Current methods for the interpretability of discriminative deep neural networks commonly rely on the model's input-gradients, i.e., the gradients of the output logits w.r.t. the inputs. The common assumption is that these input-gradients contain information regarding $p_{\theta} ( y\mid \mathbf{x} )$, the model's discriminative capabilities, thus justifying their use for interpretability. However, in this work, we show that these input-gradients can be arbitrarily manipulated as a consequence of the shift-invariance of softmax without changing the discriminative function. This leaves an open question: given that input-gradients can be arbitrary, why are they highly structured and explanatory in standard models?
In this work, we re-interpret the logits of standard softmax-based classifiers as unnormalized log-densities of the data distribution and show that input-gradients can be viewed as gradients of a class-conditional generative model $p_{\theta}(\mathbf{x} \mid y)$ implicit in the discriminative model. This leads us to hypothesize that the highly structured and explanatory nature of input-gradients may be due to the alignment of this class-conditional model $p_{\theta}(\mathbf{x} \mid y)$ with that of the ground truth data distribution $p_{\text{data}} (\mathbf{x} \mid y)$. We test this hypothesis by studying the effect of density alignment on gradient explanations. To achieve this density alignment, we use an algorithm called score-matching, and propose novel approximations to this algorithm to enable training large-scale models.
Our experiments show that improving the alignment of the implicit density model with the data distribution enhances gradient structure and explanatory power while reducing this alignment has the opposite effect. This also leads us to conjecture that unintended density alignment in standard neural network training may explain the highly structured nature of input-gradients observed in practice. Overall, our finding that input-gradients capture information regarding an implicit generative model implies that we need to re-think their use for interpreting discriminative models.

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