• Advertiser demand:
  • ROI
  • Gaining quality traffic
• Platform ecology index
  • GMV (gross merchandise volume)
    • GMV does not include discounts, costs involved and returns of products.

• expected GMV for single click is \(p(c | u, a) \times v_a\)
• Assumes that the cost of a click paid by the advertiser is \(b_a\)
  • \(roi_{(u, a)} = \frac{p(c|u,a) \times v_a}{b_a}\)
  • \(roi_a = \frac{v_a \times \sum_{u}{n_u \times p(c | u,a)}}{b_a \times \sum_{u}{n_u}} = \frac{E_u[p(c|u,a)] \times v_a}{b_a}\)
  • \(roi_a\) should keep unchanged or be improved (ROI constraint)

Notations

• Conversion: \(p(c|u, a)\) where \(u\) is a user and \(a\) is a clicked ad
• \(v_a\) is the predicted pay-per-buy(PPB) by consumers

• ROI constraint implies:
  • Raise the bid if \(\frac{p(c|u,a)}{E_u[p(c|u,a)]} \geq 1\)
  • Depress the bid if \(\frac{p(c|u,a)}{E_u[p(c|u,a)]} < 1\)

\[max_{b^*_1, ..., b^*_n} f(k,b^*_k)\] such that

• \(k = argmax_i {pctr_i \times b^*_i}\) (seller side)
• \(l(b^*_i) \leq b^*_i \leq u(b^*_i) \forall i \in A\) (buyer side)

• Gai et al. (2017)
• Calibration
  • The estimation is still biased

Group AUC

The data is grouped by user and position of ad spot

\[GAUC = \frac{\sum_{u,p}{w_{(u,p)} \times AUC_{u,p}}}{\sum_{(u,p)}{w_{(u,p)}}}\]

• \(w_{(u,p)}\) is proportional to impression times or click times

\[p(y = 1 | x) = g(\sum_{j=1}^m {\sigma(u_j^T x) \times \eta(w_j^T x)})\]

• \(\Theta = \left\{u_1, ..., um, w_1, ..., w_m\right\} \in \mathbb{R}^{d \times 2m}\) is the model parameters
• \(\sigma\) is the dividing function, divides feature space into \(m\)(hyper-parameter) different regions
• \(\eta\) is the fitting function, gives prediction in each region
• \(g\) ensures our model to satisfy the definition of probability

\[p(y=1|x) = \sum_{i=1}^m \frac{exp(u^T_i x)}{\sum_{j=1}^m{exp(u^T_j x)}} \times \frac{1}{1 + exp(-w_i^T x)}\]

Special Case

• \(\sigma\) is softmax
• \(\eta\) is sigmoid

\[argmin_{\Theta} f(\Theta) = loss(\Theta) + \lambda \left\lVert \Theta \right\rVert_{2,1} + \beta \left\lVert \Theta \right\rVert_1\]

Regularization

• \(loss(\Theta)\) : logloss ( negative log likelihood)
• \(\left\lVert \Theta \right\rVert_{2,1} = \sum_{i} \sqrt{\sum_j{\theta^2_{i,j}}}\): feature selection
• \(\left\lVert \Theta \right\rVert_1 = \sum_{i,j} \left|\theta_{i,j}\right|\): sparsity

• replace gradient with descent direction (It might be subgradient?)
• Find direction first
• Use approximated inverse-Hessian matrix to compute the step size. (Proposed by LBFGS)
• LBFGS

Data Structure (Common Feature Trick)

• In a page view, the user feature is the same with several ad features…

Only compare to LR…..