The goal of quantization is to produce a compressed model whose output distribution is as close to the original model's as possible. To do this tractably, most quantization algorithms minimize the immediate activation error of each layer as a proxy for the end-to-end error. However, this ignores the effect of future layers, making it a poor proxy. In this work, we introduce Yet Another Quantization Algorithm (YAQA), an adaptive rounding algorithm that directly considers the error at the network's output. YAQA introduces a series of theoretical results that culminate in the first end-to-end error bounds for quantization algorithms. First, we characterize the convergence time of adaptive rounding algorithms via the structure of their Hessian approximations. We then show that the end-to-end error can be bounded by the approximation's cosine similarity to the true Hessian. This admits a natural Kronecker-factored approximation with corresponding near-optimal Hessian sketches. YAQA is provably better than GPTQ/LDLQ and empirically reduces the error by $\approx 30\%$ over these methods. YAQA even achieves a lower error than quantization aware training. This translates to state of the art performance on downstream tasks, all while adding no inference overhead.

Iterative Hessian sketch (IHS) is an effective sketching method for modeling large-scale data. It was originally proposed by Pilanci and Wainwright (2016; JMLR) based on randomized sketching matrices. However, it is computationally intensive due to the iterative sketch process. In this paper, we analyze the IHS algorithm under the unconstrained least squares problem setting, then propose a deterministic approach for improving IHS via A-optimal subsampling. Our contributions are three-fold: (1) a good initial estimator based on the $A$-optimal design is suggested; (2) a novel ridged preconditioner is developed for repeated sketching; and (3) an exact line search method is proposed for determining the optimal step length adaptively. Extensive experimental results demonstrate that our proposed A-optimal IHS algorithm outperforms the existing accelerated IHS methods.

We address the statistical and optimization impacts of using classical sketch versus Hessian sketch to solve approximately the Matrix Ridge Regression (MRR) problem. Prior research has considered the effects of classical sketch on least squares regression (LSR), a strictly simpler problem. We establish that classical sketch has a similar effect upon the optimization properties of MRR as it does on those of LSR---namely, it recovers nearly optimal solutions. In contrast, Hessian sketch does not have this guarantee, instead, the approximation error is governed by a subtle interplay between the "mass" in the responses and the optimal objective value. For both types of approximations, the regularization in the sketched MRR problem gives it significantly different statistical properties from the sketched LSR problem. In particular, there is a bias-variance trade-off in sketched MRR that is not present in sketched LSR. We provide upper and lower bounds on the biases and variances of sketched MRR, these establish that the variance is significantly increased when classical sketches are used, while the bias is significantly increased when using Hessian sketches. Empirically, sketched MRR solutions can have risks that are higher by an order-of-magnitude than those of the optimal MRR solutions. We establish theoretically and empirically that model averaging greatly decreases this gap. Thus, in the distributed setting, sketching combined with model averaging is a powerful technique that quickly obtains near-optimal solutions to the MRR problem while greatly mitigating the statistical risks incurred by sketching.

Sketching techniques have become popular for scaling up machine learning algorithms by reducing the sample size or dimensionality of massive data sets, while still maintaining the statistical power of big data. In this paper, we study sketching from an optimization point of view: we first show that the iterative Hessian sketch is an optimization process with preconditioning, and develop accelerated iterative Hessian sketch via the searching the conjugate direction; we then establish primal-dual connections between the Hessian sketch and dual random projection, and apply the preconditioned conjugate gradient approach on the dual problem, which leads to the accelerated iterative dual random projection methods. Finally to tackle the challenges from both large sample size and high-dimensionality, we propose the primal-dual sketch, which iteratively sketches the primal and dual formulations. We show that using a logarithmic number of calls to solvers of small scale problem, primal-dual sketch is able to recover the optimum of the original problem up to arbitrary precision. The proposed algorithms are validated via extensive experiments on synthetic and real data sets which complements our theoretical results.