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The Sample Complexity

• f Revenue Maximization

in the Hierarchy of Deterministic Combinatorial Auctions

Ellen Vitercik

Joint work with Nina Balcan and Tuomas Sandholm

Theory Lunch 27 April 2016

Combinatorial (multi-item) auctions

Combinatorial auctions allow bidders to express preferences for bundles

• f goods

: $5 : $5 : $6

Real-world examples

• US Government wireless spectrum auctions [FCC]
• Sourcing auctions [Sandholm 2013]
• Airport time slot allocation [Rassenti 1982]
• Building development, e.g. office space in GHC (no money)
• Property sales

• Mechanism designer must determine:

– Allocation function: Who gets what? – Payment function: What does the auctioneer charge?

• Goal: design strategy-proof mechanisms

– Easy for the bidders to compute the optimal strategy – Easy for designer to analyze possible outcomes

Mechanism design

Warm-up: single-item auctions

NINA TUOMAS : $5 : $3 Second-price auction: the classic strategy-proof, single-item auction. Allocation (N:$5, T:$3) = give carrot to Nina Payment (N:$5, T:$3) = charge Nina $3 , -$3 ø, -$0 Second-Price Auction

• Standard assumptions: bidders’ valuations drawn from

distribution 𝑬, mechanism designer knows 𝑬 – Allocation and payment rules often depend on 𝑬

Revenue-maximizing combinatorial auctions

• Central problem in Automated Mechanism Design

[Conitzer and Sandholm 2002, 2003, 2004, Likhodedov and Sandholm 2004, 2005, 2015, Sandholm 2003]

Revenue-maximizing combinatorial auctions

No theory that relates the performance of the designed mechanism on the samples to that mechanism’s expected performance on 𝑬, until now.

Design Challenges Feasible Solutions Support of 𝐄 might be doubly- exponential Draw samples from 𝐄 instead NP-hard to determine the revenue-maximizing deterministic auction with respect to 𝐄 [Conitzer and Sandholm 2002] Fix a rich class of auctions. Can we learn the revenue- maximizing combinatorial auction in that class with respect to 𝐄 given samples drawn from 𝐄?

Outline

• Introduction
• Hierarchy of deterministic combinatorial auction

classes

• Our contribution: how many samples are needed to

learn over the hierarchy of auctions?

• Affine maximizer auctions and Rademacher complexity
• Mixed-bundling auctions and pseudo-dimension
• Summary and future directions

• 𝟒𝟑 possible outcomes 𝒑 = (𝒑𝟐, 𝒑𝟑)
• For example, 𝒑 = ( ,

)

Combinatorial auctions

: $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

• Social Welfare 𝒑

= SW 𝒑 = 𝒘𝒋 𝒑

𝒋∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝒑∗ maximizes SW 𝒑

• SW-i 𝒑 =

𝒘𝒌 𝒑

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕− 𝒋

𝒑−𝒋 maximizes SW-i 𝒑

• Allocation: 𝒑∗
• Payment: Nina pays SW −𝑶𝒋𝒐𝒃 𝒑−𝑶𝒋𝒐𝒃 − SW−𝑶𝒋𝒐𝒃 𝒑∗

A natural generalization of second price

The “Vickrey-Clarke-Groves mechanism” (VCG).

: $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

• 𝒑∗ =

,

• 𝒑−𝑶𝒋𝒐𝒃 = (∅,

, )

• Nina pays 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) −𝒘𝑼𝒗𝒑𝒏𝒃𝒕( ) = 0

VCG in action

How do we get the bidders to pay more?

