[Linxiushen]

After an emitted MultiSteps update, rejecting the next mini-step preserves mini_step == 0 and gradient_step > 0. has_updated therefore returns True again even though that call returned zero updates and did not advance the optimizer. Callers gating logging, evaluation or checkpoint cadence on this predicate receive a false update event.

On unmodified main 9a1f892f52e28c18eec4d72b1001731cf20da532, CPU/JIT reproduction with MultiSteps(sgd(0.1), 2, should_skip_update_fn=skip_not_finite) gives:

Input mini-step	Update	has_updated
finite	0	False
finite	-0.1	True
NaN, rejected	0	True (expected False)

Record the last skip decision independently of the arbitrary user-provided skip_state, and exclude rejected calls from has_updated. Counters, wrapped optimizer state and accumulated gradients still remain unchanged on rejection. This leaves the separate outgoing-update masking and accumulator-reset work in #1824 and #1834 untouched.

The 13 new cases cover eager/JIT execution, accumulation lengths 1 and 2, both built-in skip predicates, a custom predicate with empty diagnostic state, repeated rejection, partial accumulation, finite recovery, preserved optimizer state, and serialization of initial/emitted/skipped states.

Validation on Windows, Python 3.12, JAX 0.8.0, Flax 0.12.0 (CPU):

• python -m pytest -q optax/transforms/_accumulation_test.py optax/transforms/_masking_test.py optax/transforms/_combining_test.py: 60 passed, 29 subtests passed.
• Ruff on both changed files, Pyink on the added test range, repository license-header check and git diff --check pass.
• The same original reproduction passes with the patch. No GPU/distributed execution or performance measurement was performed.

Compatibility: MultiStepsState gains a defaulted sixth boolean field, last_step_skipped. Existing five-argument construction and its JIT update work. The serialized PyTree schema changes: strict old Flax state dictionaries require explicit migration by adding last_step_skipped=False; the old checkpoint cannot reveal its last skip event. New-schema Flax roundtrips are tested; other checkpoint engines are untested.

AI assistance was used to develop and locally validate this change.