Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these issues in the context of high-dimensional sparse linear regression, and propose novel estimators for the cases of noisy, missing and/or dependent data. Many standard approaches to noisy or missing data, such as those using the EM algorithm, lead to optimization problems that are inherently nonconvex, and it is difficult to establish theoretical guarantees on practical algorithms. While our approach also involves optimizing nonconvex programs, we are able to both analyze the statistical error associated with any global optimum, and more surprisingly, to prove that a simple algorithm based on projected gradient descent will converge in polynomial time to a small neighborhood of the set of all global minimizers. On the statistical side, we provide nonasymptotic bounds that hold with high probability for the cases of noisy, missing and/or dependent data. On the computational side, we prove that under the same types of conditions required for statistical consistency, the projected gradient descent algorithm is guaranteed to converge at a geometric rate to a near-global minimizer. We illustrate these theoretical predictions with simulations, showing close agreement with the predicted scalings.

The step-size, often denoted as α, is a key parameter for most incremental learning algorithms. Its importance is especially pronounced when performing online temporal difference (TD) learning with function approximation. Several methods have been developed to adapt the step-size online. These range from straightforward back-off strategies to adaptive algorithms based on gradient descent. We derive an adaptive upper bound on the step-size parameter to guarantee that online TD learning with linear function approximation will not diverge. We then empirically evaluate algorithms using this upper bound as a heuristic for adapting the step-size parameter online. We compare performance with related work including HL(λ) and Autostep. Our results show that this adaptive upper bound heuristic out-performs all existing methods without requiring any meta-parameters. This effectively eliminates the need to tune the learning rate of temporal difference learning with linear function approximation.

In this paper we address the following question: Can we approximately sample from a Bayesian posterior distribution if we are only allowed to touch a small mini-batch of data-items for every sample we generate?. An algorithm based on the Langevin equation with stochastic gradients (SGLD) was previously proposed to solve this, but its mixing rate was slow. By leveraging the Bayesian Central Limit Theorem, we extend the SGLD algorithm so that at high mixing rates it will sample from a normal approximation of the posterior, while for slow mixing rates it will mimic the behavior of SGLD with a pre-conditioner matrix. As a bonus, the proposed algorithm is reminiscent of Fisher scoring (with stochastic gradients) and as such an efficient optimizer during burn-in.

Kernel-based online learning has often shown state-of-the-art performance for many online learning tasks. It, however, suffers from a major shortcoming, that is, the unbounded number of support vectors, making it non-scalable and unsuitable for applications with large-scale datasets. In this work, we study the problem of bounded kernel-based online learning that aims to constrain the number of support vectors by a predefined budget. Although several algorithms have been proposed in literature, they are neither computationally efficient due to their intensive budget maintenance strategy nor effective due to the use of simple Perceptron algorithm. To overcome these limitations, we propose a framework for bounded kernel-based online learning based on an online gradient descent approach. We propose two efficient algorithms of bounded online gradient descent (BOGD) for scalable kernel-based online learning: (i) BOGD by maintaining support vectors using uniform sampling, and (ii) BOGD++ by maintaining support vectors using non-uniform sampling. We present theoretical analysis of regret bound for both algorithms, and found promising empirical performance in terms of both efficacy and efficiency by comparing them to several well-known algorithms for bounded kernel-based online learning on large-scale datasets.

Item recommendation is the task of predicting a personalized ranking on a set of items (e.g. websites, movies, products). In this paper, we investigate the most common scenario with implicit feedback (e.g. clicks, purchases). There are many methods for item recommendation from implicit feedback like matrix factorization (MF) or adaptive knearest-neighbor (kNN). Even though these methods are designed for the item prediction task of personalized ranking, none of them is directly optimized for ranking. In this paper we present a generic optimization criterion BPR-Opt for personalized ranking that is the maximum posterior estimator derived from a Bayesian analysis of the problem. We also provide a generic learning algorithm for optimizing models with respect to BPR-Opt. The learning method is based on stochastic gradient descent with bootstrap sampling. We show how to apply our method to two state-of-the-art recommender models: matrix factorization and adaptive kNN. Our experiments indicate that for the task of personalized ranking our optimization method outperforms the standard learning techniques for MF and kNN. The results show the importance of optimizing models for the right criterion.

The natural gradient allows for more efficient gradient descent by removing dependencies and biases inherent in a function's parameterization. Several papers present the topic thoroughly and precisely. It remains a very difficult idea to get your head around however. The intent of this note is to provide simple intuition for the natural gradient and its use. We review how an ill conditioned parameter space can undermine learning, introduce the natural gradient by analogy to the more widely understood concept of signal whitening, and present tricks and specific prescriptions for applying the natural gradient to learning problems.

Joint sparsity offers powerful structural cues for feature selection, especially for variables that are expected to demonstrate a "grouped" behavior. Such behavior is commonly modeled via group-lasso, multitask lasso, and related methods where feature selection is effected via mixed-norms. Several mixed-norm based sparse models have received substantial attention, and for some cases efficient algorithms are also available. Surprisingly, several constrained sparse models seem to be lacking scalable algorithms. We address this deficiency by presenting batch and online (stochastic-gradient) optimization methods, both of which rely on efficient projections onto mixed-norm balls. We illustrate our methods by applying them to the multitask lasso. We conclude by mentioning some open problems.

We introduce a new graphical model for tracking radio-tagged animals and learning their movement patterns. The model provides a principled way to combine radio telemetry data with an arbitrary set of userdefined, spatial features. We describe an efficient stochastic gradient algorithm for fitting model parameters to data and demonstrate its effectiveness via asymptotic analysis and synthetic experiments. We also apply our model to real datasets, and show that it outperforms the most popular radio telemetry software package used in ecology. We conclude that integration of different data sources under a single statistical framework, coupled with appropriate parameter and state estimation procedures, produces both accurate location estimates and an interpretable statistical model of animal movement.

Randomized algorithms that base iteration-level decisions on samples from some pool are ubiquitous in machine learning and optimization. Examples include stochastic gradient descent and randomized coordinate descent. This paper makes progress at theoretically evaluating the difference in performance between sampling with- and without-replacement in such algorithms. Focusing on least means squares optimization, we formulate a noncommutative arithmetic-geometric mean inequality that would prove that the expected convergence rate of without-replacement sampling is faster than that of with-replacement sampling. We demonstrate that this inequality holds for many classes of random matrices and for some pathological examples as well. We provide a deterministic worst-case bound on the gap between the discrepancy between the two sampling models, and explore some of the impediments to proving this inequality in full generality. We detail the consequences of this inequality for stochastic gradient descent and the randomized Kaczmarz algorithm for solving linear systems.

We analyze the convergence of gradient-based optimization algorithms whose updates depend on delayed stochastic gradient information. The main application of our results is to the development of distributed minimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible. In application to distributed optimization, we show $n$-node architectures whose optimization error in stochastic problems---in spite of asynchronous delays---scales asymptotically as $\order(1 / \sqrt{nT})$, which is known to be optimal even in the absence of delays.