SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \Delta D = B _ {a} \left(\beta_ {a} ^ {(t)} - \beta_ {a} ^ {(t - 1)}\right) + z _ {t} \\ = B _ {a} \left(\beta_ {a} ^ {(t)} - \beta_ {a} ^ {(t - 1)}\right) + \alpha_ {B} \left(w _ {B _ {a}} - \beta_ {a} ^ {(t - 1)}\right) \\ \leq B _ {a} \left(\beta_ {a} ^ {(t)} - \hat {\beta} _ {a} ^ {(t - 1)}\right) + \alpha_ {B} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ = B _ {a} \left(\left(e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1\right) \hat {\beta} _ {a} ^ {(t - 1)} + \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha}} \left(w _ {B _ {a}} - w _ {0} e _ {B _ {a}} ^ {\alpha}\right)\right) + \alpha_ {B} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ = \alpha_ {B _ {a}} \hat {\beta} _ {a} ^ {(t - 1)} + \frac {\alpha_ {B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} \left(w _ {B _ {a}} - w _ {0} e _ {B _ {a}} ^ {\alpha}\right) + \alpha_ {B} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ = \alpha_ {B _ {a}} \frac {e _ {B _ {a}} ^ {\alpha}}{e _ {B _ {a}} ^ {\alpha} - 1} \underbrace {\left(\hat {\beta} _ {a} ^ {(t - 1)} - w _ {0}\right)} _ {\geq 0} + \frac {\alpha_ {B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} \underbrace {\left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right)} _ {\geq 0} + \alpha_ {B} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ \leq \alpha_ {B} \frac {e _ {B} ^ {\alpha}}{e _ {B} ^ {\alpha} - 1} \left(\hat {\beta} _ {a} ^ {(t - 1)} - w _ {0}\right) + \frac {\alpha_ {B}}{e _ {B} ^ {\alpha} - 1} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) + \alpha_ {B} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ = \alpha_ {B} \frac {e _ {B} ^ {\alpha}}{e _ {B} ^ {\alpha} - 1} \left(\hat {\beta} _ {a} ^ {(t - 1)} - w _ {0}\right) + \alpha_ {B} \frac {e _ {B} ^ {\alpha}}{e _ {B} ^ {\alpha} - 1} \left(w _ {B _ {a}} - \hat {\beta} _ {a} ^ {(t - 1)}\right) \\ = \alpha_ {B} \frac {e _ {B} ^ {\alpha}}{e _ {B} ^ {\alpha} - 1} \left(w _ {B _ {a}} - w _ {0}\right) = B \frac {e _ {B} ^ {\alpha / B} - 1}{e _ {B} ^ {\alpha} - 1} e _ {B} ^ {\alpha} \left(w _ {B _ {a}} - w _ {0}\right) \\ \end{array}
$$
where the second inequality is due to  $\alpha_{B} \geq \alpha_{B_{a}}$  and  $e_{B} \leq e_{B_{a}}$ , as shown in Lemma 2.
A.2 Proof of Theorem 1 (Consistency)
In the following, we upper bound PRD using the comparison in Line 7 of Algorithm 1.
Lemma 3. We have
$$
\operatorname {P R D} \leq \sum_ {a} \left(\left(B _ {a} - \ell_ {a}\right) \beta_ {a} ^ {(T)} + \frac {1}{\alpha_ {B}} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} \left(w _ {a t} - \beta_ {a} ^ {(t - 1)}\right) + \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t}\right)
$$
Proof. We first split impressions  $t$  into two categories: Either the algorithm followed the prediction and assigned  $t$  to  $a^{(t)} = a_{\mathrm{(PRD)}}^{(t)}$ , or the algorithm ignored the prediction and assigned  $t$  to  $a^{(t)} = a_{\mathrm{(EXP)}}^{(t)} \neq a_{\mathrm{(PRD)}}^{(t)}$ . In the latter case, due to the selection rule in Line 7 of Algorithm 1,
$$
\alpha_ {B} \Big (w _ {a _ {(\mathrm {P R D})} ^ {(t)} t} - \beta_ {a _ {(\mathrm {P R D})} ^ {(t)}} ^ {(t - 1)} \Big) \leq w _ {a _ {(\mathrm {E X P})} ^ {(t)} t} - \beta_ {a _ {(\mathrm {E X P})} ^ {(t)}} ^ {(t - 1)}.
