Purushottam Kar

Kunal Dahiya, Nilesh Gupta, Deepak Saini et al.

Extreme Classification (XC) seeks to tag data points with the most relevant subset of labels from an extremely large label set. Performing deep XC with dense, learnt representations for data points and labels has attracted much attention due to its superiority over earlier XC methods that used sparse, hand-crafted features. Negative mining techniques have emerged as a critical component of all deep XC methods that allow them to scale to millions of labels. However, despite recent advances, training deep XC models with large encoder architectures such as transformers remains challenging. This paper identifies that memory overheads of popular negative mining techniques often force mini-batch sizes to remain small and slow training down. In response, this paper introduces NGAME, a light-weight mini-batch creation technique that offers provably accurate in-batch negative samples. This allows training with larger mini-batches offering significantly faster convergence and higher accuracies than existing negative sampling techniques. NGAME was found to be up to 16% more accurate than state-of-the-art methods on a wide array of benchmark datasets for extreme classification, as well as 3% more accurate at retrieving search engine queries in response to a user webpage visit to show personalized ads. In live A/B tests on a popular search engine, NGAME yielded up to 23% gains in click-through-rates.

Bhaskar P Mukhoty, Debojyoti Dey, Purushottam Kar

This paper presents SVAM (Sequential Variance-Altered MLE), a unified framework for learning generalized linear models under adversarial label corruption in training data. SVAM extends to tasks such as least squares regression, logistic regression, and gamma regression, whereas many existing works on learning with label corruptions focus only on least squares regression. SVAM is based on a novel variance reduction technique that may be of independent interest and works by iteratively solving weighted MLEs over variance-altered versions of the GLM objective. SVAM offers provable model recovery guarantees superior to the state-of-the-art for robust regression even when a constant fraction of training labels are adversarially corrupted. SVAM also empirically outperforms several existing problem-specific techniques for robust regression and classification. Code for SVAM is available at https://github.com/purushottamkar/svam/

Anshul Mittal, Kunal Dahiya, Shreya Malani et al.

This paper develops the MUFIN technique for extreme classification (XC) tasks with millions of labels where datapoints and labels are endowed with visual and textual descriptors. Applications of MUFIN to product-to-product recommendation and bid query prediction over several millions of products are presented. Contemporary multi-modal methods frequently rely on purely embedding-based methods. On the other hand, XC methods utilize classifier architectures to offer superior accuracies than embedding only methods but mostly focus on text-based categorization tasks. MUFIN bridges this gap by reformulating multi-modal categorization as an XC problem with several millions of labels. This presents the twin challenges of developing multi-modal architectures that can offer embeddings sufficiently expressive to allow accurate categorization over millions of labels; and training and inference routines that scale logarithmically in the number of labels. MUFIN develops an architecture based on cross-modal attention and trains it in a modular fashion using pre-training and positive and negative mining. A novel product-to-product recommendation dataset MM-AmazonTitles-300K containing over 300K products was curated from publicly available amazon.com listings with each product endowed with a title and multiple images. On the all datasets MUFIN offered at least 3% higher accuracy than leading text-based, image-based and multi-modal techniques. Code for MUFIN is available at https://github.com/Extreme-classification/MUFIN

Naishadh Parmar, Raunak Shah, Tushar Goswamy et al.

The identification and control of human factors in climate change is a rapidly growing concern and robust, real-time air-quality monitoring and forecasting plays a critical role in allowing effective policy formulation and implementation. This paper presents DELFI, a novel deep learning-based mixture model to make effective long-term predictions of Particulate Matter (PM) 2.5 concentrations. A key novelty in DELFI is its multi-scale approach to the forecasting problem. The observation that point predictions are more suitable in the short-term and probabilistic predictions in the long-term allows accurate predictions to be made as much as 24 hours in advance. DELFI incorporates meteorological data as well as pollutant-based features to ensure a robust model that is divided into two parts: (i) a stack of three Long Short-Term Memory (LSTM) networks that perform differential modelling of the same window of past data, and (ii) a fully-connected layer enabling attention to each of the components. Experimental evaluation based on deployment of 13 stations in the Delhi National Capital Region (Delhi-NCR) in India establishes that DELFI offers far superior predictions especially in the long-term as compared to even non-parametric baselines. The Delhi-NCR recorded the 3rd highest PM levels amongst 39 mega-cities across the world during 2011-2015 and DELFI's performance establishes it as a potential tool for effective long-term forecasting of PM levels to enable public health management and environment protection.

