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Online Ensemble Learning for Sector Rotation: A Gradient-Free Framework

Jiaju Miao, Pawel Polak

TL;DR

The paper tackles nonstationary sector-rotation forecasting with a gradient-free online ensemble that aggregates forecasts from a diverse model library using a Multiplicative Weights Update Method. It introduces a gain function combining exploitation and exploration, proves that the average gain converges to the out-of-sample $R^2_{oos}$ with a finite-sample bound, and adaptively tunes the learning rate to maintain performance. Empirically, PPCA-constructed sector features are forecast by 16 models, and the online ensemble outperforms individual models and offline ensembles, delivering higher predictive accuracy and stronger risk-adjusted returns across regimes, including the COVID-19 episode. The results substantiate a practical, scalable approach to ML-based asset pricing, with robust sector-rotation strategies that remain resilient to transaction costs and regime shifts.

Abstract

We propose a gradient-free online ensemble learning algorithm that dynamically combines forecasts from a heterogeneous set of machine learning models based on their recent predictive performance, measured by out-of-sample R-squared. The ensemble is model-agnostic, requires no gradient access, and is designed for sequential forecasting under nonstationarity. It adaptively reweights 16 constituent models-three linear benchmarks (OLS, PCR, LASSO) and thirteen nonlinear learners including Random Forests, Gradient-Boosted Trees, and a hierarchy of neural networks (NN1-NN12). We apply the framework to sector rotation, using sector-level features aggregated from firm characteristics. Empirically, sector returns are more predictable and stable than individual asset returns, making them suitable for cross-sectional forecasting. The algorithm constructs sector-specific ensembles that assign adaptive weights in a rolling-window fashion, guided by forecast accuracy. Our key theoretical result bounds the online forecast regret directly in terms of realized out-of-sample R-squared, providing an interpretable guarantee that the ensemble performs nearly as well as the best model in hindsight. Empirically, the ensemble consistently outperforms individual models, equal-weighted averages, and traditional offline ensembles, delivering higher predictive accuracy, stronger risk-adjusted returns, and robustness across macroeconomic regimes, including during the COVID-19 crisis.

Online Ensemble Learning for Sector Rotation: A Gradient-Free Framework

TL;DR

The paper tackles nonstationary sector-rotation forecasting with a gradient-free online ensemble that aggregates forecasts from a diverse model library using a Multiplicative Weights Update Method. It introduces a gain function combining exploitation and exploration, proves that the average gain converges to the out-of-sample with a finite-sample bound, and adaptively tunes the learning rate to maintain performance. Empirically, PPCA-constructed sector features are forecast by 16 models, and the online ensemble outperforms individual models and offline ensembles, delivering higher predictive accuracy and stronger risk-adjusted returns across regimes, including the COVID-19 episode. The results substantiate a practical, scalable approach to ML-based asset pricing, with robust sector-rotation strategies that remain resilient to transaction costs and regime shifts.

Abstract

We propose a gradient-free online ensemble learning algorithm that dynamically combines forecasts from a heterogeneous set of machine learning models based on their recent predictive performance, measured by out-of-sample R-squared. The ensemble is model-agnostic, requires no gradient access, and is designed for sequential forecasting under nonstationarity. It adaptively reweights 16 constituent models-three linear benchmarks (OLS, PCR, LASSO) and thirteen nonlinear learners including Random Forests, Gradient-Boosted Trees, and a hierarchy of neural networks (NN1-NN12). We apply the framework to sector rotation, using sector-level features aggregated from firm characteristics. Empirically, sector returns are more predictable and stable than individual asset returns, making them suitable for cross-sectional forecasting. The algorithm constructs sector-specific ensembles that assign adaptive weights in a rolling-window fashion, guided by forecast accuracy. Our key theoretical result bounds the online forecast regret directly in terms of realized out-of-sample R-squared, providing an interpretable guarantee that the ensemble performs nearly as well as the best model in hindsight. Empirically, the ensemble consistently outperforms individual models, equal-weighted averages, and traditional offline ensembles, delivering higher predictive accuracy, stronger risk-adjusted returns, and robustness across macroeconomic regimes, including during the COVID-19 crisis.
Paper Structure (5 sections, 3 theorems, 29 equations, 2 figures, 3 tables, 2 algorithms)

This paper contains 5 sections, 3 theorems, 29 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modeling Sector Returns
  3. Performance Driven Ensemble of Models
  4. Empirical Study
  5. Conclusion

Key Result

Lemma 1

Under standard linear regression assumptions, the solution to eq:opt_p_problem is where $\widehat{\mathbf{p}}_{\mathrm{OLS}} = (\widehat{\mathbf{R}}^\top \widehat{\mathbf{R}})^{-1} \widehat{\mathbf{R}}^\top \mathbf{r}$, $\widehat{\mathbf{R}} = [\widehat{\mathbf{r}}_1^\top, \ldots, \widehat{\mathbf{r}}_\tau^\top]^\top$, and $\mathbf{r} = [r_1, \ldots, r_\tau]^\top$.

Figures (2)

  • Figure 1: Comparison of standard CV (top) and MWUM (bottom). CV relies on future data and retrains models at each step using gradient- or grid-based optimization, whereas MWUM updates weights online based on recent performance, without retraining or forward-looking data.
  • Figure 2: Performance of the "Top 5" sector strategy from Jan. 1987 to Dec. 2021, using equally-weighted (left) and capital-weighted (right) returns as defined in \ref{['eq:def_sector_returns']}. Top: cumulative returns vs. S&P 500. Middle: monthly returns. Bottom: drawdowns.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2: Asymptotic equivalence of average gain and out-of-sample performance
  • proof
  • Lemma 3
  • proof