The machine learning community has become increasingly concerned with the potential for bias and discrimination in predictive models, and this has motivated a growing line of work on what it means for a classification procedure to be "fair." In particular, we investigate the tension between minimizing error disparity across different population groups while maintaining calibrated probability estimates. We show that calibration is compatible only with a single error constraint (i.e. equal false-negatives rates across groups), and show that any algorithm that satisfies this relaxation is no better than randomizing a percentage of predictions for an existing classifier. These unsettling findings, which extend and generalize existing results, are empirically confirmed on several datasets.
Jul 24 2017 cs.CV
The DenseNet architecture is highly computationally efficient as a result of feature reuse. However, a naive DenseNet implementation can require a significant amount of GPU memory: If not properly managed, pre-activation batch normalization and contiguous convolution operations can produce feature maps that grow quadratically with network depth. In this technical report, we introduce strategies to reduce the memory consumption of DenseNets during training. By strategically using shared memory allocations, we reduce the memory cost for storing feature maps from quadratic to linear. Without the GPU memory bottleneck, it is now possible to train extremely deep DenseNets. Networks with 14M parameters can be trained on a single GPU, up from 4M. A 264-layer DenseNet (73M parameters), which previously would have been infeasible to train, can now be trained on a single workstation with 8 NVIDIA Tesla M40 GPUs. On the ImageNet ILSVRC classification dataset, this large DenseNet obtains a state-of-the-art single-crop top-1 error of 20.26%.
Jun 15 2017 cs.LG
Confidence calibration -- the problem of predicting probability estimates representative of the true correctness likelihood -- is important for classification models in many applications. We discover that modern neural networks, unlike those from a decade ago, are poorly calibrated. Through extensive experiments, we observe that depth, width, weight decay, and Batch Normalization are important factors influencing calibration. We evaluate the performance of various post-processing calibration methods on state-of-the-art architectures with image and document classification datasets. Our analysis and experiments not only offer insights into neural network learning, but also provide a simple and straightforward recipe for practical settings: on most datasets, temperature scaling -- a single-parameter variant of Platt Scaling -- is surprisingly effective at calibrating predictions.
Apr 04 2017 cs.LG
Ensembles of neural networks are known to be much more robust and accurate than individual networks. However, training multiple deep networks for model averaging is computationally expensive. In this paper, we propose a method to obtain the seemingly contradictory goal of ensembling multiple neural networks at no additional training cost. We achieve this goal by training a single neural network, converging to several local minima along its optimization path and saving the model parameters. To obtain repeated rapid convergence, we leverage recent work on cyclic learning rate schedules. The resulting technique, which we refer to as Snapshot Ensembling, is simple, yet surprisingly effective. We show in a series of experiments that our approach is compatible with diverse network architectures and learning tasks. It consistently yields lower error rates than state-of-the-art single models at no additional training cost, and compares favorably with traditional network ensembles. On CIFAR-10 and CIFAR-100 our DenseNet Snapshot Ensembles obtain error rates of 3.4% and 17.4% respectively.
Mar 30 2017 cs.LG
We introduce a new convolutional neural network architecture with the ability to adapt dynamically to computational resource limits at test time. Our network architecture uses progressively growing multi-scale convolutions and dense connectivity, which allows for the training of multiple classifiers at intermediate layers of the network. We evaluate our approach in two settings: (1) anytime classification, where the network's prediction for a test example is progressively updated, facilitating the output of a prediction at any time; and (2) budgeted batch classification, where a fixed amount of computation is available to classify a set of examples that can be spent unevenly across "easier" and "harder" inputs. Experiments on three image-classification datasets demonstrate that our proposed framework substantially improves the state-of-the-art in both settings.
Nov 18 2016 cs.CV
We propose Deep Feature Interpolation (DFI), a new data-driven baseline for automatic high-resolution image transformation. As the name suggests, it relies only on simple linear interpolation of deep convolutional features from pre-trained convnets. We show that despite its simplicity, DFI can perform high-level semantic transformations like "make older/younger", "make bespectacled", "add smile", among others, surprisingly well - sometimes even matching or outperforming the state-of-the-art. This is particularly unexpected as DFI requires no specialized network architecture or even any deep network to be trained for these tasks. DFI therefore can be used as a new baseline to evaluate more complex algorithms and provides a practical answer to the question of which image transformation tasks are still challenging in the rise of deep learning.
