On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints

The problem of minimizing convex functionals of probability distributions is solved under the assumption that the density of every distribution is bounded from above and below. A system of sufficient and necessary first-order optimality conditions as well as a bound on the optimality gap of feasible candidate solutions are derived. Based on these results, two numerical algorithms are proposed that iteratively solve the system of optimality conditions on a grid of discrete points. Both algorithms use a block coordinate descent strategy and terminate once the optimality gap falls below the desired tolerance. While the first algorithm is conceptually simpler and more efficient, it is not guaranteed to converge for objective functions that are not strictly convex. This shortcoming is overcome in the second algorithm, which uses an additional outer proximal iteration, and, which is proven to converge under mild assumptions. Two examples are given to demonstrate the theoretical usefulness of the optimality conditions as well as the high efficiency and accuracy of the proposed numerical algorithms.Comment: 13 pages, 5 figures, 2 tables, published in the IEEE Transactions on Signal Processing. In previous versions, the example in Section VI.B contained some mistakes and inaccuracies, which have been fixed in this versio

A Linear Programming Approach to Sequential Hypothesis Testing

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem, depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function with respect to these multipliers coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and an be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.Comment: 25 pages, 4 figures, accepted for publication in Sequential Analysi

On the Equivalence of f-Divergence Balls and Density Bands in Robust Detection

The paper deals with minimax optimal statistical tests for two composite hypotheses, where each hypothesis is defined by a non-parametric uncertainty set of feasible distributions. It is shown that for every pair of uncertainty sets of the f-divergence ball type, a pair of uncertainty sets of the density band type can be constructed, which is equivalent in the sense that it admits the same pair of least favorable distributions. This result implies that robust tests under $f$-divergence ball uncertainty, which are typically only minimax optimal for the single sample case, are also fixed sample size minimax optimal with respect to the equivalent density band uncertainty sets.Comment: 5 pages, 1 figure, accepted for publication in the Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 201

On Optimizing the Conditional Value-at-Risk of a Maximum Cost for Risk-Averse Safety Analysis

The popularity of Conditional Value-at-Risk (CVaR), a risk functional from finance, has been growing in the control systems community due to its intuitive interpretation and axiomatic foundation. We consider a non-standard optimal control problem in which the goal is to minimize the CVaR of a maximum random cost subject to a Borel-space Markov decision process. The objective takes the form $\text{CVaR}_{\alpha}(\max_{t=0,1,\dots,N} C_t)$, where $\alpha$ is a risk-aversion parameter representing a fraction of worst cases, $C_t$ is a stage or terminal cost, and $N \in \mathbb{N}$ is the length of a finite discrete-time horizon. The objective represents the maximum departure from a desired operating region averaged over a given fraction $\alpha$ of worst cases. This problem provides a safety criterion for a stochastic system that is informed by both the probability and severity of the potential consequences of the system's trajectory. In contrast, existing safety analysis frameworks apply stage-wise risk constraints (i.e., $\rho(C_t)$ must be small for all $t$, where $\rho$ is a risk functional) or assess the probability of constraint violation without quantifying its possible severity. To the best of our knowledge, the problem of interest has not been solved. To solve the problem, we propose and study a family of stochastic dynamic programs on an augmented state space. We prove that the optimal CVaR of a maximum cost enjoys an equivalent representation in terms of the solutions to this family of dynamic programs under appropriate assumptions. We show the existence of an optimal policy that depends on the dynamics of an augmented state under a measurable selection condition. Moreover, we demonstrate how our safety analysis framework is useful for assessing the severity of combined sewer overflows under precipitation uncertainty.Comment: A shorter version is under review for IEEE Transactions on Automatic Control, submitted December 202

アクション・リサーチによる教員養成訓練

本研究論文は、日本において英語のみで授業を行う教師のための研修プログラムで用いられる実地研究プロジェクトについて述べている。研修生となる学生は、１回の教育実習を基に各自の研究テーマを選んだ。その後小規模の研究を計画し、テーマに関する調査を行った上で2 回目の授業を行った。指導者はプロジェクトに組織的な問題があることに気付いたが、学生は役に立つし有意義だと感じた。ここでは学生が行ったプロジェクトの４例を紹介する。This paper describes an action research project that was used in an English-medium teachertraining program in Japan. Trainees chose their own subjects of research, based on one teaching event. They planned a small piece of research, investigated the subject, and then taught a second class. The instructors found that, although there were logistical problems with the project, the students found it helpful and rewarding. Four examples of student projects are included