We consider the learning of algorithmic tasks by mere observation of input-output pairs. Rather than studying this as a black-box discrete regression problem with no assumption whatsoever on the input-output mapping, we concentrate on tasks that are amenable to the principle of divide and conquer, and study what are its implications in terms of learning. This principle creates a powerful inductive bias that we leverage with neural architectures that are defined recursively and dynamically, by learning two scale-invariant atomic operations: how to split a given input into smaller sets, and how to merge two partially solved tasks into a larger partial solution. Our model can be trained in weakly supervised environments, namely by just observing input-output pairs, and in even weaker environments, using a non-differentiable reward signal. Moreover, thanks to the dynamic aspect of our architecture, we can incorporate the computational complexity as a regularization term that can be optimized by backpropagation. We demonstrate the flexibility and efficiency of the Divide-and-Conquer Network on several combinatorial and geometric tasks: convex hull, clustering, knapsack and euclidean TSP. Thanks to the dynamic programming nature of our model, we show significant improvements in terms of generalization error and computational complexity.
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；Inception Institute of Artificial Intelligence（起源人工智能研究院）
Theoretical analysis of the divide-and-conquer based distributed learning with least square loss in the reproducing kernel Hilbert space (RKHS) have recently been explored within the framework of learning theory. However, the studies on learning theory for general loss functions and hypothesis spaces remain limited. To fill the gap, we study the risk performance of distributed empirical risk minimization (ERM) for general loss functions and hypothesis spaces. The main contributions are two-fold. First, we derive two tight risk bounds under certain basic assumptions on the hypothesis space, as …
We reduce a broad class of machine learning problems, usually addressed by EM or sampling, to the problem of finding the $k$ extremal rays spanning the conical hull of a data point set. These $k$ "anchors" lead to a global solution and a more interpretable model that can even outperform EM and sampling on generalization error. To find the $k$ anchors, we propose a novel divide-and-conquer learning scheme "DCA" that distributes the problem to $\mathcal O(k\log k)$ same-type sub-problems on different low-D random hyperplanes, each can be solved by any solver. For the 2D sub-problem, we present a…
Most applications demand high-performance deep neural architectures costing limited resources. Neural architecture searching is a way of automatically exploring optimal deep neural networks in a given huge search space. However, all sub-networks are usually evaluated using the same criterion; that is, early stopping on a small proportion of the training dataset, which is an inaccurate and highly complex approach. In contrast to conventional methods, here we present a divide-and-conquer (DC) approach to effectively and efficiently search deep neural architectures. Given an arbitrary search spac…
Shanghai Jiao Tong University（上海交通大学）；Zhejiang University（浙江大学）；University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Cornell University（康奈尔大学）
Divide-and-conquer is a general strategy to deal with large scale problems. It is typically applied to generate ensemble instances, which potentially limits the problem size it can handle. Additionally, the data are often divided by random sampling which may be suboptimal. To address these concerns, we propose the $DC^2$ algorithm. Instead of ensemble instances, we produce structure-preserving signature pieces to be assembled and conquered. $DC^2$ achieves the efficiency of sampling-based large scale kernel methods while enabling parallel multicore or clustered computation. The data partition …
Novartis Pharmaceuticals Corp.（诺华制药公司）；New York Institute of Technology（纽约理工学院）；Rutgers University（罗格斯大学）
In the context of big data analysis, the divide-and-conquer methodology refers to a multiple-step process: first splitting a data set into several smaller ones; then analyzing each set separately; finally combining results from each analysis together. This approach is effective in handling large data sets that are unsuitable to be analyzed entirely by a single computer due to limits either from memory storage or computational time. The combined results will provide a statistical inference which is similar to the one from analyzing the entire data set. This article reviews some recently develop…
Tsinghua University（清华大学）；The University of Arizona（亚利桑那大学）
The divide-and-conquer method has been widely used for estimating large-scale kernel ridge regression estimates. Unfortunately, when the response variable is highly skewed, the divide-and-conquer kernel ridge regression (dacKRR) may overlook the underrepresented region and result in unacceptable results. We combine a novel response-adaptive partition strategy with the oversampling technique synergistically to overcome the limitation. Through the proposed novel algorithm, we allocate some carefully identified informative observations to multiple nodes (local processors). Although the oversampli…
Active learning (AL) repeatedly trains the classifier with the minimum labeling budget to improve the current classification model. The training process is usually supervised by an uncertainty evaluation strategy. However, the uncertainty evaluation always suffers from performance degeneration when the initial labeled set has insufficient labels. To completely eliminate the dependence on the uncertainty evaluation sampling in AL, this paper proposes a divide-and-conquer idea that directly transfers the AL sampling as the geometric sampling over the clusters. By dividing the points of the clust…
We introduce an inductive logic programming approach that combines classical divide-and-conquer search with modern constraint-driven search. Our anytime approach can learn optimal, recursive, and large programs and supports predicate invention. Our experiments on three domains (classification, inductive general game playing, and program synthesis) show that our approach can increase predictive accuracies and reduce learning times.
