SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \Delta D = \beta_ {a} ^ {(t)} - \beta_ {a} ^ {(t - 1)} + z _ {t} \\ = \left(e ^ {\alpha u _ {a t}} - 1\right) \beta_ {a} ^ {(t - 1)} + \frac {\alpha}{e ^ {\alpha} - 1} \left(w _ {a t} - \int_ {U _ {a} - 1 - u _ {a t}} ^ {U _ {a} - 1} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (U _ {a} - x)} d x\right) + \alpha \left(w _ {a t} - u _ {a t} \beta_ {a} ^ {(t - 1)}\right) \\ = \alpha u _ {a t} \beta_ {a} ^ {(t - 1)} + \frac {\alpha}{e ^ {\alpha} - 1} \left(w _ {a t} - \int_ {U _ {a} - 1 - u _ {a t}} ^ {U _ {a} - 1} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (U _ {a} - x)} d x\right) + \alpha \left(w _ {a t} - u _ {a t} \beta_ {a} ^ {(t - 1)}\right) \\ = \frac {\alpha e ^ {\alpha}}{e ^ {\alpha} - 1} w _ {a t} - \frac {\alpha e ^ {\alpha}}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1 - u _ {a t}} ^ {U _ {a} - 1} \frac {w _ {x}}{u _ {x}} d x \\ = \frac {\alpha e ^ {\alpha}}{e ^ {\alpha} - 1} \Delta P. \\ \end{array}
$$
B.2 Consistency
Theorem 13. Algorithm 1 has a consistency of
$$
\frac {\mathrm {A L G}}{\mathrm {P R D}} \geq \left(1 + \frac {1}{e ^ {\alpha} - 1} \max  \left\{\frac {1}{\alpha} \left(e ^ {\alpha} - \frac {e ^ {\alpha} - 1}{\alpha}\right), \alpha \right\}\right) ^ {- 1}.
$$
As before, we split the impressions  $t$  based on whether the algorithm followed the prediction or not. If the algorithm ignores the prediction, we can use that  $\alpha\left(w_{a_{(\mathrm{PRD})}, t} - u_{a_{(\mathrm{PRD})}, t} \beta_{a_{(\mathrm{PRD})}}\right) \leq$
$w_{a(\mathrm{EXP}),t} - u_{a(\mathrm{EXP}),t}\beta_{a(\mathrm{EXP})}$  due to Line 6 in Algorithm 2. With a similar calculation, we obtain
$$
\begin{array}{l} \mathrm {P R D} = \sum_ {a} \mathrm {P R D} _ {a} \\ = \sum_ {a} \left(\sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t} + \frac {1}{\alpha} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} w _ {a t} - \frac {1}{\alpha} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} u _ {a t} \beta_ {a} ^ {(t - 1)} + \sum_ {t \in \mathbf {P} _ {a} \backslash \mathbf {X} _ {a}} u _ {a t} \beta_ {a} ^ {(t - 1)}\right). \\ \end{array}
$$
Once again, we fix an advertiser  $a$ . Let  $\rho_{a} := \sum_{t \in \mathbf{P}_{a} \cap \mathbf{X}_{a}} u_{at}$  so that we can bound
$$
\sum_ {t \in \mathbf {P} _ {a} \backslash \mathbf {X} _ {a}} u _ {a t} \beta_ {a} ^ {(t - 1)} \leq (1 - \rho_ {a}) \beta_ {a} ^ {(T)}.