Recent work has shown that convolutional networks can be substantially deeper, more accurate, and efficient to train if they contain shorter connections between layers close to the input and those close to the output. In this paper, we embrace this observation and introduce the Dense Convolutional Network (DenseNet), which connects each layer to every other layer in a feed-forward fashion. Whereas traditional convolutional networks with L layers have L connections - one between each layer and its subsequent layer - our network has L(L+1)/2 direct connections. For each layer, the feature-maps of all preceding layers are used as inputs, and its own feature-maps are used as inputs into all subsequent layers. DenseNets have several compelling advantages: they alleviate the vanishing-gradient problem, strengthen feature propagation, encourage feature reuse, and substantially reduce the number of parameters. We evaluate our proposed architecture on four highly competitive object recognition benchmark tasks (CIFAR-10, CIFAR-100, SVHN, and ImageNet). DenseNets obtain significant improvements over the state-of-the-art on most of them, whilst requiring less memory and computation to achieve high performance. Code and models are available at https://github.com/liuzhuang13/DenseNet .
Jun 07 2016 cs.CL
In recent years deep neural networks have achieved great success in sentiment classification for English, thanks in part to the availability of copious annotated resources. Unfortunately, most other languages do not enjoy such an abundance of annotated data for sentiment analysis. To tackle this problem, we propose the Adversarial Deep Averaging Network (ADAN) to transfer sentiment knowledge learned from labeled English data to low-resource languages where only unlabeled data exists. ADAN is a "Y-shaped" network with two discriminative branches: a sentiment classifier and an adversarial language identification scorer. Both branches take input from a shared feature extractor that aims to learn hidden representations that capture the underlying sentiment of the text and are invariant across languages. Experiments on Chinese and Arabic sentiment classification demonstrate that ADAN significantly outperforms several baselines, including a strong pipeline approach that relies on state-of-the-art Machine Translation.
Very deep convolutional networks with hundreds of layers have led to significant reductions in error on competitive benchmarks. Although the unmatched expressiveness of the many layers can be highly desirable at test time, training very deep networks comes with its own set of challenges. The gradients can vanish, the forward flow often diminishes, and the training time can be painfully slow. To address these problems, we propose stochastic depth, a training procedure that enables the seemingly contradictory setup to train short networks and use deep networks at test time. We start with very deep networks but during training, for each mini-batch, randomly drop a subset of layers and bypass them with the identity function. This simple approach complements the recent success of residual networks. It reduces training time substantially and improves the test error significantly on almost all data sets that we used for evaluation. With stochastic depth we can increase the depth of residual networks even beyond 1200 layers and still yield meaningful improvements in test error (4.91% on CIFAR-10).
Many tasks in computer vision can be cast as a "label changing" problem, where the goal is to make a semantic change to the appearance of an image or some subject in an image in order to alter the class membership. Although successful task-specific methods have been developed for some label changing applications, to date no general purpose method exists. Motivated by this we propose deep manifold traversal, a method that addresses the problem in its most general form: it first approximates the manifold of natural images then morphs a test image along a traversal path away from a source class and towards a target class while staying near the manifold throughout. The resulting algorithm is surprisingly effective and versatile. It is completely data driven, requiring only an example set of images from the desired source and target domains. We demonstrate deep manifold traversal on highly diverse label changing tasks: changing an individual's appearance (age and hair color), changing the season of an outdoor image, and transforming a city skyline towards nighttime.
Convolutional neural networks (CNN) are increasingly used in many areas of computer vision. They are particularly attractive because of their ability to "absorb" great quantities of labeled data through millions of parameters. However, as model sizes increase, so do the storage and memory requirements of the classifiers. We present a novel network architecture, Frequency-Sensitive Hashed Nets (FreshNets), which exploits inherent redundancy in both convolutional layers and fully-connected layers of a deep learning model, leading to dramatic savings in memory and storage consumption. Based on the key observation that the weights of learned convolutional filters are typically smooth and low-frequency, we first convert filter weights to the frequency domain with a discrete cosine transform (DCT) and use a low-cost hash function to randomly group frequency parameters into hash buckets. All parameters assigned the same hash bucket share a single value learned with standard back-propagation. To further reduce model size we allocate fewer hash buckets to high-frequency components, which are generally less important. We evaluate FreshNets on eight data sets, and show that it leads to drastically better compressed performance than several relevant baselines.
As deep nets are increasingly used in applications suited for mobile devices, a fundamental dilemma becomes apparent: the trend in deep learning is to grow models to absorb ever-increasing data set sizes; however mobile devices are designed with very little memory and cannot store such large models. We present a novel network architecture, HashedNets, that exploits inherent redundancy in neural networks to achieve drastic reductions in model sizes. HashedNets uses a low-cost hash function to randomly group connection weights into hash buckets, and all connections within the same hash bucket share a single parameter value. These parameters are tuned to adjust to the HashedNets weight sharing architecture with standard backprop during training. Our hashing procedure introduces no additional memory overhead, and we demonstrate on several benchmark data sets that HashedNets shrink the storage requirements of neural networks substantially while mostly preserving generalization performance.
