Minimax Optimal Quantization of Linear Models: Information-Theoretic Limits and Efficient Algorithms

• URL: http://arxiv.org/abs/2202.11277v1
• Date: Wed, 23 Feb 2022 02:39:04 GMT
• Title: Minimax Optimal Quantization of Linear Models: Information-Theoretic Limits and Efficient Algorithms
• Authors: Rajarshi Saha, Mert Pilanci, Andrea J. Goldsmith
• Abstract summary: We consider the problem of quantizing a linear model learned from measurements. We derive an information-theoretic lower bound for the minimax risk under this setting. We show that our method and upper-bounds can be extended for two-layer ReLU neural networks.
• Score: 59.724977092582535
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider the problem of quantizing a linear model learned from measurements $\mathbf{X} = \mathbf{W}\boldsymbol{\theta} + \mathbf{v}$. The model is constrained to be representable using only $dB$-bits, where $B \in (0, \infty)$ is a pre-specified budget and $d$ is the dimension of the model. We derive an information-theoretic lower bound for the minimax risk under this setting and show that it is tight with a matching upper bound. This upper bound is achieved using randomized embedding based algorithms. We propose randomized Hadamard embeddings that are computationally efficient while performing near-optimally. We also show that our method and upper-bounds can be extended for two-layer ReLU neural networks. Numerical simulations validate our theoretical claims.

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