ { t } } c _ { \pi } ( s _ { t } ) = 0 } \end{array}$ holds for any initial condition $s _ { 0 } \in \{ s _ { 0 } | c _ { \pi } ( s _ { 0 } ) \leq b \}$ . If $b$ is arbitrarily large then the stochastic system is globally stable in mean cost.
+
+Remark 1 Form of the cost is strictly ruled as the Euclidean norm of state or partial state, while other forms are not considered in this paper. The stability studied in this paper is a type of local asymptotic stochastic stability, which is different to the definition of mean square stability (MSS) that extensively studied on stochastic systems in control theory (Shaikhet, 1997; Huang, 2012).
+
+Before proceeding, some notations are to be defined. The closed-loop transition probability is denoted as $\begin{array} { r } { P _ { \pi } ( s ^ { \prime } | s ) \triangleq \int _ { \cal A } \pi ( a | s ) P ( s ^ { \prime } | s , a ) \mathrm { d } a } \end{array}$ . We also introduce the closed-loop state distribution at certain instant $t$ as $P ( s | \rho , \pi , t )$ , which could be defined in an iterative way: $P ( s ^ { \prime } | \rho , \pi , t + 1 ) =$ $\begin{array} { r } { \int _ { S } P _ { \pi } ( s ^ { \prime } | s ) P ( s | \rho , \pi , t ) \mathrm { d } s , \forall t \ } \end{array}$ $\forall t \in \mathbb { Z } _ { + }$ and $P ( s | \rho , \pi , 0 ) = \rho ( s )$ .
+
+# 3 MAIN RESULTS
+
+In this section, we propose the main assumptions and a new theorem.
+
+Assumption 1 The stationary distribution of state $q _ { \pi } ( s ) \triangleq \operatorname* { l i m } _ { t \infty } P ( s | \rho , \pi , t )$ exists.
+
+Assumption 2 There exists a positive constant b such that $\rho ( s ) > 0 , \forall s \in \{ s | c _ { \pi } ( s ) \leq b \} .$
+
+Our approach is to construct/find a Lyapunov function which can be used to analyze the stability of the closed-loop system. The Lyapunov method has long been used for stability analysis and controller design in control theory (Boukas & Liu, 2000), but mostly exploited along with a known model, whether deterministic or probabilistic (Corless & Leitmann, 1981; Thowsen, 1983; Huang et al., 2011).
+
+The Lyapunov function is a class of semi-positive definite functions $L : S \to \mathbb { R } _ { + }$ . The general idea of exploiting Lyapunov function is to ensure that the difference (or derivative, if the system is in continuous time) of Lyapunov function along the state trajectory is semi-negative definite, so that the state goes in the direction of decreasing the value of Lyapunov function and eventually converges to the origin or a sub-level set of Lyapunov function. Next, we give sufficient conditions for a system to be stable in mean cost in the following.
+
+Theorem 1 The stochastic system is stable in mean cost if there exists a function $L : S \to \mathbb { R _ { + } }$ and positive constants $\alpha _ { 1 } , \alpha _ { 2 }$ and $\alpha _ { 3 }$ , such that
+
+$$
+\begin{array} { c } { { \alpha _ { 1 } c _ { \pi } \left( s \right) \leq L ( s ) \leq \alpha _ { 2 } c _ { \pi } \left( s \right) } } \\ { { { \mathbb E } _ { s \sim \mu _ { \pi } } ( { \mathbb E } _ { s ^ { \prime } \sim P _ { \pi } } L ( s ^ { \prime } ) - L ( s ) ) \leq - \alpha _ { 3 } { \mathbb E } _ { s \sim \mu _ { \pi } } c _ { \pi } \left( s \right) } } \end{array}
+$$
+
+where $\begin{array} { r } { \mu _ { \pi } ( s ) \triangleq \operatorname* { l i m } _ { N \to \infty } \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s _ { t } = s | \rho , \pi , t ) } \end{array}$ is the sampling distribution.
+
+Due to space limitations, we will include the detailed proof in Appendix A. Eq. (2) is called the energy decreasing condition, i.e., requiring the expectation of Lyapunov function to be decreasing between two consecutive instants. Eq. (1) is the constraint for Lyapunov function, though a rather broad range of parameterization is covered. The sum of quadratic polynomials, e.g., $L ( s ) { \stackrel { \smile } { = } } s ^ { T } Q s$ where $Q$ is a positive definite matrix, are extensively used in the control theory. Such Lyapunov functions can be efficiently discovered by the semi-definite programming solvers and bring in limited conservatism for the control tasks where the cost are also of a quadratic form. In (Richards et al., 2018), a neural network $\phi _ { \theta } ( \cdot )$ is designed to construct the Lyapunov function, $L ( s ) = \phi _ { \theta } ( s ) ^ { T } \phi _ { \theta } ( s )$ . As explored in (Chow et al., 2018) and (Berkenkamp et al., 2017), the value function could be exploited as a Lyapunov function as well. Additionally, the sum of cost over a limited time horizon could also be employed as Lyapunov function, i.e., $\begin{array} { r } { \dot { L ( s ) } = \sum _ { t } ^ { t + N } \mathbb { E } c _ { \pi } ( s _ { t } ) } \end{array}$ , which is a valid Lyapunov candidate in model predictive control literature (Mayne & Michalska, 1990; Mayne et al., 2000).
+
+The choice of Lyapunov function candidate plays an important role in learning a policy. Value function evaluates the infinite time horizon and thus offers a better performance in general, but is rather difficult to approximate because of significant variance and bias (Schulman et al., 2015). On the other hand, the finite horizon sum of cost provides an explicit target for learning a Lyapunov function, thus inherently reduces the bias and enhances the learning process. However, as the model is unknown, predicting the future costs based on the current state and action inevitably introduces variance, which grows as the prediction horizon extends. In principle, for tasks with simple dynamics, the sum-of-cost choice enhances the convergence of learning and robustness of the trained policies, while for complicated systems the choice of value function generally produces better performance. In this paper, we use both value function and sum-of-cost over various horizons as Lyapunov function candidates in different tasks and compare their strength and weakness respectively. Now we would like to give the following two remarks.
+
+Remark 2 This remark is on Assumption 1 and sampling distribution $\mu _ { \pi }$ . If an MDP is ergodic then the existence of $q _ { \pi }$ is naturally assured, but all states have to be positive recurrent and aperiodic (Papoulis & Pillai, 2002). The existence of sampling distribution $\mu _ { \pi } ( s )$ is guaranteed by the existence of $q _ { \pi } ( s )$ . Since the sequence $\{ P ( s | \rho , \pi , t ) , t \in \mathbb { Z } _ { + } \}$ converges to $q _ { \pi } ( s )$ as $t$ approaches $\infty _ { \astrosun }$ , then by the Abelian theoThus we use , the sequence to approximat $\begin{array} { r } { \{ \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s | \rho , \pi , t ) , N \in \mathbb { Z } _ { + } \} } \end{array}$ also converges and requires data to b $\mu _ { \pi } ( s ) = q _ { \pi } ( s )$ $\mu _ { \pi }$ $q _ { \pi }$ $q _ { \pi }$
+infinite instants the episode begins. Even if the sampling period $N < < \infty$ , one can still assure that $c _ { \pi }$ converges to a neighborhood of zero, which is related to the initial state distribution and length of $N _ { i }$ , i.e., $\begin{array} { r } { \mathbb { E } _ { s _ { N } } c _ { \pi } ( s _ { N } ) \le \frac { \alpha _ { 2 } } { \alpha _ { 1 } + \alpha _ { 3 } } \mathbb { E } _ { \rho } c _ { \pi } ( s _ { 0 } ) - \frac { \alpha _ { 3 } } { \alpha _ { 1 } + \alpha _ { 3 } } \sum _ { t = 0 } ^ { N - 1 } \mathbb { E } _ { s _ { t } } c _ { \pi } ( s _ { t } ) . } \end{array}$ .
+
+Remark 3 This remark is on the connection to previous results concerning the stability of stochastic systems. It should be noted that the stability conditions of Markov chains have been reported in (Shaikhet, 1997; Meyn & Tweedie, 2012), however, of which the validation requires the full knowledge of the model, i.e., the transition probability $P ( s ^ { \prime } | s , a )$ . On the contrary, our approach solely depends on data to analyze the stability of the closed-loop system, which further enables the model-free learning algorithms with stability guarantee. However, the validation of stability through a sample-based approach theoretically requires tremendous, if not infinite, amount of samples to thoroughly estimate the distributions, which is the drawback of our approach. We would demonstrate empirically that the algorithm built upon this theorem is reliable though only limited sample is collected.
+
+# 4 ALGORITHM
+
+In this section, we propose an off-policy RL algorithm to learn stability guaranteed policies for discrete-time nonlinear stochastic system modeled by MDP. First, based on the maximum entropy actor-critic framework, we use the Lyapunov function as the critic in the policy gradient formulation. In this algorithm, a Lyapunov critic function $L _ { c }$ is needed, which satisfies $\bar { L } ( \bar { s } ) = \mathbb { E } _ { a \sim \pi } L _ { c } ( s , a )$ . The objective function $J ( \pi )$ is given as follows
+
+$$
+J ( \pi ) = \mathbb { E } _ { ( s , a , s ^ { \prime } , c ) \sim \mathcal { D } } [ \beta ( \log ( \pi _ { \theta } ( f _ { \theta } ( \epsilon , s ) | s ) ) + \mathcal { H } _ { t } ) + \lambda ( L _ { c } ( s ^ { \prime } , f _ { \theta } ( \epsilon , s ^ { \prime } ) ) - L _ { c } ( s , a ) + \alpha _ { 3 } c ) ]
+$$
+
+where the policy $\pi _ { \theta }$ is parameterized by a deep neural network $f _ { \theta } , \epsilon$ is an input vector consisted of Gaussian noise. It should be noted that the Lyapunov critic $L _ { c } ( s , a )$ will be parameterized by the square of a neural network to ensure the semi-positive definiteness of Lyapunov function required in Eq.(1), inspired by the structure explored in Richards et al. (2018). More specifically, $\overset { \cdot } { L _ { c } ( s , a ) } = \phi ^ { \hat { T } } ( s , a ) \phi ( \overset { \cdot } { s } , a )$ , where $\phi ( s , a )$ is a multi-layer fully connected neural network. $\mathcal { D }$ is the distribution of previously sampled states and actions, or a replay buffer. In the above objective, $\beta$ and $\lambda$ are Lagrange multipliers which control the relative importance of policy entropy versus energy decreasing constraint derived from Eq.(2). Similar to Haarnoja et al. (2018), the entropy of policy is expected to remain above the target entropy $\mathcal { H } _ { t }$ . The parameters of policy network are updated through gradient descent, where the gradient of Eq.(3) is approximated by
+
+$$
+\nabla _ { \theta } J ( \pi ) = \nabla _ { \theta } \beta \log ( \pi _ { \theta } ( a | s ) ) + \nabla _ { a } \beta \log ( \pi _ { \theta } ( a | s ) ) \nabla _ { \theta } f _ { \theta } ( \epsilon , s ) + \lambda \nabla _ { a ^ { \prime } } L _ { c } ( s ^ { \prime } , a ^ { \prime } ) \nabla _ { \theta } f _ { \theta } ( \epsilon , s ^ { \prime } )
+$$
+
+We use $J ( L _ { c } )$ in the following equation as the objective function to update the Lyapunov critic,
+
+$$
+J ( L _ { c } ) = \mathbb { E } _ { \mathcal { D } } \left[ \frac { 1 } { 2 } ( L _ { c } ( s , a ) - L _ { \mathrm { t a r g e t } } ( s , a ) ) ^ { 2 } \right]
+$$
+
+where $L _ { \mathrm { t a r g e t } }$ is the approximation target for $L _ { c }$
+
+If the sum of cost is chosen as Lyapunov function candidate, we have
+
+$$
+L _ { \mathrm { t a r g e t } } ( s , a ) = \Sigma _ { t } ^ { t + N } c _ { t }
+$$
+
+Here, the time horizon $N$ is a hyperparameter to be tuned, of which the influence will be demonstrated in the experiment in Section 5.5. If the value function is chosen as Lyapunov function candidate,
+
+$$
+L _ { \mathrm { t a r g e t } } ( s , a ) = c + \gamma L _ { c } ^ { \prime } ( s ^ { \prime } , f ( \epsilon , s ^ { \prime } ) )
+$$
+
+where $L _ { c } ^ { \prime }$ is the target network parameterized by $\overline { { \theta } }$ as typically used in the actor-critic methods (Haarnoja et al., 2018; Lillicrap et al., 2015b). $L _ { c } ^ { \prime }$ has the same structure with $L _ { c }$ , but the parameter is updated through exponentially moving average of weights of $L _ { c }$ controlled by a hyperparameter $\tau$ . In fact, the value function is the discounted sum of cost over infinite time horizon. Later in Section 5, we will show the influence of choosing different Lyapunov function candidates.
