Understanding Stochastic Optimization (Artificial Intelligence)
- Deep Stochastic Optimization in Finance(arXiv)
Author : A. Max Reppen, H. Mete Soner, Valentin Tissot-Daguette
Abstract : This paper outlines, and through stylized examples evaluates a novel and highly effective computational technique in quantitative finance. Empirical Risk Minimization (ERM) and neural networks are key to this approach. Powerful open source optimization libraries allow for efficient implementations of this algorithm making it viable in high-dimensional structures. The free-boundary problems related to American and Bermudan options showcase both the power and the potential difficulties that specific applications may face. The impact of the size of the training data is studied in a simplified Merton type problem. The classical option hedging problem exemplifies the need of market generators or large number of simulations.
2. Decentralized Stochastic Optimization with Inherent Privacy Protection(arXiv)
Author : Yongqiang Wang, H. Vincent Poor
Abstract : Decentralized stochastic optimization is the basic building block of modern collaborative machine learning, distributed estimation and control, and large-scale sensing. Since involved data usually contain sensitive information like user locations, healthcare records and financial transactions, privacy protection has become an increasingly pressing need in the implementation of decentralized stochastic optimization algorithms. In this paper, we propose a decentralized stochastic gradient descent algorithm which is embedded with inherent privacy protection for every participating agent against other participating agents and external eavesdroppers. This proposed algorithm builds in a dynamics based gradient-obfuscation mechanism to enable privacy protection without compromising optimization accuracy, which is in significant difference from differential-privacy based privacy solutions for decentralized optimization that have to trade optimization accuracy for privacy. The dynamics based privacy approach is encryption-free, and hence avoids incurring heavy communication or computation overhead, which is a common problem with encryption based privacy solutions for decentralized stochastic optimization. Besides rigorously characterizing the convergence performance of the proposed decentralized stochastic gradient descent algorithm under both convex objective functions and non-convex objective functions, we also provide rigorous information-theoretic analysis of its strength of privacy protection. Simulation results for a distributed estimation problem as well as numerical experiments for decentralized learning on a benchmark machine learning dataset confirm the effectiveness of the proposed approach
3. Duality in convex stochastic optimization(arXiv)
Author : Teemu Pennanen, Ari-Pekka Perkkiö
Abstract : This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in 1976. We derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new frameworks in stochastic programming and stochastic optimal control.
4. Discrete Stochastic Optimization for Public Health Interventions with Constraints(arXiv)
Author : Zewei Li, James C. Spall
Abstract : Many public health threats exist, motivating the need to find optimal intervention strategies. Given the stochastic nature of the threats (e.g., the spread of pandemic influenza, the occurrence of drug overdoses, and the prevalence of alcohol-related threats), deterministic optimization approaches may be inappropriate. In this paper, we implement a stochastic optimization method to address aspects of the 2009 H1N1 and the COVID-19 pandemics, with the spread of disease modeled by the open source Monte Carlo simulations, FluTE and Covasim, respectively. Without testing every possible option, the objective of the optimization is to determine the best combination of intervention strategies so as to result in minimal economic loss to society. To reach our objective, this application-oriented paper uses the discrete simultaneous perturbation stochastic approximation method (DSPSA), a recursive simulation-based optimization algorithm, to update the input parameters in the disease simulation software so that the output iteratively approaches minimal economic loss. Assuming that the simulation models for the spread of disease (FluTE for H1N1 and Covasim for COVID-19 in our case) are accurate representations for the population being studied, the simulation-based strategy we present provides decision makers a powerful tool to mitigate potential human and economic losses from any epidemic. The basic approach is also applicable in other public health problems, such as opioid abuse and drunk driving.
