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Abstract:We present a unified approach to obtain scaling limits of neural networks using the genus expansion technique from random matrix theory. This approach begins with a novel expansion of neural networks which is reminiscent of Butcher series for ODEs, and is obtained through a generalisation of Fa\`a di Bruno's formula to an arbitrary number of compositions. In this expansion, the role of monomials is played by random multilinear maps indexed by directed graphs whose edges correspond to random matrices, which we call operator graphs. This expansion linearises the effect of the activation functions, allowing for the direct application of Wick's principle to compute the expectation of each of its terms. We then determine the leading contribution to each term by embedding the corresponding graphs onto surfaces, and computing their Euler characteristic. Furthermore, by developing a correspondence between analytic and graphical operations, we obtain similar graph expansions for the neural tangent kernel as well as the input-output Jacobian of the original neural network, and derive their infinite-width limits with relative ease. Notably, we find explicit formulae for the moments of the limiting singular value distribution of the Jacobian. We then show that all of these results hold for networks with more general weights, such as general matrices with i.i.d. entries satisfying moment assumptions, complex matrices and sparse matrices.

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Abstract:Structured state-space models (SSMs) such as S4, stemming from the seminal work of Gu et al., are gaining popularity as effective approaches for modeling sequential data. Deep SSMs demonstrate outstanding performance across a diverse set of domains, at a reduced training and inference cost compared to attention-based transformers. Recent developments show that if the linear recurrence powering SSMs allows for multiplicative interactions between inputs and hidden states (e.g. GateLoop, Mamba, GLA), then the resulting architecture can surpass in both in accuracy and efficiency attention-powered foundation models trained on text, at scales of billion parameters. In this paper, we give theoretical grounding to this recent finding using tools from Rough Path Theory: we show that when random linear recurrences are equipped with simple input-controlled transitions (selectivity mechanism), then the hidden state is provably a low-dimensional projection of a powerful mathematical object called the signature of the input -- capturing non-linear interactions between tokens at distinct timescales. Our theory not only motivates the success of modern selective state-space models such as Mamba but also provides a solid framework to understand the expressive power of future SSM variants.

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Abstract:Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs). We show that in the infinite-width-then-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function. In the special case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021). In this setting, we also show that the width-depth limits commute. We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

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