### Common initialization schemes for recurrent neural networks are likely suboptimal

Training of recurrent neural networks (RNNs) suffers from the same kind of degeneracy problem faced by deep feedforward networks. In fact, the degeneracy problem is likely compounded in RNNs, because empirically the spectral radius of $W^k$ tends to be much larger than the spectral radius of $W_k W_{k-1} \ldots W_1$ where $W, W_1, \dots, W_k$ are random matrices drawn from the same ensemble (e.g. random Gaussian). I don’t know of a rigorous proof of this claim for random matrices (although, heuristically, it is easy to see that something like this should be true for random scalars: $\sum_i^{k} w_i \sim \mathcal{N}(0, k)$, but $kw \sim \mathcal{N}(0, k^2)$ for $w, w_1, \ldots, w_k \sim \mathcal{N}(0,1)$ –this is essentially the difference between a true random walk vs. a biased random walk (I thank Xaq Pitkow for pointing this out to me)–; exponentiating both sides, we can then see that the product of $k$ random scalars should be exponentially larger than the $k$-th power of a random scalar), but this empirical observation would explain why training linear RNNs would be harder than training deep feedforward networks and one can reasonably expect something like this to hold approximately in the nonlinear case as well.

Researchers have developed methods to deal with this degeneracy problem, hence to overcome training difficulties in RNNs. One of the most well-known of these methods is the identity initialization for the recurrent weight matrix. Others proposed constraining the weight matrix to always be orthogonal, instead of orthogonalizing it at initialization only. The logic behind both of these methods is that since orthogonal transformations are isometries of the Euclidean space, applying a bunch of these transformations in a cascade does not lead to a degeneration of the metric (by “degeneration” here, I mean the collapse of the metric along the overwhelming majority of the directions in the input space and the astronomical expansion of the metric along a very small number of remaining directions). This is guaranteed in the linear case and, again, one hopes and expects (with some justification) that things are not all that different in the nonlinear case as well. So, in other words, a sequence of orthogonal transformations propagate vectors in Euclidean space without distortion, i.e. without changing their norms or the distances between them.

This is all true and fine, however, this analysis ignores a crucial factor that is relevant in training neural networks, namely the effect of noise. Noise comes in both through the stochasticity of SGD and sometimes through direct noise injection (as in Dropout) for regularization purposes. It is a bit hard to precisely characterize the noise that arises due to SGD, but let us assume for the sake of simplicity that the noise is additive so that what we propagate in the end is some kind of “signal + noise”. Now, although it is true that orthogonal transformations propagate the signal without distortion, they also propagate the noise without distortion as well. But, ultimately, we probably want a transformation that maximizes something like the signal-to-noise ratio (SNR) of the propagated signal + noise. Then, it is no longer obvious that orthogonal transformations are optimal for this purpose, because, one can, for example, imagine transformations that would amplify the signal more than they would amplify the noise (hence distorting both the signal and the noise), thus yielding a better SNR than an orthogonal transformation.

And indeed it turns out that for linear systems with additive Gaussian noise, one can mathematically show that optimal transformations (in the sense of maximizing the total SNR of the propagated signal + noise) are not orthogonal. In fact, one can say something even stronger: any optimal transformation has to be non-normal (a normal matrix is a unitarily diagonalizable matrix; all orthogonal matrices are normal, but the reverse is not true). This is the main result of this beautiful and insightful paper by Surya Ganguli and colleagues. Perhaps the simplest example of an optimal transformation in this sense is a feedforward chain: $W_{ij} = \alpha \delta_{i,j-1}$, where $\delta$ is the Kronecker delta function. This particular example maximizes the total SNR through a mechanism known as transient amplification: it exponentially amplifies the norm of its input transiently before the norm eventually decays to zero.

This brings me to the main message of this post: that the commonly used orthogonal initializations for recurrent neural networks are likely suboptimal because of the often neglected effect of noise. Another evidence for this claim comes from looking at the trained recurrent connectivity matrices in tasks that require memory. In this work (currently under review), we have shown that the trained recurrent connectivity matrices in such tasks always end up non-normal, with a feedforward structure hidden in the recurrent connectivity, even when they are initialized with an approximately normal matrix. How non-normal the trained matrices end up depend on a wide range of factors and investigating those factors was the main motivation for our paper. So, initializing RNNs with a non-normal matrix would potentially be a useful inductive bias for these networks.

In ongoing work, I have been investigating the merits of various non-normal initialization schemes for non-linear RNNs. One particular non-normal initialization scheme that seems to work quite well (and that is very easy to implement) is combining an identity matrix (or a scaled identity matrix) with a chain structure (which was shown by Ganguli et al. to be optimal in the case of a linear model with additive Gaussian noise). More details on these results will be forthcoming in the following weeks, I hope. Another open question at this point is whether non-normal initialization schemes are also useful for the more commonly used gated recurrent architectures like LSTMs or GRUs. These often behave very differently than vanilla recurrent networks, so I am not sure whether non-normal dynamics in these architectures will be as useful as it is in vanilla RNNs.

Update (06/15/19): Our work on a new non-normal initialization scheme for RNNs described in this post is now on arxiv. The accompanying code for reproducing some of the results reported in the paper is available in this public repository.