+
+The value of Lagrange multipliers $\lambda$ and $\beta$ are adjusted by the gradient method maximizing the following two objectives respectively,
+
+$$
+\begin{array} { r l } & { J ( \beta ) = \mathbb { E } _ { \mathcal { D } } \beta \big [ \log ( \pi _ { \theta } ( a \vert s ) ) + \mathcal { H } _ { t } \big ] } \\ & { J ( \lambda ) = \mathbb { E } _ { \mathcal { D } } \lambda \big [ L _ { c } ( s ^ { \prime } , f _ { \theta } ( \epsilon , s ^ { \prime } ) ) - L _ { c } ( s , a ) + \alpha _ { 3 } c \big ] } \end{array}
+$$
+
+It should be noted that the value of $\lambda$ and $\beta$ are clipped to be positive. In addition, to prevent $\lambda$ from growing unlimitedly causing the algorithm to diverge, we set an upper bound for $\lambda$ . In our experiments, we found that 1 is a suitable value without much further tuning. Pseudo code of the proposed algorithm is shown in Algorithm 1 in Appendix B.
+
+Remark 4 The convergence of the algorithm is composed of the convergence of Lyapunov critic $L _ { c }$ and policy $\pi _ { \theta }$ respectively. Empirically, the convergence can be judged by the error of Lyapunov function approximation and value of the Lagrange multiplier (close to zero at convergence). In practice, we found that the algorithm converges well in different experiments without much tuning.
+
+# 5 EXPERIMENT
+
+In this section, we illustrate four simulated examples to demonstrate the general applicability of the proposed method. First of all, the classic control problem of CartPole balancing from control and RL literature (Barto et al., 1983) is illustrated. Then, we consider more complicated high-dimensional continuous control problem of 3D robots, e.g., HalfCheetah and FetchReach, using MuJoCo physics engine (Todorov et al., 2012). Last, we extend our approach to control robots in nanoscale, i.e., molecular robots. Specifically, we consider the problem of reference tracking for a synthetic biology gene regulatory network known as the Repressilator (Elowitz & Leibler, 2000).
+
+The proposed method is evaluated for the following aspects:
+
+• Convergence: does the proposed training algorithm converge with random parameter initialization and does the stability condition (2) hold for the learned policies; • Performance: can the goal of the task be achieved or the cumulative cost be minimized; • Robustness: how do the trained policies perform when faced with uncertainties unseen during training, such as parametric variation and external disturbances;
+
+• Generalization: can the trained policies generalize to follow reference signals that are different from the one seen during training.
+
+We compare our approach with soft actor-critic (SAC) (Haarnoja et al., 2018), one of the stateof-the-art off-policy actor-critic algorithms that outperform a series of off-policy and on-policy methods such as DDPG (Lillicrap et al., 2015a), PPO (Schulman et al., 2017) on the continuous control benchmarks. The variant of safe proximal policy optimization (SPPO) (Chow et al., 2019), a Lyapunov-based method, is also included in the comparison. The original SPPO is developed to deal with constrained MDP, where safety constraints exist. In our experiments, we modify it to apply the Lyapunov constraints on the MDP tasks and see whether it can achieve the same stability guarantee as LAC. In CartPole example, we also compare with linear quadratic regulator (LQR), a classical model-based optimal control method for stabilization.
+
+The outline of this section is as follows. In Section 5.1, a brief introduction will be given on the background and problem description of each example. Then in Section 5.2, the convergence, and performance of the proposed method is demonstrated and compared with SAC. In Section 5.4, the ability of generalization and robustness of the trained policies are evaluated and analyzed. Finally, in Section 5.5, we show the influence of choosing different Lyapunov function candidates upon the performance and robustness of trained policies.
+
+# 5.1 BACKGROUND AND PROBLEM DESCRIPTION
+
+In this section, we will give a brief introduction to the examples considered in this paper. Detailed setup information of the first three examples can be found in Appendix C.
+
+# 5.1.1 CARTPOLE
+
+This is a classical control problem. The controller is to stabilize the pole vertically at a given position. The cost is determined by the norm of the angular position of the pole and the horizontal position of the cart. The control input is the horizontal force $F \in [ - 2 0 , 2 0 ]$ applied in the cart. The agent is dead if the angle $\theta$ between pole and vertical position exceeds a threshold, and the episode ends.
+
+# 5.1.2 HALFCHEETAH
+
+The goal is to control a 17-dimensional 2-legged robot simulated in the MuJoCo simulator. The control task belongs to the reference tracking problem, i.e., to enable the robot to run at the speed of $1 \mathrm { m / s }$ in the $\mathbf { X }$ -axis direction. The cost is determined by the Euclidean difference between current speed and target speed. The control input is the torque implemented at each joint.
+
+# 5.1.3 FETCHREACH
+
+The agent is to control a simulated manipulator to track a randomly generated goal position with its end effector. The cost is determined by the Euclidean distance between end effector and goal. The control input is the torque implemented at each joint. The manipulator is also simulated in the MuJoCo simulator.
+
+# 5.1.4 REPRESSILATOR
+
+The repressilator is a synthetic biology gene regulatory network with a ring structure pioneered in Elowitz & Leibler (2000), in which each gene represses the other gene cyclically. The dynamics of temporal gene expression exhibit periodic oscillatory behavior. The dynamics of repressilator can be quantitatively described by a set of discrete-time nonlinear difference equations consisting of six states, three mRNAs for transcription and three proteins for translation, based on biochemical kinetic laws. We also include a complicated repressilator example with 4 genes to be controlled, which exhibits an unstable oscillation and is even harder to control.
+
+The objective is to force one protein concentrations to follow a priori defined reference trajectories using partially observed states. Detailed setup information of these examples are in Appendix D.
+
+# 5.1.5 MARKOVIAN JUMP SYSTEMS
+
+In addition to the systems described above, we introduce two Markovian jump systems (MJS), named MJS1 and MJS2, which contain both discrete switchings (or jumps) and continuous dynamics (Shi &
+
+Li, 2015), as test beds for the proposed and baseline methods. The objective is to force the full state to zero. Both MJSs contain unstable subsystems and the dynamics change abruptly and randomly according to the switching signal, and thus are difficult to tackle for the model-free algorithms. Moreover, MJS2 contains an unstable subsystem that is not controllable, which makes it even harder to stabilize. More details on the examples could be found in Appendix E.
+
+# 5.2 PERFORMANCE
+
+We parameterize the policy and Lyapunov critic using deep neural networks. For each example,the hyperparameters including time horizon $N$ and DNN architectures selected to construct Lyapunov functions and DNN training parameters can be found in Appendix J. For both algorithms, the hyperparameters are tuned to reach their best performance. In each task, both LAC and SAC are trained for 10 times with random initialization, average total cost and its variance during training are demonstrated in Figure 1.
+
+In the first three examples (see Figure 1(a)-(c)), SAC and LAC perform comparably in terms of the total cost at convergence and speed of convergence, while SPPO could converge in Cartpole and FetcheReach. In the Repressilator and MJS examples (see Figure 1(d,e,f)), SAC is not always able to find a policy that is capable of completing control objective, resulting in the bad average performance. On the contrary, LAC performs stably regardless of the random initialization.
+
+
+Figure 1: Cumulative control performance comparison. The Y-axis indicates the total cost during one episode and the $\mathbf { X }$ -axis indicates the total time steps in thousand. The shadowed region shows the 1-SD confidence interval over 10 random seeds. Across all trials of training, LAC converges to stabilizing solution with comparable or superior performance compared with SAC and SPPO. The experiment on Complicated-Repressilator is deferred to Appendix F.
+
+A distinguishing feature of stability assured policy is that it can force and sustain the state or tracking error to zero. This could be intuitively demonstrated by the state trajectories of closed-loop system. We evaluated this property of trained policies in the Repressilator, Complicated-Repressilator and two MJS examples. In our experiments, we found that the LAC agents stabilize the systems well in all tasks. All the state trajectories converge to the reference signal or equilibrium eventually (see Figure 11 (a,c) and Figure 12 (a,c)). On the contrary, without stability guarantee, the state trajectories either diverge (see Figure 11 b and Figure 12 d), or continuously oscillate around the reference trajectory or equilibrium (see Figure 11 d and Figure 12 b). Empirical results are deferred to Appendix F due to space limit.
+
+# 5.3 CONVERGENCE
+
+As shown in Figure 1, LAC converges stably in all experiments. Moreover, the convergence and validation of stability guarantee could also be checked by observing the value of Lagrange multipliers. When (2) is satisfied, $\lambda$ will continuously decrease until it becomes zero. Thus by checking the value and variation of $\lambda$ , the satisfaction of stability condition during training and at convergence could be validated. In Figure 2, the value of $\lambda$ during training is demonstrated. Across all training trials in the experiments, $\lambda$ converges to zero eventually, which implies that the stability guarantee is valid. A detailed discussion on this is referred to Appendix G.
+
+
+Figure 2: Value of Lagrange multiplier $\lambda$ during the training of LAC policies. The Y-axis indicates the value of $\lambda$ and the X-axis indicates the total time steps in thousand. The shadowed region shows the 1-SD confidence interval over 10 random seeds. The value of $\lambda$ gradually drops and becomes zero at convergence, which implies the satisfaction of stability condition.
+
+# 5.4 EVALUATION ON ROBUSTNESS AND GENERALIZATION
+
+It is well-known that over-parameterized policies are prone to become overfitted to a specific training environment. The ability of generalization is the key to the successful implementation of the algorithm in an uncertain real-world environment. In this part, we first evaluate the robustness of policies in the presence of system parametric uncertainties and process noise. Then, we test the robustness of controllers against external disturbances. Finally, we evaluate whether the policy is generalizable by setting different reference signals. To make a fair comparison, we removed the policies that did not converge in SAC and only evaluate the ones that perform well during training. During testing, we found that SPPO appears to be prone to variation in the environment, thus the evaluation results are referred to Appendix H.
