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Active Embedding Search via When bits meet brains: Noisy Paired Comparisons Locally competitive algorithms for sparse approximation Gregory H. Canal Georgia Institute of Technology In collaboration with: Andrew K. Massimino, Mark A.

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When bits meet brains: Locally competitive algorithms for sparse approximation

Gregory H. Canal

Georgia Institute of Technology In collaboration with: Andrew K. Massimino, Mark A. Davenport, Christopher J. Rozell

Active Embedding Search via Noisy Paired Comparisons

Sensory Information Processing Lab

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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

1 2 3 4

  • Item preferences ranked by distance to user
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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

1 2 3 4 3 4 2 1

  • Item preferences ranked by distance to user
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Active embedding search via noisy paired comparisons

Estimating preferences in similarity embedding

1 2 3 4 3 4 2 1

  • Item preferences ranked by distance to user
  • Continuous user point: hypothetical ideal item (not necessarily in

dataset)

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Active embedding search via noisy paired comparisons

Method of paired comparisons

Learn preferences via method of paired comparisons

– Direct comparisons may be explicitly solicited – Comparisons are implicitly solicited everywhere – In practice, responses are noisy, inconsistent

(David, 1963)

“Which of these two foods do you prefer to eat?”

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Active embedding search via noisy paired comparisons

  • Pairwise search: estimate user vector 𝑥 ∈ ℝ$ based on paired

comparisons between items

  • Ideal point model: continuous point 𝑥 encodes ideal item that

is preferred over all other items

Ideal point model

(Coombs, 1950)

Paired comparison (𝑞, 𝑟): user at 𝑥 ¡prefers item 𝑞 over item 𝑟 if and only if 𝑥 − 𝑞 < 𝑥 − 𝑟

𝑞 𝑟

more preferred less preferred

𝑥

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Active embedding search via noisy paired comparisons

  • Pairwise search: estimate user vector 𝑥 ∈ ℝ$ based on paired

comparisons between items

  • Ideal point model: continuous point 𝑥 encodes ideal item that

is preferred over all other items

Ideal point model

(Coombs, 1950)

Paired comparison (𝑞, 𝑟): user at 𝑥 ¡prefers item 𝑞 over item 𝑟 if and only if 𝑥 − 𝑞 < 𝑥 − 𝑟

𝑞 𝑟

more preferred less preferred

𝑥

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Active embedding search via noisy paired comparisons

– Query pairs adaptively – Add slack variables to feasible region – Repeat comparisons, take majority vote – Previous methods do not incorporate noise into pair selection

How can paired comparisons (hyperplanes) be selected?

  • Query as few pairs as possible
  • Linear models (e.g., learning to rank, latent factors)

unsuitable for nonlinear ideal point model

  • Feasible region tracking

Prior work

(Massimino & Davenport, 2018) (Jamieson & Nowak, 2011) (Wu et al., 2017; Qian et al., 2015)

𝑥

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Active embedding search via noisy paired comparisons

  • 𝑏/0 ∈ ℝ$, 𝑐/0 ∈ ℝ : weights,

threshold of hyperplane bisecting 𝑞, 𝑟

  • Model noise with logistic

response probability

– 𝑙/0: noise constant, represents signal-to-noise ratio – User estimated as posterior mean (MMSE estimator) 𝑄 𝑞 ≺ 𝑟 = 1 1 + 𝑓9:;<(=;<

> ?9@ ;<)

𝑥 A

Modeling response noise

𝑟 𝑞

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Active embedding search via noisy paired comparisons

Our contribution

  • Directly incorporate noise model into adaptive

selection of pairs

  • Strategy 1: InfoGain
  • Strategy 2: EPMV

– analytically tractable

  • Strategy 3: MCMV

– computationally tractable

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Active embedding search via noisy paired comparisons

  • 𝑍

C: binary response to ith paired comparison

  • ℎC 𝑋 : differential entropy of posterior
  • InfoGain: choose queries that maximize expected decrease

in posterior entropy i.e. information gain:

  • No closed-form expression, estimate with samples from

posterior

– Computationally expensive: scales in product of # of samples and # candidate pairs

  • Difficult to analyze convergence

Strategy 1: Maximize information gain (InfoGain)

