We derive an unbiased estimator for expectations over discrete random variables based on sampling without replacement, which reduces variance as it avoids duplicate samples. We show that our estimator can be derived as the Rao-Blackwellization of three different estimators. Combining our estimator with REINFORCE, we obtain a policy gradient estimator and we reduce its variance using a built-in control variate which is obtained without additional model evaluations. The resulting estimator is closely related to other gradient estimators. Experiments with a toy problem, a categorical Variational Auto-Encoder and a structured prediction problem show that our estimator is the only estimator that is consistently among the best estimators in both high and low entropy settings.
Program synthesis has emerged as a successful approach to the image parsing task. Most prior works rely on a two-step scheme involving supervised pretraining of a Seq2Seq model with synthetic programs followed by reinforcement learning (RL) for fine-tuning with real reference images. Fully unsupervised approaches promise to train the model directly on the target images without requiring curated pretraining datasets. However, they struggle with the inherent sparsity of meaningful programs in the search space. In this paper, we present the first unsupervised algorithm capable of parsing construc…
University of Amsterdam（阿姆斯特丹大学）；ORTEC；CIFAR（加拿大高等研究院）
The well-known Gumbel-Max trick for sampling from a categorical distribution can be extended to sample $k$ elements without replacement. We show how to implicitly apply this 'Gumbel-Top-$k$' trick on a factorized distribution over sequences, allowing to draw exact samples without replacement using a Stochastic Beam Search. Even for exponentially large domains, the number of model evaluations grows only linear in $k$ and the maximum sampled sequence length. The algorithm creates a theoretical connection between sampling and (deterministic) beam search and can be used as a principled intermediat…
Sampling is a fundamental technique, and sampling without replacement is often desirable when duplicate samples are not beneficial. Within machine learning, sampling is useful for generating diverse outputs from a trained model. We present an elegant procedure for sampling without replacement from a broad class of randomized programs, including generative neural models that construct outputs sequentially. Our procedure is efficient even for exponentially-large output spaces. Unlike prior work, our approach is incremental, i.e., samples can be drawn one at a time, allowing for increased flexibi…
Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled \emph{with} replacement. In practice, however, sampling \emph{without} replacement is very common, easier to implement in many cases, and often performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling, under various scenarios, for three types of algorithms: Any algorithm with online regret guarantees, stochastic gradient descent, and SVRG. A useful application of our SVRG analysis is a nearly-optimal algorithm for…
Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）；IT University of Copenhagen（哥本哈根信息技术大学）；BARC（哥本哈根基础算法研究中心）；CMU（卡内基梅隆大学）
Weighted sampling is a fundamental tool in data analysis and machine learning pipelines. Samples are used for efficient estimation of statistics or as sparse representations of the data. When weight distributions are skewed, as is often the case in practice, without-replacement (WOR) sampling is much more effective than with-replacement (WR) sampling: it provides a broader representation and higher accuracy for the same number of samples. We design novel composable sketches for WOR $\ell_p$ sampling, weighted sampling of keys according to a power $p\in[0,2]$ of their frequency (or for signed d…
Innopeak Technology（Innopeak科技）；Huawei Research America（华为美国研究院）；Jilin University（吉林大学）
We propose sequenced-replacement sampling (SRS) for training deep neural networks. The basic idea is to assign a fixed sequence index to each sample in the dataset. Once a mini-batch is randomly drawn in each training iteration, we refill the original dataset by successively adding samples according to their sequence index. Thus we carry out replacement sampling but in a batched and sequenced way. In a sense, SRS could be viewed as a way of performing "mini-batch augmentation". It is particularly useful for a task where we have a relatively small images-per-class such as CIFAR-100. Together wi…
Regularization is an important component of predictive model building. The hybrid bootstrap is a regularization technique that functions similarly to dropout except that features are resampled from other training points rather than replaced with zeros. We show that the hybrid bootstrap offers superior performance to dropout. We also present a sampling based technique to simplify hyperparameter choice. Next, we provide an alternative sampling technique for convolutional neural networks. Finally, we demonstrate the efficacy of the hybrid bootstrap on non-image tasks using tree-based models.
