Minimax-regret aggregator for multisite function-valued estimands

Implements the minimax-regret estimator of Zhang, Huang, and Imai (arXiv:2412.11136) for aggregating site-level function estimates. Given study-level functions $\hat f^{(i)}$, the estimator is $$\hat q = \arg\min_{q \in \Delta_{m-1}}\ q^\top \hat\Gamma\, q - \hat d^\top q, \qquad \hat\Gamma_{ij} = \sum_g w_g\, \hat f^{(i)}(x_g)\hat f^{(j)}(x_g),\ \ \hat d_i = \sum_g w_g\, (\hat f^{(i)}(x_g))^2,$$ yielding the predicted target function $\tilde f(x) = \sum_{i=1}^m \hat q_i\, \hat f^{(i)}(x)$. Unlike metahunt(), this method does not use study-level covariates; the target is the worst-case-regret aggregator over the convex hull of source functions.

Usage

minmax_regret(F_hat, grid_weights = NULL, ridge = 1e-10, wrapper = NULL)

Arguments

F_hat: An m-by-G_grid numeric matrix of source-site function evaluations on a shared grid; row i is $\hat f^{(i)}$.
grid_weights: Optional length-G_grid non-negative numeric vector defining the target measure used to compute $\hat\Gamma$ and $\hat d$. Defaults to uniform weights 1 / G_grid.
ridge: Non-negative scalar; replaces $\hat\Gamma$ with $\hat\Gamma + \text{ridge}\cdot I$ for numerical stability. Defaults to 1e-10.
wrapper: Optional reduction function (see apply_wrapper()). If NULL, prediction is the length-G_grid predicted target function; if a function, prediction is the scalar wrapper(prediction) (e.g. an ATE when wrapper = mean and rows of F_hat are CATE evaluations).

Value

An object of class "minmax_regret": a list with

prediction: Predicted target. Length-G_grid vector when wrapper = NULL; scalar otherwise.
q: Length-m simplex weights from the minimax-regret QP.
Gamma: The m-by-m Gram matrix used (post-ridge).
d: Length-m linear coefficient vector.
grid_weights: Grid weights used.
ridge: Ridge value used.
wrapper: Wrapper function or NULL.

Details

The simplex-constrained QP is solved with quadprog::solve.QP(). A small ridge is added to $\hat\Gamma$ for numerical stability when source functions are highly collinear. The resulting q is clipped to be non-negative and renormalised to sum to 1 to absorb floating-point drift.

References

Zhang, Y., Huang, M., and Imai, K. (2024). Minimax regret estimation for generalizing heterogeneous treatment effects with multisite data. arXiv:2412.11136.

Examples

set.seed(1)
G <- 30; m <- 6
x <- seq(0, 1, length.out = G)
F_hat <- rbind(
  sin(pi * x),
  cos(pi * x),
  x,
  0.5 * sin(pi * x) + 0.5 * x,
  0.3 * cos(pi * x) + 0.7 * x,
  0.4 * sin(pi * x) + 0.4 * cos(pi * x) + 0.2 * x
)

fit <- minmax_regret(F_hat)
fit$q                                        # simplex weights over sources
#> [1] 0.0 0.5 0.5 0.0 0.0 0.0
length(fit$prediction)                       # G: predicted target function
#> [1] 30

# ATE-style scalar via wrapper
minmax_regret(F_hat, wrapper = mean)$prediction
#> [1] 0.25