+
+# 5.4.1 ROBUSTNESS TO DYNAMIC UNCERTAINTY
+
+In this part, during the inference, we vary the system parameters in the model/simulator to evaluate the algorithm’s robustness against dynamic uncertainty. In the example of CartPole, we vary the length of pole $l$ . In the example of repressilator, we vary the promoter strength $a _ { i }$ and dissociation rate $K _ { i }$ . Due to stochastic nature in gene expression, we also introduce uniformly distributed noise ranging from $[ - \delta , \delta ]$ (we indicate the noise level by $\delta$ ) to the dynamic of repressilator. The stabilization performance of CartPole and tracking performance of Repressilator by LAC and SAC in the varied environment is demonstrated in Figure 3.
+
+
+Figure 3: State trajectories over time under policies trained by LAC and SAC and tested in the presence of parametric uncertainties and process noise, for CartPole and Repressilator. Solid line indicates the average trajectory and shadowed region for the 1-SD confidence interval. In (a) and (b), the pole length is varied during the inference. In (c) and (d), three parameters are selected to reflect the uncertainties in gene expression. The X-axis indicates the time and Y-axis shows the angle of pole in (a,b) and concentration of protein to be controlled in (c,d), respectively. Dashed line indicates the reference signal. The line in orange indicates the dynamic in original environment. For each curve, only the noted parameter is different with the original setting. We also show the curves in separate zoom-in view in Appendix I.1.
+
+As shown in Figure 3(a) and (c), the policies trained by LAC are very robust to parametric uncertainties of different values and achieve high tracking precision in each case. On the other hand, though SAC performs well in the original environment (Figure 3(b) and (d)), it fails to track the reference signal in all of the varied environment.
+
+# 5.4.2 ROBUSTNESS TO DISTURBANCES
+
+An inherent property of a controller for stabilization is to enable the system to recover to the normal status from perturbations such as external forces and wind. To show this, we introduce persistent external disturbances with different magnitudes in each environment and observe the performance difference between policies trained by LAC and SAC. We also include LQR as the model-based baseline. In CartPole, the agent may fall over when interfered by an external force, ending the episode in advance. Thus in this task, we measure the robustness of controller through the death-rate, i.e., the probability of falling over after being disturbed. For other tasks where the episodes are always of the same length, we measure the robustness of controller by the variation in total cost. Under each disturbance magnitude, the policies are tested for 100 trials and the performance are shown in Figure 4.
+
+
+Figure 4: Performance of policies trained by LAC, SAC and SPPO, along with controllers designed by LQR in the presence of persistent disturbances with different magnitudes. $\mathbf { X } \cdot$ -axis indicates the magnitude of the applied disturbance. For CartPole (a) the Y-axis indicates the probability of falling over and in other three examples (b)-(d) it indicates the total cost. Both policies are evaluated for 100 trials in each setting.
+
+As shown in the Figure 4, the controller trained by LAC outperforms SAC and LQR by great extent when faced with external disturbances in CartPole and repressilator (lower death rate and total cost). In the repressilator example, the policies trained by SAC are extremely vulnerable to disturbances, this is potentially due to the existence of an unstable limit cycle in the uncontrolled dynamic (Strelkowa & Barahona, 2010). In HalfCheetah, SAC and LAC are both robust to small external disturbances while LAC is more reliable to larger ones. In FetchReach, SAC and LAC are comparable with maintaining a low cost in a great range of external disturbances. This is perhaps due to the manipulator’s inherent robust mechanical design.
+
+# 5.4.3 GENERALIZATION OVER DIFFERENT TRACKING REFERENCES
+
+In this part, we introduce four different reference signals that are unseen during training in the repressilator example: sinusoids with periods of 150 (brown) and 400 (blue), and the constant reference of 8 (red) and 16 (green). We also show the original reference signal used for training (skyblue) as a benchmark. Reference signals are indicated in Figure 5 by the dashed line in respective colors. Both trained policies are evaluated to track each reference signal for 10 times, and the average dynamics of the target protein concentration are shown in Figure 5 with the solid line, while the variance of dynamic is indicated by the shadowed area.
+
+As shown in Figure 5, the policies trained by LAC could generalize well to follow previously unseen reference signals with low deviation (dynamics are very close to the dashed lines), regardless of whether they are in the same mathematical form with the one used for training or not. On the other hand, though SAC tracks the original reference signal well after the unconverged training trials being removed (see the skyblue lines), it is still unable to follow some of the reference signals (see the brown line) and possesses larger variance than LAC when following others.
+
+# 5.5 INFLUENCE OF DIFFERENT LYAPUNOV FUNCTION CANDIDATES AND STRUCTURES
+
+In this part, we evaluate the influence of choosing different Lyapunov function candidates and network structures. First, we adopt candidates of different time horizon $N \in \{ 5 , 1 0 , 1 5 , 2 0 , \infty \}$ to train policies in the CartPole example, and compare their performance in terms of total cost and robustness. Both of the Lyapunov critics are parameterized by $\begin{array} { r } { L ( s ) = \phi ( s ) ^ { T } \phi ( s ) } \end{array}$ where $\phi ( s )$ is a neural network with $m$ dimensional output. Here, $N = \infty$ implies using value function as Lyapunov candidate. For evaluation of robustness, we apply an impulsive force $F$ at $1 0 0 _ { \mathrm { t h } }$ instant and observe the death-rate of trained policies. The results are demonstrated in Figure 6 (a,b). Then we fix the horizon of candidates to be $N = 5$ but vary the structures of Lyapunov critic, and compare their perfare: $\begin{array} { r } { L ( s ) = \sum _ { j = 1 } ^ { m ^ { - } } \phi _ { j } ^ { 4 } ( s ) } \end{array}$ me metric as de(LAC-biquad); $\begin{array} { r } { L ( s ) = \sum _ { j = 1 } ^ { m } \left| \phi ( s ) \right| } \end{array}$ specifically, (LAC-abs); $\begin{array} { r } { L ( s ) = \phi ( s ) ^ { T } \phi ( s ) } \end{array}$ ctures(LACquad).
+
+
+Figure 5: State trajectories under policies trained by LAC and SAC when tracking different reference signals. Solid line indicates the average trajectory and shadowed region for the 1-SD confidence interval. The $\mathrm { X }$ -axis indicates the time and Y-axis shows the concentration of protein to be controlled. Dashed lines in different colors are the different reference signals: sinusoid with period of 150 (brown); sinusoid with period of 200 (skyblue);sinusoid with period of 400 (blue); constant reference of 8 (red); constant reference of 16 (green). We also show the curves in separate zoom-in view in Appendix I.2 .
+
+
+Figure 6: Influence of different Lyapunov function candidates and network structures. In (a) and (c), the Y-axis indicates total cost of policies during training by LAC with Lyapunov function candidates of different length of horizon $N$ and structures, and the $\mathbf { X }$ -axis indicates the total time steps in thousand. (b) and (d) shows the death-rate of policies in the presence of instant impulsive force $F$ ranging from 80 to 150 Newton.
+
+As shown in Figure 6, in the CartPole environment, both choices of Lyapunov candidates converge fast and achieve comparable total cost at convergence. However, in terms of robustness, the different choices of $N$ play an important role. As observed in Figure 6 (b), the robustness of controller decreases as the time horizon $N$ increases. On the other hand, LAC with different structures converge well and possesses similar robustness to impulsive forces. This further proves that our framework allows for a general class of Lyapunov functions, as long as the function is semi-positive definite. Besides, it is interesting to observe that LQR is more robust than SAC when faced with instant impulsive disturbance.
+
+# 6 CONCLUSIONS
+
+In this paper, we proposed a model-free approach for analyzing the stability of discrete-time nonlinear stochastic systems modeled by Markov decision process, by employing the Lyapunov function from control theory. Based on the theoretical result, a practical algorithm for designing stability assured controllers for the stabilization and tracking problems. We evaluated the proposed method in various examples and show that our method achieves not only comparable or superior performance compared with the state-of-the-art RL algorithm but also outperforms impressively in terms of robustness to uncertainties and disturbances.
+
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+
+# Appendix
+
+# A PROOF OF THEOREM 1
+
+The existence of sampling distribution $\mu _ { \pi } ( s )$ is guaranteed by the existence of $q _ { \pi } ( s )$ (Assumption 1). Since the sequence Abelian theorem, the $\{ P ( s | \rho , \pi , t ) , t \in \mathbb { Z } _ { + } \}$ $q _ { \pi } ( s )$ as also $t$ approaches onverges an $\infty$ $\begin{array} { r } { \{ \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s | \rho , \pi , t ) , N \in \mathbb { Z } _ { + } \} } \end{array}$ $\mu _ { \pi } ( s ) = q _ { \pi } ( s )$ Combined with the form of $\mu _ { \pi }$ , Eq.(2) infers that
+
+$$
+\int _ { S } \operatorname* { l i m } _ { N \infty } \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s | \rho , \pi , t ) ( \mathbb { E } _ { P _ { \pi } ( s ^ { \prime } | s ) } L ( s ^ { \prime } ) - L ( s ) ) \mathrm { d } s \leq - \alpha _ { 3 } \mathbb { E } _ { s \sim q _ { \pi } c _ { \pi } } ( s )
+$$
+
+First, on the left hand-side, according to Eq.(1), $L ( s ) \leq \alpha _ { 2 } c _ { \pi } ( s )$ for all $s \in S$ and consider that $P ( s | \rho , \pi , t ) \le 1$ ,
+
+$$
+P ( s | \rho , \pi , t ) L ( s ) \leq \alpha _ { 2 } c _ { \pi } \left( s \right) , \forall s \in \mathcal { S } , \forall t \in \mathbb { Z } _ { + }
+$$
+
+On the ofunction the sequence . According to $\begin{array} { r } { \{ \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s | \rho , \pi , t ) L ( s ) , N \in \mathbb { Z } _ { + } \} } \end{array}$ converges pointwise to thee theorem(Royden, 1968), if $q _ { \pi } ( s ) L ( s )$
+a sequence $f _ { n } ( s )$ converges pointwise to a function $f$ and is dominated by some integrable function
+$g$ in the sense that,
+
+$$
+| f _ { n } ( s ) | \leq g ( s ) , \forall s \in S , \forall n
+$$
+
+Then
+
+$$
+\operatorname* { l i m } _ { n \to \infty } \int _ { S } f _ { n } ( s ) \mathrm { d } s = \int _ { S } \operatorname* { l i m } _ { n \to \infty } f _ { n } ( s ) \mathrm { d } s
+$$
+
+Thus the left hand side of Eq.(A.1)
+
+$$
+\begin{array} { r l } & { \displaystyle \int _ { S } \displaystyle \operatorname* { l i m } _ { N \to \infty } \frac { 1 } { N } \sum _ { t = 0 } ^ { N } P ( s | \rho , \pi , t ) ( \displaystyle \int _ { S } P _ { \pi } ( s ^ { \prime } | s ) L ( s ^ { \prime } ) \mathrm { d } s ^ { \prime } - L ( s ) ) \mathrm { d } s } \\ & { = \displaystyle \operatorname* { l i m } _ { N \to \infty } \frac { 1 } { N } ( \sum _ { t = 1 } ^ { N + 1 } \mathbb { E } _ { P ( s | \rho , \pi , t ) } L ( s ) - \sum _ { t = 0 } ^ { N } \mathbb { E } _ { P ( s | \rho , \pi , t ) } L ( s ) ) } \\ & { = \displaystyle \operatorname* { l i m } _ { N \to \infty } \frac { 1 } { N } \left( \mathbb { E } _ { P ( s | \rho , \pi , N + 1 ) } L ( s ) - \mathbb { E } _ { \rho ( s ) } L ( s ) \right) } \end{array}
+$$
+
+Thus taking the relations above into consideration, Eq.(A.1) infers
+
+$$
+\operatorname* { l i m } _ { N \infty } \frac { 1 } { N } ( \mathbb { E } _ { P ( s | \rho , \pi , N + 1 ) } L ( s ) - \mathbb { E } _ { \rho ( s ) } L ( s ) ) \leq - \alpha _ { 3 } \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { P ( s | \rho , \pi , t ) } c _ { \pi } ( s )
+$$
+
+Since $\mathbb { E } _ { \rho ( s ) } L ( s )$ is a finite value and $L$ is semi-positive definite, it follows that
+
+$$
+\operatorname* { l i m } _ { t \infty } \mathbb { E } _ { P ( s | \rho , \pi , t ) } c _ { \pi } ( s ) \leq \operatorname* { l i m } _ { N \infty } \frac { 1 } { N } ( \frac { 1 } { \alpha _ { 3 } } \mathbb { E } _ { \rho ( s ) } L ( s ) ) = 0
+$$
+
+Suppose that there exists a state $s _ { 0 } \in \{ s _ { 0 } | c _ { \pi } ( s _ { 0 } ) \leq b \}$ such $\begin{array} { r } { \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { P ( s | s _ { 0 } , \pi , t ) } c _ { \pi } ( s ) = c . } \end{array}$ , $c > 0$ or $\begin{array} { r } { \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { P ( s | s _ { 0 } , \pi , t ) } c _ { \pi } ( s ) = \infty } \end{array}$ . Consider that $\rho ( s _ { 0 } ) > 0$ for all starting states in $\{ s _ { 0 } | c _ { \pi } ( s _ { 0 } ) \leq b \}$ (Assumption 2), then $\begin{array} { r } { \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { s _ { t } \sim P ( \cdot | \pi , \rho ) } c _ { \pi } ( s _ { t } ) > 0 } \end{array}$ , which is contradictory with Eq.(A.3). Thus $\forall s _ { 0 } \in \{ s _ { 0 } | c _ { \pi } ( s _ { 0 } ) \leq b \}$ , $\begin{array} { r } { \operatorname* { l i m } _ { t \infty } \mathbb { E } _ { P ( s | s _ { 0 } , \pi , t ) } c _ { \pi } ( s ) = 0 } \end{array}$ . Thus the system is stable in mean cost by Definition 1.
