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Adaptive Conditional Forest Sampling for Spectral Risk Optimisation under Decision-Dependent Uncertainty

Marcell T. Kurbucz

Abstract

Minimising a spectral risk objective, defined as a convex combination of expected cost and Conditional Value-at-Risk (CVaR), is challenging when the uncertainty distribution is decision-dependent, making both surrogate modelling and simulation-based ranking sensitive to tail estimation error. We propose Adaptive Conditional Forest Sampling (ACFS), a four-phase simulation-optimisation framework that integrates Generalised Random Forests for decision-conditional distribution approximation, CEM-guided global exploration, rank-weighted focused augmentation, and surrogate-to-oracle two-stage reranking before multi-start gradient-based refinement. We evaluate ACFS on two structurally distinct data-generating processes: a decision-dependent Student-t copula and a Gaussian copula with log-normal marginals, across three penalty-weight configurations and 100 replications per setting. ACFS achieves the lowest median oracle spectral risk on the second benchmark in every configuration, with median gaps over GP-BO ranging from 6.0% to 20.0%. On the first benchmark, ACFS and GP-BO are statistically indistinguishable in median objective, but ACFS reduces cross-replication dispersion by approximately 1.8 to 1.9 times on the first benchmark and 1.7 to 2.0 times on the second, indicating materially improved run-to-run reliability. ACFS also outperforms CEM-SO, SGD-CVaR, and KDE-SO in nearly all settings, while ablation and sensitivity analyses support the contribution and robustness of the proposed design.

Adaptive Conditional Forest Sampling for Spectral Risk Optimisation under Decision-Dependent Uncertainty

Abstract

Minimising a spectral risk objective, defined as a convex combination of expected cost and Conditional Value-at-Risk (CVaR), is challenging when the uncertainty distribution is decision-dependent, making both surrogate modelling and simulation-based ranking sensitive to tail estimation error. We propose Adaptive Conditional Forest Sampling (ACFS), a four-phase simulation-optimisation framework that integrates Generalised Random Forests for decision-conditional distribution approximation, CEM-guided global exploration, rank-weighted focused augmentation, and surrogate-to-oracle two-stage reranking before multi-start gradient-based refinement. We evaluate ACFS on two structurally distinct data-generating processes: a decision-dependent Student-t copula and a Gaussian copula with log-normal marginals, across three penalty-weight configurations and 100 replications per setting. ACFS achieves the lowest median oracle spectral risk on the second benchmark in every configuration, with median gaps over GP-BO ranging from 6.0% to 20.0%. On the first benchmark, ACFS and GP-BO are statistically indistinguishable in median objective, but ACFS reduces cross-replication dispersion by approximately 1.8 to 1.9 times on the first benchmark and 1.7 to 2.0 times on the second, indicating materially improved run-to-run reliability. ACFS also outperforms CEM-SO, SGD-CVaR, and KDE-SO in nearly all settings, while ablation and sensitivity analyses support the contribution and robustness of the proposed design.
Paper Structure (20 sections, 7 equations, 3 figures, 8 tables)

This paper contains 20 sections, 7 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem Formulation
  4. Conditional Weighted Forest Sampling
  5. The Four-Phase ACFS Algorithm
  6. Variance Reduction
  7. Competitor Methods
  8. Computational Experiments
  9. Benchmark Instances
  10. DGP1: Student-$t$ Copula with Decision-Dependent Marginals
  11. DGP2: Gaussian Copula with Log-Normal Marginals
  12. Experimental Protocol
  13. Results and Discussion
  14. Comparative Performance
  15. Statistical Significance
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Oracle objective distributions across 100 replications for all five methods, both DGPs, and all three penalty-weight settings ($\lambda\in\{0.50,0.70,0.90\}$, $\alpha=0.95$). Rows correspond to DGP1 (Student-$t$ copula, top) and DGP2 (Gaussian copula with log-normal marginals, bottom); columns correspond to $\lambda=0.50$, $\lambda=0.70$, and $\lambda=0.90$. The $y$-axis is shown on a logarithmic scale. ACFS exhibits a tighter distribution than GP-BO in every panel; the advantage is substantially larger on DGP2 where log-normal skewness amplifies tail estimation errors for GP-based surrogates.
  • Figure 2: Ablation study results for DGP1 (left) and DGP2 (right) at $\lambda=0.70$, $\alpha=0.95$, 50 replications each. Each boxplot shows the distribution of oracle $J$ across replications for one ACFS variant. The ordering of component importance differs between DGPs: removing the CEM warm-start incurs the largest penalty on DGP2 (right-skewed log-normal outcomes), while removing augmentation and two-stage reranking are jointly the most costly on DGP1.
  • Figure 3: Hyperparameter sensitivity analysis on DGP1 at $\lambda=0.70$, 20 replications per grid point. Each panel shows median oracle $J$ (dots) and inter-quartile range (bars) as one parameter varies while others are held at their defaults. The dashed vertical line marks the default value used in the main experiments.