Divyansh Chaurasia, Manoj Daram, Roshan Kumar et al.

We present a case study for the calibration of Low-cost air-quality (LCAQ) CO sensors from one of the largest multi-site-multi-season-multi-sensor-multi-pollutant mobile air-quality monitoring network deployments in India. LCAQ sensors have been shown to play a critical role in the establishment of dense, expansive air-quality monitoring networks and combating elevated pollution levels. The calibration of LCAQ sensors against regulatory-grade monitors is an expensive, laborious and time-consuming process, especially when a large number of sensors are to be deployed in a geographically diverse layout. In this work, we present the RESPIRE technique to calibrate LCAQ sensors to detect ambient CO (Carbon Monoxide) levels. RESPIRE offers specific advantages over baseline calibration methods popular in literature, such as improved prediction in cross-site, cross-season, and cross-sensor settings. RESPIRE offers a training algorithm that is provably resistant to outliers and an explainable model with the ability to flag instances of model overfitting. Empirical results are presented based on data collected during an extensive deployment spanning four sites, two seasons and six sensor packages. RESPIRE code is available at https://github.com/purushottamkar/respire.

Anshul Mittal, Kunal Dahiya, Sheshansh Agrawal et al.

Extreme multi-label classification (XML) involves tagging a data point with its most relevant subset of labels from an extremely large label set, with several applications such as product-to-product recommendation with millions of products. Although leading XML algorithms scale to millions of labels, they largely ignore label meta-data such as textual descriptions of the labels. On the other hand, classical techniques that can utilize label metadata via representation learning using deep networks struggle in extreme settings. This paper develops the DECAF algorithm that addresses these challenges by learning models enriched by label metadata that jointly learn model parameters and feature representations using deep networks and offer accurate classification at the scale of millions of labels. DECAF makes specific contributions to model architecture design, initialization, and training, enabling it to offer up to 2-6% more accurate prediction than leading extreme classifiers on publicly available benchmark product-to-product recommendation datasets, such as LF-AmazonTitles-1.3M. At the same time, DECAF was found to be up to 22x faster at inference than leading deep extreme classifiers, which makes it suitable for real-time applications that require predictions within a few milliseconds. The code for DECAF is available at the following URL https://github.com/Extreme-classification/DECAF.

Anshul Mittal, Noveen Sachdeva, Sheshansh Agrawal et al.

Deep extreme classification (XC) seeks to train deep architectures that can tag a data point with its most relevant subset of labels from an extremely large label set. The core utility of XC comes from predicting labels that are rarely seen during training. Such rare labels hold the key to personalized recommendations that can delight and surprise a user. However, the large number of rare labels and small amount of training data per rare label offer significant statistical and computational challenges. State-of-the-art deep XC methods attempt to remedy this by incorporating textual descriptions of labels but do not adequately address the problem. This paper presents ECLARE, a scalable deep learning architecture that incorporates not only label text, but also label correlations, to offer accurate real-time predictions within a few milliseconds. Core contributions of ECLARE include a frugal architecture and scalable techniques to train deep models along with label correlation graphs at the scale of millions of labels. In particular, ECLARE offers predictions that are 2 to 14% more accurate on both publicly available benchmark datasets as well as proprietary datasets for a related products recommendation task sourced from the Bing search engine. Code for ECLARE is available at https://github.com/Extreme-classification/ECLARE.

Anshul Mittal, Shikhar Mohan, Deepak Saini et al.

Deep extreme classification (XC) aims to train an encoder architecture and an accompanying classifier architecture to tag a data point with the most relevant subset of labels from a very large universe of labels. XC applications in ranking, recommendation and tagging routinely encounter tail labels for which the amount of training data is exceedingly small. Graph convolutional networks (GCN) present a convenient but computationally expensive way to leverage task metadata and enhance model accuracies in these settings. This paper formally establishes that in several use cases, the steep computational cost of GCNs is entirely avoidable by replacing GCNs with non-GCN architectures. The paper notices that in these settings, it is much more effective to use graph data to regularize encoder training than to implement a GCN. Based on these insights, an alternative paradigm RAMEN is presented to utilize graph metadata in XC settings that offers significant performance boosts with zero increase in inference computational costs. RAMEN scales to datasets with up to 1M labels and offers prediction accuracy up to 15% higher on benchmark datasets than state of the art methods, including those that use graph metadata to train GCNs. RAMEN also offers 10% higher accuracy over the best baseline on a proprietary recommendation dataset sourced from click logs of a popular search engine. Code for RAMEN will be released publicly.