+
+# B PSEUDO CODE OF ALGORITHM
+
+# Algorithm 1 Lyapunov-based Actor-Critic (LAC)
+
+| Model Type Accuracy Accuracy △PerturbedOriginal |
| ImageNet-Vid-Robust |
| Trained on ILSVRC 67.5 [64.7, 70.3] 52.5 [49.5, 55.5] 15.0 |
| + Noise Augmentation 68.8 [66.0, 71.5] 53.2 [50.2, 56.2] 15.6 |
| +lrobustness (ResNext-101) 54.3 [51.3, 57.2] 40.8 [39.0, 43.7] 12.4 |
| + FT on ImageNet- Vid 80.8 [78.3, 83.1] 65.7 [62.9, 68.5] 15.1 |
| + FT on ImageNet-Vid (ResNet-152) 84.8 [82.5, 86.8] | 70.2 [67.4, 72.8] 14.6 |
| + FT on ImageNet-Vid-Det 77.6 [75.1, 80.0] 65.4 [62.5, 68.1] 12.3 |
| YTBB-Robust |
| Trained on ILSVRC 57.0 [54.9, 59.2] | 43.8 [41.7, 46.0] 13.2 |
| +Noise Augmentation 62.3 [60.2, 64.4] | 45.7 [43.5, 47.9] | 16.6 |
| + l robustness (ResNext-101) 53.6 [51.4, 55.8] | 43.2 [41.0, 45.3] | 10.4 |
| + FT on Youtube-BB 91.4 [90.1, 92.6] | 82.0 [80.3, 83.7] | 9.4 |
| + FT on Youtube-BB (ResNet-152) 92.9 [91.6, 93.9] | 84.7 [83.0, 86.2] | 8.2 |
+
+ILSVRC Trained The WordNet hierarchy enables us to repurpose models trained for the 1,000 class ILSVRC dataset on ImageNet-Vid-Robust and YTBB-Robust (see Appendix A.1). We evaluate a wide array of ILSVRC-2012 models (available from Cadene) against our natural perturbations. Since these datasets present a substantial distribution shift from the original ILSVRC2012 validation, we expect the benign accuracy $\operatorname { a c c } _ { \mathrm { o r i g } }$ to be lower than the comparable accuracy on the ILSVRC-2012 validation set. However, our main interest here is in the difference between the original and perturbed accuracies $\operatorname { a c c } _ { \mathrm { o r i g } } - \operatorname { a c c } _ { \mathrm { p m k } }$ . A small drop in accuracy would indicate that the model is robust to small changes that occur naturally in videos. Instead, we find significant drops of $1 5 . 0 \%$ and $1 3 . 2 \%$ in accuracy on our two datasets, indicating sensitivity to such changes.
+
+Noise augmentation One hypothesis for the accuracy drop from original to perturbed accuracy is that subtle artifacts and corruptions introduced by video compression schemes could degrade performance when evaluating on these corrupted frames. The worst-case nature of the $\mathrm { p m - k }$ metric could then be focusing on these corrupted frames. One model for these corruptions are the perturbations introduced in Hendrycks and Dietterich (2019). To test this hypothesis, we evaluate models augmented with a subset of the perturbations (exactly one of: Gaussian noise, Gaussian blur, shot noise, contrast change, impulse noise, or JPEG compression). We found that these augmentation schemes did not improve robustness against our perturbations substantially, and still result in accuracy drop of $1 5 . 6 \%$ and $1 6 . 6 \%$ on the two datasets.
+
+$\ell _ { \infty }$ robustness. We evaluate the model from Xie et al. (2018), which currently performs best against $\ell _ { \infty }$ attacks on ImageNet. We find that this model has a smaller accuracy drop than the two aforementioned model types on both datasets. However, we note that the robust model achieves significantly lower original and perturbed accuracy than either of the two model types above, and the robustness gain is modest $3 \%$ compared to models of similar benign accuracy).
+
+Fine-tuning on video frames. To adapt to the new class vocabulary and the video domain, we fine-tune several network architectures on the ImageNet-Vid and Youtube-BB training sets. For Youtube-BB, we train on the anchor frames used for training in Gu et al. (2019), and for ImageNet-Vid we use all frames in the training set. We provide hyperparameters for all models in Appendix K.
+
+The resulting models significantly improve in accuracy over their ILSVRC pre-trained counterparts (e.g., $13 \%$ on ImageNet-Vid-Robust and $34 \%$ on YTBB-Robust for ResNet-50). This improvement in accuracy results in a modest improvement in the accuracy drop for YTBB-Robust, but a finetuned ResNet-50 still suffers from a significant $9 . 4 \%$ drop. On ImageNet-Vid-Robust, there is almost no change in the accuracy drop from $1 5 . 0 \%$ to $1 5 . 1 \%$ .
+
+Fine-tuning for detection on video frames. We further analyze whether additional supervision in the form of bounding box annotations improves robustness. To this end, we train the Faster R-CNN detection model Ren et al. (2015) with a ResNet-50 backbone on ImageNet-Vid. Following standard practice, the detection backbone is pre-trained on ILSVRC-2012. To evaluate this detector for classification, we assign the class with the most confident bounding box as label to the image. We find that this transformation reduces accuracy compared to the model trained for classification $( 7 7 . 6 \%$ vs. $8 0 . 8 \%$ ). While there is a slight reduction in the accuracy drop caused by natural perturbations, the reduction is well within the error bars for this test set.
+
+# 3.2 DETECTION
+
+We further study the impact of natural perturbations on object detection. Specifically, we report results for two related tasks: object localization and detection. Object detection is the standard computer vision task of correctly classifying an object and finding the coordinates of a tight bounding box containing the object. “Object localization”, meanwhile, refers to only the subtask of finding the bounding box, without attempting to correctly classify the object.
+
+We present our results on ImageNet-Vid-Robust, which contains dense bounding box labels unlike Youtube-BB, which only labels boxes at 1 frame per second. We use the popular Faster R-CNN Ren et al. (2015) and R-FCN Dai et al. (2016); Xiao and Jae Lee (2018) architectures for object detection and localization and report results in Table 3. For the R-FCN architecture, we use the model from Xiao and Jae Lee $( 2 0 1 8 ) ^ { 4 }$ . We first note the significant drop in mAP of 12 – 15 points for object detection due to perturbed frames for both the Faster R-CNN and R-FCN architectures. Next, we show that localization is indeed easier than detection, as the mAP is higher for localization than for detection (e.g., 76.6 vs 62.8 for Faster R-CNN with a ResNet-50 backbone). Perhaps surprisingly, however, switching to the localization task does not improve the drop between original and perturbed frames, indicating that natural perturbations induce both classification and localization errors. We show examples of detection failures in Figure 4.
+
+
+Figure 4: Naturally perturbed examples for detection. Red boxes indicate false positives; green boxes indicate true positives; white boxes are ground truth. Classification errors are common failures, such as the fox on the left, which is classified correctly in the anchor frame, and misclassified as a sheep in a nearby frame. However, detection models also have localization errors, where the object of interest is not correctly localized in addition to being misclassified, such as the airplane (middle) and the motorcycle (right). All visualizations show predictions with confidence greater than 0.5.
+
+# 3.3 IMPACT OF DATASET REVIEW
+
+We analyze the impact of our human review, described in Section 2.1, on the classifiers in our test bed. First, we compare the original and perturbed accuracies of a representative classifier (ResNet152 finetuned) with and without review in Table 4. Our review improves the original accuracy by $3- 4 \%$ by throwing away mislabeled or blurry anchor frames, and improves perturbed accuracy by $5- 6 \%$ by discarding pairs of dissimilar frames. Our review reduces the accuracy drop by $1 . 8 \%$ on
+
+Table 3: Detection and localization mAP for two Faster R-CNN backbones. Both detection and localization suffer from significant drops in mAP due to the perturbations. (\*Model trained on ILSVRC Det and VID 2015 datasets, and evaluated on the 2015 subset of ILSVRC-VID 2017.)
