We study the problem of batch learning from bandit feedback in the setting of extremely large action spaces. Learning from extreme bandit feedback is ubiquitous in recommendation systems, in which billions of decisions are made over sets consisting of millions of choices in a single day, yielding massive observational data. In these large-scale real-world applications, supervised learning frameworks such as eXtreme Multi-label Classification (XMC) are widely used despite the fact that they incur significant biases due to the mismatch between bandit feedback and supervised labels. Such biases can be mitigated by importance sampling techniques, but these techniques suffer from impractical variance when dealing with a large number of actions. In this paper, we introduce a selective importance sampling estimator (sIS) that operates in a significantly more favorable bias-variance regime. The sIS estimator is obtained by performing importance sampling on the conditional expectation of the reward with respect to a small subset of actions for each instance (a form of Rao-Blackwellization). We employ this estimator in a novel algorithmic procedure---named Policy Optimization for eXtreme Models (POXM)---for learning from bandit feedback on XMC tasks. In POXM, the selected actions for the sIS estimator are the top-p actions of the logging policy, where p is adjusted from the data and is significantly smaller than the size of the action space. We use a supervised-to-bandit conversion on three XMC datasets to benchmark our POXM method against three competing methods: BanditNet, a previously applied partial matching pruning strategy, and a supervised learning baseline. Whereas BanditNet sometimes improves marginally over the logging policy, our experiments show that POXM systematically and significantly improves over all baselines.

Extreme multi-label classification (XMC) is the problem of finding the relevant labels for an input, from a very large universe of possible labels. We consider XMC in the setting where labels are available only for groups of samples - but not for individual ones. Current XMC approaches are not built for such multi-instance multi-label (MIML) training data, and MIML approaches do not scale to XMC sizes. We develop a new and scalable algorithm to impute individual-sample labels from the group labels; this can be paired with any existing XMC method to solve the aggregated label problem. We characterize the statistical properties of our algorithm under mild assumptions, and provide a new end-to-end framework for MIML as an extension. Experiments on both aggregated label XMC and MIML tasks show the advantages over existing approaches.

Forecasting high-dimensional time series plays a crucial role in many applications such as demand forecasting and financial predictions. Modern real-world datasets can have millions of correlated time-series that evolve together, i.e they are extremely high dimensional (one dimension for each individual time-series). Thus there is need for exploiting these global patterns and coupling them with local calibration for better prediction. However, most recent deep learning approaches in the literature are one-dimensional, i.e, even though they are trained on the whole dataset, during prediction, the future forecast for a single dimension mainly depends on past values from the same dimension. In this paper, we seek to correct this deficiency and propose DeepGLO, a deep forecasting model which thinks globally and acts locally. In particular, DeepGLO is a hybrid model that combines a global matrix factorization model regularized by a temporal deep network with a local deep temporal model that captures patterns specific to each dimension. The global and local models are combined via a data-driven attention mechanism for each dimension. The proposed deep architecture used is a variation of temporal convolution termed as leveled network which can be trained effectively on high-dimensional but diverse time series, where different time series can have vastly different scales, without a priori normalization or rescaling. Empirical results demonstrate that DeepGLO outperforms state-of-the-art approaches on various datasets; for example, we see more than 30% improvement in WAPE over other methods on a real-world dataset that contains more than 100K-dimensional time series.

Extreme multi-label classification (XMC) aims to assign to an instance the most relevant subset of labels from a colossal label set. Due to modern applications that lead to massive label sets, the scalability of XMC has attracted much recent attention from both academia and industry. In this paper, we establish a three-stage framework to solve XMC efficiently, which includes 1) indexing the labels, 2) matching the instance to the relevant indices, and 3) ranking the labels from the relevant indices. This framework unifies many existing XMC approaches. Based on this framework, we propose a modular deep learning approach SLINMER: Semantic Label Indexing, Neural Matching, and Efficient Ranking. The label indexing stage of SLINMER can adopt different semantic label representations leading to different configurations of SLINMER. Empirically, we demonstrate that several individual configurations of SLINMER achieve superior performance than the state-of-the-art XMC approaches on several benchmark datasets. Moreover, by ensembling those configurations, SLINMER can achieve even better results. In particular, on a Wiki dataset with around 0.5 millions of labels, the precision@1 is increased from 61% to 67%.

