Near-Optimal Mean-Variance Controls under Two-time-scale Formulations and Applications
Zhixin Yang, G. Yin, Le Yi Wang, Hongwei Zhang
发表信息：
Stochastics An International Journal of Probability and Stochastic Processes: formerly Stochastics and Stochastics Reports Volume 85, Issue 4, 2013 Special Issue: Taksar Memorial Issue
Although the mean-variance control was initially formulated for financial portfolio management problems in which one wants to maximize expected return and control the risk, our motivations also stem from highway vehicle platoon controls that aim to maximize highway utility while ensuring zero accident. This paper develops near-optimal mean-variance controls of switching diffusion systems. To reduce the computational complexity, with motivations from earlier work on singularly perturbed Markovian systems \cite{SethiZ94,Yin&Zhang,GZB}, we use a two-time-scale formulation to treat the underlying systems, which is represented by use of a small parameter. As the small parameter goes to 0, we obtain a limit problem. Using the limit problem as a guide, we construct controls for the original problem, and show that the control so constructed is nearly optimal.
When we implement a portfolio selection methodology under a mean-risk formulation, it is essential to correctly model investors' risk aversion which may be time-dependent, or even state-dependent during the investment procedure. In this paper, we propose a behavior risk aversion model, which is a piecewise linear function of the current wealth level with a reference point at a preset investment target. Due to the time inconsistency of the resulting multi-period mean-variance model with an adaptive risk aversion, we investigate in this paper the time consistent behavior portfolio policy by solv…
We propose a unified framework to speed up the existing stochastic matrix factorization (SMF) algorithms via variance reduction. Our framework is general and it subsumes several well-known SMF formulations in the literature. We perform a non-asymptotic convergence analysis of our framework and derive computational and sample complexities for our algorithm to converge to an $ε$-stationary point in expectation. In addition, extensive experiments for a wide class of SMF formulations demonstrate that our framework consistently yields faster convergence and a more accurate output dictionary vis-à-v…
Visual Computing Center（视觉计算中心）；Industrial and Systems Engineering（工业与系统工程系）
Given a multivariate data set, sparse principal component analysis (SPCA) aims to extract several linear combinations of the variables that together explain the variance in the data as much as possible, while controlling the number of nonzero loadings in these combinations. In this paper we consider 8 different optimization formulations for computing a single sparse loading vector; these are obtained by combining the following factors: we employ two norms for measuring variance (L2, L1) and two sparsity-inducing norms (L0, L1), which are used in two different ways (constraint, penalty). Three …
There is a recent surge of interest in nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem in the purpose of efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the global optima effectively despite their empirical success in applications. In this paper, we provide an answer that a stochast…
Efficient sparse GP-MPC with accurate mean and variance propagation applied for quadcopter flight control
Eindhoven University of Technology（埃因霍温理工大学）；HUN-REN Institute for Computer Science and Control（匈牙利科学院计算机与控制研究所）；Széchenyi István University（塞切尼伊什特万大学）
In this work we develop a scalable computational framework for the solution of PDE-constrained optimal control under high-dimensional uncertainty. Specifically, we consider a mean-variance formulation of the control objective and employ a Taylor expansion with respect to the uncertain parameter either to directly approximate the control objective or as a control variate for variance reduction. The expressions for the mean and variance of the Taylor approximation are known analytically, although their evaluation requires efficient computation of the trace of the (preconditioned) Hessian of the …
We consider an optimal investment and risk control problem for an insurer under the mean-variance (MV) criterion. By introducing a deterministic auxiliary process defined forward in time, we formulate an alternative time-consistent problem related to the original MV problem, and obtain the optimal strategy and the value function to the new problem in closed-form. We compare our formulation and optimal strategy to those under the precommitment and game-theoretic framework. Numerical studies show that, when the financial market is negatively correlated with the risk process, optimal investment m…
Stochastic variance reduced extragradient methods for solving hierarchical variational inequalities
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机学研究所）；Mannheim University（曼海姆大学）；Technion - Israel Institute of Technology（以色列理工学院）
The Ohio State University（俄亥俄州立大学）；The University of Utah（犹他大学）；Duke University（杜克大学）
SARAH and SPIDER are two recently developed stochastic variance-reduced algorithms, and SPIDER has been shown to achieve a near-optimal first-order oracle complexity in smooth nonconvex optimization. However, SPIDER uses an accuracy-dependent stepsize that slows down the convergence in practice, and cannot handle objective functions that involve nonsmooth regularizers. In this paper, we propose SpiderBoost as an improved scheme, which allows to use a much larger constant-level stepsize while maintaining the same near-optimal oracle complexity, and can be extended with proximal mapping to handl…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
Recently, several studies (Zhou et al., 2021a; Zhang et al., 2021b; Kim et al., 2021; Zhou and Gu, 2022) have provided variance-dependent regret bounds for linear contextual bandits, which interpolates the regret for the worst-case regime and the deterministic reward regime. However, these algorithms are either computationally intractable or unable to handle unknown variance of the noise. In this paper, we present a novel solution to this open problem by proposing the first computationally efficient algorithm for linear bandits with heteroscedastic noise. Our algorithm is adaptive to the unkno…