$$
In symbols,
$$
\mathrm {P R D} = \sum_ {a} \left(\sum_ {t \in \mathbf {P} _ {a} \backslash \mathbf {X} _ {a}} w _ {a t} + \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t}\right)
$$
$$
\begin{array}{l} \leq \sum_ {a} \left(\sum_ {t \in \mathbf {P} _ {a} \backslash \mathbf {X} _ {a}} \left(\beta_ {a} ^ {(t - 1)} + \frac {1}{\alpha_ {B}} \left(w _ {a _ {\text {(E X P)}} ^ {(t)}, t} - \beta_ {a _ {\text {(E X P)}} ^ {(t)}} ^ {(t - 1)}\right)\right) + \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t}\right) \\ = \sum_{a}\left(\underbrace{\sum_{t\in\mathbf{P}_{a}\setminus\mathbf{X}_{a}}\beta_{a}^{(t - 1)}}_{(\dagger)} + \frac{1}{\alpha_{B}}\sum_{t\in \mathbf{X}_{a}\setminus \mathbf{P}_{a}}\left(w_{at} - \beta_{a}^{(t - 1)}\right) + \sum_{t\in \mathbf{P}_{a}\cap \mathbf{X}_{a}}w_{at}\right) \\ \end{array}
$$
where the last equality holds because  $\{\mathbf{P}_a\}_a$  and  $\{\mathbf{X}_a\}_a$  are both partitioning the set of all impressions due to the introduction of the dummy advertiser. For  $(\dagger)$ , we use that  $\beta_{a}$  can only increase in each round and bound
$$
\sum_ {t \in \mathbf {P} _ {a} \backslash \mathbf {X} _ {a}} \beta_ {a} ^ {(t - 1)} \leq \left(B _ {a} - \ell_ {a}\right) \beta_ {a} ^ {(T)}.
$$
For the remainder of this section, we consider a fixed advertiser  $a$ . Let us denote with  $t_i$  the  $i$ -th impression allocated to  $a$ . Let
$$
(\star) = \frac {1}{\alpha_ {B}} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} \left(w _ {a t} - \beta_ {a} ^ {(t - 1)}\right) + \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t}
$$
as part of the bound on PRD in Lemma 3.
In order to understand this bound, we make some useful observations in the following lemma to simplify the analysis. The key idea is that we may assume that impressions in  $\mathbf{X}_a$  are ordered to be non-decreasing. In particular, we need to argue that the sum  $\sum_{t\in \mathbf{X}_a\setminus \mathbf{P}_a}\beta_a^{(t - 1)}$  can only decrease (as this term is negated in  $(\star)$ ) when impressions in  $\mathbf{X}_a$  are ordered to be non-decreasing: Intuitively, each  $\beta_{a}^{(t - 1)}$  depends only on the  $B_{a}$  most valuable impressions assigned before impression  $t$ , no matter the order in which  $\mathbf{X}_a^{(t - 1)}$  arrived. We can thus minimize each  $\beta_{a}^{(t - 1)}$  if the impressions allocated prior to  $t$  are the impressions of smallest value. To simultaneously minimize each  $\beta_{a}^{(t)}$  in the sum, we order the impressions in  $\mathbf{X}_a$  to have non-decreasing value. We prove this simplification formally in the following lemma.
Lemma 4. Without loss of generality, we may assume that  $\mathbf{P}_a\cap \mathbf{X}_a$  are the most valuable impressions in  $\mathbf{X}_a$  and that impressions in  $\mathbf{X}_a$  arrive such that their values are non-decreasing.