The deep layers of modern neural networks extract a rather rich set of features as an input propagates through the network. This paper sets out to harvest these rich intermediate representations for quantization with minimal accuracy loss while significantly reducing the memory footprint and compute intensity of the DNN. This paper utilizes knowledge distillation through teacher-student paradigm (Hinton et al., 2015) in a novel setting that exploits the feature extraction capability of DNNs for higher-accuracy quantization. As such, our algorithm logically divides a pretrained full-precision D…
Spectral clustering is one of the most popular clustering methods. However, how to balance the efficiency and effectiveness of the large-scale spectral clustering with limited computing resources has not been properly solved for a long time. In this paper, we propose a divide-and-conquer based large-scale spectral clustering method to strike a good balance between efficiency and effectiveness. In the proposed method, a divide-and-conquer based landmark selection algorithm and a novel approximate similarity matrix approach are designed to construct a sparse similarity matrix within low computat…
Universal probabilistic programming systems (PPSs) provide a powerful framework for specifying rich probabilistic models. They further attempt to automate the process of drawing inferences from these models, but doing this successfully is severely hampered by the wide range of non--standard models they can express. As a result, although one can specify complex models in a universal PPS, the provided inference engines often fall far short of what is required. In particular, we show that they produce surprisingly unsatisfactory performance for models where the support varies between executions, …
In this paper, a new method is proposed for sparse PCA based on the recursive divide-and-conquer methodology. The main idea is to separate the original sparse PCA problem into a series of much simpler sub-problems, each having a closed-form solution. By recursively solving these sub-problems in an analytical way, an efficient algorithm is constructed to solve the sparse PCA problem. The algorithm only involves simple computations and is thus easy to implement. The proposed method can also be very easily extended to other sparse PCA problems with certain constraints, such as the nonnegative spa…
The divide and conquer strategy, which breaks a massive data set into a se- ries of manageable data blocks, and then combines the independent results of data blocks to obtain a final decision, has been recognized as a state-of-the-art method to overcome challenges of massive data analysis. In this paper, we merge the divide and conquer strategy with local average regression methods to infer the regressive relationship of input-output pairs from a massive data set. After theoretically analyzing the pros and cons, we find that although the divide and conquer local average regression can reach th…
Federated Learning allows training of data stored in distributed devices without the need for centralizing training data, thereby maintaining data privacy. Addressing the ability to handle data heterogeneity (non-identical and independent distribution or non-IID) is a key enabler for the wider deployment of Federated Learning. In this paper, we propose a novel Divide-and-Conquer training methodology that enables the use of the popular FedAvg aggregation algorithm by overcoming the acknowledged FedAvg limitations in non-IID environments. We propose a novel use of Cosine-distance based Weight Di…
This paper introduces a divide-and-conquer inspired adversarial learning (DACAL) approach for photo enhancement. The key idea is to decompose the photo enhancement process into hierarchically multiple sub-problems, which can be better conquered from bottom to up. On the top level, we propose a perception-based division to learn additive and multiplicative components, required to translate a low-quality image or video into its high-quality counterpart. On the intermediate level, we use a frequency-based division with generative adversarial network (GAN) to weakly supervise the photo enhancement…
Tsinghua University（清华大学）；Princeton University（普林斯顿大学）；Harbin Institute of Technology（哈尔滨工业大学）
This study presents a divide-and-conquer (DC) approach based on feature space decomposition for classification. When large-scale datasets are present, typical approaches usually employed truncated kernel methods on the feature space or DC approaches on the sample space. However, this did not guarantee separability between classes, owing to overfitting. To overcome such problems, this work proposes a novel DC approach on feature spaces consisting of three steps. Firstly, we divide the feature space into several subspaces using the decomposition method proposed in this paper. Subsequently, these…
The kernel support vector machine (SVM) is one of the most widely used classification methods; however, the amount of computation required becomes the bottleneck when facing millions of samples. In this paper, we propose and analyze a novel divide-and-conquer solver for kernel SVMs (DC-SVM). In the division step, we partition the kernel SVM problem into smaller subproblems by clustering the data, so that each subproblem can be solved independently and efficiently. We show theoretically that the support vectors identified by the subproblem solution are likely to be support vectors of the entire…
