The tremendous success of deep neural networks has motivated the need to better understand the fundamental properties of these networks, but many of the theoretical results proposed have only been for shallow networks. In this paper, we study an important primitive for understanding the meaningful input space of a deep network: span recovery. For k < n, let A ∈ R k×n be the innermost weight matrix of an arbitrary feed forward neural network M : R n → R, so M (x) can be written as M (x) = σ(Ax), for some network σ : R k → R. The goal is then to recover the row span of A given only oracle access to the value of M (x). We show that if M is a multi-layered network with ReLU activation functions, then partial recovery is possible: namely, we can provably recover k/2 linearly independent vectors in the row span of A using poly(n) non-adaptive queries to M (x). Furthermore, if M has differentiable activation functions, we demonstrate that full span recovery is possible even when the output is first passed through a sign or 0/1 thresholding function; in this case our algorithm is adaptive. Empirically, we confirm that full span recovery is not always possible, but only for unrealistically thin layers. For reasonably wide networks, we obtain full span recovery on both random networks and networks trained on MNIST data. Furthermore, we demonstrate the utility of span recovery as an attack by inducing neural networks to misclassify data obfuscated by controlled random noise as sensical inputs.
** Summary
The paper studies how to recover the span of a NN from a limited number of queries. The problem belongs to the general question of how to reconstruct functions from black-box interaction and it may find application in obfuscation attacks where very large perturbations of the input do not affect the output. The main contribution of the paper is on the theoretical analysis of a simple non-adaptive and a more sophisticated adaptive algorithm. The main finding is that under mild conditions on the structure of the NN partial recovery is possible. The empirical validation show that in practice, it is often the case the full span recover is actually possible, as the structure and weights of common NN are "friendly" enough.
** Evaluation
While the content of the paper lies a bit off my expertise, my impression is that this is a solid technical and theoretical contribution.
Detailed comments:
1- Theoretical results: The properties proved in Thm.3.4 and 4.3 are quite powerful, showing that (partial/approximate) span recovery is possible with a relatively small amount of samples (of order of n*k, where n is the original input size and k is the size of the span), in a computationally efficient way (in particular for the non-adaptive algorithm), and for Relu NN or NN with differentiable layers and final threshold function. The main question is of course the validity of the assumptions needed to prove the theorems. Asm.1 and 2 are overall reasonable and they are very well supported by Lemma 3.1. The first two assumptions in page 6 are straightforward, while I'll less convinced of 3 and 4. In fact, they need to hold for any subspace V of any dimension smaller than k. I wonder whether the assumptions may become less and less likely as the size of the subspace decrease.
2- The theorems and the paper are mostly well written but some parts may be clearer.
3- Alg.1: the computation of the gradient is never really explained apart from the high-level lemma 3.2. While an actual algorithm is reported in the appendix, it would be better to have it explained already in the main text.
4- Right after Lemma 3.2 it is said "which demonstrates the claim". I am not sure which claim it refers to.
5- In alg.1 there is a parameter r which defines the number of queries of the algorithm. Thm 3.4 provides an upper bound on the number of queries needed and it does depend on k. Since k is initially unknown, how do you actually parameterize the algorithm? is there a stopping condition that can be tested?
6- In Thm 3.4 it is said that the algorithm returns the subspace V in time that is polynomial in the main parameters of the problem. Yet, I'm not sure where such complexity comes from. In Alg.1 it seems like the subspace is the direct output of the algorithm, so the complexity is r times the cost of computing the gradient, which according to Lem3.2. is poly(n). Is this the way you finally obtain the complexity?
7- One thing I'm doubtful about is the fact that the result in Thm 3.4 seems to be independent from the depth d and width k_i of the different layers. Some conditions may be implicit in the Asm.1 and 2, though. Furthermore, in the experiments it is clearly showed that thin NNs may make the support not recoverable. Could you please make such limit more explicit in the theory?
8- In alg.4 I think lines 5-7 are just the way to execute line 4. Is that correct? If not, how do you execute line 4?
9- In alg.4 line 8 and 9 are not easy to follow and they are not really discussed in the main text. Could you please clarify?
10- The empirical validation is relatively simple but it illustrates quite well the theory. Still I wish the authors could report results that dig more in detail in the theoretical results showing how tight they are (e.g., in the dependency on n, k, and other factors). The current results provide just a hint on how accurate/informative the theory is.
11- In the empirical result, it would be great to have a much more thorough validation of the difference between the non-adaptive and the adaptive algorithms. In the current results it seems like there is very limited difference.