To prove this, we ﬁrst state the following claims.

Claim 8. kaj = γ1−j − 1 + (cid:15)/(1 − γ) ∀j ∈ [q].
Claim 9. m(cid:96)(b) = q + ψ(a∗(b)) ∀b ∈ V.
Proof of Proposition 7. As shown in Section 4.1, the setting is applicable if ka∗(b) ≤ m1(b) ∀b ∈ V.
Equivalently, according to Claims 8 and 9, the setting is applicable if γ1−j − 1 + (cid:15)/(1 − γ) ≤
q +γ1−j−j +(j−1)(γ +(cid:15)), where aj = a∗(b), for all b ∈ V. Note that (j−1)(1−γ−(cid:15))+(cid:15)/(1−γ) ≤ q
for all j ∈ [q], since the left-hand side is at most (q − 1)(1 − γ − (cid:15)) + 1 which, in turn, is at most q.
This completes the proof.

[w(o)] is w = (0, γ1−q − γj−q + γj−q · (cid:15)/(1 − γ)). Thus, kaj = γ1−j − 1 + (cid:15)/(1 − γ).

Proof of Claim 8. Recall that kaj = minw∈La
Eo∼F|aj
ψ(a(cid:48)). Equivalently, that Eo∼F|aj
(γq−j − γq−j(cid:48)
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