[Paper Review] Quantum algorithms for hedging and the Sparsitron

Yassine Hamoudi, Maharshi Ray+4 more|arXiv (Cornell University)|Feb 14, 2020

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents quantum algorithms for online learning using the Hedge algorithm, achieving polynomial speedups in estimating hedging losses and active betting via quantum sampling. It introduces a quantum analogue of the Sparsitron, preserving classical learning guarantees while offering runtime improvements of O(poly(T)√N) for T time steps and N strategies.

ABSTRACT

A paradigmatic algorithm for online learning is the Freund/Schapire Hedge algorithm with multiplicative weight updates. For multiple time steps, the algorithm constructs an allocation into different strategies or experts for which it is guaranteed that a certain regret is never much greater than the minimally achievable, even in an adversarial situation. This work presents quantum algorithms for such online learning in an oracular setting. For $T$ time steps and $N$ strategies, we exhibit run times of about $O \left ({ m poly} (T) \sqrt{N} ight)$ for passively estimating the hedging losses and for actively betting according to quantum sampling. In addition, we discuss a quantum analogue of a machine learning algorithm, the Sparsitron, which is based on the Hedge algorithm. The quantum algorithm inherits the provable learning guarantees from the classical algorithm and exhibits polynomial speedups. The speedups shown here may find relevance in both finance, for example for estimating hedging losses, and machine learning, for example for learning a generalized linear model or an Ising model.

Motivation & Objective

• To develop quantum algorithms for online learning in an oracular setting, particularly for the Hedge algorithm with multiplicative weight updates.
• To achieve provable speedups in estimating hedging losses and performing active betting using quantum sampling techniques.
• To extend the classical Sparsitron algorithm to a quantum version while maintaining its learning guarantees.
• To explore applications in finance for hedging loss estimation and in machine learning for generalized linear and Ising models.

Proposed method

• The paper designs quantum algorithms that simulate the Hedge algorithm's multiplicative weight updates using quantum oracles.
• It employs quantum amplitude estimation to passively estimate hedging losses with improved runtime scaling.
• For active betting, the algorithm uses quantum sampling to efficiently select strategies based on estimated losses.
• The quantum Sparsitron is constructed by adapting the classical Sparsitron's update rule to a quantum oracular framework.
• Runtime analysis shows a dependence on O(poly(T)√N), where T is the number of time steps and N is the number of strategies.
• The approach leverages quantum speedups in amplitude estimation and sampling to outperform classical counterparts.

Experimental results

Research questions

• RQ1 Can quantum algorithms achieve polynomial speedups in online learning tasks such as hedging loss estimation?
• RQ2 How can the classical Hedge algorithm be adapted into a quantum oracular framework with provable guarantees?
• RQ3 What is the quantum runtime complexity for active betting using quantum sampling in online learning?
• RQ4 Can the Sparsitron algorithm be generalized to a quantum version while preserving its learning guarantees?
• RQ5 What are the practical implications of these quantum algorithms in finance and machine learning applications?

Key findings

• The quantum algorithm achieves a runtime of O(poly(T)√N) for passively estimating hedging losses, offering a polynomial speedup over classical methods.
• Active betting via quantum sampling also achieves a runtime of O(poly(T)√N), significantly improving upon classical online learning algorithms.
• The quantum Sparsitron inherits the provable learning guarantees of the classical Sparsitron while benefiting from the same quantum speedup.
• The proposed algorithms are applicable to financial hedging, where accurate loss estimation is critical.
• The framework extends to learning generalized linear models and Ising models, demonstrating broad relevance in machine learning.
• The results demonstrate that quantum oracular algorithms can efficiently solve online learning problems with theoretical performance guarantees.