41 pages, 26 figures, 1 table. This work has been accepted for publication in the IEEE Journal on Selected Areas in Information Theory (JSAIT), Spl. issue on Distributed Coding and Computation
We study first-order optimization algorithms under the constraint that the descent direction is quantized using a pre-specified budget of $R$-bits per dimension, where $R \in (0 ,\infty)$. We propose computationally efficient optimization algorithms with convergence rates matching the information-theoretic performance lower bounds for: (i) Smooth and Strongly-Convex objectives with access to an Exact Gradient oracle, as well as (ii) General Convex and Non-Smooth objectives with access to a Noisy Subgradient oracle. The crux of these algorithms is a polynomial complexity source coding scheme that embeds a vector into a random subspace before quantizing it. These embeddings are such that with high probability, their projection along any of the canonical directions of the transform space is small. As a consequence, quantizing these embeddings followed by an inverse transform to the original space yields a source coding method with optimal covering efficiency while utilizing just $R$-bits per dimension. Our algorithms guarantee optimality for arbitrary values of the bit-budget $R$, which includes both the sub-linear budget regime ($R < 1$), as well as the high-budget regime ($R \geq 1$), while requiring $O\left(n^2\right)$ multiplications, where $n$ is the dimension. We also propose an efficient relaxation of this coding scheme using Hadamard subspaces that requires a near-linear time, i.e., $O\left(n \log n\right)$ additions.Furthermore, we show that the utility of our proposed embeddings can be extended to significantly improve the performance of gradient sparsification schemes. Numerical simulations validate our theoretical claims. Our implementations are available at https://github.com/rajarshisaha95/DistOptConstrComm.
Institut für Neuroinformatik, Ruhr-Universität Bochum（波鸿鲁尔大学神经信息学研究所）
We present a dual subspace ascent algorithm for support vector machine training that respects a budget constraint limiting the number of support vectors. Budget methods are effective for reducing the training time of kernel SVM while retaining high accuracy. To date, budget training is available only for primal (SGD-based) solvers. Dual subspace ascent methods like sequential minimal optimization are attractive for their good adaptation to the problem structure, their fast convergence rate, and their practical speed. By incorporating a budget constraint into a dual algorithm, our method enjoys…
University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology (KAUST)（沙特国王 Abdullah 科技大学 (KAUST)）；MIPT（莫斯科物理技术学院）
In this paper we present a convergence rate analysis of inexact variants of several randomized iterative methods. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic subspace ascent. A common feature of these methods is that in their update rule a certain sub-problem needs to be solved exactly. We relax this requirement by allowing for the sub-problem to be solved inexactly. In particular, we propose and analyze inexact randomized iterative methods for solving three closely related problems: a convex stochastic quadratic optim…
We propose novel randomized optimization methods for high-dimensional convex problems based on restrictions of variables to random subspaces. We consider oblivious and data-adaptive subspaces and study their approximation properties via convex duality and Fenchel conjugates. A suitable adaptive subspace can be generated by sampling a correlated random matrix whose second order statistics mirror the input data. We illustrate that the adaptive strategy can significantly outperform the standard oblivious sampling method, which is widely used in the recent literature. We show that the relative err…
Feature engineering plays an important role in the success of a machine learning model. Most of the effort in training a model goes into data preparation and choosing the right representation. In this paper, we propose a robust feature engineering method, Randomized Union of Locally Linear Subspaces (RULLS). We generate sparse, non-negative, and rotation invariant features in an unsupervised fashion. RULLS aggregates features from a random union of subspaces by describing each point using globally chosen landmarks. These landmarks serve as anchor points for choosing subspaces. Our method provi…
A Memory Efficient Randomized Subspace Optimization Method for Training Large Language Models
Beijing International Center for Mathematical Research, Peking University（北京大学北京数学科学中心）；Center for Data Science, Peking University（北京大学数据科学中心）；Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）
We propose a new randomized optimization method for high-dimensional problems which can be seen as a generalization of coordinate descent to random subspaces. We show that an adaptive sampling strategy for the random subspace significantly outperforms the oblivious sampling method, which is the common choice in the recent literature. The adaptive subspace can be efficiently generated by a correlated random matrix ensemble whose statistics mimic the input data. We prove that the improvement in the relative error of the solution can be tightly characterized in terms of the spectrum of the data m…
Lotus: Efficient LLM Training by Randomized Low-Rank Gradient Projection with Adaptive Subspace Switching
Hong Kong Baptist University, School of Science, Hong Kong, China（香港 Baptist 大学，科学学院，香港，中国）；Fudan University, School of Data Science, Shanghai, China（复旦大学，数据科学学院，上海，中国）；New York University, Tandon School of Engineering, New York, USA（纽约大学，Tandon 工程学院，纽约，美国）
In many artificial intelligence and computer vision systems, the same object can be observed at distinct viewpoints or by diverse sensors, which raises the challenges for recognizing objects from different, even heterogeneous views. Multi-view discriminant analysis (MvDA) is an effective multi-view subspace learning method, which finds a discriminant common subspace by jointly learning multiple view-specific linear projections for object recognition from multiple views, in a non-pairwise way. In this paper, we propose the kernel version of multi-view discriminant analysis, called kernel multi-…
We provide an exact analysis of a class of randomized algorithms for solving overdetermined least-squares problems. We consider first-order methods, where the gradients are pre-conditioned by an approximation of the Hessian, based on a subspace embedding of the data matrix. This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled randomized Hadamard transforms (SRHT). Our key technical innovation is the derivation of the limiting spectral density of SRH…
Matrix completion is a widely used technique for image inpainting and personalized recommender system, etc. In this work, we focus on accelerating the matrix completion using faster randomized singular value decomposition (rSVD). Firstly, two fast randomized algorithms (rSVD-PI and rSVD- BKI) are proposed for handling sparse matrix. They make use of an eigSVD procedure and several accelerating skills. Then, with the rSVD-BKI algorithm and a new subspace recycling technique, we accelerate the singular value thresholding (SVT) method in [1] to realize faster matrix completion. Experiments show t…
