In order to find hyperparameters for a machine learning model, algorithms such as grid search or random search are used over the space of possible values of the models' hyperparameters. These search algorithms opt the solution that minimizes a specific cost function. In language models, perplexity is one of the most popular cost functions. In this study, we propose a fractional nonlinear programming model that finds the optimal perplexity value. The special structure of the model allows us to approximate it by a linear programming model that can be solved using the well-known simplex algorithm. To the best of our knowledge, this is the first attempt to use optimization techniques to find perplexity values in the language modeling literature. We apply our model to find hyperparameters of a language model and compare it to the grid search algorithm. Furthermore, we illustrating that it results in lower perplexity values. We perform this experiment on a real-world dataset from SwiftKey to validate our proposed approach.

Matrix completion is a well-studied problem with many machine learning applications. In practice, the problem is often solved by non-convex optimization algorithms. However, the current theoretical analysis for non-convex algorithms relies heavily on the assumption that every entry is observed with exactly the same probability $p$, which is not realistic in practice. In this paper, we investigate a more realistic semi-random model, where the probability of observing each entry is at least $p$. Even with this mild semi-random perturbation, we can construct counter-examples where existing non-convex algorithms get stuck in bad local optima. In light of the negative results, we propose a pre-processing step that tries to re-weight the semi-random input, so that it becomes "similar" to a random input. We give a nearly-linear time algorithm for this problem, and show that after our pre-processing, all the local minima of the non-convex objective can be used to approximately recover the underlying ground-truth matrix.

Clustering and classification critically rely on distance metrics that provide meaningful comparisons between data points. We present mixed-integer optimization approaches to find optimal distance metrics that generalize the Mahalanobis metric extensively studied in the literature. Additionally, we generalize and improve upon leading methods by removing reliance on pre-designated "target neighbors," "triplets," and "similarity pairs." Another salient feature of our method is its ability to enable active learning by recommending precise regions to sample after an optimal metric is computed to improve classification performance. This targeted acquisition can significantly reduce computational burden by ensuring training data completeness, representativeness, and economy. We demonstrate classification and computational performance of the algorithms through several simple and intuitive examples, followed by results on real image and medical datasets.

Variational inference has experienced a recent surge in popularity owing to stochastic approaches, which have yielded practical tools for a wide range of model classes. A key benefit is that stochastic variational inference obviates the tedious process of deriving analytical expressions for closed-form variable updates. Instead, one simply needs to derive the gradient of the log-posterior, which is often much easier. Yet for certain model classes, the log-posterior itself is difficult to optimize using standard gradient techniques. One such example are random field models, where optimization based on gradient linearization has proven popular, since it speeds up convergence significantly and can avoid poor local optima. In this paper we propose stochastic variational inference with gradient linearization (SVIGL). It is similarly convenient as standard stochastic variational inference - all that is required is a local linearization of the energy gradient. Its benefit over stochastic variational inference with conventional gradient methods is a clear improvement in convergence speed, while yielding comparable or even better variational approximations in terms of KL divergence. We demonstrate the benefits of SVIGL in three applications: Optical flow estimation, Poisson-Gaussian denoising, and 3D surface reconstruction.

Many machine learning problems can be formulated as consensus optimization problems which can be solved efficiently via a cooperative multi-agent system. However, the agents in the system can be unreliable due to a variety of reasons: noise, faults and attacks. Thus, providing falsified data leads the optimization process in a wrong direction, and degrades the performance of distributed machine learning algorithms. This paper considers the problem of decentralized learning using ADMM in the presence of unreliable agents. First, we rigorously analyze the effect of falsified updates (in ADMM learning iterations) on the convergence behavior of multi-agent system. We show that the algorithm linearly converges to a neighborhood of the optimal solution under certain conditions and characterize the neighborhood size analytically. Next, we provide guidelines for network structure design to achieve a faster convergence. We also provide necessary conditions on the falsified updates for exact convergence to the optimal solution. Finally, to mitigate the influence of unreliable agents, we propose a robust variant of ADMM and show its resilience to unreliable agents.