: $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

Outcome boosting

• value ∅,

, = 𝒘𝑶𝒋𝒐𝒃 ∅ + 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) = 50¢ : $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

Outcome boosting

• value ∅,

, = 𝒘𝑶𝒋𝒐𝒃 ∅ + 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) = 50¢ + 99¢

• 𝒑∗ =

,

• 𝒑−𝑶𝒋𝒐𝒃 = (∅,

, ) : $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

Outcome boosting

• value ∅,

, = 𝒘𝑶𝒋𝒐𝒃 ∅ + 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) = 50¢ + 99¢

• 𝒑∗ =

,

• 𝒑−𝑶𝒋𝒐𝒃 = (∅,

, )

• Nina pays 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) + 99¢ −𝒘𝑼𝒗𝒑𝒏𝒃𝒕( ) = 99¢

: $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

• Boost outcomes: 𝝁(𝒑)
• Take bids 𝒘
• Compute outcome:

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚𝒑 𝑻𝑿 𝒑 + 𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝑻𝑿−𝒋 𝒑−𝒋 + 𝝁 𝒑−𝒋 − 𝑻𝑿−𝒋 𝒑∗ + 𝝁 𝒑∗

Affine maximizer auctions (AMAs)

• Boost outcomes: 𝝁(𝒑)
• Take bids 𝒘
• Compute outcome:

𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒘𝒌 𝒑 +

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝒘𝒌 𝒑−𝒋

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑−𝒋 − 𝒘𝒌 𝒑∗

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑∗

Affine maximizer auctions (AMAs)

• Boost outcomes: 𝝁(𝒑); Weight bidders: 𝒙𝒋
• Take bids 𝒘
• Compute outcome:

𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒘𝒌 𝒑 +

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝒘𝒌 𝒑−𝒋

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑−𝒋 − 𝒘𝒌 𝒑∗

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑∗

Affine maximizer auctions (AMAs)

• Boost outcomes: 𝝁(𝒑); Weight bidders: 𝒙𝒋
• Take bids 𝒘
• Compute outcome:

𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒙𝒌𝒘𝒌 𝒑 +

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝒘𝒌 𝒑−𝒋

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑−𝒋 − 𝒘𝒌 𝒑∗

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑∗

Affine maximizer auctions (AMAs)

• Boost outcomes: 𝝁(𝒑); Weight bidders: 𝒙𝒋
• Take bids 𝒘
• Compute outcome:

𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒙𝒌𝒘𝒌 𝒑 +

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝟐 𝒙𝒋 𝒙𝒌𝒘𝒌 𝒑−𝒋

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑−𝒋 − 𝒙𝒌𝒘𝒌 𝒑∗

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑∗

Affine maximizer auctions (AMAs)

Hierarchy of parameterized auction classes

Affine maximizer auctions [R79] 𝒙𝒋, 𝝁 𝒑 ∈ ℝ Virtual valuation combinatorial auctions [SL03] 𝝁 𝒑 = 𝝁𝒋 𝒑

𝒋∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁-auctions [J07]

• 𝒙𝒋 = 𝟐
• 𝝁 𝒑 ∈ ℝ

Mixed bundling auctions with reserve prices [TS12]

• 𝒙𝒋 = 𝟐
• 𝝁 𝒑 = 𝟏 except any
• utcome where a

bidder gets all items

• item reserve prices

∪ ∪ ∪ ∪ ∪

• 𝒙𝒋 = 𝟐
• 𝝁 𝒑 = 𝟏 except
• utcome where a

bidder gets all items Mixed bundling auctions [J07]

Outline

• Introduction
• Hierarchy of deterministic combinatorial auction

classes

• Our contribution: how many samples are needed to

learn over the hierarchy of auctions?

• Affine maximizer auctions and Rademacher complexity
• Mixed-bundling auctions and pseudo-dimension
• Summary and future directions

• Optimize 𝝁 𝒑 and 𝒙 given a sample 𝑻~𝑬𝑶

– (Automated Mechanism Design)

• We want:

– The auction with best revenue over the sample has almost

• ptimal expected revenue

– Any approximately revenue-maximizing auction over the sample will have approximately optimal expected revenue

• For any auction we output, we want |𝑻| large enough such that:

|empirical revenue – expected revenue| < 𝝑

• In other words, how many samples 𝐓 = 𝑶 do we need to ensure

that |empirical revenue – expected revenue| = 𝟐 𝑶 𝒔𝒇𝒘𝑩 𝒘

𝒘∈𝑻

− 𝔽𝒘~𝑬 𝒔𝒇𝒘𝑩 𝒘 < 𝝑 for all auctions 𝑩 in the class?

