Summary

Combining Tensor Network-Based Flow and Diffusion Models for enhanced portfolio optimization.

Introduction

Motivation

The progression of research from the source paper to the related papers demonstrates a growing interest in using generative models for optimization tasks, particularly in constrained environments. The source paper's introduction of flow networks for diverse candidate generation set the stage for subsequent studies to explore generative models' potential in various optimization contexts. However, a gap remains in understanding how these models can be effectively evaluated and compared, especially when considering both classical and quantum approaches. By addressing this gap, we can advance the field by providing clearer insights into the strengths and limitations of different generative models.

Hypothesis

Integrating Tensor Network-Based Flow Models with Diffusion Models will enhance solution quality and reduce generalization error in hierarchical non-convex portfolio optimization tasks compared to using either model independently.

Research Gap

Existing research has not extensively explored the integration of Tensor Network-Based Flow Models with Diffusion Models for portfolio optimization tasks, particularly in hierarchical non-convex optimization scenarios. This gap is significant as it could reveal new insights into improving solution quality and generalization error in complex financial environments.

Hypothesis Elements

Independent variable: Integration of Tensor Network-Based Flow Models with Diffusion Models

Dependent variable: Solution quality (measured by risk-adjusted returns and Sharpe Ratio) and generalization error (measured by cross-validation error and unique valid sample generation)

Comparison groups: Integrated Tensor-Diffusion Models vs. Tensor Network-Based Flow Models alone vs. Diffusion Models alone

Baseline/control: Tensor Network-Based Flow Models alone and Diffusion Models alone

Assumptions: Tensor Network-Based Flow Models can efficiently represent high-dimensional financial data; Diffusion Models can effectively refine data distributions; the integration can leverage complementary strengths of both models

Population: Financial assets (stocks, bonds, commodities) ranging from 10-100+ assets depending on experiment mode

Timeframe: Varies by experiment mode: 1-year period (MINI_PILOT), 3-year period (PILOT), 5-year period (FULL_EXPERIMENT) with daily returns

Measurement method: Risk-adjusted returns, Sharpe Ratio, cross-validation error, unique valid sample generation, and computational efficiency metrics

Proposed Method

Overview

This research investigates the integration of Tensor Network-Based Flow Models with Diffusion Models to address hierarchical non-convex optimization in portfolio management. Tensor Network-Based Flow Models leverage quantum-inspired computing to efficiently handle high-dimensional data distributions, making them ideal for capturing complex dependencies in financial data. Diffusion Models, on the other hand, excel in transforming simple noise distributions into complex data distributions through iterative refinement, which is particularly effective in modeling intricate data structures. By combining these models, the research aims to exploit their complementary strengths: the Tensor Network-Based Flow Models' ability to represent high-dimensional data efficiently and the Diffusion Models' capacity for capturing complex distributions. This integration is hypothesized to improve solution quality, as measured by risk-adjusted returns and Sharpe Ratio, and reduce generalization error, assessed through cross-validation error and unique valid sample generation. The research will employ a hierarchical non-convex optimization framework to evaluate the models' performance, providing insights into their applicability in real-world financial scenarios. The expected outcome is a more robust portfolio optimization strategy that leverages the strengths of both models to achieve superior performance in complex financial environments.

Background

Tensor Network-Based Flow Models: These models utilize tensor networks to represent and sample from complex probability distributions efficiently. In this experiment, they will be configured to capture the intricate dependencies in financial data, providing a high-dimensional representation that can be leveraged by the Diffusion Models. The choice of Tensor Network-Based Flow Models is motivated by their proven ability to handle high-dimensional data efficiently, which is crucial for modeling the complex dependencies present in financial markets. The expected role of these models is to provide a robust representation of the data distribution, enabling more accurate and diverse sampling by the Diffusion Models.

Diffusion Models: Diffusion Models will be used to iteratively refine the data distribution captured by the Tensor Network-Based Flow Models. They transform a simple noise distribution into a complex data distribution through a series of stochastic steps, effectively capturing the underlying structure of the data. The choice of Diffusion Models is based on their ability to model complex distributions and perform efficient sampling, which complements the high-dimensional representation provided by the Tensor Network-Based Flow Models. The expected role of Diffusion Models is to enhance the diversity and quality of the generated solutions, leading to improved portfolio optimization outcomes.

Hierarchical Non-Convex Optimization: This optimization framework involves solving problems with multiple levels of decision-making, where each level may have non-convex constraints or objectives. In the context of this research, it will be used to evaluate the performance of the integrated models in generating high-quality portfolio optimization solutions. The choice of this framework is based on its ability to model complex financial scenarios, providing a realistic testbed for the integrated models. The expected role of hierarchical non-convex optimization is to serve as a challenging benchmark that highlights the strengths and weaknesses of the integrated models in real-world financial applications.

Implementation

The hypothesis will be implemented by first configuring Tensor Network-Based Flow Models to capture the high-dimensional data distribution of financial markets. This involves constructing a network of tensors that approximate the joint probability distribution of the data. The parameters of the tensor network will be optimized to minimize the discrepancy between the generated and real data distributions. Next, Diffusion Models will be employed to iteratively refine this distribution, transforming a simple noise distribution into the complex data distribution captured by the tensor network. The integration will occur at the sampling stage, where the refined distribution from the Diffusion Models will be used to generate candidate solutions for hierarchical non-convex optimization tasks. The optimization framework will evaluate these solutions based on risk-adjusted returns and Sharpe Ratio, providing a comprehensive assessment of the models' performance. The implementation will involve setting up a pipeline where data flows from the Tensor Network-Based Flow Models to the Diffusion Models, with outputs from the latter being used to generate and evaluate portfolio optimization strategies. The integration logic will ensure that the strengths of both models are leveraged effectively, resulting in a robust and efficient portfolio optimization process.

