Distributed learning facilitates the scaling-up of data processing by distributing the computational burden over several nodes. Despite the vast interest in distributed learning, generalization performance of such approaches is not well understood. We address this gap by focusing on a linear regression setting. We consider the setting where the unknowns are distributed over a network of nodes. We present an analytical characterization of the dependence of the generalization error on the partitioning of the unknowns over nodes. In particular, for the overparameterized case, our results show that while the error on training data remains in the same range as that of the centralized solution, the generalization error of the distributed solution increases dramatically compared to that of the centralized solution when the number of unknowns estimated at any node is close to the number of observations. We further provide numerical examples to verify our analytical expressions.

In non-stationary environments, learning machines usually confront the domain adaptation scenario where the data distribution does change over time. Previous domain adaptation works have achieved great success in theory and practice. However, they always lose robustness in noisy environments where the labels and features of examples from the source domain become corrupted. In this paper, we report our attempt towards achieving accurate noise-robust domain adaptation. We first give a theoretical analysis that reveals how harmful noises influence unsupervised domain adaptation. To eliminate the effect of label noise, we propose an offline curriculum learning for minimizing a newly-defined empirical source risk. To reduce the impact of feature noise, we propose a proxy distribution based margin discrepancy. We seamlessly transform our methods into an adversarial network that performs efficient joint optimization for them, successfully mitigating the negative influence from both data corruption and distribution shift. A series of empirical studies show that our algorithm remarkably outperforms state of the art, over 10% accuracy improvements in some domain adaptation tasks under noisy environments.

These lecture notes endeavour to collect in one place the mathematical background required to understand the properties of kernels in general and the Random Fourier Features approximation of Rahimi and Recht (NIPS 2007) in particular. We briefly motivate the use of kernels in Machine Learning with the example of the support vector machine. We discuss positive definite and conditionally negative definite kernels in some detail. After a brief discussion of Hilbert spaces, including the Reproducing Kernel Hilbert Space construction, we present Mercer's theorem. We discuss the Random Fourier Features technique and then present, with proofs, scalar and matrix concentration results that help us estimate the error incurred by the technique. These notes are the transcription of 10 lectures given at IIT Delhi between January and April 2020.

Uplift modeling is a predictive modeling technique that estimates the user-level incremental effect of a treatment using machine learning models. It is often used for targeting promotions and advertisements, as well as for the personalization of product offerings. In these applications, there are often hundreds of features available to build such models. Keeping all the features in a model can be costly and inefficient. Feature selection is an essential step in the modeling process for multiple reasons: improving the estimation accuracy by eliminating irrelevant features, accelerating model training and prediction speed, reducing the monitoring and maintenance workload for feature data pipeline, and providing better model interpretation and diagnostics capability. However, feature selection methods for uplift modeling have been rarely discussed in the literature. Although there are various feature selection methods for standard machine learning models, we will demonstrate that those methods are sub-optimal for solving the feature selection problem for uplift modeling. To address this problem, we introduce a set of feature selection methods designed specifically for uplift modeling, including both filter methods and embedded methods. To evaluate the effectiveness of the proposed feature selection methods, we use different uplift models and measure the accuracy of each model with a different number of selected features. We use both synthetic and real data to conduct these experiments. We also implemented the proposed filter methods in an open source Python package (CausalML).

Although adaptive optimization algorithms such as Adam show fast convergence in many machine learning tasks, this paper identifies a problem of Adam by analyzing its performance in a simple non-convex synthetic problem, showing that Adam's fast convergence would possibly lead the algorithm to local minimums. To address this problem, we improve Adam by proposing a novel adaptive gradient descent algorithm named AdaX. Unlike Adam that ignores the past gradients, AdaX exponentially accumulates the long-term gradient information in the past during training, to adaptively tune the learning rate. We thoroughly prove the convergence of AdaX in both the convex and non-convex settings. Extensive experiments show that AdaX outperforms Adam in various tasks of computer vision and natural language processing and can catch up with Stochastic Gradient Descent.