$$
We still have to argue that the worst-case is when all impressions are ordered such that their value-size ratios are non-decreasing and the impressions in  $\mathbf{P}_a\cap \mathbf{X}_a$  are the ones with maximum value-size ratio among  $\mathbf{X}_a$ . The latter is obvious as it can only increase the value of PRD, so it remains to show that the non-decreasing value-size ordering minimizes the third sum in  $\mathrm{PRD}_a$  (the first two sums are invariant under reordering). To this end, note that the value of  $\beta_{a}$  for GAP is the limit of  $\beta_{a}$  for Display Ads in the following sense: For positive  $\epsilon \rightarrow 0$ , we can split each GAP-impression  $t\in \mathbf{X}_a$  into  $\frac{u_t}{\epsilon}$  identical Display Ads-impressions with value  $\frac{w_t}{u_t}$ , while assuming a budget of  $1 / \epsilon$ . Then, the GAP  $\beta_{a}$  and Display Ads  $\beta_{a}$  are identical. As we know from Display Ads, the worst case is achieved when the Display Ads-impressions with value  $\frac{w_t}{u_t}$  are in non-decreasing order. In this ordering, consecutive Display-Ads impressions with identical value  $\frac{w_t}{u_t}$  still correspond to the same GAP-impression  $t$ , so we also know that this ordering is the worst-case for GAP. We may therefore assume that the impressions are ordered such that their value-size ratios are non-decreasing. As such, we obtain
$$
\beta_ {a} ^ {(t)} = \frac {\alpha}{e ^ {\alpha} - 1} \int_ {U _ {a} ^ {(t)} - 1} ^ {U _ {a} ^ {(t)}} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (U _ {a} ^ {(t)} - x)} d x = \int_ {y - 1} ^ {y} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (y - x)} d x =: \beta_ {a} ^ {(y)}
$$
where  $y = U_{a}^{(t)}$ . Combined with the fact that  $\mathbf{P}_a \cap \mathbf{X}_a$  are last impressions in  $\mathbf{X}_a$ , we can now write
$$
\begin{array}{l} \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t} + \frac {1}{\alpha} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} w _ {a t} - \frac {1}{\alpha} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} u _ {a t} \beta_ {a} ^ {(t - 1)} \\ = \int_ {U _ {a} - \rho} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x + \frac {1}{\alpha} \int_ {0} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} d x - \frac {1}{\alpha} \int_ {0} ^ {U _ {a} - \rho_ {a}} \beta_ {a} ^ {(x)} d x \\ \end{array}
$$
This helps us to compute  $\beta_{a}^{(x)}$  in  $\mathrm{PRD}_a$  and rewrite the whole term as a linear combination of value-size ratios.
$$
\mathrm {P R D} _ {a} \leq \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \phi_ {x} d x + \int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \psi_ {x} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a}
$$
$$
\phi_ {x} := \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha (U _ {a} - x)} + \frac {1}{\alpha} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - x)}}{e ^ {\alpha} - 1}
$$
$$
\psi_ {x} := 1 + \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha \left(U _ {a} - x\right)}
$$
$$
\Omega_ {a} := \frac {1}{\alpha} \frac {1}{e ^ {\alpha} - 1} \left(\rho_ {a} e ^ {\alpha} - \frac {1}{\alpha} \left(e ^ {\alpha} - e ^ {\alpha (1 - \rho_ {a})}\right)\right)
$$
Proof. We rewrite the third sum in  $\mathrm{PRD}_a$  to
$$
\int_ {0} ^ {U _ {a} - \rho_ {a}} \beta_ {a} ^ {(x)} d x
$$
$$
\begin{array}{l} = \frac {\alpha}{e ^ {\alpha} - 1} \int_ {0} ^ {U _ {a} - \rho_ {a}} \int_ {x - 1} ^ {x} \frac {w _ {y}}{u _ {y}} e ^ {\alpha (x - y)} d y d x \\ = \frac {\alpha}{e ^ {\alpha} - 1} \int_ {0} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \int_ {0} ^ {\min  \left\{1, U _ {a} - \rho_ {a} - y \right\}} e ^ {\alpha x} d x d y \\ = \frac {\alpha}{e ^ {\alpha} - 1} \int_ {0} ^ {U _ {a} - 1 - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \int_ {0} ^ {1} e ^ {\alpha x} d x d y + \frac {\alpha}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \int_ {0} ^ {U _ {a} - \rho_ {a} - y} e ^ {\alpha x} d x d y \\ = \int_ {0} ^ {U _ {a} - 1 - \rho_ {a}} \frac {w _ {y}}{u _ {y}} d y + \frac {1}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \left(e ^ {\alpha (U _ {a} - \rho_ {a} - y)} - 1\right) d y \\ \end{array}
$$
where for the last equality, we simply evaluated the integral. Using this in place of the second sum in  $\mathrm{PRD}_a$  cancels out most of the terms of the second sum:
$$
