nsys profile of multi_fold_convergence on L40S identified compute_expected_q
as the #3 GPU consumer at 12.9% (207ms / 1382 calls — ~150 µs per call,
compute-bound at typical batch=8192 × num_atoms=51). The kernel computed each
per-action softmax over atoms in 3 passes (max → sum_exp → normalize+expected
+var+entropy+util), re-reading v_row[z]+adv_a[z] 3 times per atom for each of
the 4 × ~3.25 = 13 actions per sample.
Replaced with online softmax (running-max + running-sum) so expected_q,
sum_z_sq, and entropy all accumulate in a single forward pass over atoms.
Atom utilisation still needs a 2nd pass because it depends on the final
S = sum exp(logit-M) being known to compute prob_z = exp(logit_z-M)/S for
the threshold test.
Inner loop reduction: 3 → 2 passes, ~33% fewer global loads of branch
advantage logits (which dominate because v_row[z] is reused across 13
(action, branch) pairs and may stay in L1, while adv_a[z] flips per
action and is pure global).
Online accumulation pattern (Page-Olshen):
M, S, TZ, TZ², TLM = -inf, 0, 0, 0, 0
for z in 0..num_atoms:
logit = v_row[z] + adv_a[z]
if logit > M:
scale = exp(M - logit) # ≤ 1, no overflow
S, TZ, TZ², TLM *= scale
M = logit
w = exp(logit - M)
S += w
TZ += w * z_val
TZ² += w * z_val²
TLM += w * (logit - M)
expected_q = TZ / S
sum_z_sq = TZ² / S
entropy = log(S) - TLM / S # analytical: -sum(p log p)
Correctness: bit-stable when atoms processed in fixed order
(z = 0..num_atoms-1). The analytical entropy form is mathematically
identical to -sum(p log p):
-sum(p log p) = log(S) - (1/S) * sum(exp(logit-M) * (logit-M))
= log(S) - TLM / S
The previous code's `if (prob > 1e-10f)` underflow guard is no longer needed:
exp(logit - M) underflows cleanly to 0 when logit << M, and the analytical
form does not multiply tiny probs by very negative logs. Bit-stable per
feedback_stop_on_anomaly.md — no fuzzy tolerance change.
Atom-stat block-sum reduction unchanged (warp shfl + shared-mem bank,
deterministic).
Expected wall-clock saving: ~33% of 207ms = ~70ms across the smoke run; on
production batch=8192 × 1382 calls per fold, roughly proportional. Lower-bound
estimate — at low batch (smoke) the kernel may be launch-bound rather than
load-bound. nsys re-profile after deployment will quantify.
ABI unchanged; no Rust caller changes required. Per Invariant 7 the audit
doc dqn-gpu-hot-path-audit.md is updated with Fix 19 entry.
Build: SQLX_OFFLINE=true cargo check -p ml --lib clean (12 warnings, baseline).
Tests: cargo test -p ml --lib --no-run clean.
Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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