Definition 2’s divided conditional loss → value zero on zero-mass
groups, with mass-weighted loss as the primitive (lines 319–331).
Division is undefined there; Definition 7 already requires positive
certificate mass.
Bayes optimality and
Observation 4
Pointwise conditional-risk optimality → modelwise almost-everywhere
optimality against every admissible integrable alternative model for
arbitrary population laws (lines 333–351). A null-point change cannot
affect any subgroup integral: for example, under uniform features on
[0,1], changing an otherwise optimal binary predictor only
at zero preserves every integrated loss. For countably many labels,
action comparisons share one full-measure set; on finite positive-mass
support the pointwise conclusion follows.
Adaptive certificate
checking
Theorem 11’s undefined confidence symbol and transcript count →
failure bound 2 N exp(-n epsilon^2/32), where
n is sample size, epsilon>0 the certificate
threshold, U the submission horizon, and
N=U(U+1)^floor(2/epsilon) (lines 549–603). At failure
probability delta, it suffices that
n >= (32/epsilon^2) * (log(2U/delta) + floor(2/epsilon)*log(U+1)).
For 0 < epsilon <= 1, U >= 1, and
0 < delta <= 1, this is at most
96 epsilon^-3 log(2U/delta), the paper’s claimed sample
rate; the explicit formula is sharper in its confidence dependence.
Algorithm 2’s completed transcript may contain one more acceptance
than a queried prefix: its non-strict guard allows a final acceptance at
count floor(2/epsilon), after which no later query is made
(lines 524–547). The sparse count applies to queried prefixes.
Shared state in Algorithm 4
Interleaved submissions and repairs → one shared checker transcript,
with the protected-history scan restarted after each accepted repair
(lines 671–713). A model change can make an earlier rejected repair
improve loss; restarting supplies Theorem 14’s monotonicity and cubic
query bound.
Fresh blocks,
integer rounds, and the Lemma 15 typo
Algorithm 5’s optimization/stopping dataset D → the
current fresh block D_t; real horizon
2/epsilon → ceil(2/epsilon) rounds (lines
786–811). For 0 < epsilon <= 1, the rounded count is
at most 3/epsilon, so the oracle and sample rates remain
O(1/epsilon) in the number of blocks.
Lemma 15’s undefined g_p → the quantified group
g (lines 773–779).
Lemma 22 proof route
Disagreement-normalized counts treated as population masses → the
identity “disagreement empirical risk = group-independent constant minus
certificate objective” (lines 957–990). Minimizing the former therefore
maximizes the latter, proving the same lemma.
Algorithm 6 and Theorem 23
Full-sweep stopping → two coordinate-gap tests at the same returned
pair; arbitrary-start positive certificate → local optimality, with
positive initialization additionally giving positivity (lines 991–1025).
The Algorithm 6
note gives the exact tests, response bound, and zero-value
counterexample.