Many popular machine learning models scale poorly when deployed on CPUs. In this paper we explore the reasons why and propose a simple, yet effective approach based on the well-known Divide-and-Conquer Principle to tackle this problem of great practical importance. Given an inference job, instead of using all available computing resources (i.e., CPU cores) for running it, the idea is to break the job into independent parts that can be executed in parallel, each with the number of cores according to its expected computational cost. We implement this idea in the popular OnnxRuntime framework and…
University of California Berkeley（加州大学伯克利分校）；University of Washington Seattle（华盛顿大学西雅图分校）
Standard model-free deep reinforcement learning (RL) algorithms sample a new initial state for each trial, allowing them to optimize policies that can perform well even in highly stochastic environments. However, problems that exhibit considerable initial state variation typically produce high-variance gradient estimates for model-free RL, making direct policy or value function optimization challenging. In this paper, we develop a novel algorithm that instead partitions the initial state space into "slices", and optimizes an ensemble of policies, each on a different slice. The ensemble is grad…
University of Melbourne（墨尔本大学）；Delft University of Technology（代尔夫特理工大学）；RMIT University（皇家墨尔本理工大学）；Monash University（莫纳什大学）
The predict+optimize problem combines machine learning ofproblem coefficients with a combinatorial optimization prob-lem that uses the predicted coefficients. While this problemcan be solved in two separate stages, it is better to directlyminimize the optimization loss. However, this requires dif-ferentiating through a discrete, non-differentiable combina-torial function. Most existing approaches use some form ofsurrogate gradient. Demirovicet alshowed how to directlyexpress the loss of the optimization problem in terms of thepredicted coefficients as a piece-wise linear function. How-ever, th…
University of Virginia（弗吉尼亚大学）；Walmart Labs（沃尔玛实验室）
Online Learning to Rank (OL2R) eliminates the need of explicit relevance annotation by directly optimizing the rankers from their interactions with users. However, the required exploration drives it away from successful practices in offline learning to rank, which limits OL2R's empirical performance and practical applicability. In this work, we propose to estimate a pairwise learning to rank model online. In each round, candidate documents are partitioned and ranked according to the model's confidence on the estimated pairwise rank order, and exploration is only performed on the uncertain pair…
Hong Kong University of Science and Technology（香港科技大学）；Huazhong University of Science and Technology（华中科技大学）
We propose Deep Hierarchical Machine (DHM), a model inspired from the divide-and-conquer strategy while emphasizing representation learning ability and flexibility. A stochastic routing framework as used by recent deep neural decision/regression forests is incorporated, but we remove the need to evaluate unnecessary computation paths by utilizing a different topology and introducing a probabilistic pruning technique. We also show a specified version of DHM (DSHM) for efficiency, which inherits the sparse feature extraction process as in traditional decision tree with pixel-difference feature. …
Centro de Informática, Universidade Federal de Pernambuco（伯南布哥联邦大学信息中心）；Sungkyunkwan University Advanced Institute of Nanotechnology（成均馆大学纳米技术高级研究所）；Quantum Research Group, School of Chemistry and Physics, University of KwaZulu-Natal（夸祖鲁-纳塔尔大学化学与物理学院量子研究组）；National Institute for Theoretical Physics (NITheP)（国家理论物理研究所）
Advantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divi…
We address talker-independent monaural speaker separation from the perspectives of deep learning and computational auditory scene analysis (CASA). Specifically, we decompose the multi-speaker separation task into the stages of simultaneous grouping and sequential grouping. Simultaneous grouping is first performed in each time frame by separating the spectra of different speakers with a permutation-invariantly trained neural network. In the second stage, the frame-level separated spectra are sequentially grouped to different speakers by a clustering network. The proposed deep CASA approach opti…
Designing a good reward function is essential to robot planning and reinforcement learning, but it can also be challenging and frustrating. The reward needs to work across multiple different environments, and that often requires many iterations of tuning. We introduce a novel divide-and-conquer approach that enables the designer to specify a reward separately for each environment. By treating these separate reward functions as observations about the underlying true reward, we derive an approach to infer a common reward across all environments. We conduct user studies in an abstract grid world …
FAU Erlangen-Nuremberg（埃尔朗根-纽伦堡大学）；University of Leeds（利兹大学）
Comparison of microvascular circulation on fundoscopic images is a non-invasive clinical indication for the diagnosis and monitoring of diseases, such as diabetes and hypertensions. The differences between intra-patient images can be assessed quantitatively by registering serial acquisitions. Due to the variability of the images (i.e. contrast, luminosity) and the anatomical changes of the retina, the registration of fundus images remains a challenging task. Recently, several deep learning approaches have been proposed to register fundus images in an end-to-end fashion, achieving remarkable re…