Proof. We may assume that the impressions in  $\mathbf{P}_a\cap \mathbf{X}_a$  are the most valuable impressions in  $\mathbf{X}_a$ : this can only increase the value of  $\mathbf{P}_a$  but leaves  $\mathbf{S}_a$  unaffected, as  $\mathbf{S}_a$  are by design the  $B_{a}$  most valuable impressions in  $\mathbf{X}_a$ . All impressions in  $(\star)$  are from  $\mathbf{X}_a$ , so reordering impressions only affects  $(\star)$ . Specifically, we can show that the sum  $\sum_{t\in \mathbf{X}_a\setminus \mathbf{P}_a}\beta_a^{(t - 1)}$  in  $(\star)$  is minimized if the values in  $\mathbf{X}_a$  are ordered to be non-decreasing. Assume to the contrary that the  $i$ -th impression added to  $a$  is the last that is in order. That is  $w_{at_1}\leq w_{at_2}\leq \dots \leq w_{at_i}$  and there exists a  $j\leq i$  such that  $w_{at_{j - 1}}\leq w_{at_{i + 1}} < w_{at_j}$ . Moving  $t_{i + 1}$  ahead to its ranked position within the first  $i$  impressions allocated to  $a$  changes the ordering as follows (the first and second row show the impression values before and after changing the position of  $t_{i + 1}$ , respectively):
$$
\begin{array}{l} w _ {a t _ {1}} \leq \dots \leq w _ {a t _ {j - 1}} \leq w _ {a t _ {j}} \leq w _ {a t _ {j + 1}} \leq \dots \leq w _ {a t _ {i}} \\ w _ {a t _ {1}} \leq \dots \leq w _ {a t _ {j - 1}} \leq w _ {a t _ {i + 1}} <   w _ {a t _ {j}} \leq \dots \leq w _ {a t _ {i - 1}} \end{array}
$$
Note that each position decreases in value, even strictly at the  $j$ -th position. As such, the exponential average  $\beta_{a}^{(t-1)}$  decreases as well for  $t < t_{i+1}$ ; it remains constant for  $t \geq t_{i+1}$  as it only depends on the  $B_{a}$  most valuable impressions assigned up to  $t$  which remain the same. We can thus simultaneously minimize  $\beta_{a}^{(t)}$  for each  $t$  by putting  $\mathbf{X}_{a}$  in non-decreasing order. This reordering does not affect  $\beta_{a}^{(T)}$  or the other terms in  $(\star)$ , so we may indeed assume that values are non-decreasing.
In light of Lemma 4, we can write  $(\star)$  as follows.
Lemma 5. We have
$$
(\star) = \frac {1}{\alpha_ {B}} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \left(w _ {a t _ {i}} - \beta_ {a} ^ {(t _ {i - 1})}\right) + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}}.
$$
Proof. Impression values are non-decreasing due to Lemma 4, so  $w_{at_i}$  is the  $i$ -th least valuable impression in  $\mathbf{X}_a$ . The impressions  $\{I_a - \ell_a + 1, \dots, I_a\} = \mathbf{X}_a \cap \mathbf{P}_a$  are thus the most valuable. We can now write  $(\star)$  as
$$
\frac {1}{\alpha_ {B}} \sum_ {t \in \mathbf {X} _ {a} \backslash \mathbf {P} _ {a}} \left(w _ {a t} - \beta_ {a} ^ {(t - 1)}\right) + \sum_ {t \in \mathbf {P} _ {a} \cap \mathbf {X} _ {a}} w _ {a t} = \frac {1}{\alpha_ {B}} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \left(w _ {a t _ {i}} - \beta_ {a} ^ {(t _ {i} - 1)}\right) + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}}
$$
where  $\beta_{a}^{(t_{i} - 1)} = \beta_{a}^{(t_{i - 1})}$  as there was no change to the dual variable of advertiser  $a$  since no impression in  $\{t_{i - 1} + 1,\dots ,t_i - 1\}$  was allocated to  $a$ .
