Open- and Closed-Loop Neural Network Verification Using Polynomial Zonotopes

Abstract

We present a novel approach to efficiently compute tight non-convex enclosures of the image through neural networks with ReLU, sigmoid, or hyperbolic tangent activation functions. In particular, we abstract the input-output relation of each neuron by a polynomial approximation, which is evaluated in a set-based manner using polynomial zonotopes. While our approach can also can be beneficial for open-loop neural network verification, our main application is reachability analysis of neural network controlled systems, where polynomial zonotopes are able to capture the non-convexity caused by the neural network as well as the system dynamics. This results in a superior performance compared to other methods, as we demonstrate on various benchmarks.

Appendix A

We now provide the proof for Prop. 2. According to Def. 2, the one-dimensional polynomial zonotope $\mathcal{P}\mathcal{Z}= \langle c,G,G_I,E \rangle _{PZ}$ is defined as

$$\begin{aligned} \begin{aligned} \mathcal{P}\mathcal{Z}&= \bigg \{ c + \underbrace{\sum _{i=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{E_{(k,i)}} \bigg ) G_{(i)}}_{d(\alpha )} + \underbrace{\sum _{j=1}^{q} \beta _j G_{I(j)}}_{z(\beta )} ~ \bigg | ~ \alpha _k, \beta _j \in [-1,1] \bigg \} \\&= \big \{ c + d(\alpha ) + z(\beta )~\big |~ \alpha ,\beta \in [-\textbf{1},\textbf{1}] \big \}, \end{aligned} \end{aligned}$$

(10)

where $\alpha = [\alpha _1~\dots ~\alpha _p]^T$ and $\beta = [\beta _1~\dots ~\beta _q]^T$. To compute the image through the quadratic function $g(x)$ we require the expressions $d(\alpha )^2$, $d(\alpha ) z(\beta )$, and $z(\beta )^2$, which we derive first. For $d(\alpha )^2$ we obtain

$$\begin{aligned} \begin{aligned} d(\alpha )^2&= \bigg ( \sum _{i=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{E_{(k,i)}} \bigg ) G_{(i)} \bigg ) \bigg ( \sum _{j=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{E_{(k,j)}} \bigg ) G_{(j)} \bigg ) \\&= \sum _{i=1}^h \sum _{j=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{E_{(k,i)} + E_{(k,j)}} \bigg ) G_{(i)} G_{(j)} \\&= \sum _{i=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{2 E_{(k,i)}} \bigg ) G_{(i)}^2 + \sum _{i=1}^{h-1} \sum _{j=i+1}^h \bigg ( \prod _{k=1}^p \underbrace{\alpha _k ^{E_{(k,i)} + E_{(k,j)}}}_{\alpha _k^{\widehat{E}_{i(k,j)}}} \bigg ) 2 \underbrace{G_{(i)} G_{(j)}}_{\widehat{G}_{i(j)}} \\&\overset{\begin{array}{c} (9)\\ \vspace{-2pt} \end{array}}{=} \sum _{i=1}^{h(h+1)/2} \bigg ( \prod _{k=1}^p \alpha _k ^{\widehat{ E}_{(k,i)}} \bigg ) \widehat{G}_{(i)}, \end{aligned} \end{aligned}$$

(11)

for $d(\alpha )z(\beta )$ we obtain

$$\begin{aligned} \begin{aligned} d(\alpha )z(\beta )&= \bigg ( \sum _{i=1}^h \bigg ( \prod _{k=1}^p \alpha _k ^{E_{(k,i)}} \bigg ) G_{(i)} \bigg ) \bigg ( \sum _{j=1}^q \beta _j G_{I(j)} \bigg ) \\&= \sum _{i=1}^h \sum _{j=1}^q \underbrace{\bigg ( \beta _j \prod _{k=1}^p \alpha _k ^{E_{(k,i)}} \bigg )}_{\beta _{q + (i-1)h+j}} G_{(i)} G_{I(j)} \overset{\begin{array}{c} (9) \\ \vspace{-2pt} \end{array}}{=} \sum _{i = 1}^{hq} \beta _{q + i} \, \overline{G}_{(i)}, \end{aligned} \end{aligned}$$