• (We can only do this with high probability.)

Our contribution

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

∪ ∪ ∪ ∪ ∪

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

∪ ∪ ∪ ∪ ∪

Nearly-matching exponential lower bounds.

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

∪ ∪ ∪ ∪ ∪

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

Learning theory tool: Rademacher complexity

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

∪ ∪ ∪ ∪ ∪

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

Learning theory tool: Pseudo-dimension

Outline

• Introduction
• Hierarchy of deterministic combinatorial auction

classes

• Our contribution: how many samples are needed to

learn over the hierarchy of auctions?

• Affine maximizer auctions and Rademacher complexity
• Mixed-bundling auctions and pseudo-dimension
• Summary and future directions

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

∪ ∪ ∪ ∪ ∪

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

• Boost outcomes: 𝝁(𝒑); Weight bidders: 𝒙𝒋
• Take bids 𝒘
• Compute outcome:

𝒑∗ = 𝒃𝒔𝒉𝒏𝒃𝒚 𝒑 𝒙𝒌𝒘𝒌 𝒑 +

𝒐 𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕

𝝁 𝒑

• Compute Bidder 𝒋’s payment:

𝟐 𝒙𝒋 𝒙𝒌𝒘𝒌 𝒑−𝒋

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑−𝒋 − 𝒙𝒌𝒘𝒌 𝒑∗

𝒌∈𝑪𝒋𝒆𝒆𝒇𝒔𝒕−{𝒋}

+ 𝝁 𝒑∗

Key challenge

Our problem... Whereas typically in machine learning…

• More expressive function classes need more samples to learn
• How to measure expressivity?

– How well do functions from the class fit random noise?

• Empirical Rademacher complexity:

𝒚𝟐, … , 𝒚𝑶 ~ −𝟐, 𝟐 𝑶, 𝑻 = 𝒘𝟐, … , 𝒘𝑶 𝑺𝑻 𝓑 = 𝔽𝒚 𝒕𝒗𝒒𝑩∈𝓑

𝟐 𝑶 𝒚𝒋 ∙ 𝒔𝒇𝒘𝑩 𝒘𝒋

, where

• Rademacher complexity:

𝑺𝑶 𝓑 = 𝔽𝑻~𝑬𝑶 𝑺𝑻 𝓑

Rademacher complexity

• With probability at least 𝟐 − 𝜺, for all 𝑩 ∈ 𝓑,

|empirical revenue – expected revenue| ≤ 𝟑𝑺𝑶 𝓑 + 𝑽

𝟑 𝒎𝒐 𝟑/𝜺 𝑶

*𝑽 is the maximum revenue achievable over the support of the bidders’ valuation distributions

• More expressive function classes need more samples to learn
• How to measure expressivity?

– How well do functions from the class fit random noise?

• Empirical Rademacher complexity:

𝒚𝟐, … , 𝒚𝑶 ~ −𝟐, 𝟐 𝑶, 𝑻 = 𝒘𝟐, … , 𝒘𝑶 𝑺𝑻 𝓑 = 𝔽𝒚 𝒕𝒗𝒒𝑩∈𝓑

𝟐 𝑶 𝒚𝒋 ∙ 𝒔𝒇𝒘𝑩 𝒘𝒋

, where

• Rademacher complexity:

𝑺𝑶 𝓑 = 𝔽𝑻~𝑬𝑶 𝑺𝑻 𝓑

Rademacher complexity

𝓑 = all binary valued functions 𝑺𝑶 𝓑 = 𝟐 𝟑 𝓑 = one binary valued function 𝑺𝑶 𝓑 = 𝟏

