Unlike predictive models, for attribution methods there is no universal measure of "accuracy" or "correctness"; we can only say whether a method is intuitively "reasonable" or not, and we saw some examples of this analysis in the previous post. It is therefore common in this area to take an axiomatic approach: formalize certain "desirable" properties of attribution methods as **Axioms**, to help guide the design and evaluation of attribution methods. This post will describe a set of such axioms, based on the ICML 2017 paper of Sundararajan et al ^{[1]} (which is an excellent read on the subject).

Recall that we denote the function computed by the neural network as $F(x)$, where $x$ is the $d$-dimensional input vector. If $b$ is the baseline input vector, the attribution of the difference $F(x) - F(b)$ to feature $x_i$ is denoted by $A^F_i(x; b)$. To avoid notational clutter in this post we will assume that $b$ is the all-zeros vector, and that $F(b) = 0$, and we will simply write $A^F_i(x)$ to denote the attribution of $F(x)$ to feature $x_i$. As a running example we will continue to use the function $F(x) = \min(1, x_1 + x_2)$.

### Axiom: Completeness

We mentioned this axiom in the last two posts:

Completeness:For any input $x$, the sum of the feature attributions equals $F(x)$:\begin{equation*} F(x) = \sum_i A_i^F(x) \end{equation*}

Why is this property called "completeness"? Because any attribution method satisfying this axiom "completely" accounts for the value of $F(x)$ in terms of the feature attributions.

We saw in the last post that the *perturbation method* fails to satisfy completeness: If $F(x) = \min(1, x_1 + x_2)$, then for $x = (2,2)$, $F(x) = 1$, but $F((0,2)) = F((2,0)) = 1$, meaning that perturbing either input individually to 0 does not alter the function value, and this method would attribute 0 to both features, clearly violating Completeness.

On the other hand we saw in the last post that a certain *path method* does satisfy Completeness. Recall that the path method consists of picking a path from the origin (zero vector) to $x$, and computing how much each feature contributes along this path. In the figure below, to calculate the attributions at $P_3 = (1.2, 0.6)$, we considered two paths from the origin to $P_3$. Each segment of the path sequentially varies a *single* feature, keeping the other features (just one in this 2D case) constant. For example if we choose to do attribution using the *blue* path from the origin to $P_3$, then on the first segment only $x_2$ changes (from 0 to its final value 0.6), and causes $F$ to increase by $0.6$, and on the second segment only $x_1$ changes, and causes $F$ to increase by $0.4$, which results in attributions $(0.4, 0.6)$, which sum to 1. Similarly, the *red* path yields attributions $(1,0)$, which again sum to 1.

### Axiom: Sensitivity

There are two versions of this axiom:

Sensitivity:If $x$ has only one non-zero feature and $F(x) \neq 0$, then the attribution to that feature should be non-zero.

Insensitivity:If $F(x)$ does not mathematically depend on the value of a feature $x_i$ then the attribution to $x_i$ should always be zero.

It is easily verified that: (a) Perturbation methods satisfy Sensitivity, and (b) Completeness implies Sensitivity, so the Path method satisfies this axiom as well. However another method we saw in the last post, the *gradient method*, does not. In our example, at the point $x = (2,0)$, $F(x)=1$ but the gradient of $F(x)$ with respect to $x_1$ is 0, which would be the gradient-based attribution. The reason the gradient method violates Sensitivity is that gradients are *local*. All methods we've seen so far satisfy the Insensitivity axiom.

### Axiom: Implementation Invariance

Implementation Invariance:When two neural networks compute the same mathematical function $F(x)$, regardless of how differently they are implemented, the attributions to all features should always be identical.

This one might seem completely un-controversial, but surprisingly enough there are some attribution methods that violate this (e.g. DeepLift^{[2]} and Layer-wise Relevance Propagation or LRP^{[3]}). All the methods we have seen, however, do satisfy this axiom, since none of them refer to the implementation details of $F(x)$.

### Axiom: Linearity

Linearity:If we linearly compose two neural networks computing $F$ and $G$ to form a new neural network that computes $H = aF + bG$, then for any feature $x_i$, its attribution for $H$ is a weighted sum of the attributions for $F$ and $G$, with weights $a$ and $b$ respectively, i.e.,\begin{equation*} A_i^{H}(x) = a A_i^F(x) + b A_i^G(x) \end{equation*}In other words, this axiom says that the attribution method should preserve linearity.

### Axiom: Symmetry-Preserving

Two features are said to be * symmetric* with respect to a function $F(x)$ if interchanging them does not change the function mathematically. For example $x_1, x_2$ are symmetric in $F(x) = \min(1, x_1 + x_2)$, but they are not symmetric in $F(x) = 3x_1 + x_2$. The following axiom states that an attribution method must preserve symmetry:

Symmetry-Preserving:For any input $x$ where the values of two symmetric features are the same, their attributions should be identical as well.

### Path Methods and Aumann-Shapley Cost-Sharing

We saw a specific path method above: the path starts at the origin, and changes each feature sequentially until the specific input $x$ is reached.

For each segment where a feature $x_i$ is changed (and all others are kept constant), we then calculated how much $x_i$ contributed to the change in function value along. The specific paths we considered are *axis-parallel* paths (since each segment is parallel to some feature-axis), but there is no reason we should restrict ourselves to such paths. On a more general path from the origin to $x$, there is a way to compute the contribution of each variable, using a method called * Path Integrated Gradients*.

Path Integrated Gradients have an intriguing connection to work by two Nobel-prize winning economists Lloyd Shapley and Robert Aumann on *cost-sharing*: $F(x)$ represents a project's cost and the "features" are the demands of the participants, and attribution corresponds to cost-shares (or cost-allocations). Specifically,

Attribution based on a Path Integrated Gradient (along any fixed path from the origin to $x$) corresponds to a cost-sharing method referred to as

Aumann-Shapley^{[4]}, and it has been proved^{[5]}that this method is the only one that satisfies all of the above axioms except the Symmetry-Preserving Axiom.

And the paper by Sundararajan et. al.^{[1:1]} (on which this post is based) proposes a special case of the Path Integrated Gradient and shows:

Attribution using the Integrated Gradient along the

straight linefrom the origin to $x$, is the unique Path Method that is Symmetry-Preserving.

A future post will take a closer look at Path Integrated Gradients.

Sundararajan, Mukund, Ankur Taly, and Qiqi Yan. 2017. “Axiomatic Attribution for Deep Networks.” arXiv [cs.LG]. arXiv. http://arxiv.org/abs/1703.01365. ↩︎ ↩︎

Shrikumar, Avanti, Peyton Greenside, and Anshul Kundaje. 2017. “Learning Important Features Through Propagating Activation Differences.” arXiv [cs.CV]. arXiv. http://arxiv.org/abs/1704.02685. ↩︎

Binder, Alexander, Grégoire Montavon, Sebastian Bach, Klaus-Robert Müller, and Wojciech Samek. 2016. “Layer-Wise Relevance Propagation for Neural Networks with Local Renormalization Layers.” arXiv [cs.CV]. arXiv. http://arxiv.org/abs/1604.00825. ↩︎

Shapley, L., Robert Aumann. 1974.

*Values of Non-Atomic Games*. Princeton University Press. ↩︎Friedman, Eric J. 2004. “Paths and Consistency in Additive Cost Sharing.” International Journal of Game Theory 32 (4): 501–18. ↩︎