Argumentation theory is a powerful paradigm that formalizes a type of commonsense reasoning that aims to simulate the human ability to resolve a specific problem in an intelligent manner. A classical argumentation process takes into account only the properties related to the intrinsic logical soundness of an argument in order to determine its acceptability status. However, these properties are not always the only ones that matter to establish the argument's acceptability---there exist other qualities, such as strength, weight, social votes, trust degree, relevance level, and certainty degree, among others.

We study stochastic gradient descent {\em without replacement} (\sgdwor) for smooth convex functions. \sgdwor is widely observed to converge faster than true \sgd where each sample is drawn independently {\em with replacement}~\cite{bottou2009curiously} and hence, is more popular in practice. But it's convergence properties are not well understood as sampling without replacement leads to coupling between iterates and gradients. By using method of exchangeable pairs to bound Wasserstein distance, we provide the first non-asymptotic results for \sgdwor when applied to {\em general smooth, strongly-convex} functions. In particular, we show that \sgdwor converges at a rate of $O(1/K^2)$ while \sgd~is known to converge at $O(1/K)$ rate, where $K$ denotes the number of passes over data and is required to be {\em large enough}. Existing results for \sgdwor in this setting require additional {\em Hessian Lipschitz assumption}~\cite{gurbuzbalaban2015random,haochen2018random}. For {\em small} $K$, we show \sgdwor can achieve same convergence rate as \sgd for {\em general smooth strongly-convex} functions. Existing results in this setting require $K=1$ and hold only for generalized linear models \cite{shamir2016without}. In addition, by careful analysis of the coupling, for both large and small $K$, we obtain better dependence on problem dependent parameters like condition number.

Current approaches to website conversion testing are severely limited relying on incremental steps within a narrow range. What kind of hill climber are you? Many of us don't think of ourselves as hill climbers when we think of our daily jobs. We are marketers, product managers or digital experience professionals who are endlessly pursuing better performance (peaks) from our campaigns and work. At Sentient, we talk about the concept of the hill climbing metaphor frequently for how conversion experts and professionals do their work. When you are running experiments or trying to optimize your site or a campaign, you are looking to seek gains (higher conversion rates, increase AOV, increased sales, etc.) to drive growth.

The Expectation-Maximization (EM) algorithm is one of the most popular methods used to solve the problem of distribution-based clustering in unsupervised learning. In this paper, we propose an analysis of a generalized EM (GEM) algorithm and a designed EM-like algorithm, as linear time-invariant (LTI) systems with a feedback nonlinearity, and by leveraging tools from robust control theory, particularly integral quadratic constraints (IQCs). Towards this goal, we investigate the absolute stability of dynamical systems of the above form with a sector-bounded feedback nonlinearity, that represent the aforementioned algorithms. This analysis allows us to craft a strongly convex objective function, which led to the design of the aforementioned novel EM-like algorithm for Gaussian mixture models (GMMs). Furthermore, it allows us to establish bounds on the convergence rates of the studied algorithms. In particular, the derived bounds for our proposed EM-like algorithm generalize bounds found in the literature for the EM algorithm on GMMs, and our analysis of an existing gradient ascent GEM algorithm based on the $Q$-function allowed us to approximately recover bounds found in the literature.

Building a good predictive model requires an array of activities such as data imputation, feature transformations, estimator selection, hyper-parameter search and ensemble construction. Given the large, complex and heterogenous space of options, off-the-shelf optimization methods are infeasible for realistic response times. In practice, much of the predictive modeling process is conducted by experienced data scientists, who selectively make use of available tools. Over time, they develop an understanding of the behavior of operators, and perform serial decision making under uncertainty, colloquially referred to as educated guesswork. With an unprecedented demand for application of supervised machine learning, there is a call for solutions that automatically search for a good combination of parameters across these tasks to minimize the modeling error. We introduce a novel system called APRL (Autonomous Predictive modeler via Reinforcement Learning), that uses past experience through reinforcement learning to optimize such sequential decision making from within a set of diverse actions under a time constraint on a previously unseen predictive learning problem. APRL actions are taken to optimize the performance of a final ensemble. This is in contrast to other systems, which maximize individual model accuracy first and create ensembles as a disconnected post-processing step. As a result, APRL is able to reduce up to 71\% of classification error on average over a wide variety of problems.

