Markov Decision Processes (MDPs) are an effective model to represent decision processes in the presence of transitional uncertainty and reward tradeoffs. However, due to the difficulty in exactly specifying the transition and reward functions in MDPs, researchers have proposed uncertain MDP models and robustness objectives in solving those models. Most approaches for computing robust policies have focused on the computation of maximin policies which maximize the value in the worst case amongst all realisations of uncertainty. Given the overly conservative nature of maximin policies, recent work has proposed minimax regret as an ideal alternative to the maximin objective for robust optimization. However, existing algorithms for handling minimax regret are restricted to models with uncertainty over rewards only and they are also limited in their scalability. Therefore, we provide a general model of uncertain MDPs that considers uncertainty over both transition and reward functions. Furthermore, we also consider dependence of the uncertainty across different states and decision epochs. We also provide a mixed integer linear program formulation for minimizing regret given a set of samples of the transition and reward functions in the uncertain MDP. In addition, we provide two myopic variants of regret, namely Cumulative Expected Myopic Regret (CEMR) and One Step Regret (OSR) that can be optimized in a scalable manner. Specifically, we provide dynamic programming and policy iteration based algorithms to optimize CEMR and OSR respectively. Finally, to demonstrate the effectiveness of our approaches, we provide comparisons on two benchmark problems from literature. We observe that optimizing the myopic variants of regret, OSR and CEMR are better than directly optimizing the regret.

The goal of this tutorial is to introduce key models, algorithms, and open questions related to the use of optimization methods for solving problems arising in machine learning. It is written with an INFORMS audience in mind, specifically those readers who are familiar with the basics of optimization algorithms, but less familiar with machine learning. We begin by deriving a formulation of a supervised learning problem and show how it leads to various optimization problems, depending on the context and underlying assumptions. We then discuss some of the distinctive features of these optimization problems, focusing on the examples of logistic regression and the training of deep neural networks. The latter half of the tutorial focuses on optimization algorithms, first for convex logistic regression, for which we discuss the use of first-order methods, the stochastic gradient method, variance reducing stochastic methods, and second-order methods. Finally, we discuss how these approaches can be employed to the training of deep neural networks, emphasizing the difficulties that arise from the complex, nonconvex structure of these models.

The max-product {belief propagation} (BP) is a popular message-passing heuristic for approximating a maximum-a-posteriori (MAP) assignment in a joint distribution represented by a graphical model (GM). In the past years, it has been shown that BP can solve a few classes of linear programming (LP) formulations to combinatorial optimization problems including maximum weight matching, shortest path and network flow, i.e., BP can be used as a message-passing solver for certain combinatorial optimizations. However, those LPs and corresponding BP analysis are very sensitive to underlying problem setups, and it has been not clear what extent these results can be generalized to. In this paper, we obtain a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, traveling salesman, cycle packing, vertex/edge cover and network flow.

In this paper, we study the missing sample recovery problem using methods based on sparse approximation. In this regard, we investigate the algorithms used for solving the inverse problem associated with the restoration of missed samples of image signal. This problem is also known as inpainting in the context of image processing and for this purpose, we suggest an iterative sparse recovery algorithm based on constrained $l_1$-norm minimization with a new fidelity metric. The proposed metric called Convex SIMilarity (CSIM) index, is a simplified version of the Structural SIMilarity (SSIM) index, which is convex and error-sensitive. The optimization problem incorporating this criterion, is then solved via Alternating Direction Method of Multipliers (ADMM). Simulation results show the efficiency of the proposed method for missing sample recovery of 1D patch vectors and inpainting of 2D image signals.

Machine Learning (ML) has been successfully applied to a wide range of domains and applications. One of the techniques behind most of these successful applications is Ensemble Learning (EL), the field of ML that gave birth to methods such as Random Forests or Boosting. The complexity of applying these techniques together with the market scarcity on ML experts, has created the need for systems that enable a fast and easy drop-in replacement for ML libraries. Automated machine learning (autoML) is the field of ML that attempts to answers these needs. Typically, these systems rely on optimization techniques such as bayesian optimization to lead the search for the best model. Our approach differs from these systems by making use of the most recent advances on metalearning and a learning to rank approach to learn from metadata. We propose autoBagging, an autoML system that automatically ranks 63 bagging workflows by exploiting past performance and dataset characterization. Results on 140 classification datasets from the OpenML platform show that autoBagging can yield better performance than the Average Rank method and achieve results that are not statistically different from an ideal model that systematically selects the best workflow for each dataset. For the purpose of reproducibility and generalizability, autoBagging is publicly available as an R package on CRAN.