Debojyoti Dey, Bhaskar Mukhoty, Purushottam Kar

This paper presents AGGLIO (Accelerated Graduated Generalized LInear-model Optimization), a stage-wise, graduated optimization technique that offers global convergence guarantees for non-convex optimization problems whose objectives offer only local convexity and may fail to be even quasi-convex at a global scale. In particular, this includes learning problems that utilize popular activation functions such as sigmoid, softplus and SiLU that yield non-convex training objectives. AGGLIO can be readily implemented using point as well as mini-batch SGD updates and offers provable convergence to the global optimum in general conditions. In experiments, AGGLIO outperformed several recently proposed optimization techniques for non-convex and locally convex objectives in terms of convergence rate as well as convergent accuracy. AGGLIO relies on a graduation technique for generalized linear models, as well as a novel proof strategy, both of which may be of independent interest.

S Deepak Narayanan, Aditya Sinha, Prateek Jain et al.

Training multi-layer Graph Convolution Networks (GCN) using standard SGD techniques scales poorly as each descent step ends up updating node embeddings for a large portion of the graph. Recent attempts to remedy this sub-sample the graph that reduces compute but introduce additional variance and may offer suboptimal performance. This paper develops the IGLU method that caches intermediate computations at various GCN layers thus enabling lazy updates that significantly reduce the compute cost of descent. IGLU introduces bounded bias into the gradients but nevertheless converges to a first-order saddle point under standard assumptions such as objective smoothness. Benchmark experiments show that IGLU offers up to 1.2% better accuracy despite requiring up to 88% less compute.

Bhaskar Mukhoty, Govind Gopakumar, Prateek Jain et al.

We provide the first global model recovery results for the IRLS (iteratively reweighted least squares) heuristic for robust regression problems. IRLS is known to offer excellent performance, despite bad initializations and data corruption, for several parameter estimation problems. Existing analyses of IRLS frequently require careful initialization, thus offering only local convergence guarantees. We remedy this by proposing augmentations to the basic IRLS routine that not only offer guaranteed global recovery, but in practice also outperform state-of-the-art algorithms for robust regression. Our routines are more immune to hyperparameter misspecification in basic regression tasks, as well as applied tasks such as linear-armed bandit problems. Our theoretical analyses rely on a novel extension of the notions of strong convexity and smoothness to weighted strong convexity and smoothness, and establishing that sub-Gaussian designs offer bounded weighted condition numbers. These notions may be useful in analyzing other algorithms as well.

Darshak Chhatbar, Umair Z. Ahmed, Purushottam Kar

Automated compilation error repair, the problem of suggesting fixes to buggy programs that fail to compile, has generated significant interest in recent years. Apart from being a tool of general convenience, automated code repair has significant pedagogical applications for novice programmers who find compiler error messages cryptic and unhelpful. Existing approaches largely solve this problem using a blackbox-application of a heavy-duty generative learning technique, such as sequence-to-sequence prediction (TRACER) or reinforcement learning (RLAssist). Although convenient, such black-box application of learning techniques makes existing approaches bulky in terms of training time, as well as inefficient at targeting specific error types. We present MACER, a novel technique for accelerated error repair based on a modular segregation of the repair process into repair identification and repair application. MACER uses powerful yet inexpensive discriminative learning techniques such as multi-label classifiers and rankers to first identify the type of repair required and then apply the suggested repair. Experiments indicate that the fine-grained approach adopted by MACER offers not only superior error correction, but also much faster training and prediction. On a benchmark dataset of 4K buggy programs collected from actual student submissions, MACER outperforms existing methods by 20% at suggesting fixes for popular errors that exactly match the fix desired by the student. MACER is also competitive or better than existing methods at all error types -- whether popular or rare. MACER offers a training time speedup of 2x over TRACER and 800x over RLAssist, and a test time speedup of 2-4x over both.

Amit Chandak, Debojyoti Dey, Bhaskar Mukhoty et al.

