Machine Learning Optimization Algorithms & Portfolio Allocation (1909.10233v1)

Abstract: Portfolio optimization emerged with the seminal paper of Markowitz (1952). The original mean-variance framework is appealing because it is very efficient from a computational point of view. However, it also has one well-established failing since it can lead to portfolios that are not optimal from a financial point of view. Nevertheless, very few models have succeeded in providing a real alternative solution to the Markowitz model. The main reason lies in the fact that most academic portfolio optimization models are intractable in real life although they present solid theoretical properties. By intractable we mean that they can be implemented for an investment universe with a small number of assets using a lot of computational resources and skills, but they are unable to manage a universe with dozens or hundreds of assets. However, the emergence and the rapid development of robo-advisors means that we need to rethink portfolio optimization and go beyond the traditional mean-variance optimization approach. Another industry has faced similar issues concerning large-scale optimization problems. Machine learning has long been associated with linear and logistic regression models. Again, the reason was the inability of optimization algorithms to solve high-dimensional industrial problems. Nevertheless, the end of the 1990s marked an important turning point with the development and the rediscovery of several methods that have since produced impressive results. The goal of this paper is to show how portfolio allocation can benefit from the development of these large-scale optimization algorithms. Not all of these algorithms are useful in our case, but four of them are essential when solving complex portfolio optimization problems. These four algorithms are the coordinate descent, the alternating direction method of multipliers, the proximal gradient method and the Dykstra's algorithm.

• The paper introduces ML algorithms such as CD, ADMM, proximal operators, and Dykstra’s algorithm to address non-quadratic constraints in portfolio optimization.
• The methodology leverages coordinate descent and ADMM to efficiently solve high-dimensional, non-linear optimization problems in financial portfolios.
• The findings imply that integrating ML techniques in portfolio allocation enhances diversification, risk control, and robo-advisory strategies.

Leveraging Machine Learning Optimization Algorithms for Portfolio Allocation

Introduction

The landscape of portfolio optimization is traditionally dominated by the Mean-Variance Optimization (MVO) framework, introduced by Harry Markowitz in 1952. This quadratic programming (QP) model has been pivotal due to its computational efficiency and straightforward implementation. However, its applicability is limited by the quadratic nature of its objective function, which doesn't always accommodate the complexities of real-world financial markets. This paper explores advanced optimization algorithms from machine learning to address non-QP challenges in portfolio allocation. The algorithms discussed include Coordinate Descent (CD), Alternating Direction Method of Multipliers (ADMM), proximal operators (PO), and Dykstra's algorithm, which together offer a robust framework for solving a wide range of portfolio optimization problems beyond the traditional QP approach.

The Quadratic Programming World of Portfolio Optimization

Quadratic Programming Fundamentals

Quadratic programming problems are characterized by a quadratic objective function subject to linear constraints. The challenge within portfolio optimization is to maximize returns for a given level of risk (or minimize risk for a given level of return), which naturally aligns with the QP problem structure. The Markowitz framework efficiently solves these problems but faces criticism for lacking flexibility and robustness in handling real-world financial data.

Seeking Alternatives

Recent advances in investment strategies, such as smart beta, factor investing, and developments in robo-advisory services, demand more sophisticated optimization techniques. These can handle non-linear objective functions, complex constraint formulations, and regularization to stabilize solutions without limiting them to quadratic forms.

Leveraging Machine Learning Algorithms

Coordinate Descent and Alternating Direction Method of Multipliers (ADMM)

The coordinate descent algorithm offers a highly efficient way to tackle high-dimensional optimization problems by minimizing along one coordinate at a time. It is particularly effective for lasso regression models in statistics. Similarly, ADMM facilitates the decomposition of complex problems into smaller subproblems, promoting parallel computation. These algorithms extend the portfolio optimization toolkit beyond quadratic programming, allowing for greater flexibility in model formulation.

Proximal Operators and Dykstra's Algorithm

Proximal operators provide solutions to optimization problems involving non-smooth functions by mapping a vector into another vector that minimally contradicts a given condition. Dykstra's algorithm further allows for the resolution of constraints in a piecewise manner. These tools are instrumental in handling various portfolio constraints, like turnover limits and transaction costs, within a unified optimization framework.

Applications in Portfolio Optimization

Portfolio Allocation Models Beyond MVO

The paper discusses several portfolio allocation models that incorporate features beyond the traditional MVO, such as diversification measures and risk parity considerations. These include the Equal Risk Contribution (ERC) portfolio, diversification-constraint portfolios, and the Most Diversified Portfolio (MDP) model, each presenting a unique challenge that can be addressed through the discussed machine learning optimization algorithms.

Robo-Advisory Optimization

In the context of robo-advisory, the paper outlines a comprehensive objective function that integrates multiple aspects of portfolio optimization including benchmark-relative performance, regularization for smooth allocation paths, and risk parity. The suggested framework leverages CD, ADMM, and Dykstra's algorithm to manage complex non-linear constraints systematically, demonstrating significant potential for automated, data-driven portfolio management solutions.

Implementing Non-Linear Constraints

The discussed algorithms excel at incorporating a variety of non-linear constraints into the portfolio optimization process. These include leveraging constraints, active share constraints, and specific investment ratio limitations, illustrating the versatility and adaptability of machine learning algorithms in optimizing portfolios under realistic market conditions.

Conclusion

The exploration of machine learning optimization algorithms in portfolio allocation signifies a significant shift from the restrictive quadratic programming world. This paper successfully demonstrates the potential of CD, ADMM, proximal operators, and Dykstra's algorithm in addressing complex, non-quadratic optimization problems prevalent in modern portfolio management and investment strategies. These advancements herald a more flexible, robust, and data-intensive approach to asset allocation, catering to the evolving needs of the finance industry.