+
+| Model | Accuracy Original | Accuracy Perturbed | | △ |
| resnet152_finetuned | 84.8 [82.5, 86.8] | | 70.2 [67.4, 72.8] | 14.6 |
| resnet50_finetuned | 80.8 [78.3, 83.1] | | 65.7 [62.9, 68.5] | 15.1 |
| vgg16bn_finetuned | 78.0 [75.4, 80.4] | | 61.0 [58.1, 63.9] | 17.0 |
| nasnetalarge_imagenet_pretrained | 77.6 [75.1, 80.1] | | 62.1 [59.2, 65.0] | 15.5 |
| resnet50_detection | 77.6 [75.1, 80.1] | | 65.0 [62.1, 67.8] | 12.6 |
| inceptionresnetv2_imagenet_pretrained | 75.7 [73.1, 78.2] | | 58.7 [55.7, 61.6] | 17.0 |
| dpn107_imagenet_pretrained | 75.6 [72.9, 78.1] | | 59.1 [56.1, 62.0] | 16.5 |
| inceptionv4_imagenet_pretrained | 75.3 [72.6, 77.8] | | 59.0 [56.0, 61.9] | 16.3 |
| dpn92_imagenet_pretrained | 74.4 [71.7, 76.9] | | 56.8 [53.8, 59.7] | 17.6 |
| dpn131_imagenet_pretrained | 74.0 [71.3, 76.6] | | 59.9 [56.9, 62.8] | 14.1 |
| dpn68b_imagenet_pretrained | 73.7 [71.0, 76.2] | | 54.0 [51.0, 57.0] | 19.7 |
| resnext101_32x4d_imagenet_pretrained | 73.3 [70.6, 75.9] | | 57.2 [54.2, 60.1] | 16.1 |
| resnext101_64x4d_imagenet_pretrained | 72.9 [70.1, 75.5] | | 56.6 [53.7, 59.6] | 16.3 |
| resnet152_imagenet_pretrained | 72.8 [70.0, 75.4] | | 57.0 [54.0, 59.9] | 15.8 |
| resnet1O1_imagenet_pretrained | 71.5 [68.7, 74.1] | | 53.7 [50.8, 56.7] | 17.8 |
| fbresnet152_imagenet_pretrained | 71.5 [68.7, 74.1] | | 54.5 [51.5, 57.4] | 17.0 |
| densenet161_imagenet_pretrained | 71.4 [68.7, 74.1] | | 55.1 [52.1, 58.1] | 16.3 |
| densenet169_imagenet_pretrained | 70.2 [67.5, 72.9] | | 53.1 [50.1, 56.1] | 17.1 |
| densenet2O1_imagenet_pretrained | 70.2 [67.5, 72.9] | | 53.4 [50.4, 56.4] | 16.8 |
| dpn68_imagenet_pretrained | 69.4 [66.6, 72.1] | | 53.3 [50.3, 56.3] | 16.1 |
| bninception_imagenet_pretrained | 69.0 [66.2, 71.7] | | 49.0 [46.0, 51.9] | 20.0 |
| densenet121_imagenet_pretrained | 69.0 [66.2, 71.7] | | 50.9 [47.9, 53.8] | 18.1 |
| nasnetamobile_imagenet_pretrained | 68.8 [66.0, 71.5] | | 48.4 [45.4, 51.4] | 20.4 |
| resnet50_augment_ jpeg_compression | 68.8 [66.0, 71.5] | | 53.2 [50.2, 56.2] | 15.6 |
| resnet34_imagenet_pretrained | 68.0 [65.2, 70.7] | | 48.0 [45.0, 51.0] | 20.0 |
| resnet50_augment impulse_noise | 67.7 [64.9, 70.5] | | 50.2 [47.2, 53.2] | 17.5 |
| resnet50_augment_gaussian_blur | 67.7 [64.9, 70.5] | | 52.5 [49.5, 55.5] | 15.2 |
| resnet5O_imagenet_pretrained | 67.5 [64.7, 70.3] | | 52.5 [49.5, 55.5] | 15.0 |
| resnet50_augment gaussian_noise | 67.4 [64.5, 70.1] | | 50.6 [47.6, 53.6] | 16.8 |
| resnet50_augment shot_noise | 66.5 [63.6, 69.2] | | 51.1 [48.1, 54.1] | 15.4 |
| vgg16_bn_imagenet_pretrained | 66.4 [63.5, 69.1] | | 47.4 [44.5, 50.4] | 19.0 |
| resnet50_augment_ _defocus_blur | 66.3 [63.4, 69.1] | | 47.6 [44.6, 50.6] | 18.7 |
| vgg19_bn_imagenet_pretrained | 65.6 [62.7, 68.4] | | 46.6 [43.6, 49.6] | 19.0 |
+
+Table 5: Classification model perturbed and original accuracies for all models in our test bed evaluated on the ImageNet-Vid-Robust dataset.
+
+| Model | Accuracy Original | Accuracy Perturbed | | △ |
| resnet152_finetuned | | 92.9 [91.2, 94.3] | 84.7 [82.4, 86.8] | 8.2 |
| resnet50_finetuned | 91.4 [89.6, 93.0] | | 82.0 [79.6, 84.2] | 9.4 |
| inceptionresnetv2_finetuned | 91.3 [89.5, 92.9] | | 79.0 [76.4, 81.3] | 12.3 |
| vgg19_finetuned | 90.5 [88.6, 92.2] | | 79.1 [76.5, 81.4] | 11.4 |
| vgg16_finetuned | 89.1 [87.1, 90.8] | | 78.0 [75.4, 80.4] | 11.1 |
| inceptionv4_finetuned | 88.5 [86.5, 90.3] | | 76.3 [73.6, 78.7] | 12.2 |
| resnet18_finetuned | 88.0 [85.9, 89.8] | | 76.2 [73.6, 78.7] | 11.8 |
| alexnet_finetuned | 80.6 [78.2, 82.9] | | 64.4 [61.5, 67.3] | 16.2 |
| pnasnet5large_imagenet_pretrained | 65.2 [62.3, 68.0] | | 51.0 [48.0, 54.0] | 14.2 |
| nasnetalarge_imagenet_pretrained | 64.9 [62.0, 67.7] | | 51.4 [48.4, 54.4] | 13.5 |
| inceptionresnetv2_imagenet_pretrained | 64.5 [61.6, 67.4] | | 50.4 [47.5, 53.4] | 14.1 |
| dpn98_imagenet_pretrained | 64.1 [61.2, 66.9] | | 49.0 [46.0, 52.0] | 15.1 |
| dpn107_imagenet_pretrained | 64.1 [61.2, 66.9] | | 50.1 [47.2, 53.1] | 14.0 |
| dpn131_imagenet_pretrained | 64.0 [61.1, 66.8] | | 49.9 [46.9, 52.9] | 14.1 |
| inceptionv4_imagenet_pretrained | 63.6 [60.7, 66.4] | | 48.8 [45.8, 51.8] | 14.8 |
| Xception_imagenet_pretrained | 63.2 [60.2, 66.0] | | 47.6 [44.6, 50.6] | 15.6 |
| dpn92_imagenet_pretrained | 62.3 [59.3, 65.1] | | 47.7 [44.8, 50.7] | 14.6 |
| resnet50_augment_jpeg_compressioon | 62.3 [59.4, 65.2] | | 45.7 [42.8, 48.7] | 16.6 |
| polynet_imagenet_pretrained | 61.4 [58.4, 64.3] | | 47.3 [44.4, 50.3] | 14.1 |
| nasnetamobile_imagenet_pretrained | 61.4 [58.4, 64.3] | | 43.0 [40.1, 46.0] | 18.4 |
| resnet50_augment__shot_noise | 61.3 [58.3, 64.2] | | 46.4 [43.4, 49.3] | 14.9 |
| dpn68_imagenet_pretrained | 61.2 [58.3, 64.1] | | 44.2 [41.2, 47.2] | 17.0 |
| fbresnet152_imagenet_pretrained | 61.1 [58.1, 64.0] | | 45.9 [42.9, 48.8] | 15.2 |
| resnet152_imagenet_pretrained | 60.8 [57.8, 63.7] | | 46.5 [43.5, 49.5] | 14.3 |
| resnet101_imagenet_pretrained | 60.8 [57.8, 63.7] | | 45.2 [42.2, 48.2] | 15.6 |
| senet154_imagenet_pretrained | 60.7 [57.7, 63.6] | | 47.2 [44.3, 50.2] | 13.5 |
| resnet50_augment__impulse_noise | 60.6 [57.7, 63.5] | | 45.5 [42.6, 48.5] | 15.1 |
| se_resnet101_imagenet_pretrained | 60.5 [57.6, 63.4] | | 45.6 [42.6, 48.6] | 14.9 |
| bninception_imagenet_pretrained | 60.4 [57.4, 63.3] | | 41.8 [38.9, 44.7] | 18.6 |
| densenetl61_imagenet_pretrained | 60.2 [57.3, 63.1] | | 46.4 [43.4, 49.4] | 13.8 |
| resnet50_augment_gaussian_noise | 60.2 [57.3, 63.1] | | 45.7 [42.8, 48.7] | 14.5 |
| se_resnext50_32x4d_imagenet_pretrained | 59.9 [56.9, 62.8] | | 45.7 [42.7, 48.6] | 14.2 |
| dpn68b_imagenet_pretrained | 59.7 [56.7, 62.6] | | 45.9 [42.9, 48.8] | 13.8 |
| inceptionv3_imagenet_pretrained | 59.6 [56.6, 62.5] | | 43.8 [40.8, 46.8] | 15.8 |
| densenet121_imagenet_pretrained | 59.5 [56.5, 62.4] | | 43.1 [40.1, 46.0] | 16.4 |
| se_resnext101_32x4d_imagenet_pretrained | 59.2 [56.3, 62.1] | | 45.2 [42.3, 48.2] | 14.0 |
| densenet2O1_imagenet_pretrained | 59.2 [56.2, 62.1] | | 44.8 [41.8, 47.8] | 14.4 |
| densenet169_imagenet_pretrained | 59.2 [56.2, 62.1] | | 44.6 [41.7, 47.6] | 14.6 |
+
+Table 6: Classification model perturbed and original accuracies for all models in our test bed evaluated on the YTBB-robust dataset..
+
+| resnet50_augment__brightness_change | | 58.9 [56.0, 61.8] | | 42.6 [39.6, 45.5] | 16.3 |
| se_resnet50_imagenet_pretrained | | 58.8 [55.9, 61.7] | | 44.1 [41.1, 47.1] | 14.7 |
| se_resnet152_imagenet_pretrained | | 58.8 [55.9, 61.7] | | 44.8 [41.9, 47.8] | 14.0 |
| cafferesnetlO1_imagenet_pretrained | | 58.2 [55.2, 61.1] | | 44.3 [41.3, 47.3] | 13.9 |
| resnet50_augment__regular | | 58.0 [55.1, 61.0] | | 42.9 [39.9, 45.8] | 15.1 |
| resnet34_imagenet_pretrained | | 57.9 [55.0, 60.9] | | 42.8 [39.8, 45.7] | 15.1 |
| vgg19_imagenet_pretrained | | 57.5 [54.6, 60.5] | | 40.1 [37.2, 43.1] | 17.4 |
| resnet50_augment t_gaussian_blur | | 57.5 [54.5, 60.4] | | 41.8 [38.9, 44.7] | 15.7 |
| vgg16_bn_imagenet_pretrained | | 57.2 [54.2, 60.1] | | 39.6 [36.7, 42.6] | 17.6 |
| resnet5O_imagenet_pretrained | | 57.0 [54.1, 60.0] | | 43.8 [40.9, 46.8] | 13.2 |
| vgg19_bn_imagenet_pretrained | | 56.8 [53.9, 59.8] | | 40.6 [37.7, 43.5] | 16.2 |
| vgg16_imagenet_pretrained | | 55.4 [52.4, 58.4] | | 40.1 [37.2, 43.1] | 15.3 |
| vgg13_bn_imagenet_pretrained | | 54.8 [51.8, 57.7] | | 38.6 [35.7, 41.6] | 16.2 |
| vgg11_bn_imagenet_pretrained | | 54.8 [51.8, 57.7] | | 38.8 [35.9, 41.8] | 16.0 |
| vgg11_imagenet_pretrained | | 54.7 [51.7, 57.6] | | 38.4 [35.5, 41.3] | 16.3 |
| resnetl8_imagenet_pretrained | | 54.4 [51.4, 57.4] | | 38.1 [35.2, 41.0] | 16.3 |
| vgg13_imagenet_pretrained | | 54.2 [51.3, 57.2] | | 37.7 [34.9, 40.7] | 16.5 |
| ResNeXtDenoiseAll-101_robust_pgd | | 53.6 [50.7, 56.6] | | 43.2 [40.2, 46.1] | 10.4 |
| squeezenet1_O_imagenet_pretrained | | 51.1 [48.1, 54.1] | | 33.1 [30.3, 36.0] | 18.0 |
| squeezenetl_1_imagenet_pretrained | | 48.6 [45.6, 51.6] | | 31.3 [28.6, 34.2] | 17.3 |
| resnet50_augment__defocus_blur | | 48.4 [45.4, 51.4] | | 29.1 [26.4, 31.8] | 19.3 |
| alexnet_imagenet_pretrained | | 45.3 [42.4, 48.3] | | 30.5 [27.8, 33.3] | 14.8 |
+
+# C MODEL INDEPENDENT DISTRIBUTION SHIFT
+
+Though the distribution shift we induced in our study were model dependent because we found the worst neighbor frame for each model, we could study the same problem but impose a static set of perturbed frames across all models. In Figure 6 we study this static set of perturbations across all models and see a substantial (but smaller) drop in accuracy for both models. The static set of perturbations were chosen by choosing the neighbor frame that the largest number of models classified incorrectly.