𝐽 𝑋; 𝑍

C 𝑧C9I = ℎC9I 𝑋 − 𝐹KL[ℎC(𝑋)|𝑧C9I]

user responds

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Active embedding search via noisy paired comparisons

Information gain intuition

  • Symmetry of mutual information:
  • First term promotes selection of comparisons where
  • utcome is non-obvious, given previous responses

– Maximized when comparison response is equiprobable, i.e. probability of picking each pair item is 1/2

𝐽 𝑋;𝑍

C 𝑧C9I = 𝐼(𝑍 C|𝑧C9I) − 𝐼(𝑍 C|𝑋, 𝑧C9I)

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Active embedding search via noisy paired comparisons

Information gain intuition

  • Symmetry of mutual information:
  • Second term promotes selection of comparisons that

would have predictable outcomes if 𝑥 were known

  • Choose query where 𝑥 is far from hyperplane in

expectation

– i.e. posterior variance orthogonal to hyperplane (projected variance) is large

When 𝑥 close to hyperplane, response is unpredictable 𝑏/0

Q 𝑥− 𝑐/0

0.5

𝐽 𝑋;𝑍

C 𝑧C9I = 𝐼(𝑍 C|𝑧C9I) − 𝐼(𝑍 C|𝑋, 𝑧C9I)

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Active embedding search via noisy paired comparisons

  • Equiprobable: response is equally likely to be either item

– Determines hyperplane threshold

  • Max-variance: comparison cuts in direction of maximum

projected variance

– Determines hyperplane weights

Strategy 2: Equiprobable, max-variance (EPMV)

𝐽 𝑋;𝑍

C 𝑧C9I = 𝐼(𝑍 C|𝑧C9I) − 𝐼(𝑍 C|𝑋, 𝑧C9I)

𝑄 𝑞 ≺ 𝑟 = 1/2

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Active embedding search via noisy paired comparisons

EPMV theory

Ø EPMV approximates InfoGain Proposition

For equiprobable comparison with hyperplane weights 𝑏/0, where 𝑀I ¡is a monotonically increasing function. 𝐽 𝑋; 𝑍

C 𝑧C9I ≥ 𝑀I 𝑏/0 Q ΣZ|KL[\𝑏/0

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Active embedding search via noisy paired comparisons

EPMV theory

Theorem

For the EPMV query scheme with each selected query satisfying 𝑙/0 𝑏/0 ≥ 𝑙]C^ > 0 and stopping threshold 𝜁 > 0, consider the stopping time 𝑈

b = min 𝑗: ΣZ|hL

\ i < 𝜁 . We have

Furthermore, for any query scheme 𝐹 𝑈

b = Ω(𝑒 log I b).

𝐹 𝑈

b = O(𝑒 log I b + I b:pLq

r

𝑒slog

I b).

Ø For large noise constants (𝑙]C^ ≫ 0), EPMV reduces the posterior volume at a nearly-optimal rate.

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EPMV in practice

  • Often, one selects pair from pool, rather than

querying arbitrary hyperplanes

  • Select pair that maximizes approximate EPMV utility

function, for 𝜇 > 0

  • Computationally expensive – same utility evaluation

cost as InfoGain

Prefers max- variance queries Prefers equiprobable queries

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Active embedding search via noisy paired comparisons

Strategy 3: Mean-cut, max-variance (MCMV)

  • Computational bottleneck in EPMV is evaluating

equiprobable property

– Approximate equiprobable property with mean-cut property i.e. hyperplane passes through posterior mean 𝑏/0

Q 𝐹 𝑋 𝑍C9I − 𝑐/0 = 0

𝐽 𝑋;𝑍

C 𝑧C9I = 𝐼(𝑍 C|𝑧C9I) − 𝐼(𝑍 C|𝑋, 𝑧C9I)

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Active embedding search via noisy paired comparisons

Proposition

For mean-cut comparison with hyperplane weights 𝑏/0, where 𝑀s ¡is a monotonically increasing function.