IRIT（图卢兹计算机科学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
Minibatch decomposition methods for empirical risk minimization are commonly analysed in a stochastic approximation setting, also known as sampling with replacement. On the other hands modern implementations of such techniques are incremental: they rely on sampling without replacement, for which available analysis are much scarcer. We provide convergence guaranties for the latter variant by analysing a versatile incremental gradient scheme. For this scheme, we consider constant, decreasing or adaptive step sizes. In the smooth setting we obtain explicit complexity estimates in terms of epoch c…
Indian Institute of Technology Kanpur（印度理工学院坎普尔分校）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Google Research（谷歌研究院）
We analyze the convergence rates of stochastic gradient algorithms for smooth finite-sum minimax optimization and show that, for many such algorithms, sampling the data points without replacement leads to faster convergence compared to sampling with replacement. For the smooth and strongly convex-strongly concave setting, we consider gradient descent ascent and the proximal point method, and present a unified analysis of two popular without-replacement sampling strategies, namely Random Reshuffling (RR), which shuffles the data every epoch, and Single Shuffling or Shuffle Once (SO), which shuf…
University of Pennsylvania（宾夕法尼亚大学）；DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；Tsinghua University（清华大学）
When applying a stochastic algorithm, one must choose an order to draw samples. The practical choices are without-replacement sampling orders, which are empirically faster and more cache-friendly than uniform-iid-sampling but often have inferior theoretical guarantees. Without-replacement sampling is well understood only for SGD without variance reduction. In this paper, we will improve the convergence analysis and rates of variance reduction under without-replacement sampling orders for composite finite-sum minimization. Our results are in two-folds. First, we develop a damped variant of Fini…
Crowdsourcing platforms are now extensively used for conducting subjective pairwise comparison studies. In this setting, a pairwise comparison dataset is typically gathered via random sampling, either \emph{with} or \emph{without} replacement. In this paper, we use tools from random graph theory to analyze these two random sampling methods for the HodgeRank estimator. Using the Fiedler value of the graph as a measurement for estimator stability (informativeness), we provide a new estimate of the Fiedler value for these two random graph models. In the asymptotic limit as the number of vertices …
Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and Ré reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where $n$ positive numbers are replaced by $n$ positive definite matrices. If this inequality holds for all $n$, then without-replacement sampling indeed outperforms with…
Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to large sampling variance, the number of non-zeros is uneven among rows of the projection matrix which is generated in one trial, and more data information may be lost after dimension reduction. To break this bottleneck, based on random sampling without replacement in statistics, this…
We consider the problem of allocating samples to a finite set of discrete distributions in order to learn them uniformly well in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general optimistic tracking algorithm and analyze its sample allocation performance w.r.t.~an oracle. We then instantiate this algorithm for the four distance measures and derive bounds on the regret of their resulting allocation schemes. We verify our theoretical findings through some experiments…
We present a theoretical study of server-side optimization in federated learning. Our results are the first to show that the widely popular heuristic of scaling the client updates with an extra parameter is very useful in the context of Federated Averaging (FedAvg) with local passes over the client data. Each local pass is performed without replacement using Random Reshuffling, which is a key reason we can show improved complexities. In particular, we prove that whenever the local stepsizes are small, and the update direction is given by FedAvg in conjunction with Random Reshuffling over all c…
Understanding the basins of attraction (BoA) is often a paramount consideration for nonlinear systems. Most existing approaches to determining a high-resolution BoA require prior knowledge of the system's dynamical model (e.g., differential equation or point mapping for continuous systems, cell mapping for discrete systems, etc.), which allows derivation of approximate analytical solutions or parallel computing on a multi-core computer to find the BoA efficiently. However, these methods are typically impractical when the BoA must be determined experimentally or when the system's model is unkno…