• Key idea: split revenue function into its simpler components

– Weighted social welfare without any one bidder’s participation (𝒐 components) – Amount of revenue subtracted out to maintain strategy- proof property

• Then use compositional properties of Rademacher complexity

and other tricks, for example: If 𝑮 = 𝒈 | 𝒈 = 𝒉 + 𝒊, 𝒉 ∈ 𝑯, 𝒊 ∈ 𝑰 , then 𝑺𝑶 𝑮 ≤ 𝑺𝑶 𝑯 + 𝑺𝑶 𝑰

Rademacher complexity of AMAs

Let 𝓑 be the class of 𝒐-bidder, 𝒏-item AMA revenue functions. If 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑 ,

then with high probability over a sample 𝑻~𝑬𝑶, |empirical revenue – expected revenue| < 𝝑 for all 𝒔𝒇𝒘𝑩 ∈ 𝓑. Theorem

Outline

• Introduction
• Hierarchy of deterministic combinatorial auction

classes

• Our contribution: how many samples are needed to

learn over the hierarchy of auctions?

• Affine maximizer auctions and Rademacher complexity
• Mixed-bundling auctions and pseudo-dimension
• Summary and future directions

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

∪ ∪ ∪ ∪ ∪

• Class of auctions parameterized by a scalar 𝒅
• Boost the allocations where one bidder gets all goods by 𝒅
• value ∅,

, = 𝒘𝑶𝒋𝒐𝒃 ∅ + 𝒘𝑼𝒗𝒑𝒏𝒃𝒕( , ) = 50¢ + 99¢

• value

, , ∅ = 𝒘𝑶𝒋𝒐𝒃 , + 𝒘𝑼𝒗𝒑𝒏𝒃𝒕(∅) = 50¢ + 99¢

• How large must the sample 𝑻 be in order to ensure that for all

MBAs, |empirical revenue – expected revenue| < 𝝑?

Mixed bundling auctions (MBAs)

: $1 : $0 : $1 : 50¢ : 50¢ : 50¢ NINA TUOMAS

𝒅

Structural properties of MBA revenue functions

Fix 𝒘 ∈ 𝑻. Then 𝒔𝒇𝒘𝒘 𝒅 is piecewise linear with at most 𝒐 + 𝟐 discontinuities. Lemma revenue

• Complexity measure for binary-valued functions only
• Example: 𝑮 = {single interval on the real line}
• No set of size 3 can be labeled in all 𝟑𝟒 ways by 𝑮

VC-dimension

How can we extend VC-dim to real-valued functions?

+ +

• Class of functions 𝑮 shatters set 𝑻 = 𝒚𝟐, … , 𝒚𝑶 if

for all 𝐜 ∈ 𝟏, 𝟐 𝑶, there exists 𝒈 ∈ 𝑮 such that 𝒈 𝒚𝒋 = 𝒄𝒋

• VC-dimension of 𝑮 is the

cardinality of the largest set 𝑻 that can be shattered by 𝑮

• Sample 𝑻 = 𝒚𝟐, … , 𝒚𝑶
• Class of functions 𝑮 into −𝑽, 𝑽
• 𝒔 = 𝒔(𝟐), … , 𝒔(𝑶) ∈ ℝ𝑶

witnesses the shattering of 𝑻 by 𝑮 if for all 𝑼 ⊆ 𝑻, there exists 𝒈𝑼 ∈ 𝑮 such that 𝒈𝑼 𝒚𝒋 ≤ 𝒔(𝒋) iff 𝒚𝒋 ∈ 𝑼

• Pseudo-dimension of 𝑮 is the

cardinality of the largest sample 𝑻 that can be shattered by 𝑮

Pseudo-dimension

P-dim(𝑮) = VC-dim( (𝒚, 𝒔) ⟼ 𝟐𝒈 𝒚 −𝒔>𝟏| 𝒈 ∈ 𝑮 )

