Local Interpretable Model-agnostic Explanations

Given the original represntation of an instance being explained $x$ (the image), we can generate a binary vector $x'$ for its interpretable representation. In this case, $x'$ is an one vector of the length of number of superpixels. A perturbed sample $z'$ is a binary vector which is sampled from $x'$ randomly under the uniform distribution. Then a fake image $z$ with less superpixels is generated.

To choose the best explanation, it must reduce the cost between the true model and approximate function, and the complexity penalty of approximate function.

$$\xi(x) = \arg\min_{g\in G}\underbrace{\mathcal{L}(f,g,\pi_x)}_{\text{cost}}+\underbrace{\Omega(g)}_{\text{complexity}}$$

where

$$ \begin{align*} \mathcal{L}(f,g,\pi_x) &= \sum_{z,z'\in\mathcal{Z}}\pi_x(z)(f(z)-g(z'))^2 \\ g(z') &= w_g\cdot z'\\ \Omega(g) &= \infty\mathcal{1}[\|w_g\|_0\gt K] \\ \pi_x(z)&= e^{-\frac{1}{\sigma^2}\|x-z\|^2_2} \end{align*} $$

where $f(x)$ is the neural network model which predicts the probability of $x$ belonging to a certain class, $g$ an interpretable model, $\pi_x(z)$ the proximity measure between an instance $z$(fake image) to $x$(real image), $\mathcal{L}$ the fidelity function and $\Omega$ the complexity measure (restricting the number of superpixels used to explain the label).

$\mathcal{L}(f,g,\pi_x)$ measures how unfaithful $g$ is approximating $f$ in the locality defined by $\pi_x$. If this term is small, it represents $g$ faithfully approximates $f$.

$\Omega(g)$ is similar to a regularization term, it limits the complexity of the approximate model $g$. If $g$ is a random forest algorithm, it limits the depth by $K$. If $g$ is a linear model, it limits the number of features. Our aim is to approximate $f(z)$ by $g(z')$ as follows,

$$f(z)\approx g(z') = w_g\cdot z'$$

Least absolute shrinkage and selection operator (lasso) is applied to compute the representation of $w_g$. The lasso estimate ($\hat{w}_g$) is defined by,

$$\hat{w}_g = \arg\min(f(z)-w_g\cdot z')^2\quad\text{subject to}\quad |w_g|\lt K \\ =\arg\min_{w_g}\sum_{z,z'\in\mathcal{Z}}(f(z)-w_g\cdot z')+\lambda\|w_g\|_1$$

Compared with ridge regression which only shrinks coefficients, lasso can set some coefficients to 0, retain the relatively important varaible and give an easily interpretable model.

Local Interpretable Model-agnostic Explanations

References