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

02/23/2022

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by   Rajarshi Saha, et al.

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We consider the problem of quantizing a linear model learned from measurements 𝐗 = 𝐖θ + 𝐯. The model is constrained to be representable using only dB-bits, where B ∈ (0, ∞) 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.