+
+
+Figure 6: Model accuracy on original vs. perturbed images for a static set of perturbed frames across all models. The grey points and grey linear fit correspond to the perturbed accuracies of models evaluated on per model perturbations studied in Figure 3
+
+# D PER CLASS ANALYSIS
+
+We study the effect of our perturbations on the 30 classes in ImageNet-Vid-Robust and YTBB-Robust to determine whether the performance drop was concentrated in a few “hard” classes.
+
+
+Figure 7: Per-class accuracy statistics for our best performing classification model (fine-tuned ResNet152) on ImageNet-Vid-Robust and YTBB-Robust. For Youtube-BB, note that ‘zebra’ is the least common label, present in only 24 anchor frames sampled by Gu et al. (2019), of which 4 are included in our dataset.
+
+Figure 7 shows the original and perturbed accuracies across classes for our best performing model (a fine-tuned ResNet-152). Although there are a few particularly difficult classes for perturbed accuracy (e.g., lion or monkey on ImageNet-Vid-Robust), the accuracy drop is spread across most classes. On ImageNet-Vid-Robust, this model saw a total drop of $1 4 . 4 \%$ between original and perturbed images and a median drop of $1 4 . 0 \%$ in per-class accuracy. On YTBB-Robust, the total drop was $8 . 9 \%$ and the median drop was $6 . 7 \%$ .
+
+# E PER-FRAME CONDITIONAL ROBUSTNESS METRIC INTRODUCED IN GU ET AL. (2019)
+
+
+Figure 8: Conditional robustness metric from Gu et al. (2019) on perturbed frames as a function of perturbation distance on ImageNet-Vid-Robust and YTBB-Robust. Model accuracies from five different model types and the best performing model are shown. The model architecture is ResNet-50 unless otherwise mentioned.
+
+In concurrent work, the authors of Gu et al. (2019) considered a different metric of robustness. In this section, we compute this metric on all models in our test bed to compare our findings to $\mathrm { G u }$ et al. (2019). There are two main differences between PM- $\mathbf { \nabla } \cdot \mathbf { k }$ and the robustness metric in Gu et al. (2019).
+
+1. For two visually similar “neighbor” frames $I _ { 0 }$ and $I _ { 1 }$ with true label $Y$ and classifier $f$ , Gu et al. (2019) studies the conditional probability $P ( f ( I _ { 1 } ) = y | f ( I _ { 0 } ) = y )$ 2. While PM-k looks for errors in all neighbor frames in a neighborhood of $k$ frames away from the anchor frame (so this would include frames $1 , 2 , \ldots , \mathbf { k }$ frames away), Gu et al. (2019) only considers errors from exactly $\mathbf { k }$ frames away.
+
+In Fig. 9 we illustrate simple example where two videos can have the same behavior for the metric introduced by Gu et al. (2019) but drastically different behavior for the PM-kmetric.
+
+
+Figure 9: For the two example videos above the score from Gu et al. (2019) metric (Accuracy $\ @ \mathrm { ~ K ~ }$ ) is identical, but the PM- $\mathbf { \nabla } \cdot \mathbf { k }$ metric behaves substantially differently when the errors are spread across many independent videos, as shown in the right example
+
+# F $\ell _ { \infty }$ DISTANCE VS PM-K ACCURACY
+
+$\ell _ { \infty }$ adversarial examples are well studied in the robustness community, yet the connection between $\ell _ { \infty }$ and other forms of more “natural” robustness is unclear. Here, we plot the cumulative distribution of the $\ell _ { \infty }$ distance between pairs of nearby frames in our datasets. In Figure 10, we show the CDF of $\ell _ { \infty }$ distance for all pairs, all reviewed pairs, and mistakes made by 3 indicative models. Note the fbrobust model is trained specifically to be robust to $\ell _ { \infty }$ adversaries.
+
+
+Figure 10: CDF showing the $\ell _ { \infty }$ distance between pairs of frames from different distributions.
+
+Table 7: Analyzing results based on frame-type in video compression. See Appendix H.1 for details.
+
+ | Methods | Input | Computer | Photo | CS | Physics |
| Supervised GCN 22] Supervised GAT [47] | X,A,Y X,A,Y | 86.51 ± 0.54 86.93 ± 0.29 | 92.42 ± 0.22 92.56 ± 0.35 | 93.03 ± 0.31 92.31 ± 0.24 | 95.65 ± 0.16 95.47 ± 0.15 |
| prrrrssnren | Raw Features 4 Linear CCA [18] DeepWalk [32 DeepWalk + features GAE [21 DGI MVGRL [15] GRACE1 四 GCA1 [58] X,A | X X A X,A X,A X,A X,S,A X,A | 73.81 ± 0.00 79.84 ± 0.53 85.68 ± 0.06 86.28 ± 0.07 85.27 ± 0.19 83.95 ± 0.47 87.52 ± 0.11 86.25 ± 0.25 | 78.53 ± 0.00 86.92 ± 0.72 89.44 ± 0.11 90.05 ± 0.08 91.62 ± 0.13 91.61 ± 0.22 91.74 ± 0.07 92.15 ± 0.24 | 90.37 ± 0.00 93.13 ± 0.18 84.61 ± 0.22 87.70 ± 0.04 90.01 ± 0.71 92.15 ± 0.63 92.11 ± 0.12 | 93.58 ± 0.00 95.04 ± 0.17 91.77 ± 0.15 94.90 ± 0.09 94.92 ± 0.07 94.51 ± 0.52 95.33 ± 0.03 |
+
+1 GCA is essentially an enhanced version of GRACE by adopting adaptive augmentations. Both GRACE and GCA would suffer from out of memory on Coauthor-Physics using a GPU wth 16GB memory. The reported results are from authors’ papers using a 32GB GPU.
+
+# 5.2 Ablation Study and Scalability Comparison
+
+Effectiveness of invariance/decorrelation terms. We alter our loss by removing the invariance/decorrelation term respectively to study the effects of each component, with results reported in Table 4. We find that only using the invariance term will lead to merely performance drop instead of completely collapsed solutions. This is because node embeddings are normalized along the instance dimension to have a zero-mean and fixed-standard deviation, and the worst solution is no worse than dimensional collapse (i.e., all the embeddings lie in an line, and our decorrelation term can help to prevent it) instead of complete collapse (i.e., all the embeddings degenerate into a single point). As expected, only optimizing the decorrelation term will lead to poor result, as the model learns nothing meaningful but disentangled representation. In Appendix $\bigtriangledown$ we discuss the relationship between complete/dimensional collapse, when the two cases happen and how to avoid them.
+
+Effect of decorrelation intensity. We study how the intensity of feature decorrelation improves/degrades the performance by increasing the trade-off hyper-parameter $\lambda$ . Fig. $2$ shows test accuracy w.r.t. different $\lambda$ ’s on Cora, Citeseer and Pubmed. The performance benefits from a proper selection of $\lambda$ $( \mathrm { f r o m 0 . 0 0 0 5 }$ to 0.001 in our experiments). When $\lambda$ is too small, the decorrelation term does not work; if it is too large, the invariance term would be neglected, leading to serious performance degrade. An interesting finding is that even when $\lambda$ is very small or even equals to 0 (w/o $\mathcal { L } _ { d e c }$ in Table $\textcircled{4}$ , the test accuracy on Citeseer does not degrade as much as that on Cora and Citeseer. The reason is that node embeddings of Citeseer is already highly uncorrelated even without the decorrelation term. Appendix F visualizes the correlation matrices without/with decorrelations.
+
+Effect of embedding dimension. Fig. 3 shows the effect of the embedding dimension. Similar to contrastive methods [48, 15, 57, 58], CCA-SSG benefits from a large embedding dimension (compared with supervised learning), while the optimal embedding dimension of CCA-SSG (512 on most benchmarks) is a bit larger than other methods (usually 128 or 256). Yet, we notice a performance drop as the embedding dimension increases. We conjecture that the CCA is essentially a dimension-reduction method, the ideal embedding dimension ought to be smaller than the dimension of input. Hence we do not apply it on well-compressed datasets (e.g. ogbn-arXiv and ogbn-product).
+
+Scalability Comparison. Table 5 compares model size, training time (till the epoch that gives the highest evaluation accuracy) and memory cost of CCA-SSG with other methods, on Cora, Pubmed and Amazon-Computers. Overall, our method has fewer parameters, shorter training time, and fewer memory cost than MVGRL, GRACE and GCA in most cases. DGI is another simple and efficient model, but it yields much poorer performance. The results show that despite its simplicity and efficiency, our method achieves even better (or competitive) performance.
+
+Table 4: Ablation study of node classification accuracy $( \% )$ on the key components of CCA-SSG.