Applications in robotics, such as multi-robot target tracking, involve the execution of information acquisition tasks by teams of mobile robots. However, in failure-prone or adversarial environments, robots get attacked, their communication channels get jammed, and their sensors fail, resulting in the withdrawal of robots from the collective task, and, subsequently, the inability of the remaining active robots to coordinate with each other. As a result, traditional design paradigms become insufficient and, in contrast, resilient designs against system-wide failures and attacks become important. In general, resilient design problems are hard, and even though they often involve objective functions that are monotone and (possibly) submodular, scalable approximation algorithms for their solution have been hitherto unknown. In this paper, we provide the first algorithm, enabling the following capabilities: minimal communication, i.e., the algorithm is executed by the robots based only on minimal communication between them, system-wide resiliency, i.e., the algorithm is valid for any number of denial-of-service attacks and failures, and provable approximation performance, i.e., the algorithm ensures for all monotone and (possibly) submodular objective functions a solution that is finitely close to the optimal. We support our theoretical analyses with simulated and real-world experiments, by considering an active information acquisition application scenario, namely, multi-robot target tracking.

Two popular examples of first-order optimization methods over linear spaces are coordinate descent and matching pursuit algorithms, with their randomized variants. While the former targets the optimization by moving along coordinates, the latter considers a generalized notion of directions. Exploiting the connection between the two algorithms, we present a unified analysis of both, providing affine invariant sublinear $\mathcal{O}(1/t)$ rates on smooth objectives and linear convergence on strongly convex objectives. As a byproduct of our affine invariant analysis of matching pursuit, our rates for steepest coordinate descent are the tightest known. Furthermore, we show the first accelerated convergence rate $\mathcal{O}(1/t^2)$ for matching pursuit on convex objectives.

Population risk---the expectation of the loss over the sampling mechanism---is always of primary interest in machine learning. However, learning algorithms only have access to empirical risk, which is the average loss over training examples. Although the two risks are typically guaranteed to be pointwise close, for applications with nonconvex nonsmooth losses (such as modern deep networks), the effects of sampling can transform a well-behaved population risk into an empirical risk with a landscape that is problematic for optimization. The empirical risk can be nonsmooth, and it may have many additional local minima. This paper considers a general optimization framework which aims to find approximate local minima of a smooth nonconvex function $F$ (population risk) given only access to the function value of another function $f$ (empirical risk), which is pointwise close to $F$ (i.e., $\|F-f\|_{\infty} \le \nu$). We propose a simple algorithm based on stochastic gradient descent (SGD) on a smoothed version of $f$ which is guaranteed to find an $\epsilon$-second-order stationary point if $\nu \le O(\epsilon^{1.5}/d)$, thus escaping all saddle points of $F$ and all the additional local minima introduced by $f$. We also provide an almost matching lower bound showing that our SGD-based approach achieves the optimal trade-off between $\nu$ and $\epsilon$, as well as the optimal dependence on problem dimension $d$, among all algorithms making a polynomial number of queries. As a concrete example, we show that our results can be directly used to give sample complexities for learning a ReLU unit, whose empirical risk is nonsmooth.

The natural gradient method has been used effectively in conjugate Gaussian process models, but the non-conjugate case has been largely unexplored. We examine how natural gradients can be used in non-conjugate stochastic settings, together with hyperparameter learning. We conclude that the natural gradient can significantly improve performance in terms of wall-clock time. For ill-conditioned posteriors the benefit of the natural gradient method is especially pronounced, and we demonstrate a practical setting where ordinary gradients are unusable. We show how natural gradients can be computed efficiently and automatically in any parameterization, using automatic differentiation. Our code is integrated into the GPflow package.

Machine Learning (ML) is one of the most exciting and dynamic areas of modern research and application. The purpose of this review is to provide an introduction to the core concepts and tools of machine learning in a manner easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as the bias-variance tradeoff, overfitting, regularization, and generalization before moving on to more advanced topics in both supervised and unsupervised learning. Topics covered in the review include ensemble models, deep learning and neural networks, clustering and data visualization, energy-based models (including MaxEnt models and Restricted Boltzmann Machines), and variational methods. Throughout, we emphasize the many natural connections between ML and statistical physics. A notable aspect of the review is the use of Python notebooks to introduce modern ML/statistical packages to readers using physics-inspired datasets (the Ising Model and Monte-Carlo simulations of supersymmetric decays of proton-proton collisions). We conclude with an extended outlook discussing possible uses of machine learning for furthering our understanding of the physical world as well as open problems in ML where physicists maybe able to contribute. (Notebooks are available at https://physics.bu.edu/~pankajm/MLnotebooks.html )