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Generalized Random Forests Using Fixed-Point Trees
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Proposes GRF-FPT, a gradient-free fixed-point single-step pseudo-outcome splitting rule for generalized random forests that avoids Jacobian inversion, yielding multi-fold speedups while keeping GRF's consistency and asymptotic normality.
Problem
Generalized random forests (GRFs) use gradient-based splitting that requires estimating and inverting a local Jacobian, which is computationally expensive and unstable in high dimensions and under correlated regressors.
Approach
Replace the Jacobian-based first-order child approximation with a single fixed-point update from the parent solution, giving pseudo-outcomes -xi^T psi(O_i) that are passed to standard multivariate CART; the split ranking is invariant to the scale factor eta, and the Stage II GRF estimator is unchanged.
Key results
- GRF-FPT achieves speedups of up to 3.5x over GRF-grad at dimension K=256 in simulations.
- GRF-FPT matches GRF-grad estimation MSE across 50 replications while reducing computation time.
- Fixed-point splits are far more stable than gradient-based splits under highly correlated regressors (correlation 0.80-0.99).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace gradient-based tree splitting with fixed-point tree splitting | GRF · VCM and HTE simulations | computation time: up to 3.5× faster | Helped |
| Replace gradient-based tree splitting with fixed-point tree splitting | GRF · VCM and HTE simulations | MSE | No clear effect |
| Use the accelerated pseudo-outcome approximation instead of the exact fixed-point pseudo-outcomes (ablation) | GRF-FPT · VCM and HTE | estimation accuracy | No clear effect |
| Change the fixed-point pseudo-outcome scale factor η (ablation) | GRF-FPT tree splitting · Candidate tree splits | split selection | No clear effect |
Surprising findings
- The split chosen by the fixed-point criterion is invariant to the scale factor eta, so the choice of step size does not affect the tree.
Limitations
- Speedups are reported primarily for simulated VCM/HTE designs; real-data evaluation is a single application to California housing.
- Theoretical guarantees rely on inherited GRF assumptions and the approximation holds asymptotically (o_P terms).
Experimental setup
Datasets: California housing (1990 census)
Benchmarks: California housing
Metrics: fit time speedup factor mean squared error
Baselines: GRF-gradient (GRF-grad)
Builds on: generalized random forests honest random forests causal forests
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