We study online learning under logarithmic loss with regular parametric models. In this setting, each strategy corresponds to a joint distribution on sequences. The minimax optimal strategy is the normalized maximum likelihood (NML) strategy. We show that the sequential NML (SNML) strategy predicts minimax optimally (i.e., as NML) if and only if the joint distribution on sequences defined by SNML is exchangeable. This property also characterizes the optimality of a Bayesian prediction strategy. In that case, the optimal prior distribution is Jeffreys prior for a broad class of parametric models for which the maximum likelihood estimator is asymptotically normal. The optimal prediction strategy, NML, depends on the number n of rounds of the game, in general. However, when a Bayesian strategy is optimal, NML becomes independent of n. Our proof uses this to exploit the asymptotics of NML. The asymptotic normality of the maximum likelihood estimator is responsible for the necessity of Jeffreys prior.

In recent times we have witnessed the emergence of large online markets with two-sided preferences that are responsible for businesses worth billions of dollars. Recommendation systems are critical components of such markets. It is to be noted that the matching in such a market depends on the preferences of both sides, consequently, the construction of a recommendation system for such a market calls for consideration of preferences of both sides. The online dating market, and the online freelancer market are examples of markets with two-sided preferences. Recommendation systems for such markets are fundamentally different from typical rating based product recommendations. We pose this problem as a bipartite ranking problem. There has been extensive research on bipartite ranking algorithms. Typically, generalized linear regression models are popular methods of constructing such ranking on account of their ability to be learned easily from big data, and their computational simplicity on engineering platforms. However, we show that for markets with two sided preferences, one can improve the AUC (Area Under the receiver operator Curve) score by considering separate models for preferences of both the sides and constructing a two layer architecture for ranking. We call this a two-level model algorithm. For both synthetic and real data we show that the two-level model algorithm has a better AUC performance than the direct application of a generalized linear model such as L1logistic regression or an ensemble method such as random forest algorithm. We provide a theoretical justification of AUC optimality of two-level model and pose a theoretical problem for a more general result.

We propose two ensemble regression algorithms for forecasting the daily price of crude oil from features extracted from the U.S. Energy Administration and some international news agencies. An ensemble regression model consists of a group of homogeneous regressors with varying parameters, e.g. Linear regression models with different ridge regularization parameters. The first ensemble method called "recent leader" picks the individual regressor with least mean square error over recent data. The second model called "exponentially weighted ensemble" combines individual regressors in a linear fashion with weights of constituent models decaying exponentially with the mean square error over past predictions. These two methods were tested with linear regression, support vector regression, decision trees and Gaussian processes. Exponentially weighted ensemble with support vector regression had the best performance.

We study online learning under logarithmic loss with regular parametric models. Hedayati and Bartlett (2012) showed that a Bayesian prediction strategy with Jeffreys prior and sequential normalized maximum likelihood (SNML) coincide and are optimal if and only if the latter is exchangeable, which occurs if and only if the optimal strategy can be calculated without knowing the time horizon in advance. They put forward the question what families have exchangeable SNML strategies. We answer this question for one-dimensional exponential families: SNML is exchangeable only for three classes of natural exponential family distributions,namely the Gaussian, the gamma, and the Tweedie exponential family of order 3/2.