Mehmet A. Donmez

Optimal Investment Under Transaction Costs: A Threshold Rebalanced Portfolio Approach

We study how to invest optimally in a financial market having a finite number of assets from a signal processing perspective. Specifically, we investigate how an investor should distribute capital over these assets and when he/she should reallocate the distribution of the funds over these assets to maximize the expected cumulative wealth over any investment period. In particular, we introduce a portfolio selection algorithm that maximizes the expected cumulative wealth in i.i.d. two-asset discrete-time markets where the market levies proportional transaction costs in buying and selling stocks. We achieve this using “threshold rebalanced portfolios”, where trading occurs only if the portfolio breaches certain thresholds. Under the assumption that the relative price sequences have log-normal distribution from the Black-Scholes model, we evaluate the expected wealth under proportional transaction costs and find the threshold rebalanced portfolio that achieves the maximal expected cumulative wealth over any investment period. Our derivations can be readily extended to markets having more than two stocks, where these extensions are provided in the paper. As predicted from our derivations, we significantly improve the achieved wealth with respect to the portfolio selection algorithms from the literature on historical data sets under both mild and heavy transaction costs.

A Deterministic Analysis of an Online Convex Mixture of Experts Algorithm

We analyze an online learning algorithm that adaptively combines outputs of two constituent algorithms (or the experts) running in parallel to estimate an unknown desired signal. This online learning algorithm is shown to achieve and in some cases outperform the mean-square error (MSE) performance of the best constituent algorithm in the steady state. However, the MSE analysis of this algorithm in the literature uses approximations and relies on statistical models on the underlying signals. Hence, such an analysis may not be useful or valid for signals generated by various real-life systems that show high degrees of nonstationarity, limit cycles and that are even chaotic in many cases. In this brief, we produce results in an individual sequence manner. In particular, we relate the time-accumulated squared estimation error of this online algorithm at any time over any interval to the one of the optimal convex mixture of the constituent algorithms directly tuned to the underlying signal in a deterministic sense without any statistical assumptions. In this sense, our analysis provides the transient, steady-state, and tracking behavior of this algorithm in a strong sense without any approximations in the derivations or statistical assumptions on the underlying signals such that our results are guaranteed to hold. We illustrate the introduced results through examples.

Adaptive mixture methods based on Bregman divergences

We investigate adaptive mixture methods that linearly combine outputs of m constituent filters running in parallel to model a desired signal. We use Bregman divergences and obtain certain multiplicative updates to train the linear combination weights under an affine constraint or without any constraints. We use unnormalized relative entropy and relative entropy to define two different Bregman divergences that produce an unnormalized exponentiated gradient update and a normalized exponentiated gradient update on the mixture weights, respectively. We then carry out the mean and the mean-square transient analysis of these adaptive algorithms when they are used to combine outputs of m constituent filters. We illustrate the accuracy of our results and demonstrate the effectiveness of these updates for sparse mixture systems.

Steady State and Transient MSE Analysis of Convexly Constrained Mixture Methods

We investigate convexly constrained mixture methods to adaptively combine outputs of two adaptive filters running in parallel to model a desired unknown system. We compare several algorithms with respect to their mean-square error in the steady state, when the underlying unknown system is nonstationary with a random walk model. We demonstrate that these algorithms are universal such that they achieve the performance of the best constituent filter in the steady state if certain algorithmic parameters are chosen properly. We also demonstrate that certain mixtures converge to the optimal convex combination filter such that their steady-state performances can be better than the best constituent filter. We also perform the transient analysis of these updates in the mean and mean-square error sense. Furthermore, we show that the investigated convexly constrained algorithms update certain auxiliary variables through sigmoid nonlinearity, hence, in this sense, related.

