Nir Endy, Idan Daniel Grosbard, Yuval Ran-Milo, Yonatan Slutzky, Itay Tshuva, Raja Giryes. Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2025.

Conventional wisdom attributes the mysterious generalization abilities of overparameterized neural networks to gradient descent (and its variants). The recent volume hypothesis challenges this view: it posits that these generalization abilities persist even when gradient descent is replaced by Guess & Check (G&C), i.e., by randomly drawing weight settings until one that fits the training data is found. The validity of the volume hypothesis for wide and deep neural networks remains an open question. In this paper, we theoretically investigate this question for matrix factorization (with linear and non-linear activation): a canonical testbed in neural network theory. We first prove that generalization under G&C deteriorates with increasing width, establishing what is, to our knowledge, the first canonical case where G&C is provably inferior to gradient descent. Conversely, we prove that generalization under G&C improves with increasing depth, revealing a stark contrast between wide and deep networks, which we further validate empirically. These findings suggest that even in simple settings, there may not be a simple answer to the question of whether neural networks need gradient descent to generalize well.

Structured state space models (SSMs), the core engine behind prominent neural networks such as S4 and Mamba, are linear dynamical systems adhering to a specified structure, most notably diagonal. In contrast to typical neural network modules, whose parameterizations are real, SSMs often use complex parameterizations. Theoretically explaining the benefits of complex parameterizations for SSMs is an open problem. The current paper takes a step towards its resolution, by establishing formal gaps between real and complex diagonal SSMs. Firstly, we prove that while a moderate dimension suffices in order for a complex SSM to express all mappings of a real SSM, a much higher dimension is needed for a real SSM to express mappings of a complex SSM. Secondly, we prove that even if the dimension of a real SSM is high enough to express a given mapping, typically, doing so requires the parameters of the real SSM to hold exponentially large values, which cannot be learned in practice. In contrast, a complex SSM can express any given mapping with moderate parameter values. Experiments corroborate our theory, and suggest a potential extension of the theory that accounts for selectivity, a new architectural feature yielding state of the art performance.