Mass public quarantining, colloquially known as a lock-down, is a non-pharmaceutical intervention to check spread of disease. This paper presents ESOP (Epidemiologically and Socio-economically Optimal Policies), a novel application of active machine learning techniques using Bayesian optimization, that interacts with an epidemiological model to arrive at lock-down schedules that optimally balance public health benefits and socio-economic downsides of reduced economic activity during lock-down periods. The utility of ESOP is demonstrated using case studies with VIPER (Virus-Individual-Policy-EnviRonment), a stochastic agent-based simulator that this paper also proposes. However, ESOP is flexible enough to interact with arbitrary epidemiological simulators in a black-box manner, and produce schedules that involve multiple phases of lock-downs.

Ankit Jalan, Purushottam Kar

Extreme classification seeks to assign each data point, the most relevant labels from a universe of a million or more labels. This task is faced with the dual challenge of high precision and scalability, with millisecond level prediction times being a benchmark. We propose DEFRAG, an adaptive feature agglomeration technique to accelerate extreme classification algorithms. Despite past works on feature clustering and selection, DEFRAG distinguishes itself in being able to scale to millions of features, and is especially beneficial when feature sets are sparse, which is typical of recommendation and multi-label datasets. The method comes with provable performance guarantees and performs efficient task-driven agglomeration to reduce feature dimensionalities by an order of magnitude or more. Experiments show that DEFRAG can not only reduce training and prediction times of several leading extreme classification algorithms by as much as 40%, but also be used for feature reconstruction to address the problem of missing features, as well as offer superior coverage on rare labels.

Vaibhav B Sinha, Sneha Kudugunta, Adepu Ravi Sankar et al.

We present DANTE, a novel method for training neural networks using the alternating minimization principle. DANTE provides an alternate perspective to traditional gradient-based backpropagation techniques commonly used to train deep networks. It utilizes an adaptation of quasi-convexity to cast training a neural network as a bi-quasi-convex optimization problem. We show that for neural network configurations with both differentiable (e.g. sigmoid) and non-differentiable (e.g. ReLU) activation functions, we can perform the alternations effectively in this formulation. DANTE can also be extended to networks with multiple hidden layers. In experiments on standard datasets, neural networks trained using the proposed method were found to be promising and competitive to traditional backpropagation techniques, both in terms of quality of the solution, as well as training speed.

Dheeraj Mekala, Vivek Gupta, Purushottam Kar et al.

Hierarchical classification is supervised multi-class classification problem over the set of class labels organized according to a hierarchy. In this report, we study the work by Ramaswamy et. al. on hierarchical classification over symmetric tree distance loss. We extend the consistency of hierarchical classification algorithm over asymmetric tree distance loss. We design a $\mathcal{O}(nk\log{}n)$ algorithm to find Bayes optimal classification for a k-ary tree as a hierarchy. We show that under reasonable assumptions over asymmetric loss function, the Bayes optimal classification over this asymmetric loss can be found in $\mathcal{O}(k\log{}n)$. We exploit this insight and attempt to extend the Ova-Cascade algorithm \citet{ramaswamy2015convex} for hierarchical classification over the asymmetric loss.

Amartya Sanyal, Pawan Kumar, Purushottam Kar et al.

We present a class of algorithms capable of directly training deep neural networks with respect to large families of task-specific performance measures such as the F-measure and the Kullback-Leibler divergence that are structured and non-decomposable. This presents a departure from standard deep learning techniques that typically use squared or cross-entropy loss functions (that are decomposable) to train neural networks. We demonstrate that directly training with task-specific loss functions yields much faster and more stable convergence across problems and datasets. Our proposed algorithms and implementations have several novel features including (i) convergence to first order stationary points despite optimizing complex objective functions; (ii) use of fewer training samples to achieve a desired level of convergence, (iii) a substantial reduction in training time, and (iv) a seamless integration of our implementation into existing symbolic gradient frameworks. We implement our techniques on a variety of deep architectures including multi-layer perceptrons and recurrent neural networks and show that on a variety of benchmark and real data sets, our algorithms outperform traditional approaches to training deep networks, as well as some recent approaches to task-specific training of neural networks.

Prateek Jain, Purushottam Kar

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. The monograph will lead the reader through several widely used non-convex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Claudio Gentile, Shuai Li, Purushottam Kar et al.