𝒚𝟐 𝒈 𝒚𝟐 ≤ 𝒔(𝟐) 𝟏 𝒚𝟑 𝒈 𝒚𝟑 ≤ 𝒔(𝟑) 𝟏 𝒚𝟒 𝒈 𝒚𝟒 > 𝒔(𝟒) 𝟐 𝒚𝟓 𝒈 𝒚𝟓 ≤ 𝒔(𝟓) 𝟏 𝒚𝟔 𝒈 𝒚𝟔 > 𝒔(𝟔) 𝟐 𝒚𝟕 𝒈 𝒚𝟕 ≤ 𝒔(𝟕) 𝟏 𝒚𝟖 𝒈 𝒚𝟖 > 𝒔(𝟖) 𝟐 𝒚𝟗 𝒈 𝒚𝟗 > 𝒔(𝟗) 𝟐 𝒚𝟘 𝒈 𝒚𝟘 ≤ 𝒔(𝟘) 𝟏

• Set of auction revenue functions 𝓑 with range in 𝟏, 𝑽 ,

distribution 𝑬 over valuations 𝒘.

• For every 𝝑 > 𝟏, 𝜺 ∈ 𝟏, 𝟐 , if

𝑶 = 𝑷

𝑽 𝝑 𝟑

P−dim(𝓑) ∗ 𝐦𝐨

𝑽 𝝑 + 𝐦𝐨 𝟐 𝜺

, then with probability at least 𝟐 − 𝜺 over a sample 𝑻~𝑬𝑶, |empirical revenue – expected revenue| < 𝝑 for every 𝒔𝒇𝒘𝑩 ∈ 𝓑.

How many samples do we need?

Pseudo-dimension allows us to derive strong sample complexity bounds.

How many samples do we need?

Affine maximizer auctions [R79] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Virtual valuation combinatorial auctions [SL03] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

𝝁-auctions [J07] 𝑶 = 𝑷 𝑽𝒐𝒏 𝒏 𝑽 + 𝒐𝒏/𝟑 /𝝑

𝟑

Mixed bundling auctions [J07] 𝑶 = 𝑷 𝑽/𝝑 𝟑 Mixed bundling auctions with reserve prices [TS12] 𝑶 = 𝑷 𝑽/𝝑 𝟑𝒏𝟒

Variables 𝑶: sample size 𝒐: # bidders 𝒏: # items 𝑽: maximum revenue achievable over the support of the bidders’ valuation distributions

∪ ∪ ∪ ∪ ∪

2-bidder MBA pseudo-dimension

Let 𝓑 = 𝒔𝒇𝒘𝒅 𝒅≥𝟏 be the class of 𝒐-bidder, 𝒏-item mixed bundling auction revenue functions. Then P-dim(𝓑) = 𝑷 𝐦𝐩𝐡 𝒐 . Theorem

Proof sketch.

• Fact: there exists a set of 2 samples that is shattered by 𝓑.
• Suppose, for a contradiction, that 𝑻 = 𝒘𝟐, 𝒘𝟑, 𝒘𝟒 is shatterable.
• Recall 𝒘𝟐 = 𝒘𝟐

𝟐, 𝒘𝟑 𝟐

• This means:

– There exists 𝒔 = 𝒔𝟐, 𝒔𝟑, 𝒔𝟒 ∈ ℝ𝟒 and 𝟑|𝑻| = 𝟗 MBA parameters 𝐃 = 𝒅𝟐, … , 𝒅𝟗 such that 𝒔𝒇𝒘𝒅𝟐, … , 𝒔𝒇𝒘𝒅𝟗 induce all 8 binary labelings on 𝑻 with respect to 𝒔.

