DiscQuant: A Quantization Method for Neural Networks Inspired by Discrepancy Theory

Quantizing the weights of a neural network has two steps: (1) Finding a good low bit-complexity representation for weights (which we call the quantization grid) and (2) Rounding the original weights to values in the quantization grid. In this paper, we study the problem of rounding optimally given any quantization grid. The simplest and most commonly used way to round is Round-to-Nearest (RTN). By rounding in a data-dependent way instead, one can improve the quality of the quantized model significantly. We study the rounding problem from the lens of \emph{discrepancy theory}, which studies how well we can round a continuous solution to a discrete solution without affecting solution quality too much. We prove that given $m=\mathrm{poly}(1/\epsilon)$ samples from the data distribution, we can round all but $O(m)$ model weights such that the expected approximation error of the quantized model on the true data distribution is $\le \epsilon$ as long as the space of gradients of the original model is approximately low rank (which we empirically validate). Our proof, which is algorithmic, inspired a simple and practical rounding algorithm called \emph{DiscQuant}. In our experiments, we demonstrate that DiscQuant significantly improves over the prior state-of-the-art rounding method called GPTQ and the baseline RTN over a range of benchmarks on Phi3mini-3.8B and Llama3.1-8B. For example, rounding Phi3mini-3.8B to a fixed quantization grid with 3.25 bits per parameter using DiscQuant gets 64\% accuracy on the GSM8k dataset, whereas GPTQ achieves 54\% and RTN achieves 31\% (the original model achieves 84\%). We make our code available at https://github.com/jerry-chee/DiscQuant.

Stronger Coreset Bounds for Kernel Density Estimators via Chaining

We apply the discrepancy method and a chaining approach to give improved bounds on the coreset complexity of a wide class of kernel functions. Our results give randomized polynomial time algorithms to produce coresets of size $O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$ for the Gaussian and Laplacian kernels in the case that the data set is uniformly bounded, an improvement that was not possible with previous techniques. We also obtain coresets of size $O\big(\frac{1}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$ for the Laplacian kernel for $d$ constant. Finally, we give the best known bounds of $O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log(2\max\{1,\alpha\})}\big)$ on the coreset complexity of the exponential, Hellinger, and JS Kernels, where $1/\alpha$ is the bandwidth parameter of the kernel.