Combining Lemmas 3 and 5, we obtain:
Lemma 6. PRD  $\leq \sum_{a}\mathrm{PRD}_{a}$  where
$$
\mathrm {P R D} _ {a} := \frac {1}{\alpha_ {B}} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \left(w _ {a t _ {i}} - \beta_ {a} ^ {(t _ {i - 1})}\right) + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} + (I _ {a} - \ell_ {a}) \beta_ {a} ^ {(T)}.
$$
In the following, we use the non-decreasing ordering of impressions in  $\mathbf{X}_a$  to compute  $\beta_{a}^{(t_{i - 1})}$  and bound  $\mathrm{PRD}_a$  with a linear combination of values  $w_{at_i}$ . Consider the  $j$ -th impression  $t_j$  allocated to  $a$ . Since we assume that impression values are non-decreasing, we know that  $t_j$  becomes the most valuable impression right after it is allocated. After the allocation of the  $(j + 1)$ -th impression to  $a$ , it becomes the second most valuable impression, and so forth, until it is disposed after the allocation of the  $(j + B_a)$ -th impression. The value  $w_{at_j}$  therefore appears alongside each coefficient in the convex combination that defines  $\beta_{a}^{(t_{i - 1})}$  for  $i \in \{j + 1, \dots, j + B_a\}$ . Expanding each  $\beta_{a}^{(t_{i - 1})}$  in the sum  $\sum_{i = 1}^{I_a - \ell_a} \beta_{a}^{(t_{i - 1})}$  in  $\mathrm{PRD}_a$ , we thus observe that the coefficients of values  $w_{at_j}$  for  $j \leq I_a - \ell_a - B_a$  sum up to 1. We use this fact to cancel out most of the values in  $\sum_{i = 1}^{I_a - \ell_a} w_{at_i}$ . What remains are only the values  $w_{at_i}$  for  $i \in \{I_a - \ell_a - B_a + 1, \dots, I_a - \ell_a\}$ . For  $i \in \{I_a - \ell_a - B_a + 1, \dots, I_a - B_a\}$ , we bound  $w_{at_i}$  by  $w_{at_{I_a - B_a}}$  which is really the best we can hope for. Formally, we show:
$$
\mathrm {P R D} _ {a} \leq \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} \phi_ {i} w _ {a t _ {i}} + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} \psi_ {i} w _ {a t _ {i}} + w _ {a t I _ {a} - B _ {a}} \Omega_ {a}
$$
with coefficients
$$
\phi_ {i} := (B _ {a} - \ell_ {a}) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha}} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}} + \frac {1}{\alpha B} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1}
$$
$$
\psi_ {i} := 1 + (B _ {a} - \ell_ {a}) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}}
$$
$$
\Omega_ {a} := \frac {1}{\alpha_ {B}} \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \left(\ell_ {a} e _ {B _ {a}} ^ {\alpha} - \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (B _ {a} - \ell_ {a}) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}\right).