Jan 27 2015 cs.LG
Support vector machines (SVM) can classify data sets along highly non-linear decision boundaries because of the kernel-trick. This expressiveness comes at a price: During test-time, the SVM classifier needs to compute the kernel inner-product between a test sample and all support vectors. With large training data sets, the time required for this computation can be substantial. In this paper, we introduce a post-processing algorithm, which compresses the learned SVM model by reducing and optimizing support vectors. We evaluate our algorithm on several medium-scaled real-world data sets, demonstrating that it maintains high test accuracy while reducing the test-time evaluation cost by several orders of magnitude---in some cases from hours to seconds. It is fair to say that most of the work in this paper was previously been invented by Burges and Schölkopf almost 20 years ago. For most of the time during which we conducted this research, we were unaware of this prior work. However, in the past two decades, computing power has increased drastically, and we can therefore provide empirical insights that were not possible in their original paper.
Covariance and histogram image descriptors provide an effective way to capture information about images. Both excel when used in combination with special purpose distance metrics. For covariance descriptors these metrics measure the distance along the non-Euclidean Riemannian manifold of symmetric positive definite matrices. For histogram descriptors the Earth Mover's distance measures the optimal transport between two histograms. Although more precise, these distance metrics are very expensive to compute, making them impractical in many applications, even for data sets of only a few thousand examples. In this paper we present two methods to compress the size of covariance and histogram datasets with only marginal increases in test error for k-nearest neighbor classification. Specifically, we show that we can reduce data sets to 16% and in some cases as little as 2% of their original size, while approximately matching the test error of kNN classification on the full training set. In fact, because the compressed set is learned in a supervised fashion, it sometimes even outperforms the full data set, while requiring only a fraction of the space and drastically reducing test-time computation.
The past years have witnessed many dedicated open-source projects that built and maintain implementations of Support Vector Machines (SVM), parallelized for GPU, multi-core CPUs and distributed systems. Up to this point, no comparable effort has been made to parallelize the Elastic Net, despite its popularity in many high impact applications, including genetics, neuroscience and systems biology. The first contribution in this paper is of theoretical nature. We establish a tight link between two seemingly different algorithms and prove that Elastic Net regression can be reduced to SVM with squared hinge loss classification. Our second contribution is to derive a practical algorithm based on this reduction. The reduction enables us to utilize prior efforts in speeding up and parallelizing SVMs to obtain a highly optimized and parallel solver for the Elastic Net and Lasso. With a simple wrapper, consisting of only 11 lines of MATLAB code, we obtain an Elastic Net implementation that naturally utilizes GPU and multi-core CPUs. We demonstrate on twelve real world data sets, that our algorithm yields identical results as the popular (and highly optimized) glmnet implementation but is one or several orders of magnitude faster.
Apr 04 2014 cs.LG
In this paper, we evaluate the performance of various parallel optimization methods for Kernel Support Vector Machines on multicore CPUs and GPUs. In particular, we provide the first comparison of algorithms with explicit and implicit parallelization. Most existing parallel implementations for multi-core or GPU architectures are based on explicit parallelization of Sequential Minimal Optimization (SMO)---the programmers identified parallelizable components and hand-parallelized them, specifically tuned for a particular architecture. We compare these approaches with each other and with implicitly parallelized algorithms---where the algorithm is expressed such that most of the work is done within few iterations with large dense linear algebra operations. These can be computed with highly-optimized libraries, that are carefully parallelized for a large variety of parallel platforms. We highlight the advantages and disadvantages of both approaches and compare them on various benchmark data sets. We find an approximate implicitly parallel algorithm which is surprisingly efficient, permits a much simpler implementation, and leads to unprecedented speedups in SVM training.
Feb 28 2014 cs.LG
The goal of machine learning is to develop predictors that generalize well to test data. Ideally, this is achieved by training on an almost infinitely large training data set that captures all variations in the data distribution. In practical learning settings, however, we do not have infinite data and our predictors may overfit. Overfitting may be combatted, for example, by adding a regularizer to the training objective or by defining a prior over the model parameters and performing Bayesian inference. In this paper, we propose a third, alternative approach to combat overfitting: we extend the training set with infinitely many artificial training examples that are obtained by corrupting the original training data. We show that this approach is practical and efficient for a range of predictors and corruption models. Our approach, called marginalized corrupted features (MCF), trains robust predictors by minimizing the expected value of the loss function under the corruption model. We show empirically on a variety of data sets that MCF classifiers can be trained efficiently, may generalize substantially better to test data, and are also more robust to feature deletion at test time.