Deep learning models have become state of the art for natural language processing (NLP) tasks, however deploying these models in production system poses significant memory constraints. Existing compression methods are either lossy or introduce significant latency. We propose a compression method that leverages low rank matrix factorization during training,to compress the word embedding layer which represents the size bottleneck for most NLP models. Our models are trained, compressed and then further re-trained on the downstream task to recover accuracy while maintaining the reduced size. Empirically, we show that the proposed method can achieve 90% compression with minimal impact in accuracy for sentence classification tasks, and outperforms alternative methods like fixed-point quantization or offline word embedding compression. We also analyze the inference time and storage space for our method through FLOP calculations, showing that we can compress DNN models by a configurable ratio and regain accuracy loss without introducing additional latency compared to fixed point quantization. Finally, we introduce a novel learning rate schedule, the Cyclically Annealed Learning Rate (CALR), which we empirically demonstrate to outperform other popular adaptive learning rate algorithms on a sentence classification benchmark.

In many applications such as recommender systems and multi-label learning the task is to complete a partially observed binary matrix. Such PU learning (positive-unlabeled) problems can be solved by one-class matrix factorization (MF). In practice side information such as user or item features in recommender systems are often available besides the observed positive user-item connections. In this work we consider a generalization of one-class MF so that two types of side information are incorporated and a general convex loss function can be used. The resulting optimization problem is very challenging, but we derive an efficient and effective alternating minimization procedure. Experiments on large-scale multi-label learning and one-class recommender systems demonstrate the effectiveness of our proposed approach.

In this paper, we investigate a divide and conquer approach to Kernel Ridge Regression (KRR). Given n samples, the division step involves separating the points based on some underlying disjoint partition of the input space (possibly via clustering), and then computing a KRR estimate for each partition. The conquering step is simple: for each partition, we only consider its own local estimate for prediction. We establish conditions under which we can give generalization bounds for this estimator, as well as achieve optimal minimax rates. We also show that the approximation error component of the generalization error is lesser than when a single KRR estimate is fit on the data: thus providing both statistical and computational advantages over a single KRR estimate over the entire data (or an averaging over random partitions as in other recent work, [30]). Lastly, we provide experimental validation for our proposed estimator and our assumptions.

Given a symmetric nonnegative matrix $A$, symmetric nonnegative matrix factorization (symNMF) is the problem of finding a nonnegative matrix $H$, usually with much fewer columns than $A$, such that $A \approx HH^T$. SymNMF can be used for data analysis and in particular for various clustering tasks. In this paper, we propose simple and very efficient coordinate descent schemes to solve this problem, and that can handle large and sparse input matrices. The effectiveness of our methods is illustrated on synthetic and real-world data sets, and we show that they perform favorably compared to recent state-of-the-art methods.

Several learning applications require solving high-dimensional regression problems where the relevant features belong to a small number of (overlapping) groups. For very large datasets and under standard sparsity constraints, hard thresholding methods have proven to be extremely efficient, but such methods require NP hard projections when dealing with overlapping groups. In this paper, we show that such NP-hard projections can not only be avoided by appealing to submodular optimization, but such methods come with strong theoretical guarantees even in the presence of poorly conditioned data (i.e. say when two features have correlation $\geq 0.99$), which existing analyses cannot handle. These methods exhibit an interesting computation-accuracy trade-off and can be extended to significantly harder problems such as sparse overlapping groups. Experiments on both real and synthetic data validate our claims and demonstrate that the proposed methods are orders of magnitude faster than other greedy and convex relaxation techniques for learning with group-structured sparsity.