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The standard model and the bandit model are two generalizations of themistake-bound model to online multiclass classification. In both models thelearner guesses a classification in each round, but in the standard model thelearner recieves the correct classification after each guess, while in thebandit model the learner is only told whether or not their guess is correct ineach round. For any set $F$ of multiclass classifiers, define $opt_{std}(F)$and $opt_{bandit}(F)$ to be the optimal worst-case number of predictionmistakes in the standard and bandit models respectively.  Long (Theoretical Computer Science, 2020) claimed that for all $M > 2$ andinfinitely many $k$, there exists a set $F$ of functions from a set $X$ to aset $Y$ of size $k$ such that $opt_{std}(F) = M$ and $opt_{bandit}(F) \ge (1 -o(1))(|Y|\ln{|Y|})opt_{std}(F)$. The proof of this result depended on thefollowing lemma, which is false e.g. for all prime $p \ge 5$, $s = \mathbf{1}$(the all $1$ vector), $t = \mathbf{2}$ (the all $2$ vector), and all $z$.  Lemma: Fix $n \ge 2$ and prime $p$, and let $u$ be chosen uniformly at randomfrom $\left\{0, \dots, p-1\right\}^n$. For any $s, t \in \left\{1, \dots,p-1\right\}^n$ with $s \neq t$ and for any $z \in \left\{0, \dots,p-1\right\}$, we have $\Pr(t \cdot u = z \mod p \text{ } | \text{ } s \cdot u =z \mod p) = \frac{1}{p}$.  We show that this lemma is false precisely when $s$ and $t$ are multiples ofeach other mod $p$. Then using a new lemma, we fix Long's proof.

A note on the price of bandit feedback for mistake-bounded online learning