Dynamical Isometry and a Mean Field Theory of LSTMs and GRUs

Abstract

Training recurrent neural networks (RNNs) on long sequence tasks is plagued with difficulties arising from the exponential explosion or vanishing of signals as they propagate forward or backward through the network. Many techniques have been proposed to ameliorate these issues, including various algorithmic and architectural modifications. Two of the most successful RNN architectures, the LSTM and the GRU, do exhibit modest improvements over vanilla RNN cells, but they still suffer from instabilities when trained on very long sequences. In this work, we develop a mean field theory of signal propagation in LSTMs and GRUs that enables us to calculate the time scales for signal propagation as well as the spectral properties of the state-to-state Jacobians. By optimizing these quantities in terms of the initialization hyperparameters, we derive a novel initialization scheme that eliminates or reduces training instabilities. We demonstrate the efficacy of our initialization scheme on multiple sequence tasks, on which it enables successful training while a standard initialization either fails completely or is orders of magnitude slower. We also observe a beneficial effect on generalization performance using this new initialization.

The edge of chaos: quantum field theory and deep neural networks

“…Turning to the titular edge of chaos, we are inspired by the aforementioned works [15,16,[27][28][29][30] examining criticality in various deep network architectures. However, while many of these papers used the phrase "mean-field theory", they did not actually rely on any MFT analysis: as mentioned above, Gaussianity arises simply as a consequence of the central limit theorem (CLT).…”

“…As discussed in the introduction and in more detail below, the result holds only at weak 't Hooft coupling, 29 and the perturbative expansion assumes T /N < 1; the latter is the regime of practical relevance for modern deep neural networks [2]. 30 It would be interesting to explore these connections to O(N ) theory in more detail, e.g., to see whether the analysis can be extended beyond the perturbative (weak-coupling) regime we consider here.…”

“…It has since appeared in many fields ranging from theoretical neuroscience and neurophysiology [21][22][23], to biological and complex systems [24,25], and was also explored in an early model of bioplausible neural networks in [26]. More recently, [15,16,[27][28][29][30] demonstrated that networks initialized at criticality are trainable to far greater depths than those lying further into either phase. To see this, one identifies a correlation length that sets the scale at which correlations between local degrees of freedom -e.g., the activations of two different neurons, or the spins of two different magnetic dipoles -decay with separation or depth through the network.…”

“…In fact, we shall require both σw and σ b to be sufficiently small relative to γ, as will be explained in more detail below 30. Note that the authors of[2] referred to T /N → ∞ as the "chaotic limit", which differs from the use of that terminology here, but the important point is that the perturbative expansion cannot be performed if T /N is large.…”

The edge of chaos: quantum field theory and deep neural networks

“…It has since appeared in many fields ranging from theoretical neuroscience and neurophysiology [23][24][25], to biological and complex systems [26,27], and was also explored in an early model of bioplausible neural networks in [28]. More recently, [17,18,[29][30][31][32] demonstrated that networks initialized at criticality are trainable to far greater depths than those lying further into either phase. To see this, one identifies a correlation length that sets the scale at which correlations between local degrees of freedom -e.g., the activations of two different neurons, or the spins of two different magnetic dipoles -decay with separation or depth through the network.…”