Learning the Valuations of a k-demand Agent Hanrui Zhang - - PowerPoint PPT Presentation

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Learning the Valuations of a k-demand Agent Hanrui Zhang - - PowerPoint PPT Presentation

Dec 25, 2022 219 likes •575 views

Learning the Valuations of a k-demand Agent Hanrui Zhang Vincent Conitzer Duke University this talk: optimal (up to lower order terms) algorithm for actively learning the valuations of a k- demand agent algorithm with

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Learning the Valuations

  • f a k-demand Agent

Hanrui Zhang Vincent Conitzer

  • Duke University
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this talk:

  • optimal (up to lower order terms) algorithm

for actively learning the valuations of a k- demand agent

  • algorithm with polynomial time & sample

complexity for passively learning the valuations of a k-demand agent

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k-demand agent: demands a set of items

  • f size <=k maximizing her utility, i.e.,

total value - total price

  • demand set: the set of items the agent

demands

k-demand agents and demand sets

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Unit-demand agents

value: price: $10 $12 $8 $6 $5 $5 surplus: $4 $7 $3 agent buys: ✘ ✔ ✘

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k-demand agents and demand sets

value: price: $5 $6 $4 $3 $4 $3 $2 $2 agent is 2-demand — they want no more than 2 items

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k-demand agents and demand sets

value: price: $5 surplus: 2-demand agent buys: $6 $4 $3 $4 $3 $2 $2 $1 $3 $2 $1 ✘ ✔ ✔ ✘

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k-demand agents and demand sets

value: price: $5 surplus: 2-demand agent buys: $6 $4 $3 $4 $3 $2 $2 $1 $3 $2 $1 ✘ ✔ ✔ ✘ demand set

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Demand queries

demand query: given a vector of prices, returns a demand set (which may not be unique)

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value: price: v1 = $5 2-demand agent buys: v2 = $6 v3 = $4 v4 = $3 p1 = $4 p2 = $2 p3 = $2 p4 = $2 ✘ ✔ ✔ ✘

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value: price: 2-demand agent buys: p1 = $4 p2 = $2 p3 = $2 p4 = $2 ✘ ✔ ✔ ✘ price: 2-demand agent buys: p1 = $2 p2 = $5 p3 = $3 p4 = $1.5 ✔ ✘ ✘ ✔ v1 = $5 v2 = $6 v3 = $4 v4 = $3

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value: price: 2-demand agent buys: p1 = $7 p2 = $3.5 p3 = $5.5 p4 = $4 ✘ ✔ ✘ ✘ price: 2-demand agent buys: p1 = $4 p2 = $2 p3 = $2 p4 = $2 ✘ ✔ ✔ ✘ price: 2-demand agent buys: p1 = $2 p2 = $5 p3 = $3 p4 = $1.5 ✔ ✘ ✘ ✔ v1 = $5 v2 = $6 v3 = $4 v4 = $3

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Actively learning the valuations

  • suppose there are n items, and the value vi of each

item is an integer between 1 and W

  • how many demand queries suffice to learn the full

valuations (i.e., (vi)i) of a k-demand agent?

  • spoiler: optimal number of queries is

(n log W) / (k log (n / k)) + n / k ± o(…)

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(n log W) / (k log (n / k)) + n / k ± o(…)

Sketch of lower bound

amount of information encoded in (vi)i maximum amount

  • f information

per query

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(n log W) / (k log (n / k)) + n / k ± o(…) necessary in the following case:

  • exactly one item is special, which has value 0
  • all other items have value 1
  • the special item is chosen uniformly at random

Sketch of lower bound

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Sketch of upper bound

  • warmup: n = k = 1
  • need to learn: a single number v1 in {1, 2, …, W}
  • query: given p, returns whether p < v1
  • optimal solution: binary search — log W queries
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Sketch of upper bound

  • slight generalization: n = k (= 1)
  • need to learn: a vector (vi)i of integers in {1, 2, …, W}
  • query: given (pi)i, returns, for each item i, whether pi < vi
  • optimal solution: simultaneous binary search — log W

queries

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Sketch of upper bound

  • general case: n ≥ k ≥ 1
  • straightforward solution: (1) divide items into groups of

size k, and (2) perform simultaneous binary search for each group sequentially

  • (n / k) log W queries
  • LB is (n log W) / (k log (n / k)) — can we do better?
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Sketch of upper bound

idea: biased binary search

  • learn v1 using log W queries, use item 1 as reference
  • in each query, post p1 = v1 - 0.5, so item 1 is marginally

attractive

  • for all other items, post biased (rather than middle-of-

possible-range) prices

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

n = 4 k = 1 v1

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 prices biased toward higher end of possible ranges

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Sketch of upper bound

  • in each query, post p1 = v1 - 0.5, so item 1 is marginally

attractive

  • for all other items, post biased (rather than middle-of-

possible range) prices

  • if item 1 in demand set: many items are overpriced;

shrink their possible ranges by a little

  • if item 1 not in demand set: a few items are underpriced;

shrink their possible ranges by a lot

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 in demand set: many items are overpriced; shrink their possible ranges by a little

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 in demand set: many items are overpriced; shrink their possible ranges by a little

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 in demand set: many items are overpriced; shrink their possible ranges by a little

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 not in demand set: a few items are underpriced; shrink their possible ranges by a lot

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 not in demand set: a few items are underpriced; shrink their possible ranges by a lot

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Sketch of upper bound

25 50 75 100 item 1 item 2 item 3 item 4

p1 = v1 - 0.5 p2 p3 p4 n = 4 k = 1 if item 1 not in demand set: a few items are underpriced; shrink their possible ranges by a lot

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Sketch of upper bound

  • if item 1 in demand set: many items are overpriced;

shrink their possible ranges by a little

  • if item 1 not in demand set: a few items are underpriced;

shrink their possible ranges by a lot

  • adjust bias to equalize information gain
  • larger information gain (~ k log (n / k)) in both cases!
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  • so far: tight UB & LB for active learning
  • next: (very brief discussion of)

computation & sample efficient algorithm for passive learning

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Passively learning valuations

  • prices are distributed according to a distribution 𝒠
  • true valuations v: a vector of real numbers
  • algorithm observes m iid sample price vectors pj

together with demand set Sj under pj

  • given {(Sj, pj)}, algorithm outputs a hypothesis vector h

which recovers v in a PAC sense — algorithm succeeds with probability 1 - 𝜀, in which case with probability 1 - 𝛇, demand set under (v, p) = demand set under (h, p)

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Passively learning valuations

  • idea: empirical risk minimization
  • tool: multiclass ERM principle & Natarajan dimension
  • treat problem as multiclass classification with < nk labels
  • hypothesis class has Natarajan dimension n
  • sample complexity is poly(n, k, log(1 / 𝜀), 1 / 𝛇)
  • solving ERM = finding a feasible solution to an LP
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Future directions

  • more general valuations, e.g., matroid-demand
  • tighter sample complexity bounds for passive learning
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Thanks for your attention!

Questions?

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  • in economic theory: learning utility functions from

revealed preferences (Samuelson, 1938; Afriat, 1967; Beigman & Vohra, 2006; …)

  • in CS: preference elicitation (Blum et al., 2004; Lahaie

& Parkes, 2004; Sandholm & Boutilier, 2006; …)

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