(12)

and for $z(\beta )^2$ we obtain

$$\begin{aligned} z(\beta )^2&= \bigg ( \sum _{i=1}^q \beta _i G_{I(i)} \bigg ) \bigg ( \sum _{j=1}^q \beta _j G_{I(j)} \bigg ) = \sum _{i=1}^q \sum _{j=1}^q \beta _i \beta _j \, G_{I(i)} G_{I(j)} \nonumber \\&= \sum _{i=1}^q \beta _i^2 G_{I(i)}^2 + \sum _{i=1}^{q-1} \sum _{j=i+1}^q \beta _i \beta _j \, 2 \, G_{I(i)} G_{I(j)} \nonumber \\&= 0.5 \sum _{i=1}^q G_{I(i)}^2 + \sum _{i=1}^q \underbrace{(2 \beta _i^2 - 1)}_{\beta _{(h+1)q+i}} 0.5 \, G_{I(i)}^2 + \sum _{i=1}^{q-1} \sum _{j=i+1}^q \underbrace{\beta _i \beta _j}_{\beta _{a(i,j)}} 2 \, \underbrace{G_{I(i)} G_{I(j)}}_{\check{G}_{i(j)}}\nonumber \\&\overset{\begin{array}{c} (9)\\ \vspace{-2pt} \end{array}}{=} 0.5 \sum _{i=1}^q G_{I(i)}^2 + \sum _{i=1}^{q(q+1)/2} \beta _{(h+1)q + i} \, \check{G}_{(i)}, \end{aligned}$$

(13)

where the function a(i, j) maps indices i, j to a new index:

$$\begin{aligned} a(i,j) = (h+2)q + j-i + \sum _{k=1}^{i-1} q-k. \end{aligned}$$

In (12) and (13), we substituted the expressions $\beta _j \prod _{k=1}^p \alpha _k^{E_{(k,i)}}$, $2\beta _i^2 -1$, and $\beta _i\beta _j$ containing polynomial terms of the independent factors $\beta$ by new independent factors, which results in an enclosure due to the loss of dependency. The substitution is possible since

$$\begin{aligned} \beta _j \prod _{k=1}^p \alpha _k^{E_{(k,i)}} \in [-1,1],~~ 2\beta _i^2 -1 \in [-1,1],~~ \text {and} ~~\beta _i\beta _j \in [-1,1]. \end{aligned}$$

Finally, we obtain for the image

$$\begin{aligned} \begin{aligned}&\big \{ g(x)~\big |~ x \in \mathcal{P}\mathcal{Z} \big \} = \big \{a_1\,x^2 + a_2\,x + a_3~ \big | ~ x \in \mathcal{P}\mathcal{Z} \big \}\\&~ \\ \overset{\begin{array}{c} (10) \\ \vspace{-2pt} \end{array}}{=}&\big \{ a_1( c + d(\alpha ) + z(\beta ))^2 + a_2(c + d(\alpha ) + z(\beta )) + a_3~\big | ~ \alpha ,\beta \in [-\textbf{1},\textbf{1}] \big \}\\&~ \\ =&\big \{ a_1c^2 + a_2c + a_3+ (2 a_1c + a_2) d(\alpha ) + a_1d(\alpha )^2\\&~~ + (2 a_1c + a_2) z(\beta ) + 2 a_1d(\alpha )z(\beta ) + a_1z(\beta )^2 ~ \big | ~ \alpha ,\beta \in [-\textbf{1},\textbf{1}] \big \}\\&~ \\ \overset{\begin{array}{c} (11),(12),(13)\\ \vspace{-2pt} \end{array}}{\subseteq }&\bigg \langle a_1c^2 + a_2c + a_3+ 0.5 \, a_1\sum _{i=1}^q G_I^2, \big [(2 a_1c + a_2)G ~~ a_1\widehat{G} \big ], \\&~~~~~~~~~~~~~~~~~~~~~ \big [ (2 a_1c + a_2)G_I ~~ 2a_1\overline{G} ~~a_1\check{G} \big ], \big [E ~~ \widehat{E} \big ] \bigg \rangle _{PZ} \overset{\begin{array}{c} (8) \\ \vspace{-2pt} \end{array}}{=} \langle c_q,G_q,G_{I,q},E_q \rangle _{PZ}, \end{aligned} \end{aligned}$$

which concludes the proof.