+
+| Duluth.Q485708 the_Grateful_Dead.Q212533 Polar_Music_Prize.Q754841 ： award_received.P166 award_received.P166 Sara_Lownds.Q457433 | spouse.P26 vork_location.P937 residence.P551 United_States_of_America.Q30 country_of_citizenship.P27 | residence.P551 tour_with country_of_citizenship.P27 | Harmonica.Q51290 instrument.P1303 BobDy an.Q392 | tour_with | and_the_Heartbreakers.Q2117272 member_of.P463 | genre.P136 record_label.P264 genre. | Rock_music.Q11399 Columbia_ tour_with _Records.Q183387 e.P136 | genre.P136 the_Band.Q600344 Urban_contemporary_gospel.Q5967604 . |
| Jnited_Against_Apartheid.Q126977 |
| New_York_City.Q60 Minneapolis.Q36091 Features new knowledge not in | | Grammy_Lifetime_Achievement_Award.Q935843 | award_received.P166 | genre.P136 unmarried_partner.P451 | award_received.P166 Folk_music.Q235858 | occupation.P106 residence.P551 | songwriter.Q753110 Nashville.Q23197 |
| O | /Kennedy_Center_Honors.414306 | O (in blue) |
| existing KGs (in yellow) | award_received.P166/ genre.P136 | Joan_Baez.Q131725_award_received.P166 American_Folk_Music.Q1541229 ： | singer-songwriter.Q488205 | |
| occupation.P106 record_label.P264 | Proper_Records.Q7250169 | Contains knowledge in existing KGs |
| Thomas_Mertoh_Award.Q337620 |
+
+#
\ No newline at end of file
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+# GROUNDING PHYSICAL CONCEPTS OF OBJECTS AND EVENTS THROUGH DYNAMIC VISUAL REASONING
+
+Zhenfang Chen The University of Hong Kong
+
+Jiayuan Mao MIT CSAIL
+
+Jiajun Wu Stanford University
+
+Kwan-Yee K. Wong The University of Hong Kong
+
+Joshua B. Tenenbaum MIT BCS, CBMM, CSAIL
+
+Chuang Gan MIT-IBM Watson AI Lab
+
+# ABSTRACT
+
+We study the problem of dynamic visual reasoning on raw videos. This is a challenging problem; currently, state-of-the-art models often require dense supervision on physical object properties and events from simulation, which are impractical to obtain in real life. In this paper, we present the Dynamic Concept Learner (DCL), a unified framework that grounds physical objects and events from dynamic scenes and language. DCL first adopts a trajectory extractor to track each object over time and to represent it as a latent, object-centric feature vector. Building upon this object-centric representation, DCL learns to approximate the dynamic interaction among objects using graph networks. DCL further incorporates a semantic parser to parse question into semantic programs and, finally, a program executor to run the program to answer the question, levering the learned dynamics model. After training, DCL can detect and associate objects across the frames, ground visual properties and physical events, understand the causal relationship between events, make future and counterfactual predictions, and leverage these extracted presentations for answering queries. DCL achieves state-of-the-art performance on CLEVRER, a challenging causal video reasoning dataset, even without using ground-truth attributes and collision labels from simulations for training. We further test DCL on a newly proposed video-retrieval and event localization dataset derived from CLEVRER, showing its strong generalization capacity.
+
+# 1 INTRODUCTION
+
+Visual reasoning in dynamic scenes involves both the understanding of compositional properties, relationships, and events of objects, and the inference and prediction of their temporal and causal structures. As depicted in Fig. 1, to answer the question “What will happen next?” based on the observed video frames, one needs to detect the object trajectories, predict their dynamics, analyze the temporal structures, and ground visual objects and events to get the answer “The blue sphere and the yellow object collide”.
+
+Recently, various end-to-end neural network-based approaches have been proposed for joint understanding of video and language (Lei et al., 2018; Fan et al., 2019). While these methods have shown great success in learning to recognize visually complex concepts, such as human activities $\mathrm { { X u } }$ et al., 2017; Ye et al., 2017), they typically fail on benchmarks that require the understanding of compositional and causal structures in the videos and text (Yi et al., 2020). Another line of research has been focusing on building modular neural networks that can represent the compositional structures in scenes and questions, such as object-centric scene structures and multi-hop reasoning (Andreas et al., 2016; Johnson et al., 2017b; Hudson & Manning, 2019). However, these methods are designed for static images and do not handle the temporal and causal structure in dynamic scenes well, leading to inferior performance on video causal reasoning benchmark CLEVRER (Yi et al., 2020).
+
+To model the temporal and causal structures in dynamic scenes, Yi et al. (2020) proposed an oracle model to combine symbolic representation with video dynamics modeling and achieved state-ofthe-art performance on CLEVRER. However, this model requires videos with dense annotations for visual attributes and physical events, which are impractical or extremely labor-intensive in real scenes.
+
+
+Question: What will happen next? Answer: The blue sphere and the yellow object collide.
+Figure 1: The process to handle visual reasoning in dynamic scenes. The trajectories of the target blue and yellow spheres are marked by the sequences of bounding boxes. Object attributes of the blue sphere and yellow sphere and the collision event are marked by blue, yellow and purple colors. Stroboscopic imaging is applied for motion visualization.
+
+We argue that such dense explicit video annotations are unnecessary for video reasoning, since they are naturally encoded in the question answer pairs associated with the videos. For example, the question answer pair and the video in Fig. 1 can implicitly inform a model what the concepts “sphere”, “blue”, “yellow” and “collide” really mean. However, a video may contain multiple fast-moving occluded objects and complex object interactions, and the questions and answers have diverse forms. It remains an open and challenging problem to simultaneously represent objects over time, train an accurate dynamic model from raw videos, and align objects with visual properties and events for accurate temporal and causal reasoning, using vision and language as the only supervision.
+
+Our main ideas are to factorize video perception and reasoning into several modules: object tracking, object and event concept grounding, and dynamics prediction. We first detect objects in the video, associating them into object tracks across the frames. We can then ground various object and event concepts from language, train a dynamic model on top of object tracks for future and counterfactual predictions, analyze relationships between events, and answer queries based on these extracted representations. All these modules can be trained jointly by watching videos and reading paired questions and answers.
+
+To achieve this goal, we introduce Dynamic Concept Learner (DCL), a unified neural-symbolic framework for recognizing objects and events in videos and analyzing their temporal and causal structures, without explicit annotations on visual attributes and physical events such as collisions during training. To facilitate model training, a multi-step training paradigm has been proposed. We first run an object detector on individual frames and associate objects across frames based on a motion-based correspondence. Next, our model learns concepts about object properties, relationships, and events by reading paired questions and answers that describe or explain the events in the video. Then, we leverage the acquired visual concepts in the previous steps to refine the object association across frames. Finally, we train a dynamics prediction network (Li et al., 2019b) based on the refined object trajectories and optimize it jointly with other learning parts in this unified framework. Such a training paradigm ensures that all neural modules share the same latent space for representing concepts and they can bootstrap the learning of each other.
+
+We evaluate DCL’s performance on CLEVRER, a video reasoning benchmark that includes descriptive, explanatory, predictive, and counterfactual reasoning with a uniform language interface. DCL achieves state-of-the-art performance on all question categories and requires no scene supervision such as object properties and collision events. To further examine the grounding accuracy and transferability of the acquired concepts, we introduce two new benchmarks for video-text retrieval and spatial-temporal grounding and localization on the CLEVRER videos, namely CLEVRERRetrieval and CLEVRER-Grounding. Without any further training, our model generalizes well to these benchmarks, surpassing the baseline by a noticeable margin.
+
+# 2 RELATED WORK
+
+Our work is related to reasoning and answering questions about visual content. Early studies like (Wu et al., 2016; Zhu et al., 2016; Gan et al., 2017) typically adopted monolithic network architectures and mainly focused on visual understanding. To perform deeper visual reasoning, neural module networks were extensively studied in recent works (Johnson et al., 2017a; Hu et al., 2018; Hudson & Manning, 2018; Amizadeh et al., 2020), where they represent symbolic operations with small neural networks and perform multi-hop reasoning. Some previous research has also attempted to learn visual concepts through visual question answering (Mao et al., 2019). However, it mainly focused on learning static concepts in images, while our DCL aims at learning dynamic concepts like moving and collision in videos and at making use of these concepts for temporal and causal reasoning.
+
+
+Figure 2: DCL’s architecture for counterfactual questions during inference. Given an input video and its corresponding question and choice, we first use a program parser to parse the question and the choice into executable programs. We adopt an object trajectory detector to detect trajectories of all objects. Then, the extracted objects are sent to a dynamic predictor to predict their dynamics. Next, the extracted objects are sent to the feature extractor to extract latent representations for objects and events. Finally, we feed the parsed programs and latent representation to the symbolic executor to answer the question and optimize concept learning.
+
+Later, visual reasoning was extended to more complex dynamic videos (Lei et al., 2018; Fan et al., 2019; Li et al., 2020; 2019a; Huang et al., 2020). Recently, Yi et al. (2020) proposed CLEVRER, a new video reasoning benchmark for evaluating computational models’ comprehension of the causal structure behind physical object interaction. They also developed an oracle model, combining the neuro-symbolic visual question-answering model (Yi et al., 2018) with the dynamics prediction model (Li et al., 2019b), showing competitive performance. However, this model requires explicit labels for object attributes, masks, and spatio-temporal localization of events during training. Instead, our DCL has no reliance on any labels for objects and events and can learn these concepts through natural supervision (i.e., videos and question-answer pairs).
+
+Our work is also related to temporal and relational reasoning in videos via neural networks (Wang & Gupta, 2018; Materzynska et al., 2020; Ji et al., 2020). These works typically rely on specific action annotations, while our DCL learns to ground object and event concepts and analyze their temporal relations through question answering. Recently, various benchmarks (Riochet et al., 2018; Bakhtin et al., 2019; Girdhar & Ramanan, 2020; Baradel et al., 2020; Gan et al., 2020) have been proposed to study dynamics and reasoning in physical scenes. However, these datasets mainly target at pure video understanding and do not contain natural language question answering. Much research has been studying dynamic modeling for physical scenes (Lerer et al., 2016; Battaglia et al., 2013; Mottaghi et al., 2016; Finn et al., 2016; Shao et al., 2014; Fire & Zhu, 2015; Ye et al., 2018; Li et al., 2019b). We adopt PropNet (Li et al., 2019b) for dynamics prediction and feed the predicted scenes to the video feature extractor and the neuro-symbolic executor for event prediction and question answering.
+
+While many works (Zhou et al., 2019; 2018; Gan et al., 2015) have been studying on the problems of understanding human actions and activities (e.g., running, cooking, cleaning) in videos, our work’s primary goal is to design a unified framework for learning physical object and event concepts (e.g., collision, falling, stability). These tasks are of great importance in practical applications such as industrial robot manipulation which requires AI systems with human-like physical common sense.
+
+# 3 DYNAMIC CONCEPT LEARNER
+
+In this section, we introduce a new video reasoning model, Dynamic Concept Learner (DCL), which learns to recognize video attributes, events, and dynamics and to analyze their temporal and causal structures, all through watching videos and answering corresponding questions. DCL contains five modules, 1) an object trajectory detector, 2) video feature extractor, 3) a dynamic predictor, 4) a language program parser, and 5) a neural symbolic executor. As shown in Fig. 2, given an input video, the trajectory detector detects objects in each frame and associates them into trajectories; the feature extractor then represents them as latent feature vectors. After that, DCL quantizes the objects’ static concepts (i.e., color, shape, and material) by matching the latent object features with the corresponding concept embeddings in the executor. As these static concepts are motion-independent, they can be adopted as an additional criteria to refine the object trajectories. Based on the refined trajectories, the dynamics predictor predicts the objects’ movement and interactions in future and counterfactual scenes. The language parser parses the question and choices into functional programs, which are executed by the program executor on the latent representation space to get answers.
+
+The object and event concept embeddings and the object-centric representation share the same latent space; answering questions associated with videos can directly optimize them through backpropagation. The object trajectories and dynamics can be refined by the object static attributes predicted by DCL. Our framework enjoys the advantages of both transparency and efficiency, since it enables step-by-step investigations of the whole reasoning process and has no requirements for explicit annotations of visual attributes, events, and object masks.
+
+# 3.1 MODEL DETAILS
+
+Object Detection and Tracking. Given a video, the object trajectory detector detects object proposals in each frame and connects them into object trajectories $O \doteq \{ o ^ { n } \} _ { n = 1 } ^ { N }$ n}Nn=1, where on $o ^ { n } =$ $\{ b _ { t } ^ { n } \} _ { t = 1 } ^ { T }$ and $N$ is the number of objects in the video. $b _ { t } = [ x _ { t } ^ { n } , y _ { t } ^ { n } , w _ { t } ^ { n } , h _ { t } ^ { n } ]$ is an object proposal at frame $t$ and $T$ is the frame number, where $( x _ { t } ^ { n } , y _ { t } ^ { n } )$ denotes the normalized proposal coordinate center and $w _ { t } ^ { n }$ and $h _ { t } ^ { n }$ denote the normalized width and height, respectively.
