Second, the measure of “features per dimension” used by Elhage et al. (2022) might be misleading. See the paper for details of how they arrived at this quantity. But as shown in the figure above, “features per dimension” is defined as the Frobenius norm of the weight matrix before the layer divided by the number of neurons in the layer. But there is a simple sanity check that this doesn’t pass. In the case of a ReLU network without bias terms, multiplying a weight matrix by a constant factor will cause the “features per dimension” to be increased by that factor squared while leaving the activations in the forward pass unchanged up to linearity until a non-ReLU operation (like a softmax) is performed. And since each component of a softmax’s output is strictly increasing in that component of the input, scaling weight matrices will not affect the classification.
It’s worth noting that Elhage+2022 studied an autoencoder with tied weights and no softmax, so there isn’t actually freedom to rescale the weight matrix without affecting the loss in their model, making the scale of the weights meaningful. I agree that this measure doesn’t generalize to other models/tasks though.
They also define a more fine-grained measure (the dimensionality of each individual feature) in a way that is scale-invariant and which broadly agrees with their coarser measure...
It’s worth noting that Elhage+2022 studied an autoencoder with tied weights and no softmax, so there isn’t actually freedom to rescale the weight matrix without affecting the loss in their model, making the scale of the weights meaningful. I agree that this measure doesn’t generalize to other models/tasks though.
They also define a more fine-grained measure (the dimensionality of each individual feature) in a way that is scale-invariant and which broadly agrees with their coarser measure...
Thanks, +1 to the clarification value of this comment. I appreciate it. I did not have the tied weights in mind when writing this.