Robust estimation in flat fading channels under bounded channel uncertainties

We investigate channel equalization problem for time-varying flat fading channels under bounded channel uncertainties. We analyze three robust methods to estimate an unknown signal transmitted through a time-varying flat fading channel. These methods are based on minimizing certain mean-square error criteria that incorporate the channel uncertainties into their problem formulations instead of directly using the inaccurate channel information that is available. We present closed-form solutions to the channel equalization problems for each method and for both zero mean and nonzero mean signals. We illustrate the performances of the equalization methods through simulations.

Adaptive mixture methods using Bregman divergences

We investigate affinely constrained mixture methods adaptively combining outputs of m constituent filters running in parallel to model a desired signal. We use Bregman divergences and obtain multiplicative updates to train these linear combination weights under the affine constraints. We use the unnormalized relative entropy and the relative entropy that produce the exponentiated gradient update with unnormalized weights (EGU) and the exponentiated gradient update with positive and negative weights (EG), respectively. We carry out the mean and the mean-square transient analysis of the affinely constrained mixtures of m filters using the EGU or EG algorithms. We compare performances of different algorithms through our simulations and illustrate the accuracy of our results.

Steady state MSE analysis of convexly constrained mixture methods

We study the steady-state performances of four convexly constrained mixture algorithms that adaptively combine outputs of two adaptive filters running in parallel to model an unknown system. We demonstrate that these algorithms are universal such that they achieve the performance of the best constituent filter in the steady-state if certain algorithmic parameters are chosen properly. We also demonstrate that certain mixtures converge to the optimal convex combination filter such that their steady-state performances can be better than the best constituent filter. Furthermore, we show that the investigated convexly constrained algorithms update certain auxiliary variables through sigmoid nonlinearity, hence, in this sense, related.

Robust least squares methods under bounded data uncertainties

We study the problem of estimating an unknown deterministic signal that is observed through an unknown deterministic data matrix under additive noise. In particular, we present a minimax optimization framework to the least squares problems, where the estimator has imperfect data matrix and output vector information. We define the performance of an estimator relative to the performance of the optimal least squares (LS) estimator tuned to the underlying unknown data matrix and output vector, which is defined as the regret of the estimator. We then introduce an efficient robust LS estimation approach that minimizes this regret for the worst possible data matrix and output vector, where we refrain from any structural assumptions on the data. We demonstrate that minimizing this worst-case regret can be cast as a semi-definite programming (SDP) problem. We then consider the regularized and structured LS problems and present novel robust estimation methods by demonstrating that these problems can also be cast as SDP problems. We illustrate the merits of the proposed algorithms with respect to the well-known alternatives in the literature through our simulations. Introducing robust estimation algorithms based on a novel regret formulation.Performance tradeoff between best-case and worst-case optimal estimators owing to the regret formulation.Demonstrating the superior performance of the introduced novel algorithms.

Competitive and online piecewise linear classification

In this paper, we study the binary classification problem in machine learning and introduce a novel classification algorithm based on the “Context Tree Weighting Method”. The introduced algorithm incrementally learns a classification model through sequential updates in the course of a given data stream, i.e., each data point is processed only once and forgotten after the classifier is updated, and asymptotically achieves the performance of the best piecewise linear classifiers defined by the “context tree”. Since the computational complexity is only linear in the depth of the context tree, our algorithm is highly scalable and appropriate for real time processing. We present experimental results on several benchmark data sets and demonstrate that our method provides significant computational improvement both in the test (5 ~ 35×) and training phases (40 ~ 1000×), while achieving high classification accuracy in comparison to the SVM with RBF kernel.

Transient analysis of convexly constrained mixture methods

We study the transient performances of three convexly constrained adaptive combination methods that combine outputs of two adaptive filters running in parallel to model a desired unknown system. We propose a theoretical model for the mean and mean-square convergence behaviors of each algorithm. Specifically, we provide expressions for the time evolution of the mean and the variance of the combination parameters, as well as for the mean square errors. The accuracy of the theoretical models are illustrated through simulations in the case of a mixture of two LMS filters with different step sizes.