Budget Pacing

Budget pacing control:

  • take the daily budget as input
  • calculate the delivery schedule
  • based on the schedule, DSP tries to spread the actions throughout the day

Notation:

  • $n$ ad requests
  • $x_i = \{0, 1\}$, decision whether to bid on $i$ or not
  • $v_i$ - value if we win on $i$ and show it to user
  • $c_i$ - cost of showing $i$
  • $\hat{c}_i$ - how much we are willing to bid
    • for second price option, $\hat{c}_i = c_i + \epsilon_i$, and $\epsilon_i$ is unknown to the bidder at bid time
  • $B$ total budget, allocated in $T$ slots $b_t$ s.t. $\sum b_t = B$

Goal:

  • maximize $\sum v_i x_i$
  • s.t. $\sum_j c_j x_j \leqslant b_t$ for all $j$ of $b_t$


Optimization

Metrics:

  • eCPC or eCPA, want to minimize them

More notation:

  • let $s(t) = \sum c_j x_j$ - how much we actually spent during $t$
  • want $\sum s(t)$ be as close to $B$ as possible
  • and $s(t)$ to be close to $b_t$


Assumption:

  • $s(t)$ is proportional to the number of impressions served at the time
  • it means that the price of individual impressions are approximately the same during this time slot
  • the length of the time slot can be chosen s.t. this assumption is not violated

Setting Bid Rate

For each ad spot for time $t$ we assign a bid rate (pacing rate):

  • $s(t)$ is proportional to number of impressions served
  • $\text{requests}(t)$: # of received bid requests
  • $\text{bids}(t)$: # of times we decided to bid
  • $\text{imps}(t)$: # of shown impressions at $t$
  • $\text{bid_rate}(t)$: rate at which we decide to bid on the request
    • $\text{bid_rate}(t) = \text{bids}(t) \, / \, \text{requests}(t)$
  • $\text{win_rate}(t)$: rate at which we win the bid
    • $\text{win_rate}(t) = \text{imps}(t) \, / \, \text{bids}(t)$
  • $s(t) \sim \text{request}(t) \cdot \text{bid_rate}(t) \cdot \text{win_rate}(t)$

So we can control $s(t)$ by changing our $\text{bid_rate}(t)$

Changing bid_rate:

Adjusting bid rate for slot $t+1$:

  • take feedback from slot $t$
  • use this for $t+1$

we can control $\text{bid_rate}(t+1)$


Budget Schedule

Selecting $b_t$

  • uniform - not the best one
  • traffic is different during each hour
  • should allocate more budget for periods with better quality traffic
  • i.e. to the time where the target audience is more active

History-based:

  • for each $b_t$ calculate $p_t$
  • $p_t$ is probability of success (click or conversion)
  • $\sum p_t = 1$

so, ideal spending for the next time slot can be calculated as

$$\left(B - \sum_{m=1}^{t} s(m) \right) \cdot \cfrac{p_{t+1}}{\sum_{m=t+1}^{T} p_m}$$

two parts

  • remaining budget: how much money we still have
  • how good is the next slot compared to all other remaining slots

note that if $p_t = 0$ for some $t$, the system will never explore this time slots, so should always give it some chance

Once we calculate the desired bid rate, we can

  • select top quality ad requests to bid on
  • choose the price we are willing to bid for these requests


Selecting Good Quality Ad Requests

for each impression $i$ we can choose the price to bid with

construct bidding histogram $c^*$

  • this is historical average of costs $c_i$ in $b_t$
  • now we know which $b_t$ is cheapest
  • take base price and scale it up/down considering bid rate

Bid Price

bid_rate vs bid_price:

  • bid_rate controls the frequency of bidding
  • but bid price is important
  • if it's not high enough, we don't win the impression
  • if it's too high, CPA rises


bid price adjustments

  • split requests into 3 groups based on bid rate:
  • define $\beta_1$, $\beta_2$ s.t. $0 < \beta_1 < \beta_2 < 1$

bid_rate can be in one of these 3 groups:

  • safe: $0 < br \leqslant \beta_1$, no delivery issues
  • critical: $\beta_1 < br \leqslant \beta_2$, normal delivery
  • danger: $\beta_2 < br \leqslant 1$ and cannot win enough impressions

Let $u_i$ be the base bid price however set based on CTR/AR model

  • (AR - action rate or conversion rate)


Safe region: learning $\hat{c}_i$

  • look at submitted bid price $\hat{c}_i$ and actual price $c_i$
  • let $\theta_i = c_i \, / \, \hat{c}_i$
  • build a histogram of $\theta_i$, choose $\theta^*$ as the bottom 1-2 percentile
  • submit $\hat{c}_i = \theta^* \cdot u_i$

Critical region:

  • bid $\hat{c}_i = u_i$

Danger region:

  • Reasons for being in this region:
  • audience is too specific, not enough bid requests that satisfy all the criteria
  • bid price is too low to win impressions
  • Can increase the bid price by some coefficient $\rho \geqslant 1$


Models

Estimation of CTR/AR

Why estimate AR?

  • to predict the quality of an ad request
  • to set the bid price: e.g. AR * CPA goal

Ways to do it:


Problems

Cold Start Problem

Ideas:

  • use content features for models to estimate CTR/AR
  • epsilon-greedy strategy for online bid optimization


Overspending

Because of unusual unexpected activity spikes we might spend all the budget earlier than expected

solution:

  • monitor total spend and stop the campaign when we spend more than B
  • monitor spend at $t$ and allow to spend no more than $b_t + \delta$, and pause until $t+1$ if the limit is exceeded


System

Ad request -> check bid rate -> evaluate CTR/AR -> bid -> ...
... -> SSP -> ...
... -> bid log with win -> save to db
db <-> train CTR/AR models
   <-> compute good bid rate


Sources

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