perf(scm): add FISTA adaptive restart to simplex QP solvers

The inner SCM/SDID QP solver (solve_simplex_qp / solve_simplex_qp_lr)
is first-order FISTA, whose iteration count scales with the condition
number of Q = X0'VX0. Pre-treatment outcome panels are highly collinear
(condition number ~1e6), and the outer V optimisation skews V toward a
few periods, pushing each inner solve from ~0.2ms to ~10ms. With ~10^4
inner solves per fit this produced multi-second runtimes (10-20x slower
than tidysynth, which uses a fixed-iteration interior-point solver).

Add the O'Donoghue & Candes (2015) gradient-scheme adaptive restart:
reset the momentum term t_acc whenever dot(grad, x - x_prev) > 0. This
removes stale momentum on ill-conditioned Q and converges to the same
optimum (verified to match quadprog within 1e-10). Typical SCM fits
speed up 10-20x (e.g. N_co=39, T_pre=40: 2250ms -> 220ms).

Also fix a fragile SDID covariate test that asserted a time-invariant
covariate changes the ATT at tol=1e-6. Time-invariant covariates are
absorbed by the two-way fixed effects, so the true change is ~0; the
test only passed on the old solver's numerical residual. Rewrite it to
use a genuinely time-varying covariate.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

Author: yo5uke

src/optim.cpp

@@ -49,21 +49,31 @@ arma::vec solve_simplex_qp(const arma::mat& Q, const arma::vec& c, int max_iter
 
   arma::vec y = x;
   double t_acc = 1.0;
-  
+
   for(int iter = 0; iter < max_iter; iter++) {
     arma::vec x_prev = x;
-    
+
     // Gradient step
     arma::vec grad = Q * y - c;
-    
+
     // Projection step
     x = proj_simplex(y - t * grad);
-    
+
+    // Adaptive restart (O'Donoghue & Candes 2015, gradient scheme): when the
+    // momentum direction disagrees with the descent direction, the
+    // acceleration is overshooting and slowing convergence on ill-conditioned
+    // Q (the dominant cost in SCM, where Q = X0'VX0 is highly collinear).
+    // Resetting t_acc removes the stale momentum. This only changes the path
+    // to the optimum, not the optimum itself.
+    if (arma::dot(grad, x - x_prev) > 0.0) {
+      t_acc = 1.0;
+    }
+
     // FISTA acceleration
     double t_acc_next = (1.0 + std::sqrt(1.0 + 4.0 * t_acc * t_acc)) / 2.0;
     y = x + ((t_acc - 1.0) / t_acc_next) * (x - x_prev);
     t_acc = t_acc_next;
-    
+
     if(arma::norm(x - x_prev, 2) < tol) {
       break;
     }
@@ -97,6 +107,10 @@ arma::vec solve_simplex_qp_lr(const arma::mat& B, const arma::vec& b,
     arma::vec x_prev = x;
     arma::vec grad   = B.t() * (B * y - b);         // O(k * N_co)
     x = proj_simplex(y - t * grad);
+    // Adaptive restart (gradient scheme) -- see solve_simplex_qp for rationale.
+    if (arma::dot(grad, x - x_prev) > 0.0) {
+      t_acc = 1.0;
+    }
     double t_acc_next = (1.0 + std::sqrt(1.0 + 4.0 * t_acc * t_acc)) / 2.0;
     y = x + ((t_acc - 1.0) / t_acc_next) * (x - x_prev);
     t_acc = t_acc_next;

tests/testthat/test-integration.R

@@ -1452,9 +1452,17 @@ test_that("Phase 13a: SDID without covariates — beta_hat is length-0", {
 })
 
 test_that("Phase 13a: covariate partial-out changes the ATT estimate", {
-  fit_no  <- scm_fit(y ~ d | id + time, data = panel_cov, method = "sdid")
-  fit_cov <- scm_fit(y ~ d | id + time, data = panel_cov, method = "sdid",
-                     covariates = "cov1")
+  # cov1/cov2 in panel_cov are time-invariant, so they are absorbed by the
+  # two-way fixed effects and (correctly) leave the SDID ATT essentially
+  # unchanged. A meaningful partial-out test needs a genuinely time-varying
+  # covariate that drives the outcome.
+  set.seed(99)
+  pc <- panel_cov
+  pc$covtv <- rnorm(nrow(pc), 0, 1) + 0.5 * pc$time
+  pc$y     <- pc$y + 0.8 * pc$covtv
+  fit_no  <- scm_fit(y ~ d | id + time, data = pc, method = "sdid")
+  fit_cov <- scm_fit(y ~ d | id + time, data = pc, method = "sdid",
+                     covariates = "covtv")
   expect_false(isTRUE(all.equal(fit_no$estimate, fit_cov$estimate, tol = 1e-6)))
 })