$$
Proof. We start by rewriting the terms in  $\mathrm{PRD}_a$  individually. Since we assume that the values are non-decreasing, we can express  $\beta_{a}^{(t_{i - 1})}$  as the exponential average of values  $w_{at_i - B_a}, w_{at_i - B_a + 1}, \ldots, w_{at_i - 1}$  of the last  $B_a$  impressions (for simplicity, we set  $w_{at_j} = 0$  for  $j \leq 0$ ). Summing over multiple iterations, we thus obtain for the sum over the dual variables that
$$
\sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \beta_ {a} ^ {(t _ {i - 1})} = \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \sum_ {j = i - B _ {a}} ^ {i - 1} w _ {a t _ {j}} e _ {B _ {a}} ^ {\alpha (i - j - 1) / B _ {a}}
$$
$$
\begin{array}{l} = \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {j = 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {j}} \sum_ {i = j + 1} ^ {\min  \{j + B _ {a}, I _ {a} - \ell_ {a} \}} e _ {B _ {a}} ^ {\alpha (i - j - 1) / B _ {a}} \\ = \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {j = 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {j}} \sum_ {i = 1} ^ {\min  \{B _ {a}, I _ {a} - \ell_ {a} - j \}} e _ {B _ {a}} ^ {\alpha (i - 1) / B _ {a}} \\ = \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {j = 1} ^ {I _ {a} - B _ {a} - \ell_ {a}} w _ {a t _ {j}} \sum_ {i = 1} ^ {I _ {a}} e _ {B _ {a}} ^ {\alpha (i - 1) / B _ {a}} \\ + \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {j = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a} - j} e _ {B _ {a}} ^ {\alpha (i - 1) / B _ {a}} \\ = \sum_ {i = 1} ^ {I _ {a} - B _ {a} - \ell_ {a}} w _ {a t _ {i}} + \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} \left(e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}} - 1\right). \\ \end{array}
$$
where for the last equality, we use that the two inner sums are geometric. We can use this expression to cancel out most of the terms of the first sum in  $\mathrm{PRD}_a$ :
$$
\begin{array}{l} \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} \left(w _ {a t _ {i}} - \beta_ {a} ^ {(t _ {i - 1})}\right) \\ = \sum_ {i = 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} - \sum_ {i = 1} ^ {I _ {a} - B _ {a} - \ell_ {a}} w _ {a t _ {i}} - \frac {1}{\alpha_ {B} \left(e _ {B _ {a}} ^ {\alpha} - 1\right)} \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} \left(e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}} - 1\right) \\ = \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} \left(1 - \frac {e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1}\right) \\ = \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t i} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} \\ = \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t i} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha \left(I _ {a} - \ell_ {a} - i\right) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} + \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - B _ {a}} w _ {a t i} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha \left(I _ {a} - \ell_ {a} - i\right) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1}. \tag {3} \\ \end{array}
$$
We use that  $w_{at_i} \leq w_{atI_a - B_a}$  for all  $i \leq I_a - B_a$  to upper bound the second sum, divided by  $\alpha_B$ , in (3) to
$$
\begin{array}{l} \frac {1}{\alpha_ {B}} \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - B _ {a}} w _ {a t _ {i}} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} \\ \leq w _ {a t I _ {a} - B _ {a}} \frac {1}{\alpha_ {B}} \sum_ {i = I _ {a} - B _ {a} - \ell_ {a} + 1} ^ {I _ {a} - B _ {a}} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} (4) \\ = w _ {a t _ {I _ {a} - B _ {a}}} \frac {1}{\alpha_ {B}} \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \left(\ell_ {a} e _ {B _ {a}} ^ {\alpha} - \sum_ {i = B _ {a} - \ell_ {a}} ^ {B _ {a} - 1} e _ {B _ {a}} ^ {\alpha i / B _ {a}}\right) \\ = w _ {a t _ {I _ {a} - B _ {a}}} \underbrace {\frac {1}{\alpha_ {B}} \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \left(\ell_ {a} e _ {B _ {a}} ^ {\alpha} - \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (B _ {a} - \ell_ {a}) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}\right)} _ {= \Omega_ {a}}. (5) \\ \end{array}
$$
Furthermore, by definition of  $\beta_{a}^{(T)} = \beta_{a}^{(t_{I_{a}})}$ ,
$$
\left(B _ {a} - \ell_ {a}\right) \beta_ {a} ^ {(T)} = \left(B _ {a} - \ell_ {a}\right) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}}. \tag {6}
$$
We combine (3), (5), and (6) and group terms to obtain the desired bound
$$
\begin{array}{l} \mathrm {P R D} _ {a} \leq \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} + \frac {1}{\alpha_ {B}} \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} + w _ {a t I _ {a} - B _ {a}} \Omega_ {a} \\ + \left(B _ {a} - \ell_ {a}\right) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}} \\ = \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} w _ {a t _ {i}} \underbrace {\left((B _ {a} - \ell_ {a}) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}} + \frac {1}{\alpha_ {B}} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1}\right)} _ {= \phi_ {i}} \\ + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} \underbrace {\left(1 + (B _ {a} - \ell_ {a}) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}}\right)} _ {= \psi_ {i}} + w _ {I _ {a} - B _ {a}} \Omega_ {a} \\ \end{array}
$$
We can express ALG analogously:
Lemma 8. We have  $\mathrm{ALG} = \sum_{a}\mathrm{ALG}_{a}$  where
$$
\mathrm {A L G} _ {a} := \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}}.
