# Learning Threshold-Type Investment Strategies with Stochastic Gradient Method

## TL;DR

This paper introduces a stochastic gradient-based learning algorithm for threshold-type investment strategies in online portfolio optimization, demonstrating its convergence and effectiveness across various stock price models.

## Contribution

It is the first systematic study applying the Kiefer--Wolfowitz stochastic gradient method to learn optimal threshold strategies in portfolio optimization.

## Key findings

- The algorithm converges to the log-optimal threshold strategy.
- Optimal threshold strategies exist across diverse stock price models.
- Hyperparameter tuning can be effectively performed with limited data.

## Abstract

In online portfolio optimization the investor makes decisions based on new, continuously incoming information on financial assets (typically their prices). In our study we consider a learning algorithm, namely the Kiefer--Wolfowitz version of the Stochastic Gradient method, that converges to the log-optimal solution in the threshold-type, buy-and-sell strategy class. The systematic study of this method is novel in the field of portfolio optimization; we aim to establish the theory and practice of Stochastic Gradient algorithm used on parametrized trading strategies. We demonstrate on a wide variety of stock price dynamics (e.g. with stochastic volatility and long-memory) that there is an optimal threshold type strategy which can be learned. Subsequently, we numerically show the convergence of the algorithm. Furthermore, we deal with the typically problematic question of how to choose the hyperparameters (the parameters of the algorithm and not the dynamics of the prices) without knowing anything about the price other than a small sample.