Matrix completion is a ubiquitous tool in machine learning and data analysis. Most work in this area has focused on the number of observations necessary to obtain an accurate low-rank approximation. In practice, however, the cost of observations is an important limiting factor, and experimentalists may have on hand multiple modes of observation with differing noise-vs-cost trade-offs. This paper considers matrix completion subject to such constraints: a budget is imposed and the experimentalist's goal is to allocate this budget between two sampling modalities in order to recover an accurate low-rank approximation. Specifically, we consider that it is possible to obtain low noise, high cost observations of individual entries or high noise, low cost observations of entire columns. We introduce a regression-based completion algorithm for this setting and experimentally verify the performance of our approach on both synthetic and real data sets. When the budget is low, our algorithm outperforms standard completion algorithms. When the budget is high, our algorithm has comparable error to standard nuclear norm completion algorithms and requires much less computational effort.
We propose to focus on the problem of discovering neural network architectures efficient in terms of both prediction quality and cost. For instance, our approach is able to solve the following tasks: learn a neural network able to predict well in less than 100 milliseconds or learn an efficient model that fits in a 50 Mb memory. Our contribution is a novel family of models called Budgeted Super Networks (BSN). They are learned using gradient descent techniques applied on a budgeted learning objective function which integrates a maximum authorized cost, while making no assumption on the nature …
We seek decision rules for prediction-time cost reduction, where complete data is available for training, but during prediction-time, each feature can only be acquired for an additional cost. We propose a novel random forest algorithm to minimize prediction error for a user-specified {\it average} feature acquisition budget. While random forests yield strong generalization performance, they do not explicitly account for feature costs and furthermore require low correlation among trees, which amplifies costs. Our random forest grows trees with low acquisition cost and high strength based on gre…
University of Science and Technology of China（中国科学技术大学）；University of Chinese Academy of Sciences（中国科学院大学）；Microsoft Research（微软研究院）
Thompson sampling is one of the earliest randomized algorithms for multi-armed bandits (MAB). In this paper, we extend the Thompson sampling to Budgeted MAB, where there is random cost for pulling an arm and the total cost is constrained by a budget. We start with the case of Bernoulli bandits, in which the random rewards (costs) of an arm are independently sampled from a Bernoulli distribution. To implement the Thompson sampling algorithm in this case, at each round, we sample two numbers from the posterior distributions of the reward and cost for each arm, obtain their ratio, select the arm …
Bayesian optimization (BO) is a sample-efficient approach to optimizing costly-to-evaluate black-box functions. Most BO methods ignore how evaluation costs may vary over the optimization domain. However, these costs can be highly heterogeneous and are often unknown in advance. This occurs in many practical settings, such as hyperparameter tuning of machine learning algorithms or physics-based simulation optimization. Moreover, those few existing methods that acknowledge cost heterogeneity do not naturally accommodate a budget constraint on the total evaluation cost. This combination of unknown…
Technion Israel Institute of Technology（以色列理工学院）；Indian Institute of Science（印度科学学院）；Adobe Research（Adobe研究院）
Learning good interventions in a causal graph can be modelled as a stochastic multi-armed bandit problem with side-information. First, we study this problem when interventions are more expensive than observations and a budget is specified. If there are no backdoor paths from an intervenable node to the reward node then we propose an algorithm to minimize simple regret that optimally trades-off observations and interventions based on the cost of intervention. We also propose an algorithm that accounts for the cost of interventions, utilizes causal side-information, and minimizes the expected cu…
Frequently, acquiring training data has an associated cost. We consider the situation where the learner may purchase data during training, subject TO a budget. IN particular, we examine the CASE WHERE each feature label has an associated cost, AND the total cost OF ALL feature labels acquired during training must NOT exceed the budget.This paper compares methods FOR choosing which feature label TO purchase next, given the budget AND the CURRENT belief state OF naive Bayes model parameters.Whereas active learning has traditionally focused ON myopic(greedy) strategies FOR query selection, this p…
Google DeepMind（谷歌DeepMind）；Harvard University（哈佛大学）；Google Research（谷歌研究院）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）