For solving finite-sum optimization problems, SGD without replacement sampling is empirically shown to outperform SGD. Denoting by $n$ the number of components in the cost and $K$ the number of epochs of the algorithm , several recent works have shown convergence rates of without-replacement SGD that have better dependency on $n$ and $K$ than the baseline rate of $O(1/(nK))$ for SGD. However, there are two main limitations shared among those works: the rates have extra poly-logarithmic factors on $nK$, and denoting by $κ$ the condition number of the problem, the rates hold after $κ^c\log(nK)$ …
Stochastic gradient descent without replacement sampling is widely used in practice for model training. However, the vast majority of SGD analyses assumes data is sampled with replacement, and when the function minimized is strongly convex, an $\mathcal{O}\left(\frac{1}{T}\right)$ rate can be established when SGD is run for $T$ iterations. A recent line of breakthrough works on SGD without replacement (SGDo) established an $\mathcal{O}\left(\frac{n}{T^2}\right)$ convergence rate when the function minimized is strongly convex and is a sum of $n$ smooth functions, and an $\mathcal{O}\left(\frac{…
University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
The problem of adaptive sampling for estimating probability mass functions (pmf) uniformly well is considered. Performance of the sampling strategy is measured in terms of the worst-case mean squared error. A Bayesian variant of the existing upper confidence bound (UCB) based approaches is proposed. It is shown analytically that the performance of this Bayesian variant is no worse than the existing approaches. The posterior distribution on the pmfs in the Bayesian setting allows for a tighter computation of upper confidence bounds which leads to significant performance gains in practice. Using…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Mila, Université de Montréal（蒙特利尔大学米拉研究所）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）；Princeton University（普林斯顿大学）
Gradient compression is a popular technique for improving communication complexity of stochastic first-order methods in distributed training of machine learning models. However, the existing works consider only with-replacement sampling of stochastic gradients. In contrast, it is well-known in practice and recently confirmed in theory that stochastic methods based on without-replacement sampling, e.g., Random Reshuffling (RR) method, perform better than ones that sample the gradients with-replacement. In this work, we close this gap in the literature and provide the first analysis of methods w…
Princeton University（普林斯顿大学）；Weizmann Institute of Science（魏茨曼科学研究所）
Recently, there has been much interest in studying the convergence rates of without-replacement SGD, and proving that it is faster than with-replacement SGD in the worst case. However, known lower bounds ignore the problem's geometry, including its condition number, whereas the upper bounds explicitly depend on it. Perhaps surprisingly, we prove that when the condition number is taken into account, without-replacement SGD \emph{does not} significantly improve on with-replacement SGD in terms of worst-case bounds, unless the number of epochs (passes over the data) is larger than the condition n…
Microsoft Research（微软研究院）；Massachusetts Institute of Technology（麻省理工学院）
We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement} \cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strong…
Support Vector Data Description (SVDD) is a popular outlier detection technique which constructs a flexible description of the input data. SVDD computation time is high for large training datasets which limits its use in big-data process-monitoring applications. We propose a new iterative sampling-based method for SVDD training. The method incrementally learns the training data description at each iteration by computing SVDD on an independent random sample selected with replacement from the training data set. The experimental results indicate that the proposed method is extremely fast and prov…
Rutgers University – New Brunswick（罗格斯大学新不伦瑞克分校）；Yale University（耶鲁大学）
Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman \cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated by Frank \cite{Frank1978} of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the \emph{sublinear} regime where only a vanishing fraction of the vertices are sampled. We show…
Stanford University（斯坦福大学）；Carnegie Mellon University（卡内基梅隆大学）；Google（谷歌）；Argo AI
For machine learning models trained with limited labeled training data, validation stands to become the main bottleneck to reducing overall annotation costs. We propose a statistical validation algorithm that accurately estimates the F-score of binary classifiers for rare categories, where finding relevant examples to evaluate on is particularly challenging. Our key insight is that simultaneous calibration and importance sampling enables accurate estimates even in the low-sample regime (< 300 samples). Critically, we also derive an accurate single-trial estimator of the variance of our method …
Quantum Algorithms for Sampling Log-Concave Distributions and Estimating Normalizing Constants
Joint Center for Quantum Information and Computer Science（量子信息与计算机科学联合中心）；University of Maryland（马里兰大学）；Center on Frontiers of Computing Studies（计算前沿研究中心）；Peking University（北京大学）；Simons Institute（西蒙斯研究所）；UC Berkeley（加州大学伯克利分校）；Pennsylvania State University（宾夕法尼亚州立大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