We investigate a novel cluster-of-bandit algorithm CAB for collaborative recommendation tasks that implements the underlying feedback sharing mechanism by estimating the neighborhood of users in a context-dependent manner. CAB makes sharp departures from the state of the art by incorporating collaborative effects into inference as well as learning processes in a manner that seamlessly interleaving explore-exploit tradeoffs and collaborative steps. We prove regret bounds under various assumptions on the data, which exhibit a crisp dependence on the expected number of clusters over the users, a natural measure of the statistical difficulty of the learning task. Experiments on production and real-world datasets show that CAB offers significantly increased prediction performance against a representative pool of state-of-the-art methods.

Kush Bhatia, Prateek Jain, Parameswaran Kamalaruban et al.

We study the problem of robust time series analysis under the standard auto-regressive (AR) time series model in the presence of arbitrary outliers. We devise an efficient hard thresholding based algorithm which can obtain a consistent estimate of the optimal AR model despite a large fraction of the time series points being corrupted. Our algorithm alternately estimates the corrupted set of points and the model parameters, and is inspired by recent advances in robust regression and hard-thresholding methods. However, a direct application of existing techniques is hindered by a critical difference in the time-series domain: each point is correlated with all previous points rendering existing tools inapplicable directly. We show how to overcome this hurdle using novel proof techniques. Using our techniques, we are also able to provide the first efficient and provably consistent estimator for the robust regression problem where a standard linear observation model with white additive noise is corrupted arbitrarily. We illustrate our methods on synthetic datasets and show that our methods indeed are able to consistently recover the optimal parameters despite a large fraction of points being corrupted.

Purushottam Kar, Shuai Li, Harikrishna Narasimhan et al.

The estimation of class prevalence, i.e., the fraction of a population that belongs to a certain class, is a very useful tool in data analytics and learning, and finds applications in many domains such as sentiment analysis, epidemiology, etc. For example, in sentiment analysis, the objective is often not to estimate whether a specific text conveys a positive or a negative sentiment, but rather estimate the overall distribution of positive and negative sentiments during an event window. A popular way of performing the above task, often dubbed quantification, is to use supervised learning to train a prevalence estimator from labeled data. Contemporary literature cites several performance measures used to measure the success of such prevalence estimators. In this paper we propose the first online stochastic algorithms for directly optimizing these quantification-specific performance measures. We also provide algorithms that optimize hybrid performance measures that seek to balance quantification and classification performance. Our algorithms present a significant advancement in the theory of multivariate optimization and we show, by a rigorous theoretical analysis, that they exhibit optimal convergence. We also report extensive experiments on benchmark and real data sets which demonstrate that our methods significantly outperform existing optimization techniques used for these performance measures.

Shuai Li, Purushottam Kar

We propose an efficient Context-Aware clustering of Bandits (CAB) algorithm, which can capture collaborative effects. CAB can be easily deployed in a real-world recommendation system, where multi-armed bandits have been shown to perform well in particular with respect to the cold-start problem. CAB utilizes a context-aware clustering augmented by exploration-exploitation strategies. CAB dynamically clusters the users based on the content universe under consideration. We give a theoretical analysis in the standard stochastic multi-armed bandits setting. We show the efficiency of our approach on production and real-world datasets, demonstrate the scalability, and, more importantly, the significant increased prediction performance against several state-of-the-art methods.

Kush Bhatia, Himanshu Jain, Purushottam Kar et al.

The objective in extreme multi-label learning is to train a classifier that can automatically tag a novel data point with the most relevant subset of labels from an extremely large label set. Embedding based approaches make training and prediction tractable by assuming that the training label matrix is low-rank and hence the effective number of labels can be reduced by projecting the high dimensional label vectors onto a low dimensional linear subspace. Still, leading embedding approaches have been unable to deliver high prediction accuracies or scale to large problems as the low rank assumption is violated in most real world applications. This paper develops the X-One classifier to address both limitations. The main technical contribution in X-One is a formulation for learning a small ensemble of local distance preserving embeddings which can accurately predict infrequently occurring (tail) labels. This allows X-One to break free of the traditional low-rank assumption and boost classification accuracy by learning embeddings which preserve pairwise distances between only the nearest label vectors. We conducted extensive experiments on several real-world as well as benchmark data sets and compared our method against state-of-the-art methods for extreme multi-label classification. Experiments reveal that X-One can make significantly more accurate predictions then the state-of-the-art methods including both embeddings (by as much as 35%) as well as trees (by as much as 6%). X-One can also scale efficiently to data sets with a million labels which are beyond the pale of leading embedding methods.