Experiments Plan

Operationalization Information

Please implement an experiment to test the hypothesis that integrating Tensor Network-Based Flow Models with Diffusion Models will enhance solution quality and reduce generalization error in hierarchical non-convex portfolio optimization tasks compared to using either model independently.

Experiment Overview

This experiment will compare three approaches to portfolio optimization:
1. Tensor Network-Based Flow Models alone (Baseline 1)
2. Diffusion Models alone (Baseline 2)
3. Integrated Tensor-Diffusion Models (Experimental)

The experiment should evaluate these approaches on hierarchical non-convex portfolio optimization tasks, measuring performance through risk-adjusted returns, Sharpe Ratio, cross-validation error, and unique valid sample generation.

Implementation Details

Data Preparation

Use financial time series data for a diverse set of assets (stocks, bonds, commodities, etc.)
For the MINI_PILOT, use 10 assets over a 1-year period with daily returns
For the PILOT, use 50 assets over a 3-year period with daily returns
For the FULL_EXPERIMENT, use 100+ assets over a 5-year period with daily returns
Split the data into training (60%), validation (20%), and test (20%) sets
Preprocess the data to handle missing values, normalize returns, and calculate covariance matrices

Model Implementations

Tensor Network-Based Flow Model

Implement a Matrix Product State (MPS) tensor network structure to represent the joint probability distribution of asset returns
Configure the bond dimension to control the expressivity of the model (start with bond dimension=10 for MINI_PILOT, 20 for PILOT, 50 for FULL_EXPERIMENT)
Optimize the tensor network parameters using maximum likelihood estimation on the training data
Implement sampling methods to generate portfolio weight configurations from the learned distribution

Diffusion Model

Implement a diffusion model that gradually transforms a simple noise distribution into a complex distribution of portfolio weights
Use a forward diffusion process that gradually adds noise to the data
Implement a reverse diffusion process that learns to denoise the data
Train the model to generate realistic portfolio weight configurations
For MINI_PILOT, use 50 diffusion steps; for PILOT, use 100 steps; for FULL_EXPERIMENT, use 200 steps

Integrated Tensor-Diffusion Model

Use the Tensor Network-Based Flow Model to capture the high-dimensional representation of asset return distributions
Feed the output distribution from the Tensor Network as the starting point for the Diffusion Model
Implement the integration at the sampling stage, where the Tensor Network provides the initial distribution and the Diffusion Model refines it
Ensure that the output of the integrated model is a set of portfolio weights that can be evaluated in the optimization framework

Hierarchical Non-Convex Optimization Framework

Implement a hierarchical portfolio optimization framework with multiple levels of constraints
Include non-convex constraints such as cardinality constraints (limiting the number of assets), minimum position sizes, and sector exposure limits
Define a hierarchical structure where top-level decisions (e.g., asset class allocation) constrain lower-level decisions (e.g., individual asset selection)
For each model, generate candidate portfolio weight configurations and evaluate them within this framework

Evaluation Metrics

Risk-adjusted returns: Calculate the returns of each portfolio adjusted for risk
Sharpe Ratio: Measure the excess return per unit of risk
Cross-validation error: Assess how well the models generalize to unseen data
Unique valid sample generation: Count the number of unique, valid portfolio configurations generated by each model
Computational efficiency: Measure the time required to generate and evaluate solutions

Experiment Structure

Implement three experiment modes controlled by a global variable PILOT_MODE:

MINI_PILOT Mode

Use 10 assets over a 1-year period
Run 5 optimization trials for each model
Use simplified hierarchical constraints (2 levels only)
Generate 20 portfolio configurations per model
Use smaller model sizes (bond dimension=10 for tensor networks, 50 diffusion steps)
Purpose: Quick verification that code runs correctly and produces sensible results

PILOT Mode

Use 50 assets over a 3-year period
Run 10 optimization trials for each model
Use moderate hierarchical constraints (3 levels)
Generate 100 portfolio configurations per model
Use medium model sizes (bond dimension=20, 100 diffusion steps)
Purpose: Verify that the experimental approach shows promise and differences between models are emerging

FULL_EXPERIMENT Mode

Use 100+ assets over a 5-year period
Run 30 optimization trials for each model
Use complex hierarchical constraints (4+ levels)
Generate 500 portfolio configurations per model
Use full model sizes (bond dimension=50, 200 diffusion steps)
Purpose: Complete evaluation with statistical significance

Analysis and Reporting

For each model and experiment mode, report:
Mean and standard deviation of risk-adjusted returns
Mean and standard deviation of Sharpe Ratio
Cross-validation error on validation set
Number of unique valid portfolio configurations generated
Computational time required

Statistical analysis:
Perform paired t-tests to compare the performance of the three models
Use bootstrap resampling to estimate confidence intervals for performance differences
Create box plots showing the distribution of performance metrics across trials

Visualization:
Generate plots showing the distribution of portfolio weights for each model
Create efficiency frontier plots comparing the three approaches
Visualize the hierarchical structure of the generated portfolios

Implementation Notes

Set PILOT_MODE to "MINI_PILOT" initially and run the experiment
If the MINI_PILOT runs successfully, set PILOT_MODE to "PILOT" and run again
After the PILOT completes, stop and do not run the FULL_EXPERIMENT (this will be manually triggered after human verification)
Ensure all random processes use fixed seeds for reproducibility
Log intermediate results to allow for experiment resumption if interrupted
Save model checkpoints after training each component

Please implement this experiment with careful attention to the integration between the Tensor Network-Based Flow Model and the Diffusion Model, as this integration is the key novel aspect being tested.

Paper ID

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