Model B’s undefined normalized-gradient step at zero gradient →
remain at the current point (Definitions 1–3 and Algorithm 1).
Definition 2
weights and Appendix Theorem 5 bias
Appendix Theorem 5’s almost-sure summability of a possibly adapted
random bias → a deterministic summable bias sequence in the current
theorem. This suffices for the main-text executions, whose bias is zero;
it does not establish the broader random-bias statement.
Appendix Theorem 5:
nonunique minimizers
The main proofs’ invocation of a unique-minimizer theorem →
convergence to a point in the minimizer set. C1–C3 allow the
one-dimensional ideal law with density one on [-1,-1/2] and
[1/2,1] and zero elsewhere; every point of
[-1/2,1/2] minimizes expected absolute distance. The
replacement uses convergence of squared-distance potentials, compactness
to obtain a minimizer subsequence, and that potential to force
convergence of the full path. Finitely many affine-spanning minimizers
supply a common probability-one event, avoiding an uncountable
intersection.
Appendix C.4 Lemma 2:
the infinity-one case
The printed heading (p=1,q=infinity) →
(p=infinity,q=1), matching the displayed utility and query
norms (p. 351).
Near ties alone → near ties or active-coordinate crossing. If
i0 maximizes |x_i-v_i|, with current point
x, ideal v, and query radius r,
the bad event additionally includes |x_i0-v_i0|<r.
Outside it and the near-tie event
exists j!=i0, |x_i0-v_i0|<|x_j-v_j|+r, moving the active
coordinate distance r toward its ideal minimizes
infinity-norm cost on the L1 query ball. In one dimension
the near-tie event is empty, but an overshooting step need not minimize
cost. Bounded density and support give probability O(r) for
the added crossing slabs and near-tie strips, preserving the convergence
estimate.
Proposition 1 and
Appendix C.6 Lemma 4
The Appendix identifies “some coordinate block is within radius
r” with a full-vector Euclidean ball → containment in the
union of coordinate slabs |x_i-z_i|<r, where
x is the ideal point and z the current point.
Bounded support (C1) and bounded density (C3) bound the union’s
probability by a constant times r, supplying the estimate
needed for Proposition 1.
Proposition 2’s
coordinatewise Model B
The general normalized-gradient ray → the Appendix C.7 proof’s
coordinatewise rule: move each non-tied decomposition coordinate the
full L-infinity radius toward its sampled ideal. The target
is the ordinary coordinatewise median. A literal normalized ray can
weight coordinates differently: derivative magnitudes 1 or
1/2 in the first coordinate and 2 in the
second produce a weighted-median first-coordinate target.
C1’s bounded closed convex feasible set → a set additionally closed
under replacing one coordinate of a feasible point by the corresponding
coordinate of another feasible point. The triangle
x>=0, y>=0, x+y<=1 satisfies C1 but replacement of
the first coordinate of (0,1) by that of (1,0)
leaves it. This shows the added geometric restriction is stronger; it
does not show the restriction is necessary for Proposition 2 or refute
the broader conclusion.
Theorem 3’s
full-space and projected readings
Theorem 3’s zero aggregate directional field at a convergent trace’s
limit under C1–C3 → zero field under an additional condition making
every aggregate direction feasible; on a constrained projected space,
the checked conclusion is “zero field or no feasible aggregate
direction.” A binding constraint can prevent motion in a preferred
direction. These statements neither prove nor refute zero field on every
bounded convex set allowed by C1.