\begin{array}{l} \int_ {0} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} d x - \int_ {0} ^ {U _ {a} - \rho_ {a}} \beta_ {a} ^ {(x)} d x \\ = \int_ {0} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} d x - \int_ {0} ^ {U _ {a} - 1 - \rho_ {a}} \frac {w _ {y}}{u _ {y}} d y - \frac {1}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \left(e ^ {\alpha (U _ {a} - \rho_ {a} - y)} - 1\right) d y \\ = \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \left(1 - \frac {e ^ {\alpha (U _ {a} - \rho_ {a} - y)} - 1}{e ^ {\alpha} - 1}\right) d y \\ = \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - y)}}{e ^ {\alpha} - 1} d y \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \frac {e ^ {\alpha} - e ^ {\alpha \left(U _ {a} - \rho_ {a} - y\right)}}{e ^ {\alpha} - 1} d y + \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - 1} \frac {w _ {y}}{u _ {y}} \frac {e ^ {\alpha} - e ^ {\alpha \left(U _ {a} - \rho_ {a} - y\right)}}{e ^ {\alpha} - 1} d y. \tag {13} \\ \end{array}
$$
We upper bound the third sum
$$
\begin{array}{l} \frac {1}{\alpha} \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - 1} \frac {w _ {y}}{u _ {y}} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - y)}}{e ^ {\alpha} - 1} d y \leq \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \frac {1}{\alpha} \int_ {U _ {a} - 1 - \rho_ {a}} ^ {U _ {a} - 1} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - y)}}{e ^ {\alpha} - 1} d y \\ = \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \frac {1}{\alpha} \frac {1}{e ^ {\alpha} - 1} \left(\rho_ {a} e ^ {\alpha} - \int_ {1 - \rho_ {a}} ^ {1} e ^ {\alpha y} d y\right) \\ = \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \underbrace {\frac {1}{\alpha} \frac {1}{e ^ {\alpha} - 1} \left(\rho_ {a} e ^ {\alpha} - \frac {1}{\alpha} \left(e ^ {\alpha} - e ^ {\alpha \left(1 - \rho_ {a}\right)}\right)\right)} _ {= \Omega_ {a}} \tag {14} \\ \end{array}
$$
$$
\left(1 - \rho_ {a}\right) \beta_ {a} ^ {(U _ {a})} = \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (U _ {a} - x)} d x \tag {15}
$$
Combining (13), (14), and (15) and grouping terms yields
$$
\begin{array}{l} \int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x + \frac {1}{\alpha} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {y}}{u _ {y}} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - y)}}{e ^ {\alpha} - 1} d y + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ + \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} \int_ {U _ {a} - 1} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} e ^ {\alpha (U _ {a} - x)} d x \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \underbrace {\left((1 - \rho_ {a}) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha (U _ {a} - x)} + \frac {1}{\alpha} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - x)}}{e ^ {\alpha} - 1}\right)} _ {= \phi_ {x}} d x \\ + \int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \underbrace {\left(1 + (1 - \rho_ {a}) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha (U _ {a} - x)}\right)} _ {= \psi_ {x}} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a}. \\ \end{array}
$$
Analogously to Display Ads, we define
$$
\Phi_ {a} := \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \phi_ {x} d x \quad \text {a n d} \quad \Psi_ {a} := \int_ {U _ {a} - \rho} ^ {U _ {a}} \psi_ {x} d x
$$
and the total coefficient  $\tau_{a} \coloneqq \Phi_{a} + \Psi_{a} + \Omega_{a}$  which by a calculation similar to Lemma 10 can be shown to be
$$
\tau_ {a} = 1 + \frac {1}{e ^ {\alpha} - 1} \frac {1}{\alpha} \left(e ^ {\alpha} - \frac {e ^ {\alpha} - 1}{\alpha}\right).
$$
Lemma 15. We have
$$
\mathrm {P R D} \leq \max  \left\{\tau_ {a}, \frac {\Psi_ {a}}{\rho_ {a}} \right\} \mathrm {A L G}
$$
if  $\rho_{a} > 0$  and otherwise,
$$
\mathrm {P R D} \leq \tau_ {a} \mathrm {A L G}.
$$
Proof. Again, let
$$
\bar {w} _ {\Phi} := \frac {1}{1 - \rho_ {a}} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} d x
$$
$$
\bar {w} _ {\Psi} := \frac {1}{\rho_ {a}} \int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x
$$
be the average coefficients on the intervals  $[U_a - 1, U_a - \rho_a]$  and  $[U_a - \rho_a, U_a]$ , respectively. The latter coefficients are still decreasing as
$$
\psi_{x} = 1 + \underbrace{\left(1 - \rho_{a}\right)\frac{\alpha}{e^{\alpha} - 1}}_{\geq 0}e^{\alpha (U_{a} - x)}
$$
so we can bound the linear combination
$$
\int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \psi_ {x} d x \leq \bar {w} _ {\Psi} \int_ {U _ {a} - \rho} ^ {U _ {a}} \psi_ {x} d x = \bar {w} _ {\Psi} \Psi_ {a}.