Learning to defer uncertain predictions to costly experts offers a powerful strategy for improving the accuracy and efficiency of machine learning systems. However, standard training procedures for deferral algorithms typically require querying all experts for every training instance, an approach that becomes prohibitively expensive when expert queries incur significant computational or resource costs. This undermines the core goal of deferral: to limit unnecessary expert usage. To overcome this challenge, we introduce the budgeted deferral framework, which aims to train effective deferral alg…
Department of Computer Science（计算机科学系）；University of California, Los Angeles (UCLA)（加利福尼亚大学洛杉矶分校（UCLA））；Los Angeles, CA 90095, USA（美国加利福尼亚州洛杉矶市 90095）
In many real-world learning scenarios, features are only acquirable at a cost constrained under a budget. In this paper, we propose a novel approach for cost-sensitive feature acquisition at the prediction-time. The suggested method acquires features incrementally based on a context-aware feature-value function. We formulate the problem in the reinforcement learning paradigm, and introduce a reward function based on the utility of each feature. Specifically, MC dropout sampling is used to measure expected variations of the model uncertainty which is used as a feature-value function. Furthermor…
Washington University in St. Louis（圣路易斯华盛顿大学）；Microsoft Research（微软研究院）
As machine learning transitions increasingly towards real world applications controlling the test-time cost of algorithms becomes more and more crucial. Recent work, such as the Greedy Miser and Speedboost, incorporate test-time budget constraints into the training procedure and learn classifiers that provably stay within budget (in expectation). However, so far, these algorithms are limited to the supervised learning scenario where sufficient amounts of labeled data are available. In this paper we investigate the common scenario where labeled data is scarce but unlabeled data is available in …
Classifiers are often utilized in time-constrained settings where labels must be assigned to inputs quickly. To address these scenarios, budgeted multi-stage classifiers (MSC) process inputs through a sequence of partial feature acquisition and evaluation steps with early-exit options until a confident prediction can be made. This allows for fast evaluation that can prevent expensive, unnecessary feature acquisition in time-critical instances. However, performance of MSCs is highly sensitive to several design aspects -- making optimization of these systems an important but difficult problem. T…
University of Washington（华盛顿大学）；CNRS（法国国家科学研究中心）；Univ. Lille（里尔大学）；Inria Lille（里尔法国国家信息与自动化研究所）；Université Paris Descartes（巴黎笛卡尔大学）
We study a generalization of the multi-armed bandit problem with multiple plays where there is a cost associated with pulling each arm and the agent has a budget at each time that dictates how much she can expect to spend. We derive an asymptotic regret lower bound for any uniformly efficient algorithm in our setting. We then study a variant of Thompson sampling for Bernoulli rewards and a variant of KL-UCB for both single-parameter exponential families and bounded, finitely supported rewards. We show these algorithms are asymptotically optimal, both in rateand leading problem-dependent consta…
The typical algorithmic problem in viral marketing aims to identify a set of influential users in a social network, who, when convinced to adopt a product, shall influence other users in the network and trigger a large cascade of adoptions. However, the host (the owner of an online social platform) often faces more constraints than a single product, endless user attentions, unlimited budget and unbounded time; in reality, multiple products need to be advertised, each user can tolerate only a small number of recommendations, influencing user has a cost and advertisers have only limited budgets,…
Institute for Basic Science（基础科学研究院）；London School of Economics（伦敦政治经济学院）；KAIST（韩国科学技术院）
Motivated by recent developments in designing algorithms based on individual item scores for solving utility maximization problems, we study the framework of using test scores, defined as a statistic of observed individual item performance data, for solving the budgeted stochastic utility maximization problem. We extend an existing scoring mechanism, namely the replication test scores, to incorporate heterogeneous item costs as well as item values. We show that a natural greedy algorithm that selects items solely based on their replication test scores outputs solutions within a constant factor…
Stepsize anything: A unified learning rate schedule for budgeted-iteration training
State Key Lab of General AI, School of Intelligence Science and Technology, Peking University（人工智能通用基础研究国家重点实验室，智能科学与技术学院，北京大学）；ByteDance Seed（字节跳动种子）；Institute for Artificial Intelligence, Peking University（人工智能研究院，北京大学）；Pazhou Laboratory (Huangpu), Guangzhou, Guangdong, China（琶洲实验室（黄埔），广州，广东，中国）