Increasing technological sophistication and widespread use of smartphones and wearable devices provide opportunities for innovative and highly personalized health interventions. A Just-In-Time Adaptive Intervention (JITAI) uses real-time data collection and communication capabilities of modern mobile devices to deliver interventions in real-time that are adapted to the in-the-moment needs of the user. The lack of methodological guidance in constructing data-based JITAIs remains a hurdle in advancing JITAI research despite the increasing popularity of JITAIs among clinical scientists. In this article, we make a first attempt to bridge this methodological gap by formulating the task of tailoring interventions in real-time as a contextual bandit problem. Interpretability requirements in the domain of mobile health lead us to formulate the problem differently from existing formulations intended for web applications such as ad or news article placement. Under the assumption of linear reward function, we choose the reward function (the "critic") parameterization separately from a lower dimensional parameterization of stochastic policies (the "actor"). We provide an online actor-critic algorithm that guides the construction and refinement of a JITAI. Asymptotic properties of the actor-critic algorithm are developed and backed up by numerical experiments. Additional numerical experiments are conducted to test the robustness of the algorithm when idealized assumptions used in the analysis of contextual bandit algorithm are breached.

Generalized linear model with $L_1$ and $L_2$ regularization is a widely used technique for solving classification, class probability estimation and regression problems. With the numbers of both features and examples growing rapidly in the fields like text mining and clickstream data analysis parallelization and the use of cluster architectures becomes important. We present a novel algorithm for fitting regularized generalized linear models in the distributed environment. The algorithm splits data between nodes by features, uses coordinate descent on each node and line search to merge results globally. Convergence proof is provided. A modifications of the algorithm addresses slow node problem. For an important particular case of logistic regression we empirically compare our program with several state-of-the art approaches that rely on different algorithmic and data spitting methods. Experiments demonstrate that our approach is scalable and superior when training on large and sparse datasets.

We develop a simple routine unifying the analysis of several important recently-developed stochastic optimization methods including SAGA, Finito, and stochastic dual coordinate ascent (SDCA). First, we show an intrinsic connection between stochastic optimization methods and dynamic jump systems, and propose a general jump system model for stochastic optimization methods. Our proposed model recovers SAGA, SDCA, Finito, and SAG as special cases. Then we combine jump system theory with several simple quadratic inequalities to derive sufficient conditions for convergence rate certifications of the proposed jump system model under various assumptions (with or without individual convexity, etc). The derived conditions are linear matrix inequalities (LMIs) whose sizes roughly scale with the size of the training set. We make use of the symmetry in the stochastic optimization methods and reduce these LMIs to some equivalent small LMIs whose sizes are at most 3 by 3. We solve these small LMIs to provide analytical proofs of new convergence rates for SAGA, Finito and SDCA (with or without individual convexity). We also explain why our proposed LMI fails in analyzing SAG. We reveal a key difference between SAG and other methods, and briefly discuss how to extend our LMI analysis for SAG. An advantage of our approach is that the proposed analysis can be automated for a large class of stochastic methods under various assumptions (with or without individual convexity, etc).

When we use the 1st, 2nd, and nth (with a large n) order polynomial regression, we may get the following results. As we can see, the 1st order model is too simple to fit the data, the 2nd order model has reasonable fitting, and the nth order model is very complicated and can fit every single sample very well. But the problem with the 3rd case is that machine learning is not about fitting the training data, but about generalizing beyond the training data. While it can achieve small training error, it will have large test error.

Mutual Information (MI) is often used for feature selection when developing classifier models. Estimating the MI for a subset of features is often intractable. We demonstrate, that under the assumptions of conditional independence, MI between a subset of features can be expressed as the Conditional Mutual Information (CMI) between pairs of features. But selecting features with the highest CMI turns out to be a hard combinatorial problem. In this work, we have applied two unique global methods, Truncated Power Method (TPower) and Low Rank Bilinear Approximation (LowRank), to solve the feature selection problem. These algorithms provide very good approximations to the NP-hard CMI based feature selection problem. We experimentally demonstrate the effectiveness of these procedures across multiple datasets and compare them with existing MI based global and iterative feature selection procedures.