2-bidder MBA pseudo-dimension

Let 𝓑 = 𝒔𝒇𝒘𝒅 𝒅≥𝟏 be the class of 𝟑-bidder, 𝒏-item mixed bundling auction revenue functions. Then P-dim(𝓑) = 𝟑. Theorem We need to show that no set of 3 samples can be shattered by 𝓑.

𝐧𝐣𝐨 𝒘𝟐

𝒋

𝒏 , 𝒘𝟑

𝒋

𝒏 𝒅𝒋 𝒅

2-bidder MBA pseudo-dimension

Fix 𝒘𝒋 ∈ 𝑻. Then 𝒔𝒇𝒘𝒘𝒋 𝒅 is piecewise linear with one discontinuity, with a slope of 2 followed by a constant function with value 𝐧𝐣𝐨 𝒘𝟐

𝒋

𝒏 , 𝒘𝟑

𝒋

𝒏 . Lemma revenue

𝐧𝐣𝐨 𝒘𝟑

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏 𝒅𝒋 𝒅

Case 1: 𝒔𝟒 < 𝒏𝒋𝒐 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏

𝒔𝟒 revenue

Case 1: 𝒔𝟒 < 𝒏𝒋𝒐 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏

𝒔𝒇𝒘𝒘𝟒(𝒅) increasing 𝒔𝒇𝒘𝒘𝟒 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏 𝒅𝟒 𝒅

Case 1: 𝒔𝟒 < 𝒏𝒋𝒐 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏

𝒔𝒇𝒘𝒘𝟒(𝒅) increasing 𝒔𝒇𝒘𝒘𝟒 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏 𝒔𝒇𝒘𝒘𝟑(𝒅) increasing 𝒔𝒇𝒘𝒘𝟑 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟑

𝒏 , 𝒘𝟑

𝟑

𝒏 𝒅𝟒 𝒅𝟑 𝒅

Case 1: 𝒔𝟒 < 𝒏𝒋𝒐 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏

𝒔𝒇𝒘𝒘𝟒(𝒅) increasing 𝒔𝒇𝒘𝒘𝟒 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏 𝒔𝒇𝒘𝒘𝟑(𝒅) increasing 𝒔𝒇𝒘𝒘𝟑 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟑

𝒏 , 𝒘𝟑

𝟑

𝒏 𝒔𝒇𝒘𝒘𝟐(𝒅) increasing 𝒔𝒇𝒘𝒘𝟐 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟐

𝒏 , 𝒘𝟑

𝟐

𝒏 𝒅𝟒 𝒅𝟑 𝒅𝟐 𝒅

𝒔𝒇𝒘𝒘𝟒(𝒅) increasing 𝒔𝒇𝒘𝒘𝟒 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏 𝒔𝒇𝒘𝒘𝟑(𝒅) increasing 𝒔𝒇𝒘𝒘𝟑 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟑

𝒏 , 𝒘𝟑

𝟑

𝒏 𝒔𝒇𝒘𝒘𝟐(𝒅) increasing 𝒔𝒇𝒘𝒘𝟐 𝒅 = 𝐧𝐣𝐨 𝒘𝟐

𝟐

𝒏 , 𝒘𝟑

𝟐

𝒏 𝒅𝟒 𝒅𝟑 𝒅𝟐 𝒅

Case 1: 𝒔𝟒 < 𝒏𝒋𝒐 𝒘𝟐

𝟒

𝒏 , 𝒘𝟑

𝟒

𝒏

𝒔𝒇𝒘𝒘𝟑(𝒅) increasing 𝒔𝒇𝒘𝒘𝟐(𝒅) increasing We need: This is impossible, so we reach

a contradiction. Therefore, no set of size 3 can be shattered by the class of 2-bidder MBA revenue functions, so the pseudo-dimension is at most 2.

𝒔𝒇𝒘𝒘𝟒(𝒅) increasing

Summary

• Analyzed the sample complexity of learning over a

hierarchy of deterministic combinatorial auctions

• Uncovered structural properties of these auctions’

revenue functions along the way – Of independent interest beyond sample complexity results