Tuning Frequency Bias of State Space Models

ICLR 2025 Conference Submission3382 Authors

Keywords: state-space models, sequence models, Long-Range Arena, frequency bias

TL;DR: We propose two mechanisms to diminish or increase the learning rate of high-frequency components relative to low-frequency ones in a state space model (SSM).

Abstract: State space models (SSMs) leverage linear, time-invariant (LTI) systems to effectively learn sequences with long-range dependencies. By analyzing the transfer functions of LTI systems, we find that SSMs exhibit an implicit bias toward capturing low-frequency components more effectively than high-frequency ones. This behavior aligns with the broader notion of frequency bias in deep learning model training. We show that the initialization of an SSM assigns it an innate frequency bias and that training the model in a conventional way does not alter this bias. Based on our theory, we propose two mechanisms to tune frequency bias: either by scaling the initialization to tune the inborn frequency bias; or by applying a Sobolev-norm-based filter to adjust the sensitivity of the gradients to high-frequency inputs, which allows us to change the frequency bias via training. Using an image-denoising task, we empirically show that we can strengthen, weaken, or even reverse the frequency bias using both mechanisms. By tuning the frequency bias, we can also improve SSMs' performance on learning long-range sequences, averaging an $88.26\\%$ accuracy on the Long-Range Arena (LRA) benchmark tasks.

Primary Area: other topics in machine learning (i.e., none of the above)

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Submission Number: 3382