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Suppose that we take a neural network of a given topology, and run it through two training processes, obtaining two different sets of converged weights at the end of the training.

What is a good way to measure the difference between the two sets of weights?

If this were a curve-fit where the parameters were largely orthogonal, we could simply treat the parameters as a cartesian space and compute the length of the vector difference between the sets.

However, the weights in a neural network are often not at all orthogonal. There are trivial redundancies, in that we can reorder nodes within a layer, or multiply the input weights to a given node by a constant factor and the output weights by its inverse. Even if we remove these by putting the weights in some canonical form (normalizing and sorting them, for instance), I can't see any reason to assume that we won't have plenty of remaining non-orthogonality. As a result, two sets of neural-network weights that have a large vector-difference length may be essentially equivalent in practice.

An alternate conceptual approach would be to numerically integrate the difference between the output values over the entire input space, but for typical neural networks this space has high enough dimension for such an integration to be intractable. So we'd need to do some sort of sampling-based approach, which introduces more handwaving about whether it's a meaningful measure.

Is there any existing consensus on what's a good measure? Or, for that matter, any existing practice for what other researchers have used?

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    $\begingroup$ Welcome to CS.SE! There are several possible answers, but before I suggest some, let me ask you this: How do you plan to use the resulting similarity/difference value? What properties do you want the similarity measure to have? What purpose will you use it for? Different measures might be appropriate for different purposes. $\endgroup$ – D.W. Apr 25 '17 at 0:59
  • $\begingroup$ Good questions! I had to think a bit. :) The main thing I want to use this for is heuristic understanding of whether the "solutions" are meaningfully different (which I think I can refine to: are they likely to have different behavior for some set of inputs?) The use I'm imagining is: If I run with N sets of random initial weights and get N "different" solutions, even if they have similar average error, I may want to keep looking for more. Whereas if I get N solutions that are all "not different", I may conclude that further runs are unlikely to find anything that's significantly better. $\endgroup$ – Brooks Moses Apr 25 '17 at 1:16
  • $\begingroup$ A related use is: If I have a converged set of weights, and I randomly perturb them and then re-converge to a new set, have I got back to the "same" solution or a different one? $\endgroup$ – Brooks Moses Apr 25 '17 at 1:18
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    $\begingroup$ "putting the weights in some canonical form (normalizing and sorting them" - this is not a stable solution. A very small change in one wight might flip the sorting order. $\endgroup$ – MSalters Apr 25 '17 at 7:03
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    $\begingroup$ @BrooksMoses: Given the lack of answers, apparently not. I know from practical experience it's non-trivial. Even with a single training process, the mere random initialisation tends to result in vastly different networks that perform surprisingly similar (given the numerical differences) and also surprisingly different (given the identical training data). $\endgroup$ – MSalters Apr 25 '17 at 18:36
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One way to compare two neural networks is to compare how similar their predictions are, on typical instances.

Ideally, we'd like to compute the expected value of this similarity, taken over the distribution on instances. However, as you say, the input space is high-dimensional, so the integral is hard to compute. Also, the distribution on instances typically isn't known in any analytical or useful form.

Therefore, the standard trick is to estimate this expected value, using sampling. We draw a sample of instances, compute the similarity of the predictions from the two networks on each instance, and take the average of those similarity values. If the sample is large enough, this should be a good approximation to the expected value (to the integral). So, you'll set aside a separate validation set, and then compute the average similarity between the two networks' predictions on this validation set.

I can formalize this more precisely. Let $f,g$ be the two classifiers. Suppose there are $k$ classes. Assume you are using a softmax layer at the output of the neural network. Then the output of a classifier $f$ on instance $x$ is a $k$-vector $f(x)$ whose $i$th component can be interpreted as the probability that $x$ has class $c$ (as predicted by the classifier). So, we want some way to measure the similarity of the two $k$-vectors $f(x),g(x)$, i.e., the similarity of those two probability distributions on the $k$ classes. You can use any distance metric or similarity measure on probability distributions.

Let $d(\cdot,\cdot)$ represent a distance measure on two distributions (or, more generally, a dissimilarity measure -- it need not be a metric). Ideally, we'd like to compute $\mathbb{E}[d(f(X),g(X))]$ where the expectation is taken with respect to a random variable $X$ that is distributed according to the input distribution on instances. To approximate this, we compute

$$D = {1 \over n} \sum_{j=1}^n d(f(x_j),g(x_j)),$$

where $x_1,\dots,x_n$ is a validation set that is held out and set aside for this purpose. The validation set should be separate from the training set.

You can now instantiate $d$ with any distance measure on distributions of your choice. Some reasonable choices include: KL divergence, or the Jensen-Shannon divergence (a symmetric version of the KL divergence); total variation distance; earth mover's distance; or a very simple measure where $d(u,v) = 1$ if $\arg\max_i u_i \ne \arg\max_i v_i$, else $d(u,v)=0$ (i.e., compare whether both networks predicts the same class as the most likely class).

This approach is suitable if you want to measure how similar the predictions of the two neural networks are, without regard to their internal weights -- i.e., treating them as a black box. I don't know how to compare how similar their internal structure is; there are probably many non-trivial transformations to networks that compute essentially the same thing, but with very different-looking weights.

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I found an altenative method from a 2009 paper by Montreal researchers Bengio. This is similar that they used the prediction output. The diff is that they seemed to concat the predictions as a big vector, and then embed this using t-sne. So they can visualize multiple model as points embedded in a 2d plain.

This is great for visualizing many many models, and the Euclidean distance may also be good “distance” estimate.

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    $\begingroup$ Would you mind providing the name of the 2009 paper by Bengio, who is quite a prolific author? A link would also be great. $\endgroup$ – Yuval Filmus Feb 6 at 6:52
  • $\begingroup$ static.googleusercontent.com/media/research.google.com/en//pubs/… This is old paper so some of the specifics may no longer be true. But techniques and some inquires are still fresh $\endgroup$ – kechan Feb 8 at 8:38
  • $\begingroup$ Please update your answer with this information. Also include the title of the paper and a complete list of authors. $\endgroup$ – Yuval Filmus Feb 8 at 8:57
  • $\begingroup$ “Why does Unsupervised Pre-training Help Deep Learning?” Dumitru Erhan Yoshua Bengio , Aaron Courville Pierre-Antoine Manzagol Pascal Vincent Samy Bengi $\endgroup$ – kechan Feb 23 at 23:25

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