$$
However,  $\phi_x$  is not always decreasing which can be seen by rearranging
$$
\begin{array}{l} \phi_ {x} = \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha \left(U _ {a} - x\right)} + \frac {1}{\alpha} \frac {e ^ {\alpha} - e ^ {\alpha \left(U _ {a} - \rho_ {a} - x\right)}}{e ^ {\alpha} - 1} \\ = \frac {1}{e ^ {\alpha} - 1} \left((1 - \rho_ {a}) \alpha - \frac {1}{\alpha} e ^ {- \alpha \rho_ {a}}\right) e ^ {\alpha (U _ {a} - x)} + \frac {1}{\alpha} \frac {1}{e ^ {\alpha} - 1} e ^ {\alpha} \\ \end{array}
$$
We observe that  $\phi_x$  is decreasing if  $(1 - \rho_a)\alpha$  is at least  $\frac{1}{\alpha} e^{-\alpha \rho_a}$ , and we analyze two cases based on the relationship of both terms:
-  $(1 - \rho_{a})\alpha \geq \frac{1}{\alpha} e^{-\alpha \rho_{a}}$ : We have  $\int_{U_a - 1}^{U_a - \rho_a}\frac{w_x}{u_x}\phi_xdx\leq \bar{w}_{\Phi}\Phi_a$  and thus
$$
\begin{array}{l} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \phi_ {x} d x + \int_ {U _ {a} - \rho} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \psi_ {x} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ \leq \bar {w} _ {\Phi} \Phi_ {a} + \bar {w} _ {\Psi} \Psi_ {a} + w _ {a, I _ {a} - B _ {a}} \Omega_ {a} \\ \leq \bar {w} _ {\Phi} \left(\Phi_ {a} + \Omega_ {a}\right) + \bar {w} _ {\Psi} \Psi_ {a} \\ = \bar {w} _ {\Phi} (1 - \rho_ {a}) \tau_ {a} + \bar {w} _ {\Phi} (\Phi_ {a} + \Omega_ {a} - (1 - \rho_ {a}) \tau_ {a}) + \bar {w} _ {\Psi} \Psi_ {a} \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {s}} \tau_ {a} d x + \bar {w} _ {\Phi} \left(\Phi_ {a} + \Omega_ {a} - \left(1 - \rho_ {a}\right) \tau_ {a}\right) + \bar {w} _ {\Psi} \Psi_ {a} \tag {16} \\ \end{array}
$$
-  $(1 - \rho_{a})\alpha \leq \frac{1}{\alpha} e^{-\alpha \rho_{a}}$ : We can still show that  $\phi_x \leq \tau_a$  as
$$
\begin{array}{l} \phi_ {x} = \left(1 - \rho_ {a}\right) \frac {\alpha}{e ^ {\alpha} - 1} e ^ {\alpha (U _ {a} - x)} + \frac {1}{\alpha} \frac {e ^ {\alpha} - e ^ {\alpha (U _ {a} - \rho_ {a} - x)}}{e ^ {\alpha} - 1} \\ \leq 1 + \frac {1}{e ^ {\alpha} - 1} \frac {1}{\alpha} \left(e ^ {\alpha} - \frac {e ^ {\alpha} - 1}{\alpha}\right) = \tau_ {a} \\ \end{array}
$$
$$
\Longleftrightarrow \underbrace {\left(1 - \rho_ {a}\right) \alpha - \frac {1}{\alpha} e ^ {- \alpha \rho_ {a}}} _ {\leq 0} \underbrace {e ^ {\alpha (U _ {a} - x)}} _ {\geq 0} \leq e ^ {\alpha} - 1 - \frac {1}{\alpha} \frac {e ^ {\alpha} - 1}{\alpha} = \underbrace {\left(1 - \frac {1}{\alpha^ {2}}\right)} _ {\geq 0} \underbrace {(e ^ {\alpha} - 1)} _ {\geq 0}.