$$
Proof. We have  $\mathrm{ALG} = \sum_{a}\sum_{t\in \mathbf{S}_{a}}w_{at}$ . As we always dispose of the least valuable impression in Algorithm 1,  $\mathbf{S}_a$  are the  $B_{a}$  most valuable impressions in  $\mathbf{X}_a$ . Due to Lemma 4, these are  $\mathbf{S}_a = \{I_a - B_a + 1,\dots ,I_a\}$  and hence  $\sum_{t\in \mathbf{S}_a}w_{at} = \sum_{i = I_a - B_a + 1}^{I_a}w_{at_i} = \mathrm{ALG}_a$ .
We upper bound the ratio PRD/ALG by  $\max_{a}\mathrm{PRD}_{a} / \mathrm{ALG}_{a}$ . To this end, we fix an advertiser  $a$  and upper bound the ratio  $\mathrm{PRD}_a / \mathrm{ALG}_a$ . Recall from Lemmas 7 and 8 that we can express  $\mathrm{PRD}_a$  and  $\mathrm{ALG}_a$  as linear combination over impression values. We could obtain a natural upper bound by comparing impression value coefficients. However, in the following lemma, we show how to use the non-decreasing ordering due to Lemma 4 to obtain a tighter bound.
$$
\Phi_ {a} := \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} \phi_ {i} \quad \text {a n d} \quad \Psi_ {a} := \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} \psi_ {i}
$$
as the total factor mass on values  $w_{at_i}$  for  $\phi_i$  and  $\psi_i$ , respectively. Let  $\tau_a \coloneqq \left(\Phi_a + \Psi_a + \Omega_a\right) / B_a$  be the average factor. Recall that
$$
\mathrm {P R D} _ {a} \leq \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a} - \ell_ {a}} \phi_ {i} w _ {a t _ {i}} + \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} \psi_ {i} w _ {a t _ {i}} + w _ {a t I _ {a} - B _ {a}} \Omega_ {a} \tag {7}
$$
$$
\mathrm {A L G} _ {a} = \sum_ {i = I _ {a} - B _ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}}.
$$
In the following lemma, we use that  $w_{at_i} \leq w_{at_j}$  for  $i \leq j$  due to Lemma 4, to further upper bound the RHS of 7 by a linear combination of the values, where we move mass from coefficients on  $w_{at_i}$  to coefficients on  $w_{at_j}$ . Additionally, we move mass from  $\Omega_a$  to coefficients  $\phi_i$  for  $i \in \{I_a - B_a + 1, \dots, I_a - \ell_a\}$  and from  $\phi_i$  to  $\psi_j$  for  $j \in \{I_a - \ell_a + 1, \dots, I_a\}$ . In the best case, we are able to redistribute mass equally across all values, in which case the consistency is given as the average factor  $\tau_a$ . Otherwise, the factors on the largest values dominate, giving us a consistency of  $\Psi_a / \ell_a$ .
