ripr

ripr computes reverse information projections onto nulls that take the form of a union of convex parts, and then proves an upper bound on the expectation over the null of whatever random variable you get. The projection is fitted via Frank–Wolfe (Jaggi 2013) and/or EM; bounds are proved by a branch-and-bound algorithm (currently for multinomial families, via the Bernstein enclosure of Garloff (1985) and Leroy (2012)). At the end, you get a random variable that is genuinely an e-variable for the null.

Fitting and certification are deliberately separate: any approximate projection can be rescaled (by an upper bound on the expectation over the null) to get a valid e-variable. This way, a very difficult, non-convex optimisation problem splits into two easier problems: approximate a distribution, then compute a bound. The better the fit, the higher the e-power, and validity is guaranteed.

Installation

# install.packages("remotes")
remotes::install_github("fleverest/ripr")

The usual workflow

Take a three-category multinomial distribution (n = 20 samples) and the plurality null: candidate 1 does not win outright, H0 = union_j {theta : theta_1 <= theta_j}. This null is a union of two convex parts, so it’s built with union():

library(ripr)

K <- 3L
n <- 20L
family <- multinomial_family(n_trials = n, k = K)

# Defines {theta : theta_1 <= theta_j}
plurality_part <- function(j) {
  vertices <- diag(K)
  vertices[, 1L] <- replace(numeric(K), c(1L, j), 0.5)
  simplex_region(vertices = vertices)
}

plurality <- null_model(
  family,
  union(plurality_part(2), plurality_part(3))
)

We nominate a discrete mixture-multinomial alternative (here a single dirac at q = (0.40, 0.35, 0.25)), so Q is just Multinomial(20, q):

Q <- family(c(0.40, 0.35, 0.25))

Fitting is a sequence of step verbs: fw_step() for Frank–Wolfe, em_step() for EM. Here we run Frank–Wolfe for 40 iterations, then ripr_finish() solves the weights and drops any low-mass atoms:

set.seed(1)
state <- ripr_init(Q, plurality)
state <- fw_step(state, times = 40L, until = gap_below(1e-10))
fit <- ripr_finish(state, reoptimise = TRUE, identify = TRUE, record_gap = TRUE)

c(kl = fit$kl, gap = fit$gap_final, atoms = n_atoms(fit$W0))
#>           kl          gap        atoms 
#>  0.028247494  0.000134276 30.000000000

Plurality null, with the support of the alternative mixing distribution W1 and the fitted null mixture W0.

The fit gives a candidate e-variable: the mixture likelihood ratio Q / P*. random_variables are callable, mapping outcomes to their realisation:

X <- likelihood(Q, label = "Q") /
  likelihood(fit$P_star, label = "P*")

outcomes <- rbind(
  c(10L, 10L, 0L),
  c(8L, 7L, 5L)
)
X(outcomes)
#> [1] 0.8470767 1.0779181

certify() proves an upper bound on expectation of X over the null, including a lower-bound that is the largest value the search actually attained:

cert <- certify(X, plurality, tol = 1e-9)
c(
  upper = cert$sup_ub,
  attained = cert$sup_lb,
  width = cert$sup_ub - cert$sup_lb
)
#>        upper     attained        width 
#> 1.000134e+00 1.000134e+00 9.437988e-10

Rescaling by the upper bound turns X into a bona fide e-variable for the plurality null:

E <- X / cert$sup_ub
print(E)
#> <random_variable> Q / P* / 1.000134 
#>   on count_space, dimension 3
print(E(outcomes))
#> [1] 0.846963 1.077773

Learn more

vignette("ripr") walks through the same example in more depth: why validity survives a deliberately bad fit, how a bound from certify() differs from sup_lb(), and what happens when no certification method exists for a family/geometry pair. vignette("regions") covers the region interface and set algebra (union(), intersect(), setdiff(), disjoin(), …) used to build nulls out of convex pieces.

References

Garloff, Jürgen. 1985. “Convergent Bounds for the Range of Multivariate Polynomials.” International Symposium on Interval Mathematics, 37–56.

Jaggi, Martin. 2013. “Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization.” International Conference on Machine Learning, 427–35.

Leroy, Richard. 2012. “Convergence Under Subdivision and Complexity of Polynomial Minimization in the Simplicial Bernstein Basis.” Reliable Computing 17: 11–21.