$$
$$
\begin{array}{l} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \phi_ {x} d x + \int_ {U _ {a} - \rho} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \psi_ {x} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ \leq \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \phi_ {x} d x + \bar {w} _ {\Psi} \Psi_ {a} + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x - \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \left(\tau_ {a} - \phi_ {x}\right) d x + \bar {w} _ {\Psi} \Psi_ {a} + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ \leq \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x - \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \left(\tau_ {a} - \phi_ {x}\right) d x + \bar {w} _ {\Psi} \Psi_ {a} + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \left(\Phi_ {a} + \Omega_ {a} - (1 - \rho_ {a}) \tau_ {a}\right) + \bar {w} _ {\Psi} \Psi_ {a} \tag {17} \\ \end{array}
$$
In both cases (16) and (17), we have shown that
$$
\begin{array}{l} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \phi_ {x} d x + \int_ {U _ {a} - \rho} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} \psi_ {x} d x + \frac {w _ {U _ {a} - 1}}{u _ {U _ {a} - 1}} \Omega_ {a} \\ \leq \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + v (\Phi_ {a} + \Omega_ {a} - (1 - \rho_ {a}) \tau_ {a}) + \bar {w} _ {\Psi} \Psi_ {a} \\ \end{array}
$$
for a  $v\leq \bar{w}_{\Phi}$
$$
\begin{array}{l} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + v \left(\Phi_ {a} + \Omega_ {a} - \left(1 - \rho_ {a}\right) \tau_ {a}\right) + \bar {w} _ {\Psi} \Psi_ {a} \\ \leq \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + \bar {w} _ {\Psi} \max  \left\{\Phi_ {a} + \Omega_ {a} - (1 - \rho_ {a}) \tau_ {a}, 0 \right\} + \bar {w} _ {\Psi} \Psi_ {a} \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + \bar {w} _ {\Psi} \max \left\{\Phi_ {a} + \Psi_ {a} + \Omega_ {a} - (1 - \rho_ {a}) \tau_ {a}, \Psi_ {a} \right\} \\ = \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} \tau_ {a} d x + \bar {w} _ {\Psi} \max  \left\{\rho_ {a} \tau_ {a}, \Psi_ {a} \right\} \\ \leq \tau_ {a} \int_ {U _ {a} - 1} ^ {U _ {a} - \rho_ {a}} \frac {w _ {x}}{u _ {x}} d x + \max  \left\{\tau_ {a}, \frac {\Psi_ {a}}{\rho_ {a}} \right\} \int_ {U _ {a} - \rho_ {a}} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x \\ \leq \max  \left\{\tau_ {a}, \frac {\Psi_ {a}}{\rho_ {a}} \right\} \int_ {U _ {a} - 1} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x \\ \end{array}
$$
$$
\operatorname {o r} \leq \tau_ {a} \int_ {U _ {a} - 1} ^ {U _ {a}} \frac {w _ {x}}{u _ {x}} d x \text {i f} \rho_ {a} = 0
$$
Note that for the bound of Lemma 11, we did not require that  $\ell_{a}$  is integral. We can thus apply Lemma 11 to bound  $\max \left\{\tau_{a}, \frac{\Psi_{a}}{\rho_{a}}\right\}$  and obtain the same result, which proves Theorem 13.
Figure 6: Experimental results for varying values of  $\alpha$  on synthetic data with 12 advertisers and 2000 impressions of 10 types, where we report the same quantities as in Figure 3. We use Dual Base predictions for different  $\sigma$  and  $\epsilon$ . Note that there are two black lines indicating the performance of the worst-case algorithm without predictions, corresponding to the datasets with differing  $\sigma$ .
Figure 7: Performance for varying prediction quality with the data from Figure 6 (top) for  $\alpha = 2$  (top) and  $\alpha = 5$  (bottom).
C Further Experimental Results
C.1 Real-World Data
Description of the Yahoo Dataset: The original dataset contains impression allocations to 16268 advertisers throughout 123 days, each tagged with the advertiser that bought the impression and a set of keyphrases that categorize the user for whom the impression is displayed. Lavastida et al. (2021) then consider the 20 most common keyphrases and create an impression type for each non-empty subset thereof. Whenever an advertiser buys an impression with a certain set of keyphrases, we assume that all impression types that correspond to a superset of these keyphrases are relevant for this advertiser, and that it derives some constant value (say, 1) from this allocation. At the same time, the number of impressions we create from each impression type (i.e. the supply) is the number of impression allocations in the original dataset that show that the impression type is relevant for an advertiser. As such, we obtain around 2 million impressions. Lavastida et al. (2021) try multiple
Figure 8: Performance on a worst-case instance with different predictors.
impression orders and budgets for the advertisers, but due to space constraints we restrict ourselves to display all impressions of a type at once, in supply-ascending order. We determine advertisers' budgets by allocating each impression to one of the advertisers with non-zero valuation uniformly at random and taking the number of allocated impressions at the end to be the advertiser's budget.