Kush Bhatia, Prateek Jain, Purushottam Kar

We study the problem of Robust Least Squares Regression (RLSR) where several response variables can be adversarially corrupted. More specifically, for a data matrix X \in R^{p x n} and an underlying model w*, the response vector is generated as y = X'w* + b where b \in R^n is the corruption vector supported over at most C.n coordinates. Existing exact recovery results for RLSR focus solely on L1-penalty based convex formulations and impose relatively strict model assumptions such as requiring the corruptions b to be selected independently of X. In this work, we study a simple hard-thresholding algorithm called TORRENT which, under mild conditions on X, can recover w* exactly even if b corrupts the response variables in an adversarial manner, i.e. both the support and entries of b are selected adversarially after observing X and w*. Our results hold under deterministic assumptions which are satisfied if X is sampled from any sub-Gaussian distribution. Finally unlike existing results that apply only to a fixed w*, generated independently of X, our results are universal and hold for any w* \in R^p. Next, we propose gradient descent-based extensions of TORRENT that can scale efficiently to large scale problems, such as high dimensional sparse recovery and prove similar recovery guarantees for these extensions. Empirically we find TORRENT, and more so its extensions, offering significantly faster recovery than the state-of-the-art L1 solvers. For instance, even on moderate-sized datasets (with p = 50K) with around 40% corrupted responses, a variant of our proposed method called TORRENT-HYB is more than 20x faster than the best L1 solver.

Purushottam Kar, Harikrishna Narasimhan, Prateek Jain

The problem of maximizing precision at the top of a ranked list, often dubbed Precision@k (prec@k), finds relevance in myriad learning applications such as ranking, multi-label classification, and learning with severe label imbalance. However, despite its popularity, there exist significant gaps in our understanding of this problem and its associated performance measure. The most notable of these is the lack of a convex upper bounding surrogate for prec@k. We also lack scalable perceptron and stochastic gradient descent algorithms for optimizing this performance measure. In this paper we make key contributions in these directions. At the heart of our results is a family of truly upper bounding surrogates for prec@k. These surrogates are motivated in a principled manner and enjoy attractive properties such as consistency to prec@k under various natural margin/noise conditions. These surrogates are then used to design a class of novel perceptron algorithms for optimizing prec@k with provable mistake bounds. We also devise scalable stochastic gradient descent style methods for this problem with provable convergence bounds. Our proofs rely on novel uniform convergence bounds which require an in-depth analysis of the structural properties of prec@k and its surrogates. We conclude with experimental results comparing our algorithms with state-of-the-art cutting plane and stochastic gradient algorithms for maximizing prec@k.

Harikrishna Narasimhan, Purushottam Kar, Prateek Jain

Modern classification problems frequently present mild to severe label imbalance as well as specific requirements on classification characteristics, and require optimizing performance measures that are non-decomposable over the dataset, such as F-measure. Such measures have spurred much interest and pose specific challenges to learning algorithms since their non-additive nature precludes a direct application of well-studied large scale optimization methods such as stochastic gradient descent. In this paper we reveal that for two large families of performance measures that can be expressed as functions of true positive/negative rates, it is indeed possible to implement point stochastic updates. The families we consider are concave and pseudo-linear functions of TPR, TNR which cover several popularly used performance measures such as F-measure, G-mean and H-mean. Our core contribution is an adaptive linearization scheme for these families, using which we develop optimization techniques that enable truly point-based stochastic updates. For concave performance measures we propose SPADE, a stochastic primal dual solver; for pseudo-linear measures we propose STAMP, a stochastic alternate maximization procedure. Both methods have crisp convergence guarantees, demonstrate significant speedups over existing methods - often by an order of magnitude or more, and give similar or more accurate predictions on test data.