In text mining, information retrieval, and machine learning, text documents are commonly represented through variants of sparse Bag of Words (sBoW) vectors (e.g. TF-IDF). Although simple and intuitive, sBoW style representations suffer from their inherent over-sparsity and fail to capture word-level synonymy and polysemy. Especially when labeled data is limited (e.g. in document classification), or the text documents are short (e.g. emails or abstracts), many features are rarely observed within the training corpus. This leads to overfitting and reduced generalization accuracy. In this paper we propose Dense Cohort of Terms (dCoT), an unsupervised algorithm to learn improved sBoW document features. dCoT explicitly models absent words by removing and reconstructing random sub-sets of words in the unlabeled corpus. With this approach, dCoT learns to reconstruct frequent words from co-occurring infrequent words and maps the high dimensional sparse sBoW vectors into a low-dimensional dense representation. We show that the feature removal can be marginalized out and that the reconstruction can be solved for in closed-form. We demonstrate empirically, on several benchmark datasets, that dCoT features significantly improve the classification accuracy across several document classification tasks.
Recently, machine learning algorithms have successfully entered large-scale real-world industrial applications (e.g. search engines and email spam filters). Here, the CPU cost during test time must be budgeted and accounted for. In this paper, we address the challenge of balancing the test-time cost and the classifier accuracy in a principled fashion. The test-time cost of a classifier is often dominated by the computation required for feature extraction-which can vary drastically across eatures. We decrease this extraction time by constructing a tree of classifiers, through which test inputs traverse along individual paths. Each path extracts different features and is optimized for a specific sub-partition of the input space. By only computing features for inputs that benefit from them the most, our cost sensitive tree of classifiers can match the high accuracies of the current state-of-the-art at a small fraction of the computational cost.
Recent work in metric learning has significantly improved the state-of-the-art in k-nearest neighbor classification. Support vector machines (SVM), particularly with RBF kernels, are amongst the most popular classification algorithms that uses distance metrics to compare examples. This paper provides an empirical analysis of the efficacy of three of the most popular Mahalanobis metric learning algorithms as pre-processing for SVM training. We show that none of these algorithms generate metrics that lead to particularly satisfying improvements for SVM-RBF classification. As a remedy we introduce support vector metric learning (SVML), a novel algorithm that seamlessly combines the learning of a Mahalanobis metric with the training of the RBF-SVM parameters. We demonstrate the capabilities of SVML on nine benchmark data sets of varying sizes and difficulties. In our study, SVML outperforms all alternative state-of-the-art metric learning algorithms in terms of accuracy and establishes itself as a serious alternative to the standard Euclidean metric with model selection by cross validation.
As machine learning algorithms enter applications in industrial settings, there is increased interest in controlling their cpu-time during testing. The cpu-time consists of the running time of the algorithm and the extraction time of the features. The latter can vary drastically when the feature set is diverse. In this paper, we propose an algorithm, the Greedy Miser, that incorporates the feature extraction cost during training to explicitly minimize the cpu-time during testing. The algorithm is a straightforward extension of stage-wise regression and is equally suitable for regression or multi-class classification. Compared to prior work, it is significantly more cost-effective and scales to larger data sets.
Jun 19 2012 cs.LG
Stacked denoising autoencoders (SDAs) have been successfully used to learn new representations for domain adaptation. Recently, they have attained record accuracy on standard benchmark tasks of sentiment analysis across different text domains. SDAs learn robust data representations by reconstruction, recovering original features from data that are artificially corrupted with noise. In this paper, we propose marginalized SDA (mSDA) that addresses two crucial limitations of SDAs: high computational cost and lack of scalability to high-dimensional features. In contrast to SDAs, our approach of mSDA marginalizes noise and thus does not require stochastic gradient descent or other optimization algorithms to learn parameters ? in fact, they are computed in closed-form. Consequently, mSDA, which can be implemented in only 20 lines of MATLAB^TM, significantly speeds up SDAs by two orders of magnitude. Furthermore, the representations learnt by mSDA are as effective as the traditional SDAs, attaining almost identical accuracies in benchmark tasks.
We investigate unsupervised pre-training of deep architectures as feature generators for "shallow" classifiers. Stacked Denoising Autoencoders (SdA), when used as feature pre-processing tools for SVM classification, can lead to significant improvements in accuracy - however, at the price of a substantial increase in computational cost. In this paper we create a simple algorithm which mimics the layer by layer training of SdAs. However, in contrast to SdAs, our algorithm requires no training through gradient descent as the parameters can be computed in closed-form. It can be implemented in less than 20 lines of MATLABTMand reduces the computation time from several hours to mere seconds. We show that our feature transformation reliably improves the results of SVM classification significantly on all our data sets - often outperforming SdAs and even deep neural networks in three out of four deep learning benchmarks.
Feb 13 2009 cs.AI
Empirical evidence suggests that hashing is an effective strategy for dimensionality reduction and practical nonparametric estimation. In this paper we provide exponential tail bounds for feature hashing and show that the interaction between random subspaces is negligible with high probability. We demonstrate the feasibility of this approach with experimental results for a new use case -- multitask learning with hundreds of thousands of tasks.