Matrix completion has attracted much interest in the past decade in machine learning and computer vision. For low-rank promotion in matrix completion, the nuclear norm penalty is convenient due to its convexity but has a bias problem. Recently, various algorithms using nonconvex penalties have been proposed, among which the proximal gradient descent (PGD) algorithm is one of the most efficient and effective. For the nonconvex PGD algorithm, whether it converges to a local minimizer and its convergence rate are still unclear. This work provides a nontrivial analysis on the PGD algorithm in the nonconvex case. Besides the convergence to a stationary point for a generalized nonconvex penalty, we provide more deep analysis on a popular and important class of nonconvex penalties which have discontinuous thresholding functions. For such penalties, we establish the finite rank convergence, convergence to restricted strictly local minimizer and eventually linear convergence rate of the PGD algorithm. Meanwhile, convergence to a local minimizer has been proved for the hard-thresholding penalty. Our result is the first shows that, nonconvex regularized matrix completion only has restricted strictly local minimizers, and the PGD algorithm can converge to such minimizers with eventually linear rate under certain conditions. Illustration of the PGD algorithm via experiments has also been provided. Code is available at https://github.com/FWen/nmc.

Summary: True prescriptive analytics requires the use of real optimization techniques that very few applications actually use. Here's a refresher on optimization with examples of where and how they're best used. Predictive analytics and optimization have gone hand in hand since the very beginning. But in 2014 some erudite journal decided we needed another phrase for this combo and it became Prescriptive Analytics, theoretically differentiating what could happen (predictive) from what should happen (prescriptive) through the application of optimization. Originally I felt strongly that this was a distinction without a difference and only served to confuse our customers who were having a hard enough time five years ago understanding why they should even be doing predictive.

Previous work has shown that popular trending events are important external factors which pose significant influence on user search behavior and also provided a way to computationally model this influence. However, their problem formulation was based on the strong assumption that each event poses its influence independently. This assumption is unrealistic as there are many correlated events in the real world which influence each other and thus, would pose a joint influence on the user search behavior rather than posing influence independently. In this paper, we study this novel problem of Modeling the Joint Influences posed by multiple correlated events on user search behavior. We propose a Joint Influence Model based on the Multivariate Hawkes Process which captures the inter-dependency among multiple events in terms of their influence upon user search behavior. We evaluate the proposed Joint Influence Model using two months query-log data from https://search.yahoo.com/. Experimental results show that the model can indeed capture the temporal dynamics of the joint influence over time and also achieves superior performance over different baseline methods when applied to solve various interesting prediction problems as well as real-word application scenarios, e.g., query auto-completion.

We introduce the active exploration problem in Markov decision processes (MDPs). Each state of the MDP is characterized by a random value and the learner should gather samples to estimate the mean value of each state as accurately as possible. Similarly to active exploration in multi-armed bandit (MAB), states may have different levels of noise, so that the higher the noise, the more samples are needed. As the noise level is initially unknown, we need to trade off the exploration of the environment to estimate the noise and the exploitation of these estimates to compute a policy maximizing the accuracy of the mean predictions. We introduce a novel learning algorithm to solve this problem showing that active exploration in MDPs may be significantly more difficult than in MAB. We also derive a heuristic procedure to mitigate the negative effect of slowly mixing policies. Finally, we validate our findings on simple numerical simulations.

The predictive quality of machine learning models is typically measured in terms of their (approximate) expected prediction accuracy or the so-called Area Under the Curve (AUC). Minimizing the reciprocals of these measures are the goals of supervised learning. However, when the models are constructed by the means of empirical risk minimization (ERM), surrogate functions such as the logistic loss or hinge loss are optimized instead. In this work, we show that in the case of linear predictors, the expected error and the expected ranking loss can be effectively approximated by smooth functions whose closed form expressions and those of their first (and second) order derivatives depend on the first and second moments of the data distribution, which can be precomputed. Hence, the complexity of an optimization algorithm applied to these functions does not depend on the size of the training data. These approximation functions are derived under the assumption that the output of the linear classifier for a given data set has an approximately normal distribution. We argue that this assumption is significantly weaker than the Gaussian assumption on the data itself and we support this claim by demonstrating that our new approximation is quite accurate on data sets that are not necessarily Gaussian. We present computational results that show that our proposed approximations and related optimization algorithms can produce linear classifiers with similar or better test accuracy or AUC, than those obtained using state-of-the-art approaches, in a fraction of the time.