C.2 Synthetic Data
Results: Figure 5 shows consistency and robustness of our algorithm on synthetic data on  $T = 2000$  impressions of 10 types and  $k = 12$  advertisers, for a variation of predictions. The plot shows the performance for predictions from the optimum solutions (with varying corruption) and the dual base prediction. Our algorithm converges to almost perfect consistency and robustness for  $\alpha = 10$ , given the optimum solution. At the same time, we observe that the algorithm is robust against both random and biased corruption, as the robustness does not drop to the prediction's low competitiveness of around 0.7. Furthermore, the algorithm performs well in combination with the dual base prediction for  $\epsilon = 0.1$  even though the first 200 impressions are clearly not representative of all synthetically generated impressions.
To investigate the our algorithm in conjunction with an easily available prediction, we also analyze the behavior of the dual base algorithm for different values of  $\sigma$  and  $\epsilon$  in Figure 6. The performance of our algorithm under dual base predictions clearly improves for increasing values of  $\sigma$  as impressions become more evenly distributed across the day. Generally, sampling more impressions helps but dual base predictions may also lead to a drop in robustness, and more samples can even lead to a more adversarial prediction, as we explore further below. Yet, the robustness does still stays above the prediction's competitiveness in these cases.
Figure 7 shows consistency and robustness for different predictions with varying competitiveness on  $\alpha \in \{2,5\}$ . We achieve this by varying the fraction  $\epsilon \in [0,1]$  of samples for the dual base algorithm and the corruption rate  $p \in [0,1]$  for random and biased corruptions. For  $\alpha = 2$ , the consistency exceeds 1 if the prediction is not very good (competitiveness below 0.9). The algorithm is not heavily influenced by a bad prediction since  $\alpha = 2$  is low, so the total obtained value remains relatively constant. For  $\alpha = 5$ , the algorithm might however follow the bad choices of the prediction, so the competitiveness varies more. As expected, the average robustness decreases for increasing  $\alpha$ , but the dual base prediction starts out with a much lower robustness than the corrupted predictions. The reason for that is that both the dual base algorithm and exponential averaging make their decisions based on the discounted gain. Our algorithm might therefore easily disregard a corrupted prediction as its discounted gain is low (or even negative), but the dual base prediction looks like a sensible choice. The dual base algorithm therefore manages to fool the algorithm for low  $\alpha$ , while a biased corruption leads to the worst corruption for larger values of  $\alpha$ .
**Hard Instances:** We consider the worst-case instance for the Display Ads problem described in Mehta et al. (2007). For  $k$  advertisers, we create impressions of types  $r \in \{1, \dots, k\}$ . An impression  $t$  of type  $r$  has zero value for the first  $r - 1$  advertisers  $w_{1,t} = \dots = w_{r - 1,t} = 0$  and value 1 for the following advertisers  $w_{r,t} = \dots = w_{k,t} = 1$ . We first show all impressions of type 1, then all impressions of type 2, and so forth. The instance is difficult as the algorithm—not knowing about future impressions—has to allocate impressions of a type equally among advertisers that can
Figure 9: Performance on synthetic Ad Words instances, compared to the algorithm of Mehta et al. (2007). The black lines show the robustness of two worst-case algorithms without predictions: The algorithm due to Feldman et al. (2009a) which is the basis for our algorithm, and the algorithm of Mehta et al. (2007), which serves as a basis for the algorithm of Mahdian et al. (2007).
derive value from this impression type. As shown by Mehta et al. (2007), the competitiveness of the exponential averaging algorithm reaches  $1 - \frac{1}{e}$  for  $k \to \infty$  on this instance.
We evaluate the performance of our algorithm on this worst-case instance in Figure 8. Providing the optimum solution as prediction allows the algorithm to quickly ascend to a perfect robustness of 1. We also consider two (biased) corrupted versions of this prediction with  $p \in \{50\%, 75\% \}$ . In both cases, the algorithm still achieves a robustness above the competitiveness of the prediction. The dual base algorithm cannot deliver meaningful predictions as it only sees impressions of the first type, which are clearly not representative of the following impressions by construction.
C.3 Evaluation of GAP on an Ad Words Instance