Research map / Learning theory and algorithms / Sample complexity
Probabilistic Factorial Experimental Design for Combinatorial Interventions
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Introduces probabilistic factorial experimental design, where each unit randomly receives a combination of treatments drawn from a product Bernoulli distribution set by per-treatment dosages, and derives near-optimal passive (dosage 1/2) and active (numerically optimized acquisition) designs for learning bounded-order Boolean-function interaction models.
Problem
Designing factorial experiments that learn combinatorial intervention effects (bounded-order treatment interactions) when testing all 2^p combinations is infeasible, and fractional designs require careful, prior-dependent subset selection.
Approach
Replace deterministic combination selection with per-treatment dosages that induce random product-Bernoulli assignments of combinations to units; show uniform dosage 1/2 is near-optimal for estimating any k-way interaction model passively, and derive a numerically optimizable acquisition function for multi-round active designs.
Key results
- In passive simulation with p=8, degree-1 model, n=64, the fractional 2^(8-2) Resolution V design gave MSE 0.14 ± 0.062 versus 0.16 ± 0.078 for half dosage (300 trials).
- Theory: half dosage is optimal up to factor 1+O(ln(n)/n) for any k-way interaction model; sample complexity O(k p^{3k} ln p).
- Active setting (p=15, k=2, n=75) and small-n high-noise setting (p=5, k=1, n=16, sigma=5) show the optimal acquisition outperforms half dosage when n is small relative to K.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the half-dosage design instead of a Resolution V fractional design | Passive probabilistic factorial design · Simulated degree-1 Boolean function (p=8, 64 samples) | mean squared error: 0.14 ± 0.062 -> 0.16 ± 0.078 | Hurt |
| use the optimal active dosage rather than randomly sampled dosages each round | Active probabilistic factorial design · Simulated active setting, 10 sequential rounds | approximation error | Helped |
| use the optimal active dosage rather than half dosage each round | Active probabilistic factorial design · Simulated active setting, 10 sequential rounds | approximation error | Mixed |
| use the optimal active dosage rather than a partial factorial design | Active probabilistic factorial design · Simulated active setting, 10 sequential rounds | approximation error | Mixed |
| use half dosage rather than dosages farther from 0.5 | Passive probabilistic factorial design · Passive simulations (p=10, 20, or 30; k=2) | coefficient estimation error | Helped |
| use uniform dosage L/p rather than deviating from it under the limited-supply constraint | Probabilistic factorial design with limited supply · Limited-supply simulations (L=2, p=8, 9, or 10; k=2) | approximation error | Helped |
| use half dosage rather than dosages farther from 0.5 under a misspecified interaction degree | Passive probabilistic factorial design · Misspecified model (p=5; assumed k=2 to 4; true k=5) | estimation error | Helped |
Surprising findings
- In the single-round passive setting, a fractional factorial design can outperform half dosage at small sample sizes, though the gap is small.
Limitations
- Assumes a product Bernoulli (no interference or censoring) infection mechanism.
- Model assumes outcomes depend only on received treatments, with no unit-specific covariates.
- Misspecified-model experiments are empirical only; theory does not cover misspecification.
Experimental setup
Datasets: simulated Boolean function outcomes with Fourier coefficients sampled uniformly
Benchmarks: simulated Boolean function experiments
Metrics: mean squared error of estimated Fourier coefficients
Baselines: fractional factorial design random dosage half dosage partial design
Builds on: fractional factorial design Boolean function Fourier analysis generalized response surface model
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