Lemma 9. We have
$$
\frac {\mathrm {P R D} _ {a}}{\mathrm {A L G} _ {a}} \leq \left\{ \begin{array}{l l} \max  \left\{\tau_ {a}, \frac {\Psi_ {a}}{\ell_ {a}} \right\} & i f \ell_ {a} > 0 \\ \tau_ {a} & o t h e r w i s e \end{array} \right.
$$
$$
\tau_ {a} = 1 + \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \frac {1}{\alpha_ {B}} \left(e _ {B _ {a}} ^ {\alpha} - \frac {e _ {B _ {a}} ^ {\alpha} - 1}{\alpha_ {B _ {a}}}\right)
$$
$$
\frac {\Psi_ {a}}{\ell_ {a}} = 1 + \left(\frac {B _ {a}}{\ell_ {a}} - 1\right) \frac {e _ {B _ {a}} ^ {\alpha \ell_ {a} / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1}.
$$
Proof. We calculate  $\tau_{a}$  and  $\Psi_{a} / \ell_{a}$  separately in Lemma 10 below. Our main goal is to distribute mass from the factors  $\phi_{i},\psi_{i}$ , and from  $\Omega_{a}$  equally to the values  $w_{I_a - B_a + 1},\ldots ,w_{I_a}$ . We begin by taking a closer look at the factors  $\phi_{i}$  and  $\psi_{i}$ . First, note that  $\psi_{i}$  is always decreasing in  $i$  as
$$
\psi_{i} = 1 + \underbrace{(B_{a} - \ell_{a})}_{\geq 0}\underbrace{\frac{e^{\alpha / B_{a}} - 1}{e^{\alpha}_{B_{a}} - 1}}_{\geq 0}e^{\alpha (I_{a} - i) / B_{a}}.
$$
We can therefore bound the linear combination over values in  $\{I_a - \ell_a + 1, \dots, I_a\}$  using the average value  $\bar{w}_{\Psi} \coloneqq \frac{1}{\ell_a} \sum_{i = I_a - \ell_a + 1}^{I_a} w_{at_i}$  as
$$
\sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} w _ {a t _ {i}} \psi_ {i} \leq \sum_ {i = I _ {a} - \ell_ {a} + 1} ^ {I _ {a}} \bar {w} _ {\Psi} \psi_ {i} = \bar {w} _ {\Psi} \Psi_ {a}. \tag {8}
$$
However,  $\phi_{i}$  is not always decreasing which can be seen by rearranging
$$
\begin{array}{l} \phi_ {i} = \left(B _ {a} - \ell_ {a}\right) \frac {e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1}{e _ {B _ {a}} ^ {\alpha} - 1} e ^ {\alpha (I _ {a} - i) / B _ {a}} + \frac {1}{\alpha_ {B}} \frac {e _ {B _ {a}} ^ {\alpha} - e _ {B _ {a}} ^ {\alpha (I _ {a} - \ell_ {a} - i) / B _ {a}}}{e _ {B _ {a}} ^ {\alpha} - 1} \\ = \frac {1}{e _ {B _ {a}} ^ {\alpha} - 1} \left((B _ {a} - \ell_ {a}) (e _ {B _ {a}} ^ {\alpha / B _ {a}} - 1) - \frac {1}{\alpha_ {B}} e _ {B _ {a}} ^ {- \alpha \ell_ {a} / B _ {a}}\right) e _ {B _ {a}} ^ {\alpha (I _ {a} - i) / B _ {a}} + \frac {1}{\alpha_ {B}} \frac {e _ {B _ {a}} ^ {\alpha}}{e _ {B _ {a}} ^ {\alpha} - 1}. \\ \end{array}
$$
We observe that  $\phi_{i}$  is decreasing if  $(B_{a} - \ell_{a})\left(e_{B_{a}}^{\alpha /B_{a}} - 1\right)$  is at least  $\frac{1}{\alpha_B} e_{B_a}^{-\alpha \ell_a / B_a}$ , and we analyze two cases based on the relationship of both terms.