Towards provably efficient quantum algorithms for large-scale machine-learning models
The University of Chicago（芝加哥大学）；qBraid Co.（qBraid公司）；SeQure（SeQure公司）；Argonne National Laboratory（阿贡国家实验室）；University of California, Berkeley（加州大学伯克利分校）；Massachusetts Institute of Technology（麻省理工学院）；Brandeis University（布兰迪斯大学）；Free University Berlin（柏林自由大学）
大型机器学习模型是人工智能的革命性技术，其瓶颈包括预训练和微调过程中消耗的巨大计算开销、功耗和时间。在这项工作中，我们证明容错量子计算有可能为通用（随机）梯度下降算法提供可证明的高效解决方案，其复杂度为 O(T^2 polylog(n))，其中 n 是模型规模，T 是训练迭代次数，只要模型具有足够的耗散性和稀疏性，并且学习率较小即可。基于先前针对耗散微分方程的高效量子算法，我们发现并证明了类似算法适用于（随机）梯度下降——机器学习的主要算法。在实验中，我们对参数量从 700 万到 1.03 亿的大规模机器学习模型实例进行了基准测试。我们发现，在稀疏训练的背景下，模型剪枝后的学习早期阶段有可能实现量子加速，这促使我们提出一种稀疏参数下载与重新上传方案。我们的工作有力地表明，容错量子算法有可能为大多数最先进的大规模机器学习问题做出贡献。
英文摘要：
Large machine learning models are revolutionary technologies of artificial intelligence whose bottlenecks include huge computational expenses, power, and time used both in the pre-training and fine-tuning process. In this work, we show that fault-tolerant quantum computing could possibly provide provably efficient resolutions for generic (stochastic) gradient descent algorithms, scaling as O(T^2 polylog(n)), where n is the size of the models and T is the number of iterations in the training, as long as the models are both sufficiently dissipative and sparse, with small learning rates. Based on earlier efficient quantum algorithms for dissipative differential equations, we find and prove that similar algorithms work for (stochastic) gradient descent, the primary algorithm for machine learning. In practice, we benchmark instances of large machine learning models from 7 million to 103 million parameters. We find that, in the context of sparse training, a quantum enhancement is possible at the early stage of learning after model pruning, motivating a sparse parameter download and re-upload scheme. Our work shows solidly that fault-tolerant quantum algorithms could potentially contribute to most state-of-the-art, large-scale machine-learning problems.
We address an inherent difficulty in welfare-theoretic fair machine learning by proposing an equivalently axiomatically-justified alternative and studying the resulting computational and statistical learning questions. Welfare metrics quantify overall wellbeing across a population of one or more groups, and welfare-based objectives and constraints have recently been proposed to incentivize fair machine learning methods to produce satisfactory solutions that consider the diverse needs of multiple groups. Unfortunately, many machine-learning problems are more naturally cast as loss minimization …
Provably Correct Learning Algorithms in the Presence of Time-Varying Features Using a Variational Perspective
Massachusetts Institute of Technology（麻省理工学院）；Brigham and Women’s Hospital and Harvard Medical School（布里奇沃特医院和哈佛医学院）；Air Force Research Laboratory（空军研究实验室）
The University of Chicago（芝加哥大学）；Northwestern University（西北大学）；The University of Chicago Booth School of Business（芝加哥大学布斯商学院）
Training a classifier under non-convex constraints has gotten increasing attention in the machine learning community thanks to its wide range of applications such as algorithmic fairness and class-imbalanced classification. However, several recent works addressing non-convex constraints have only focused on simple models such as logistic regression or support vector machines. Neural networks, one of the most popular models for classification nowadays, are precluded and lack theoretical guarantees. In this work, we show that overparameterized neural networks could achieve a near-optimal and nea…
We describe a layer-by-layer algorithm for training deep convolutional networks, where each step involves gradient updates for a two layer network followed by a simple clustering algorithm. Our algorithm stems from a deep generative model that generates mages level by level, where lower resolution images correspond to latent semantic classes. We analyze the convergence rate of our algorithm assuming that the data is indeed generated according to this model (as well as additional assumptions). While we do not pretend to claim that the assumptions are realistic for natural images, we do believe …
We develop necessary and sufficient conditions and a novel provably consistent and efficient algorithm for discovering topics (latent factors) from observations (documents) that are realized from a probabilistic mixture of shared latent factors that have certain properties. Our focus is on the class of topic models in which each shared latent factor contains a novel word that is unique to that factor, a property that has come to be known as separability. Our algorithm is based on the key insight that the novel words correspond to the extreme points of the convex hull formed by the row-vectors …
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform the required learning tasks. For hierarchical data, the space of choice is a hyperbolic space because it guarantees low-distortion embeddings for tree-like structures. The geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. We propose a unified framework…
Carnegie Mellon University（卡内基梅隆大学）；School of Computer Science（计算机科学学院）；Machine Learning Department（机器学习系）；Forbes Avenue, Pittsburgh, PA 15213, USA（5000 Forbes Avenue, Pittsburgh, PA 15213, USA（地址非机构，此处按要求应忽略，但原文本中该地址与机构关联，需确认））
We consider the problem of matrix column subset selection, which selects a subset of columns from an input matrix such that the input can be well approximated by the span of the selected columns. Column subset selection has been applied to numerous real-world data applications such as population genetics summarization, electronic circuits testing and recommendation systems. In many applications the complete data matrix is unavailable and one needs to select representative columns by inspecting only a small portion of the input matrix. In this paper we propose the first provably correct column …
University of Toronto（多伦多大学）；Georgia Institute of Technology（佐治亚理工学院）
Linear regression in $\ell_p$-norm is a canonical optimization problem that arises in several applications, including sparse recovery, semi-supervised learning, and signal processing. Generic convex optimization algorithms for solving $\ell_p$-regression are slow in practice. Iteratively Reweighted Least Squares (IRLS) is an easy to implement family of algorithms for solving these problems that has been studied for over 50 years. However, these algorithms often diverge for p > 3, and since the work of Osborne (1985), it has been an open problem whether there is an IRLS algorithm that is guaran…
Provably Efficient Off-Policy Adversarial Imitation Learning with Convergence Guarantees