+
+The object detector first uses a pre-trained region proposal network (Ren et al., 2015) to generate object proposals in all frames, which are further linked across connective frames to get all objects’ trajectories. Let $\{ b _ { t } ^ { i } \} _ { i = 1 } ^ { N }$ and $\{ b _ { t + 1 } ^ { j } \} _ { j = 1 } ^ { N }$ to be two sets of proposals in two connective frames. Inspired by Gkioxari & Malik (2015), we define a connection score $s _ { l }$ between $b _ { t } ^ { i }$ and $b _ { t + 1 } ^ { j }$ to be
+
+$$
+s _ { l } ( b _ { t } ^ { i } , b _ { t + 1 } ^ { j } ) = s _ { c } ( b _ { t } ^ { i } ) + s _ { c } ( b _ { t + 1 } ^ { j } ) + \lambda _ { 1 } \cdot I o U ( b _ { t } ^ { i } , b _ { t + 1 } ^ { j } ) ,
+$$
+
+where $s _ { c } ( b _ { t } ^ { i } )$ is the confidence score of the proposal $b _ { t } ^ { i }$ , $I o U$ is the intersection over union and $\lambda _ { 1 }$ is a scalar. Gkioxari $\&$ Malik (2015) adopts a greedy algorithm to connect the proposals without global optimization. Instead, we assign boxes $\{ b _ { t + 1 } ^ { j } \} _ { j = 1 } ^ { N }$ at the $t + 1$ frame to $\{ b _ { t } ^ { i } \} _ { i = 1 } ^ { N }$ by a linear sum assignment.
+
+Video Feature Extraction. Given an input video and its detected object trajectories, we extract three kinds of latent features for grounding object and event concepts. It includes 1) the average visual feature $f ^ { v } \in \mathbb { R } ^ { N \times D _ { 1 } }$ for static attribute prediction, 2) temporal sequence feature $\bar { f } ^ { s } \in \mathbb { R } ^ { N \times \breve { 4 } T }$ for dynamic attribute and unary event prediction, and 3) interactive feature $f ^ { c } \in \mathbb { R } ^ { K \times N \times N \times D _ { 2 } }$ for collision event prediction, where $D _ { 1 }$ and $D _ { 2 }$ denote dimensions of the features and $K$ is the number of sampled frames. We give more details on how to extract these features in Appendix B.
+
+Grounding Object and Event Concepts. Video Reasoning requires a model to ground object and event concepts in videos. DCL achieves this by matching object and event representation with object and event embeddings in the symbolic executor. Specifically, DCL calculates the confidence score that the $n$ -th object is moving by $\left[ \cos ( s ^ { \mathrm { m o v i n g } } , m _ { d a } ( f _ { n } ^ { s } ) ) - \stackrel { . } { \delta } \right] / \lambda .$ , where $f _ { n } ^ { s }$ denotes the temporal sequence feature for the $n$ -th object, $s ^ { \mathrm { m o v i n g } }$ denotes a vector embedding for concept moving, and $m _ { d a }$ denotes a linear transformation, mapping $f _ { n } ^ { s }$ into the dynamic concept representation space. $\delta$ and $\lambda$ are the shifting and scaling scalars, and $\cos ( )$ calculates the cosine similarity between two vectors. DCL grounds static attributes and the collision event similarly, matching average visual features and interactive features with their corresponding concept embeddings in the latent space. We give more details on the concept and event quantization in Appendix E.
+
+Trajectory Refinement. The connection score in Eq. 1 ensures the continuity of the detected object trajectories. However, it does not consider the objects’ visual appearance; therefore, it may fail to track the objects and may connect inconsistent objects when different objects are close to each other and moving rapidly. To detect better object trajectories and to ensure the consistency of visual appearance along the track, we add a new term to Eq. 1 and re-define the connection score to be
+
+$$
+s _ { l } ( \{ b _ { m } ^ { i } \} _ { m = 0 } ^ { t } , b _ { t + 1 } ^ { j } ) = s _ { c } ( b _ { t } ^ { i } ) + s _ { c } ( b _ { t + 1 } ^ { j } ) + \lambda _ { 1 } \cdot I o U ( b _ { t } ^ { i } , b _ { t + 1 } ^ { j } ) + \lambda _ { 2 } \cdot f _ { \mathrm { a p p e a r } } ( \{ b _ { m } ^ { i } \} _ { m = 0 } ^ { t } , b _ { t + 1 } ^ { j } ) ,
+$$
+
+where $f _ { \mathrm { a p p e a r } } \big ( \{ b _ { m } ^ { i } \} _ { m = 0 } ^ { t } , b _ { t + 1 } ^ { j } \big )$ measures the attribute similarity between the newly added proposal $b _ { t + 1 } ^ { j }$ and all proposals $( \{ b _ { m } ^ { i } \} _ { m = 0 } ^ { t }$ in previous frames. We define $f _ { \mathrm { a p p e a r } }$ as
+
+$$
+f _ { \mathrm { a p p e a r } } \bigl ( \bigl \{ b _ { m } ^ { i } \bigr \} _ { m = 0 } ^ { t } , b _ { t + 1 } ^ { j } \bigr ) = \frac { 1 } { 3 \times t } \sum _ { a t t r } \sum _ { m = 0 } ^ { t } f _ { \mathrm { a t t r } } \bigl ( b _ { m } ^ { i } , b _ { t + 1 } ^ { j } \bigr ) ,
+$$
+
+where attr $\in \{ \mathrm { c o l o r }$ , material, shape}. $f _ { \mathrm { a t t r } } ( \boldsymbol { b } _ { m } , \boldsymbol { b } _ { t + 1 } )$ equals to 1 when $b _ { m } ^ { i }$ and $b _ { t + 1 } ^ { j }$ have the same attribute, and 0 otherwise. In Eq. 2, $f _ { \mathrm { a p p e a r } }$ ensures that the detected trajectories have consistent visual appearance and helps to distinguish the correct object when different objects are close to each other in the same frame. These additional static attributes, including color, material, and shape, are extracted without explicit annotation during training. Specifically, we quantize the attributes by choosing the concept whose concept embedding has the best cosine similarity with the object feature. We iteratively connect proposals at the $t + 1$ frame to proposals at the $t$ frame and get a set of object trajectories $O = \{ o ^ { \bar { n } } \} _ { n = 1 } ^ { \bar { N } }$ , where $o ^ { n } = \{ b _ { t } ^ { n } \} _ { t = 1 } ^ { T }$ .
+
+Dynamic Prediction. Given an input video and the refined trajectories of objects, we predict the locations and RGB patches of the objects in future or counterfactual scenes with a Propagation Network (Li et al., 2019b). We then generate the predicted scenes by pasting RGB patches into the predicted locations. The generated scenes are fed to the feature extractor to extract the corresponding features. Such a design enables the question answer pairs associated with the predicted scenes to optimize the concept embeddings and requires no explicit labels for collision prediction, leading to better optimization. This is different from Yi et al. (2020), which requires dense collision event labels to train a collision classifier.
+
+To predict the locations and RGB patches, the dynamic predictor maintains a directed graph $\left. V , D \right. =$ $\left. \{ v _ { n } \} _ { n = 1 } ^ { N } , \{ d _ { n _ { 1 } , n _ { 2 } } \} _ { n _ { 1 } = 1 , n _ { 2 } = 1 } ^ { N , N } \right.$ . The $n$ -th vertex $v _ { n }$ is represented by a concatenation of tuple $\langle b _ { t } ^ { n } , p _ { t } ^ { n } \rangle$ over a small time window $w$ , where $b _ { t } ^ { n } = [ x _ { t } ^ { n } , y _ { t } ^ { n } , w _ { t } ^ { n } , h _ { t } ^ { n } ]$ is the $n$ -th object’s normalized coordinates and $p _ { t } ^ { n }$ is a cropped RGB patch centering at $( x _ { t } ^ { n } , y _ { t } ^ { n } )$ . The edge $d _ { n _ { 1 } , n _ { 2 } }$ denotes the relation between the $n _ { 1 }$ -th and $n _ { 2 }$ -th objects and is represented by the concatenation of the normalized coordinate difference $b _ { t } ^ { n _ { 1 } } - b _ { t } ^ { n _ { 2 } }$ . The dynamic predictor performs multi-step message passing to simulate instantaneous propagation effects.
+
+During inference, the dynamics predictor predicts the locations and patches at frame $k + 1$ using the features of the last $w$ observed frames in the original video. We get the predictions at frame $k + 2$ by autoregressively feeding the predicted results at frame $k + 1$ as the input to the predictor. To get the counterfactual scenes where the $n$ -th object is removed, we remove the $n$ -th vertex and its associated edges from thnormalized coordinates $\{ \hat { b } _ { k ^ { \prime } } ^ { n } \} _ { n = 1 , k ^ { \prime } = 1 } ^ { N , K ^ { \prime } ^ { - } }$ ict counterfactual and RGB patches $\{ \hat { p } _ { k ^ { \prime } } ^ { n } \} _ { n = 1 , k ^ { \prime } = 1 } ^ { N , K }$ tively, we get that all predicted $K ^ { \prime }$ redictedframes. We give more details on the dynamic predictor at Appendix C.
+
+Language Program Parsing. The language program parser aims to translate the questions and choices into executable symbolic programs. Each executable program consists of a series of operations like selecting objects with certain properties, filtering events happening at a specific moment, finding the causes of an event, and eventually enabling transparent and step-by-step visual reasoning. Moreover, these operations are compositional and can be combined to represent questions with various compositionality and complexity. We adopt a seq2seq model (Bahdanau et al., 2015) with an attention mechanism to translate word sequences into a set of symbolic programs and treat questions and choices, separately. We give detailed implementation of the program parser in Appendix D.
+
+Symbolic Execution. Given a parsed program, the symbolic executor explicitly runs it on the latent features extracted from the observed and predicted scenes to answer the question. The executor consists of a series of functional modules to realize the operators in symbolic programs. The last operator’s output is the answer to the question. Similar to Mao et al. (2019), we represent all object states, events, and results of all operators in a probabilistic manner during training. This makes the whole execution process differential w.r.t. the latent representations from the observed and predicted scenes. It enables the optimization of the feature extractor and concept embeddings in the symbolic executor. We provide the implementation of all the operators in Appendix E.
+
+# 3.2 TRAINING AND INFERENCE
+
+Training. We follow a multi-step training paradigm to optimize the model: 1) We first extract object trajectories with the scoring function in Eq. 1 and optimize the video feature extractor and concept embeddings in the symbolic executor with only descriptive and explanatory questions; 2) We quantize the static attributes for all objects with the feature extractor and the concept embeddings learned in Step 1) and refine object trajectories with the scoring function Eq. 2; 3) Based on the refined trajectories, we train the dynamic predictor and predict dynamics for future and counterfactual scenes; 4) We train the full DCL with all the question answer pairs and get the final model. The program executor is fully differentiable w.r.t. the feature extractor and concept embeddings. We use cross-entropy loss to supervise open-ended questions and use mean square error loss to supervise counting questions. We provide specific loss functions for each module in Appendix H.
+
+