Weights, constraints and validation stopping

Observation weights and offsets

weights describes observation contributions; offset supplies a known contribution on the linear-predictor scale. Ordinary multichannel models accept shared observation vectors or channel-specific matrices. Scalar weights and offsets are expanded across observations and channels.

set.seed(11)
X <- matrix(rnorm(300*12), 300, 12)
offset <- rep(log(2), 300)
y <- rpois(300, exp(.5 + .8*X[, 1] + offset))
fit <- L0glm.fit(X, y, family=poisson("log"), intercept=TRUE,
  weights=seq(.8, 1.2, length.out=300), offset=offset,
  lambda="gic", relaxed=TRUE, maxit=100)
predict(fit, X[1:3, ], offset=offset[1:3])
#>      [,1]
#> [1,] 2.30
#> [2,] 3.70
#> [3,] 1.01

Multinomial models use observation weights with category-vector offsets. Cox uses observation weights and one predictor offset per observation. See the family tutorials for details.

Nonnegative coefficients and mandatory terms

Set nonnegative=TRUE for a nonnegative fit. Intercepts are constrained too, except that binomial and quasibinomial intercepts remain unrestricted. always.include identifies predictor columns that remain eligible and unpenalized; an added intercept is also unpenalized.

Eigen solvers enforce nonnegativity by clamping negative proposals to zero. This works well in practice when the true active effects are positive and the regularization level is well chosen, for example lambda="gic". Ridge shrinkage stabilizes the updates and often keeps genuine active effects positive. OSQP quadratic programming and glmnet/bigGlm coordinate descent also provide formally constrained ridge solves. In the package’s deconvolution solver benchmarks, Eigen Cholesky LLT generally gives better support recovery and coefficient accuracy than these alternatives, together with much faster fits. The native L0nnpois() backend specializes in nonnegative identity-link Poisson models.

Starts and normalization

start=NULL uses a training-data ridge initializer. Scalar start=1 initializes ones in the normalized coefficient coordinates. Other explicit coefficient vectors and matrices are interpreted on the original scale.

Explicit zero starts exclude the corresponding coefficients; rows that are zero across all channels are omitted from the solves. Intercepts and always.include terms remain eligible. Returned coefficients retain the original dimensions. For multinomial models, starts are CLR-centered and all-zero starting rows are excluded.

normalize="auto" uses RMS scaling for signed regression and preserves the scale of a fully nonnegative design with a constrained intercept. An unrestricted intercept permits implicit centering without densifying a sparse matrix. Other choices are "none", "l2", "rms" and "sd". RMS scaling gives centered columns L2 norm \sqrt n when centering is used. All coefficients are transformed back to the original scale.

Dispersion

dispersion	Behaviour for families with an estimated scale
NULL	Initialize and estimate the dispersion
A positive value	Keep the supplied value fixed
A negative value	Start at its absolute value, then estimate it

Poisson and binomial dispersion is fixed at one. Negative-binomial shape uses mgcv::nb(theta=...); see its tutorial.

Held-out early stopping

Validation selects a checkpoint during fitting. It is particularly useful when prediction favours a point before the final sparse-selection iteration. Coefficient updates continue to use training observations only.

sim <- simul(n=300L, p=300L, n_nonzero=20L, m=4L, seed=7L)
set.seed(8)
v <- matrix(rpois(length(sim$y_true), sim$y_true), 300L, 4L)

validated <- L0nnpois(sim$X, sim$y, lambda="gic", ncores=2L,
  validation=list(X=sim$X, y=v, patience=3L, minit=3L))
validated$validation$deviance
#> [1] 0.499
validated$validation$selection$history
#>   iteration deviance selected dispersion
#> 1         1    0.503       48         NA
#> 2         2    0.499       35         NA
#> 3         3    0.499       34         NA
#> 4         4    0.499       34         NA
#> 5         5    0.499       34         NA
#> 6         6    0.499       34         NA

Here validation counts are an independent repeat measurement of the same latent signals. In another application, pass a separate design and response.

The native Poisson backend scores validation deviance in C++, shares the design across OpenMP workers and restores each channel’s best checkpoint. Grouped selection also retains the best shared-support pass. Relaxed refitting can keep its initial selected coefficients when subsequent steps increase validation loss.

The default controls are patience=3L, minit=3L and tol=1e-8. An improvement must exceed tol * max(1, best_deviance) to reset patience. Validation weights and offsets follow the response formats; mu_floor=1e-8 is used only for positive-mean validation scoring. Diagnostics include selection, relaxed, iteration histories and the selected start/batch candidate.

The same interface works through L0glm.fit() for all supported families. validation$rule="1se" selects a simpler candidate within one empirical standard error of the best candidate. Gaussian identity fits can compare default and conditional marginal starts; an explicit start fixes that choice.

Use a separate test set to assess the model after selecting its stopping point.