Continuous time Hamiltonian Monte Carlo is introduced, as a powerful alternative to Markov chain Monte Carlo methods for continuous target distributions. The method is constructed in two steps: First Hamiltonian dynamics are chosen as the deterministic dynamics in a continuous time piecewise deterministic Markov process. Under very mild restrictions, such a process will have the desired target distribution as an invariant distribution. Secondly, the numerical implementation of such processes, based on adaptive numerical integration of second order ordinary differential equations is considered. The numerical implementation yields an approximate, yet highly robust algorithm that, unlike conventional Hamiltonian Monte Carlo, enables the exploitation of the complete Hamiltonian trajectories (and hence the title). The proposed algorithm may yield large speedups and improvements in stability relative to relevant benchmarks, while incurring numerical errors that are negligible relative to the overall Monte Carlo errors.

In this work, we propose Sum-Product-Transform Networks (SPTN), an extension of sum-product networks that uses invertible transformations as additional internal nodes. The type and placement of transformations determine properties of the resulting SPTN with many interesting special cases. Importantly, SPTN with Gaussian leaves and affine transformations pose the same inference task tractable that can be computed efficiently in SPNs. We propose to store affine transformations in their SVD decompositions using an efficient parametrization of unitary matrices by a set of Givens rotations. Last but not least, we demonstrate that G-SPTNs achieve state-of-the-art results on the density estimation task and are competitive with state-of-the-art methods for anomaly detection.

We extend the framework of Boltzmann machines to a network of complex-valued neurons with variable amplitudes, referred to as Complex Amplitude-Phase Boltzmann machine (CAP-BM). The model is capable of performing unsupervised learning on the amplitude and relative phase distribution in complex data. The sampling rule of the Gibbs distribution and the learning rules of the model are presented. Learning in a Complex Amplitude-Phase restricted Boltzmann machine (CAP-RBM) is demonstrated on synthetic complex-valued images, and handwritten MNIST digits transformed by a complex wavelet transform. Specifically, we show the necessity of a new amplitude-amplitude coupling term in our model. The proposed model is potentially valuable for machine learning tasks involving complex-valued data with amplitude variation, and for developing algorithms for novel computation hardware, such as coupled oscillators and neuromorphic hardware, on which Boltzmann sampling can be executed in the complex domain.

We study the problem of high-dimensional robust mean estimation in the presence of a constant fraction of adversarial outliers. A recent line of work has provided sophisticated polynomial-time algorithms for this problem with dimension-independent error guarantees for a range of natural distribution families. In this work, we show that a natural non-convex formulation of the problem can be solved directly by gradient descent. Our approach leverages a novel structural lemma, roughly showing that any approximate stationary point of our non-convex objective gives a near-optimal solution to the underlying robust estimation task. Our work establishes an intriguing connection between algorithmic high-dimensional robust statistics and non-convex optimization, which may have broader applications to other robust estimation tasks.

Machine-learning algorithms trained on features extracted from static code analysis can successfully detect Android malware. However, these approaches can be evaded by sparse evasion attacks that produce adversarial malware samples in which only few features are modified. This can be achieved, e.g., by injecting a small set of fake permissions and system calls into the malicious application, without compromising its intrusive functionality. To improve adversarial robustness against such sparse attacks, learning algorithms should avoid providing decisions which only rely upon a small subset of discriminant features; otherwise, even manipulating some of them may easily allow evading detection. Previous work showed that classifiers which avoid overemphasizing few discriminant features tend to be more robust against sparse attacks, and have developed simple metrics to help identify and select more robust algorithms. In this work, we aim to investigate whether gradient-based attribution methods used to explain classifiers' decisions by identifying the most relevant features can also be used to this end. Our intuition is that a classifier providing more uniform, evener attributions should rely upon a larger set of features, instead of overemphasizing few of them, thus being more robust against sparse attacks. We empirically investigate the connection between gradient-based explanations and adversarial robustness on a case study conducted on Android malware detection, and show that, in some cases, there is a strong correlation between the distribution of such explanations and adversarial robustness. We conclude the paper by discussing how our findings may thus enable the development of more efficient mechanisms both to evaluate and to improve adversarial robustness.