Dechen Zhang

Dechen Zhang, Xuan Tang, Yingyu Liang et al.

Low-precision training is critical for optimizing the trade-off between model quality and training costs, necessitating the joint allocation of model size, dataset size, and numerical precision. While empirical scaling laws suggest that quantization impacts effective model and data capacities or acts as an additive error, the theoretical mechanisms governing these effects remain largely unexplored. In this work, we initiate a theoretical study of scaling laws for low-precision training within a high-dimensional sketched linear regression framework. By analyzing multiplicative (signal-dependent) and additive (signal-independent) quantization, we identify a critical dichotomy in their scaling behaviors. Our analysis reveals that while both schemes introduce an additive error and degrade the effective data size, they exhibit distinct effects on effective model size: multiplicative quantization maintains the full-precision model size, whereas additive quantization reduces the effective model size. Numerical experiments validate our theoretical findings. By rigorously characterizing the complex interplay among model scale, dataset size, and quantization error, our work provides a principled theoretical basis for optimizing training protocols under practical hardware constraints.

Dechen Zhang, Junwei Su, Difan Zou

The use of low-bit quantization has emerged as an indispensable technique for enabling the efficient training of large-scale models. Despite its widespread empirical success, a rigorous theoretical understanding of its impact on learning performance remains notably absent, even in the simplest linear regression setting. We present the first systematic theoretical study of this fundamental question, analyzing finite-step stochastic gradient descent (SGD) for high-dimensional linear regression under a comprehensive range of quantization targets: data, labels, parameters, activations, and gradients. Our novel analytical framework establishes precise algorithm-dependent and data-dependent excess risk bounds that characterize how different quantization affects learning: parameter, activation, and gradient quantization amplify noise during training; data quantization distorts the data spectrum; and data and label quantization introduce additional approximation and quantized error. Crucially, we prove that for multiplicative quantization (with input-dependent quantization step), this spectral distortion can be eliminated, and for additive quantization (with constant quantization step), a beneficial scaling effect with batch size emerges. Furthermore, for common polynomial-decay data spectra, we quantitatively compare the risks of multiplicative and additive quantization, drawing a parallel to the comparison between FP and integer quantization methods. Our theory provides a powerful lens to characterize how quantization shapes the learning dynamics of optimization algorithms, paving the way to further explore learning theory under practical hardware constraints.

Dechen Zhang, Zhenmei Shi, Yi Zhang et al.

Kernel ridge regression (KRR) is a foundational tool in machine learning, with recent work emphasizing its connections to neural networks. However, existing theory primarily addresses the i.i.d. setting, while real-world data often exhibits structured dependencies - particularly in applications like denoising score learning where multiple noisy observations derive from shared underlying signals. We present the first systematic study of KRR generalization for non-i.i.d. data with signal-noise causal structure, where observations represent different noisy views of common signals. By developing a novel blockwise decomposition method that enables precise concentration analysis for dependent data, we derive excess risk bounds for KRR that explicitly depend on: (1) the kernel spectrum, (2) causal structure parameters, and (3) sampling mechanisms (including relative sample sizes for signals and noises). We further apply our results to denoising score learning, establishing generalization guarantees and providing principled guidance for sampling noisy data points. This work advances KRR theory while providing practical tools for analyzing dependent data in modern machine learning applications.