Purushottam Kar, Harikrishna Narasimhan, Prateek Jain

Modern applications in sensitive domains such as biometrics and medicine frequently require the use of non-decomposable loss functions such as precision@k, F-measure etc. Compared to point loss functions such as hinge-loss, these offer much more fine grained control over prediction, but at the same time present novel challenges in terms of algorithm design and analysis. In this work we initiate a study of online learning techniques for such non-decomposable loss functions with an aim to enable incremental learning as well as design scalable solvers for batch problems. To this end, we propose an online learning framework for such loss functions. Our model enjoys several nice properties, chief amongst them being the existence of efficient online learning algorithms with sublinear regret and online to batch conversion bounds. Our model is a provable extension of existing online learning models for point loss functions. We instantiate two popular losses, prec@k and pAUC, in our model and prove sublinear regret bounds for both of them. Our proofs require a novel structural lemma over ranked lists which may be of independent interest. We then develop scalable stochastic gradient descent solvers for non-decomposable loss functions. We show that for a large family of loss functions satisfying a certain uniform convergence property (that includes prec@k, pAUC, and F-measure), our methods provably converge to the empirical risk minimizer. Such uniform convergence results were not known for these losses and we establish these using novel proof techniques. We then use extensive experimentation on real life and benchmark datasets to establish that our method can be orders of magnitude faster than a recently proposed cutting plane method.

Prateek Jain, Ambuj Tewari, Purushottam Kar

The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard $L_0$ constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for IHT-style methods in the high dimensional statistical setting. Our bounds are tight and match known minimax lower bounds. Our results rely on a general analysis framework that enables us to analyze several popular hard thresholding style algorithms (such as HTP, CoSaMP, SP) in the high dimensional regression setting. We also extend our analysis to a large family of "fully corrective methods" that includes two-stage and partial hard-thresholding algorithms. We show that our results hold for the problem of sparse regression, as well as low-rank matrix recovery.

Hsiang-Fu Yu, Prateek Jain, Purushottam Kar et al.

The multi-label classification problem has generated significant interest in recent years. However, existing approaches do not adequately address two key challenges: (a) the ability to tackle problems with a large number (say millions) of labels, and (b) the ability to handle data with missing labels. In this paper, we directly address both these problems by studying the multi-label problem in a generic empirical risk minimization (ERM) framework. Our framework, despite being simple, is surprisingly able to encompass several recent label-compression based methods which can be derived as special cases of our method. To optimize the ERM problem, we develop techniques that exploit the structure of specific loss functions - such as the squared loss function - to offer efficient algorithms. We further show that our learning framework admits formal excess risk bounds even in the presence of missing labels. Our risk bounds are tight and demonstrate better generalization performance for low-rank promoting trace-norm regularization when compared to (rank insensitive) Frobenius norm regularization. Finally, we present extensive empirical results on a variety of benchmark datasets and show that our methods perform significantly better than existing label compression based methods and can scale up to very large datasets such as the Wikipedia dataset.

Purushottam Kar, Bharath K Sriperumbudur, Prateek Jain et al.

In this paper, we study the generalization properties of online learning based stochastic methods for supervised learning problems where the loss function is dependent on more than one training sample (e.g., metric learning, ranking). We present a generic decoupling technique that enables us to provide Rademacher complexity-based generalization error bounds. Our bounds are in general tighter than those obtained by Wang et al (COLT 2012) for the same problem. Using our decoupling technique, we are further able to obtain fast convergence rates for strongly convex pairwise loss functions. We are also able to analyze a class of memory efficient online learning algorithms for pairwise learning problems that use only a bounded subset of past training samples to update the hypothesis at each step. Finally, in order to complement our generalization bounds, we propose a novel memory efficient online learning algorithm for higher order learning problems with bounded regret guarantees.

Purushottam Kar, Harish Karnick

We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Bochner's theorem for translation invariant positive definite kernels on the real line (Rudin, 1962).

Purushottam Kar

We present generalization bounds for the TS-MKL framework for two stage multiple kernel learning. We also present bounds for sparse kernel learning formulations within the TS-MKL framework.

Purushottam Kar, Prateek Jain

We address the problem of general supervised learning when data can only be accessed through an (indefinite) similarity function between data points. Existing work on learning with indefinite kernels has concentrated solely on binary/multi-class classification problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classification. We give a "goodness" criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efficient algorithms for supervised learning using "good" similarity functions. We demonstrate the effectiveness of our model on three important super-vised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness definition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves significantly higher accuracies than the baseline methods while offering reduced computational costs.

Purushottam Kar, Harish Karnick

Approximating non-linear kernels using feature maps has gained a lot of interest in recent years due to applications in reducing training and testing times of SVM classifiers and other kernel based learning algorithms. We extend this line of work and present low distortion embeddings for dot product kernels into linear Euclidean spaces. We base our results on a classical result in harmonic analysis characterizing all dot product kernels and use it to define randomized feature maps into explicit low dimensional Euclidean spaces in which the native dot product provides an approximation to the dot product kernel with high confidence.