Paper:
Contextual Pricing for Lipschitz Buyers
We investigate the problem of learning a Lipschitz function from binary feedback. In this problem, a learner is trying to learn a Lipschitz function f : [0,1] d → [0, 1] over the course of T rounds. On round t, an adversary provides the learner with an input x t , the learner submits a guess y t for f (x t ), and learns whether y t > f (x t ) or y t ≤ f (x t ). The learner's goal is to minimize their total loss t (f (x t ), y t ) (for some loss function ). The problem is motivated by contextual dynamic pricing, where a firm must sell a stream of differentiated products to a collection of buyers with non-linear valuations for the items and observes only whether the item was sold or not at the posted price. For the symmetric loss (f (x t ), y t ) = |f (x t ) − y t |, we provide an algorithm for this problem achieving total loss O(log T ) when d = 1 and O(T (d−1)/d ) when d > 1, and show that both bounds are tight (up to a factor of √ log T ). For the pricing loss function (f (x t ), y t ) = f (x t ) − y t 1{y t ≤ f (x t )} we show a regret bound of O(T d/(d+1) ) and show that this bound is tight. We present improved bounds in the special case of a population of linear buyers.

Review:
This  paper studies the problem of learning  a  Lipschitz  function  $f(x)$  from  binary  feedback.  Two  types of loss  are  considered,  the  symmetric  ell-1  loss  and  the  unsymmetric  and  discontinuous  pricing  loss.  To  minimize  the  losses,  four  algorithms  are  proposed.      Alg. 1  maintains  a  set  $S  =  {  (X_j,  Y_j)  }$,  where  $X_j$  is  a  partition  interval  of  feasible set  $[0,1]$  and  $Y_j$  contains  the  lower  and  upper  bounds  of  $f(x)$.  At  each  iteration,  an  adversary  queries  a  point  $x_t$.  The  learner  response  with  estimate  $y  =  mid(Y_j)$,  where  $X_j$  is  the  interval  containing  $x_t$.  Base  on  the  feedback,  the  learner  shrinks  $Y_j$  by  intersecting  itself  with  a  set  derived  from  Lipschitz  property  of  $f(x)$.  The  iteration  is  finalized  by  splitting  $X_j$  into  two  new  intervals  when  the  length  of  $Y_j$  can  hardly  be  reduced.  Alg. 2  encodes  the  upper  bound  and  lower  bound  of  $f(x)$  in  sets  $S+$  and  $S-$  respectively.  At each  iteration,  the  learner  receives  a  query  $x_t$,  composing  an  interval  containing  $f(x_t)$  by  solving  a linear  optimization  constrained  on  $S+$  and  $S-$. And it uses  the  mid-point  of  the  interval  as  the  current  guess.  The   $S+$  and  $S-$  are  updated  based  on  the  binary  feedback. Alg. 3  and  Alg. 4  are  extensions  of  Alg. 1  to  the  case  of  multi-dimensional  Lipschitz  functions.  The  author  proves  that  Alg. 3  and  Alg.  4  can  learn  $f(x)$  with  sublinear  regret  for  the  ell-1  loss  and  the  pricing  loss  respectively.  The  two  algorithm  are  optimal  by  giving  the  lower  bound  of  Lipschitz  learning  problem.  The problem is similar to learning a Lipchitz  function  from  binary  feedback  with  ell-1  loss  and  pricing  loss  by  showing  the  optimality  of  their  methods.  The following questions need to be clarified. 1)  All  the  proposed  algorithms  depend  on  the  Lipschitz  constant  $L$.  However,  in  practice,  $L$  may not be satisfied. How  to  adapt  the  proposed  methods  when the condition is not meet ?  2)  Alg.  2  only  supports  the one-dimensional  learning  problem.  How  to  extend  it  to  the  multi-dimension  learning  problem ?  3)  No  empirical  studies  are  performed.  Though  the  author  argues  that  the  model  can  be  used  in  dynamic  pricing  problem,  no  empirical  evidence  validates  the performance.