Division of Systems Engineering（系统工程系）；Department of Computer Science（计算机科学系）；Amazon Robotics（亚马逊机器人）；Department of Electrical and Computer Engineering（电气与计算机工程系）
We investigate the role of noise in optimization algorithms for learning over-parameterized models. Specifically, we consider the recovery of a rank one matrix $Y^*\in R^{d\times d}$ from a noisy observation $Y$ using an over-parameterization model. We parameterize the rank one matrix $Y^*$ by $XX^\top$, where $X\in R^{d\times d}$. We then show that under mild conditions, the estimator, obtained by the randomly perturbed gradient descent algorithm using the square loss function, attains a mean square error of $O(σ^2/d)$, where $σ^2$ is the variance of the observational noise. In contrast, the …
Stanford University（斯坦福大学）；Massachusetts Institute of Technology（麻省理工学院）
We consider the problem of explaining the predictions of an arbitrary blackbox model $f$: given query access to $f$ and an instance $x$, output a small set of $x$'s features that in conjunction essentially determines $f(x)$. We design an efficient algorithm with provable guarantees on the succinctness and precision of the explanations that it returns. Prior algorithms were either efficient but lacked such guarantees, or achieved such guarantees but were inefficient. We obtain our algorithm via a connection to the problem of {\sl implicitly} learning decision trees. The implicit nature of this …
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Many high-dimensional and large-volume data sets of practical relevance have hierarchical structures induced by trees, graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform required learning tasks. For hierarchical data, the space of choice is a hyperbolic space since it guarantees low-distortion embeddings for tree-like structures. Unfortunately, the geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorit…
Institute for Advanced Study（高等研究院）；Carnegie Mellon University（卡内基梅隆大学）；Princeton University（普林斯顿大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
A fundamental challenge in artificial intelligence is to build an agent that generalizes and adapts to unseen environments. A common strategy is to build a decoder that takes the context of the unseen new environment as input and generates a policy accordingly. The current paper studies how to build a decoder for the fundamental continuous control task, linear quadratic regulator (LQR), which can model a wide range of real-world physical environments. We present a simple algorithm for this problem, which uses upper confidence bound (UCB) to refine the estimate of the decoder and balance the ex…
In many real world applications of machine learning, models have to meet certain domain-based requirements that can be expressed as constraints (e.g., safety-critical constraints in autonomous driving systems). Such constraints are often handled by including them in a regularization term, while learning a model. This approach, however, does not guarantee 100% satisfaction of the constraints: it only reduces violations of the constraints on the training set rather than ensuring that the predictions by the model will always adhere to them. In this paper, we present a framework for learning model…
Mean-Field is an efficient way to approximate a posterior distribution in complex graphical models and constitutes the most popular class of Bayesian variational approximation methods. In most applications, the mean field distribution parameters are computed using an alternate coordinate minimization. However, the convergence properties of this algorithm remain unclear. In this paper, we show how, by adding an appropriate penalization term, we can guarantee convergence to a critical point, while keeping a closed form update at each step. A convergence rate estimate can also be derived based on…
We consider large-scale Markov decision processes (MDPs) with an unknown cost function and employ stochastic convex optimization tools to address the problem of imitation learning, which consists of learning a policy from a finite set of expert demonstrations. We adopt the apprenticeship learning formalism, which carries the assumption that the true cost function can be represented as a linear combination of some known features. Existing inverse reinforcement learning algorithms come with strong theoretical guarantees, but are computationally expensive because they use reinforcement learning o…
Laboratoire Hubert Curien UMR CNRS 5516（于贝尔·居里安实验室（法国国家科学研究中心 UMR 5516））
In recent years, the crucial importance of metrics in machine learning algorithms has led to an increasing interest for optimizing distance and similarity functions. Most of the state of the art focus on learning Mahalanobis distances (requiring to fulfill a constraint of positive semi-definiteness) for use in a local k-NN algorithm. However, no theoretical link is established between the learned metrics and their performance in classification. In this paper, we make use of the formal framework of good similarities introduced by Balcan et al. to design an algorithm for learning a non PSD linea…
Provably Robust Training of Quantum Circuit Classifiers Against Parameter Noise
Computer Science, University of California Los Angeles（计算机科学，加州大学洛杉矶分校）；Electrical and Computer Engineering, University of California Los Angeles（电气与计算机工程，加州大学洛杉矶分校）
Johannes Kepler University Linz（林茨约翰内斯·开普勒大学）；AWS Center for Quantum Computing（AWS量子计算中心）；National Institute of Standards and Technology（美国国家标准与技术研究院）；University of Maryland（马里兰大学）
Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not le…
Enigma machines are devices that perform cryptography using pseudo-random numbers. The original enigma machine code was broken by detecting hidden patterns in these pseudo-random numbers. This paper proposes a model for a quantum optical enigma machine and shows that the phenomenon of quantum data locking makes such quantum enigma machines provably secure even in the presence of noise and loss.
In modelling complex processes, the potential past data that influence future expectations are immense. Models that track all this data are not only computationally wasteful but also shed little light on what past data most influence the future. There is thus enormous interest in dimensional reduction-finding automated means to reduce the memory dimension of our models while minimizing its impact on its predictive accuracy. Here we construct dimensionally reduced quantum models by machine learning methods that can achieve greater accuracy than provably optimal classical counterparts. We demons…
