• 可以開始研究如何把我們在Second Price Auction的應用拓展到First Price
  • Challenge: Where is the data of first price auction?
• 我認為在 KDD 2018 Deep Censored Learning 之中，我們測試三種 parametric distribution 不是對的方向
  • 我希望可以有更有彈性的，對於 distribution 的描述

• \(y\) 是 winning price
• \(X\) 是 features
• 給定一個數列\(0 < tau_1 < \tau_2 < ... < \tau_k < 1\)
• 如果用\(q_{\tau}\)代表\(y\)的\(\tau\)-quantile，我希望找到\(q_{\tau_1} \leq q_{\tau_2} \leq ... \leq q_{\tau_k} | X\) 的估計
• 只要\(\tau\) 夠密，這就是一個描述CDF的方式，可能會比現有的parametric way 更有彈性

• Deep Quantile Regression, A blog posted at Mar. 25.
• \(y\) is an observed winning price, \(x\) is the features, \(f(\tau, x)\)表示基於\(x\)預測的\(\tau\)-quantile
• Loss:

\[\mathbb{L}(\delta = y - f(\tau, x)) = \left\{\begin{array}{lc} \tau \delta & \text{ if } \delta \geq 0, \\ (\tau - 1) \delta & \text{ if } \delta < 0. \end{array}\right.\]

• It is trivial to implement \(\tau\)-quantile regression in Keras
• However, can we estimate a series of quantile at the same time?
  • Yes, multi-task learning
  • Challenge: cross-over \(\hat{q}_{\tau_1} > \hat{q}_{\tau_2}\)
• Existed Implementation: Keras Quantile Model
• Quantile Loss in Tensorflow

\[\mathbb{L}(\delta = y - max(C, f(\tau, x))) = \left\{\begin{array}{lc} \tau \delta & \text{ if } \delta \geq 0, \\ (\tau - 1) \delta & \text{ if } \delta < 0. \end{array}\right.\]