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# Myersonian Regression

NIPS 2020, (2020)

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摘要

Motivated by pricing applications in online advertising, we study a variant of linear regression with a discontinuous loss function that we term Myersonian regression. In this variant, we wish to find a linear function f : Rd → R that well approximates a set of points (xi, vi) ∈ Rd × [0, 1] in the following sense: we receive a loss of vi ...更多

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简介

- The Myerson price of a distribution is the price that maximizes the revenue when selling to a buyer whose value is drawn from that distribution.
- In many modern applications such as online marketplaces and advertising, the seller doesn’t just set one price p but must instead price a variety of differentiated products.
- In these settings, a seller must design a policy to price items based on their features in order to optimize revenue.
- One would train a pricing policy on historical bids and apply this policy on future products

重点内容

- In economics, the Myerson price of a distribution is the price that maximizes the revenue when selling to a buyer whose value is drawn from that distribution
- While it is not surprising that solving Myersonian regression exactly is NP-hard given the discontinuity in the reward function, this has been left open by several previous works
- The same reduction implies that under the Exponential Time Hypothesis (ETH) any algorithm approximating it within an m additive factor must run in time at least eΩ(poly(1/ )), ruling out a fully-polynomial time approximation scheme (FPTAS) for the problem
- We show that (UMR) is unstable in the sense that arbitrarily small perturbations in the input can lead to completely different solutions
- We show that under the Exponential Time Hypothesis (ETH), any algorithm that achieves a m-additive approximation for Myersonian regression must run in time at least exp(O −1/6 )
- A interesting avenue of investigation for future work is to understand how strategic buyers would change their bids in response to a contextual batch learning algorithm and how to design algorithms that are aware of strategic response

结果

- The authors' main result is a polynomial time approximation scheme (PTAS) using dimensionality reduction.
- The same reduction implies that under the Exponential Time Hypothesis (ETH) any algorithm approximating it within an m additive factor must run in time at least eΩ(poly(1/ )), ruling out a fully-polynomial time approximation scheme (FPTAS) for the problem
- This hardness of approximation perfectly complements the algorithmic results, showing that the guarantees are essentially the best that one can hope for

结论

- The authors give the first approximation algorithm for learning a linear pricing function without any assumption on the data other than normalization.
- A interesting avenue of investigation for future work is to understand how strategic buyers would change their bids in response to a contextual batch learning algorithm and how to design algorithms that are aware of strategic response
- This is a well studied problem in non-contextual online learning (Amin et al [2013], Medina and Mohri [2014b], Drutsa [2017], Vanunts and Drutsa [2019], Nedelec et al [2019]) as well as in online contextual learning (Amin et al [2014], Golrezaei et al [2019]).
- Formulating a model of strategic response to batch learning algorithms is itself open

总结

## Introduction:

The Myerson price of a distribution is the price that maximizes the revenue when selling to a buyer whose value is drawn from that distribution.- In many modern applications such as online marketplaces and advertising, the seller doesn’t just set one price p but must instead price a variety of differentiated products.
- In these settings, a seller must design a policy to price items based on their features in order to optimize revenue.
- One would train a pricing policy on historical bids and apply this policy on future products
## Results:

The authors' main result is a polynomial time approximation scheme (PTAS) using dimensionality reduction.- The same reduction implies that under the Exponential Time Hypothesis (ETH) any algorithm approximating it within an m additive factor must run in time at least eΩ(poly(1/ )), ruling out a fully-polynomial time approximation scheme (FPTAS) for the problem
- This hardness of approximation perfectly complements the algorithmic results, showing that the guarantees are essentially the best that one can hope for
## Conclusion:

The authors give the first approximation algorithm for learning a linear pricing function without any assumption on the data other than normalization.- A interesting avenue of investigation for future work is to understand how strategic buyers would change their bids in response to a contextual batch learning algorithm and how to design algorithms that are aware of strategic response
- This is a well studied problem in non-contextual online learning (Amin et al [2013], Medina and Mohri [2014b], Drutsa [2017], Vanunts and Drutsa [2019], Nedelec et al [2019]) as well as in online contextual learning (Amin et al [2014], Golrezaei et al [2019]).
- Formulating a model of strategic response to batch learning algorithms is itself open

相关工作

- Our work is in the broad area of learning for revenue optimization. The papers in this area can be categorized along two axis: online vs batch learning and contextual vs non-contextual. In the online non-contextual setting, Kleinberg and Leighton [2003] give the optimal algorithm for a single buyer which was later extended to optimal reserve pricing in auctions in Cesa-Bianchi et al [2013]. In the online contextual setting there is a stream of recent work deriving optimal regret bounds for pricing (Amin et al [2014], Cohen et al [2016], Javanmard and Nazerzadeh [2016], Javanmard [2017], Lobel et al [2017], Mao et al [2018], Leme and Schneider [2018], Shah et al [2019]). For batch learning in non-contextual settings there is a long line of work establishing tight sample complexity bounds for revenue optimization (Cole and Roughgarden [2014], Morgenstern and Roughgarden [2015, 2016]) as well as approximation algorithms to reserve price optimization (Paes Leme et al [2016], Roughgarden and Wang [2019], Derakhshan et al [2019]).

Our paper is in the setting of contextual batch learning. Medina and Mohri [2014a] started the work on this setting by showing generalization bounds via Rademacher complexity. They also observe that the loss function is discontinuous and non-convex and propose the use of a surrogate loss. They bound the difference between the pricing loss and the surrogate loss and design algorithms for minimizing the surrogate loss. Medina and Vassilvitskii [2017] design a pricing algorithm based on clustering, where first features are clustered and then a non-contextual pricing algorithm is used on each cluster. Shen et al [2019] replaces the pricing loss by a convex loss function derived from the theory of market equilibrium and argue that the clearing price is a good approximation of the optimal price in real datasets. A common theme in the previous papers is to replace the pricing loss by a more amenable loss function and give conditions under which the new loss approximates the pricing loss. Instead here we study the pricing loss directly. We give the first hardness proof in this setting and also give a (1 − )-approximation without any conditions on the data other than bounded norm.

引用论文

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