In order to solve large matrix completion problems with practical computational cost, an approximate approach based on matrix factorization has been widely used. Alternating least squares (ALS) and stochastic gradient descent (SGD) are two major algorithms to this end. In this study, we propose a new algorithm, namely cavity-based matrix factorization (CBMF) and approximate cavity-based matrix factorization (ACBMF), which are developed based on the cavity method from statistical mechanics. ALS yields solutions with less iterations when compared to those of SGD. This is because its update rules…
The matrix completion problem has been studied broadly under many underlying conditions. The problem has been explored under adaptive or non-adaptive, exact or estimation, single-phase or multi-phase, and many other categories. In most of these cases, the observation cost of each entry is uniform and has the same cost across the columns. However, in many real-life scenarios, we could expect elements from distinct columns or distinct positions to have a different cost. In this paper, we explore this generalization under adaptive conditions. We approach the problem under two different cost model…
ETH Zürich（苏黎世联邦理工学院）；MPI-IS（马克斯·普朗克智能系统研究所）；KU Leuven（鲁汶大学）
Lidar is a vital sensor for estimating the depth of a scene. Typical spinning lidars emit pulses arranged in several horizontal lines and the monetary cost of the sensor increases with the number of these lines. In this work, we present the new problem of optimizing the positioning of lidar lines to find the most effective configuration for the depth completion task. We propose a solution to reduce the number of lines while retaining the up-to-the-mark quality of depth completion. Our method consists of two components, (1) line selection based on the marginal contribution of a line computed vi…
Institut für Neuroinformatik, Ruhr-Universität Bochum（波鸿鲁尔大学神经信息学研究所）
Budgeted Stochastic Gradient Descent (BSGD) is a state-of-the-art technique for training large-scale kernelized support vector machines. The budget constraint is maintained incrementally by merging two points whenever the pre-defined budget is exceeded. The process of finding suitable merge partners is costly; it can account for up to 45% of the total training time. In this paper we investigate computationally more efficient schemes that merge more than two points at once. We obtain significant speed-ups without sacrificing accuracy.
CMLA, ENS Cachan（卡尚高等师范学校CMLA）；University of Minnesota（明尼苏达大学）
A general framework based on Gaussian models and a MAP-EM algorithm is introduced in this paper for solving matrix/table completion problems. The numerical experiments with the standard and challenging movie ratings data show that the proposed approach, based on probably one of the simplest probabilistic models, leads to the results in the same ballpark as the state-of-the-art, at a lower computational cost.
Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；RIKEN Center for AIP（理化学研究所人工智能项目中心）；University of Tokyo（东京大学）
Feature missing is a serious problem in many applications, which may lead to low quality of training data and further significantly degrade the learning performance. While feature acquisition usually involves special devices or complex process, it is expensive to acquire all feature values for the whole dataset. On the other hand, features may be correlated with each other, and some values may be recovered from the others. It is thus important to decide which features are most informative for recovering the other features as well as improving the learning performance. In this paper, we try to …
University of Liège（列日大学）；University of Cambridge（剑桥大学）
We exploit the versatile framework of Riemannian optimization on quotient manifolds to develop R3MC, a nonlinear conjugate-gradient method for low-rank matrix completion. The underlying search space of fixed-rank matrices is endowed with a novel Riemannian metric that is tailored to the least-squares cost. Numerical comparisons suggest that R3MC robustly outperforms state-of-the-art algorithms across different problem instances, especially those that combine scarcely sampled and ill-conditioned data.
Estimating scene geometry from data obtained with cost-effective sensors is key for robots and self-driving cars. In this paper, we study the problem of predicting dense depth from a single RGB image (monodepth) with optional sparse measurements from low-cost active depth sensors. We introduce Sparse Auxiliary Networks (SANs), a new module enabling monodepth networks to perform both the tasks of depth prediction and completion, depending on whether only RGB images or also sparse point clouds are available at inference time. First, we decouple the image and depth map encoding stages using spars…
This paper addresses the large-scale acquisition of end-to-end network performance. We made two distinct contributions: ordinal rating of network performance and inference by matrix completion. The former reduces measurement costs and unifies various metrics which eases their processing in applications. The latter enables scalable and accurate inference with no requirement of structural information of the network nor geometric constraints. By combining both, the acquisition problem bears strong similarities to recommender systems. This paper investigates the applicability of various matrix fac…
