Jiale Chen, Torsten Hoefler, Dan Alistarhcs.DS cs.LG
Adaptive rounding methods such as GPTQ, or equivalently Babai's nearest plane algorithm, round a real matrix to integers under a quadratic metric. They process the entries in a fixed order, one at a time, propagating each rounding error to the entries not yet processed through a triangular feedback matrix. We study the two-sided version of this task, in which fixed nonsingular basis matrices act on both the left and the right of the residual; the familiar one-sided case is the special case of an identity right basis. Vectorizing the matrix turns the two-sided objective into a quadratic metric whose Gram matrix is a Kronecker product, so the one-dimensional algorithm applies verbatim, but takes quartic time in the matrix dimension. We present GPTQ-2D, which produces the identical rounded matrix in cubic time. It rounds the entries anti-diagonal by anti-diagonal; entries on the same anti-diagonal are independent and are rounded in parallel.
When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\vartheta}+D^{-1})$ under Holder dependence of exponent $\vartheta$. Execution arithmetic can reverse this conclusion: full-state write-back introduces a $Dρ_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=Θ(L)$. Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.
The Strong Lottery Ticket Hypothesis (SLTH) asserts that sufficiently overparameterized, randomly initialized neural networks contain sparse subnetworks that, even without any training, can match the performance of a small trained network on a given dataset. A key mathematical tool in the theoretical study of SLTH has been the Random Subset Sum Problem (RSSP). The SLTH has recently been extended to the quantized setting, where the network weights are sampled from a discrete set rather than from a continuous interval. These new results are however far from those in arbitrary-precision setting in several ways. In this work, we provide an analysis of the RSSP in the discrete setting, and use it to derive tight SLTH guarantees in the quantized case. Our analysis obtain tight bounds on the failure probability of finding a strong lottery ticket in the quantized regime, providing an exponential improvement over previous results. Most importantly, it unifies the literature by showing that both approximate representations in the continuous setting and exact representations in quantized settings naturally emerge as limiting cases of our results. This perspective not only sharpens existing bounds but also provides a cohesive framework that simultaneously handles approximation and rounding errors.
In recent years, weight quantization that encodes the learnable parameters of large language models in an $n$-bit format has garnered significant attention due to its potential for model compression and inference acceleration. Many practical techniques have been developed; however, the theoretical understanding of many aspects, especially the approximation and degradation of expressive power as the number of quantization bits decreases, remains unclear. In this paper, we provide a theoretical investigation into the expressive capability of large language models relative to the number of quantization bits. We argue that 1.58-bit is the limiting precision for weight quantization by establishing the universal approximation and expressive collapse properties of weight-quantized models with respect to the number of quantization bits. Additionally, we confirm that weight quantization leads to expressive degradation, in which the expressive capacity of weight-quantized models degrades polynomially as the number of quantization bits decreases. These theoretical findings provide a solid foundation for advancing weight quantization in the context of scaling laws and shed insights for future research in model compression and inference acceleration.
We study distributed optimization with stochastic gradients and finite-bit communication modeled by random (unbiased) quantization. We propose q-PDGD, a quantized stochastic primal-dual method, and analyze it under relaxed global geometry. Under restricted secant inequality (RSI), a constant step-size yields linear contraction to an explicit neighborhood determined by gradient noise, quantization distortion, and network connectivity, while a diminishing step-size achieves O(1/k) convergence without shared-minimizer assumptions. Under Polyak-Lojasiewicz (PL) inequality, we obtain linear-to-neighborhood convergence in the same stochastic quantized setting. Our results match the best-known centralized stochastic rates in oracle complexity, and are supported by experiments demonstrating the predicted tradeoffs between quantization level, step-size choice, and graph structure.
SignSGD compresses each stochastic gradient coordinate to a single bit, offering substantial memory and communication savings, but its 1-bit quantization removes magnitude information and is known to leave a generalization gap relative to well-tuned SGD. We revisit SignSGD from a 1-bit quantization and dithering perspective and contribute three improvements. First, we derive a small-batch convergence rate for SignSGD under unimodal symmetric gradient noise using a signal-to-noise weighted stationarity measure, removing the large-batch assumption of prior analyses. Second, we inject annealed Gaussian noise before the sign operator, which acts as a classical dithering mechanism and probabilistically restores magnitude information lost to hard thresholding. Third, we adapt the SWATS strategy to sign-based updates with a projection-based learning-rate calibration that smoothly transitions from SignSGD to SGD. Single-worker experiments on ResNet-18 isolate optimizer effects from communication aspects: pre-sign dithering surpasses Adam on CIFAR-100, and the calibrated switch reaches 92.18% test accuracy on CIFAR-10, outperforming both pure SGD 91.38% and pure SignSGD with momentum 90.82%.