Proposition

For mean-cut comparisons with 𝑏/0

Q ΣZ|KL[\𝑏/0 ≫ 0,

MCMV theory

Ø MCMV approximates InfoGain

𝐽 𝑋; 𝑍

C 𝑧C9I ≥ 𝑀s 𝑏/0 Q ΣZ|KL[\𝑏/0

Ø MCMV approximates EPMV

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MCMV in practice

  • Select pair that maximizes utility function, for 𝜇 > 0
  • Computational cost is much cheaper than InfoGain

and EPMV

– Scales with sum of # number of posterior samples and # candidate pairs, rather than product

Prefers max- variance queries Prefers mean-cut queries

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Active embedding search via noisy paired comparisons

Methods overview

Method Advantages Limitations InfoGain Directly minimizes posterior volume Computationally expensive Difficult to analyze EPMV Convergence guarantee Computationally expensive MCMV Computationally cheap No convergence guarantee (future work)

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Simulated results

  • Item embedding constructed from Yummly Food-10k

dataset

– 10,000 food items – ~1 million human comparisons between items

  • Simulated pairwise search

– Noise constant 𝑙/0 estimated from training comparisons – User preference point drawn uniformly from hypercube, 𝑒 = 4

(Wilber et al., 2015; 2014)

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Active embedding search via noisy paired comparisons

Simulated results – baseline methods

  • Random

– pairs selected uniformly at random – user estimated as posterior mean

  • GaussCloud

– pairs chosen to approximate Gaussian point cloud around estimate, shrinks over multiple stages – user estimated by approximately solving non-convex program

  • ActRank

– pairs selected that intersect feasible region of preference points – query repeated multiple times, majority vote taken – user estimated as Chebyshev center*

(Jamieson & Nowak, 2011) (Massimino & Davenport, 2018)

* our addition

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Active embedding search via noisy paired comparisons

Simulated results - MSE

  • Mean-squared error (MSE)

measures accuracy in estimating user preference point

𝑥 𝑥 A

error

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Active embedding search via noisy paired comparisons

Simulated results - MSE

  • Mean-squared error (MSE)

measures accuracy in estimating user preference point

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Active embedding search via noisy paired comparisons

Simulated results - MSE

  • Mean-squared error (MSE)

measures accuracy in estimating user preference point

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Active embedding search via noisy paired comparisons

Simulated results - MSE

  • Mean-squared error (MSE)

measures accuracy in estimating user preference point

MCMV: 7.5x speedup

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Active embedding search via noisy paired comparisons

Simulated results - MSE

  • Mean-squared error (MSE)

measures accuracy in estimating user preference point

MCMV: 7.5x speedup

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Active embedding search via noisy paired comparisons

Simulated results – Kendall’s Tau distance

  • Normalized Kendall’s Tau

distance for preference ranking

  • f 15 randomly selected items

1 2 3 4

𝑥 𝑥 A

3 4 2 1

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Active embedding search via noisy paired comparisons

Simulated results – Kendall’s Tau distance

  • Normalized Kendall’s Tau

distance for preference ranking

  • f 15 randomly selected items
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Active embedding search via noisy paired comparisons

Simulated results – Kendall’s Tau distance

  • Normalized Kendall’s Tau

distance for preference ranking

  • f 15 randomly selected items
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Active embedding search via noisy paired comparisons

Simulated results – Kendall’s Tau distance

  • Normalized Kendall’s Tau

distance for preference ranking

  • f 15 randomly selected items
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Active embedding search via noisy paired comparisons

Simulated results – Kendall’s Tau distance

  • Normalized Kendall’s Tau

distance for preference ranking

  • f 15 randomly selected items
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Active embedding search via noisy paired comparisons

Takeaways

  • First effort to directly model noise in active pairwise

preference learning for ideal point model

– InfoGain – Equiprobable max-variance (EPMV) – Mean-cut max-variance (MCMV)

  • Preliminary support for robustness to noise mismatch
  • Potential applications

– Advertising, online shopping – Parameter settings – Product customization, recipe generation – Database search (medical records, faces)

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Active embedding search via noisy paired comparisons

Sensory Information Processing Lab

gregory.canal@gatech.edu

http://siplab.gatech.edu

@GregHCanal

Code available at: https://github.com/siplab-gt/pairsearch

POSTER #260 TODAY, 6:30 – 9:00 PM, Pacific Ballroom

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Active embedding search via noisy paired comparisons

Simulated mismatched noise - MSE

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Active embedding search via noisy paired comparisons

Simulated mismatched noise – Kendall’s Tau