Source Formula Domains

Source: Freund and Schapire, A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting.

Lemma 1 and Theorem 2: the Hedge parameter endpoints

• Algorithm-box $\beta\in[0,1]$ → $0<\beta\leq1$ in Lemma 1 and $0<\beta<1$ in Theorem 2. At $\beta=0$, a positive-loss round can zero every weight, leaving the normalized allocation undefined. Theorem 2 also uses $\log\beta$ and divides by $1-\beta$.

Lemma 4: the zero-loss-bound extension

• Undefined tuning formula at $\overline L=0$ → its continuous extension $\beta=0$. Here $\overline L$ bounds cumulative loss. The inequality in this branch reduces to $R\leq R$, where $R$ is the comparator term. The source specifies no endpoint convention for its division or $L\log\beta$.

Theorem 5: the tuned Hedge parameter

• Unqualified $\beta=g(\widetilde L/\log N)$ → $N\geq2$ and $\widetilde L>0$, where $N$ is the expert count, $\widetilde L$ the loss bound, and $g(z)=1/(1+\sqrt{2/z})$. Otherwise a denominator in the tuning formula is zero. These restrictions are needed for that displayed formula; necessity for the regret guarantee is not claimed (one expert is trivial).

Theorem 6: endpoint errors

• Prose error range $\epsilon_t\in[0,1]$ → $0<\epsilon_t<1$ for executed Figure 2 rounds. At zero error, $\log(1/\beta_t)$ is not finite and normalization can fail; at unit error, $\beta_t=\epsilon_t/(1-\epsilon_t)$ divides by zero. An endpoint execution needs a stopping or limiting rule absent from the display. This is a definedness issue, not a counterexample to the bound with a suitable endpoint convention.

Theorems 10–12: endpoint errors in the variant algorithms

• Undefined executed-round endpoints in Figures 3–5 → $0<\epsilon_t\leq1/2$ for M1 and AdaBoost.R, and $0<\epsilon_t<1$ for M2. The excluded endpoints have the quotient/logarithm problems above; the upper-half stopping regime of M1 and AdaBoost.R is already in the source.