optimizing generalized rate metrics with three players · optimizing generalized rate metrics with...

Optimizing Generalized Rate Metrics with Three Players

Harikrishna Narasimhan, Andrew Cotter, Maya Gupta

Constrained Learning ProblemsGeneral evaluation metric

Complexpolicy / fairness goal

Constrained Learning Problems

Example: Fair Hiring

G1

G2

Constrained Learning ProblemsF-measure


G1

G2


Equal opportunity


G1

G2



G1

G2

Equal precision



G1

G2

Equal precision

G-mean



G1

G2

Equal precision

Match distribution

G-mean



G1

G2

Equal precision

Match Distribution

G-mean

How does one design learning algorithms to handle general performance metrics and constraints?

Problem Setup

where there are K prediction rates

Problem Setup

where there are K prediction rates:

for some Expectations of counts E.g. false positive rate, coverage

Problem Setup

where there are K prediction rates:

for some Expectations of counts E.g. false positive rate, coverage

Non-continuous in 𝜃

Non-decomposable: not simple averages of pointwise errors

Prior Methods ● Solution 1: Use Surrogates‡

○ Relax indicators with continuous surrogates

‡Joachims, 15; Kar et al. 14; 16; N et al. 15; Goh et al. 16; Zafar et al. 17



○ Relaxing constraints with surrogates may make the problem infeasible

○ Surrogates may output values outside the range of

‡Joachims, 15; Kar et al. 14; 16; N et al. 15; Goh et al. 16; Zafar et al. 17

Defined for [0, 1]





● Solution 2: Cost-weighted Minimization Oracle*

○ Sequence of weighted objectives and use an oracle to solve sub-problem. Strong oracle assumption!

Defined for [0, 1]

*Parambath et al. 14; Koyejo et al. 14; N et al. 15; Yan et al. 18; N 18; Alabi et al. 18





● Solution 2: Linear Minimization Oracle*

○ Sequence of weighted objectives and use an oracle to solve sub-problem

Defined for [0, 1]

*Parambath et al. 14; Koyejo et al. 14; N et al. 15; Yan et al. 18; N 18; Alabi et al. 18

This Paper

● General framework that recovers prior methods as special cases● Practical algorithms with minimal use of surrogates and tighter handling

of constraints

Min-max Game Formulation

Convex, monotonic

Unconstrained problem; same ideas apply with constraints


Slack variables to decouple rates


Lagrangian min-max problem

Three-player Viewpoint

● 𝝀-player: Linear


● 𝝀-player: Linear● ξ-player: Convex


● 𝝀-player: Linear● ξ-player: Convex● θ-player: Non-continuous due to indicators

Expectation of Indicator

Choosing Player Strategiesξ-player 𝝀-player θ-player

Best Response SGDwith Indicators

Oracle-based Alg.

(analytical solution)



Best Response with Indicators

Cost-weighted minimization oracle

Oracle-based Alg.

(analytical solution)





SGD with Surrogates

Idea: Use original objective for 𝝀 and surrogates for θExtends work of Cotter et al.‘19

Replace with convex surrogate

Oracle-based Alg.

Surrogate-based Alg.

Enables tighter handling of constraints

Players Find Equilibriumξ-player 𝝀-player θ-player




SGD with Surrogates

Oracle-based Alg.


● Iterative approach returns a stochastic classifier





SGD with Surrogates

Oracle-based Alg.


● Iterative approach returns a stochastic classifier○ Near-optimality & near-feasibility guarantees (convex )





SGD with Surrogates

Oracle-based Alg.



Equilibrium

Mixed Nash

Mixed Coarse Correlated





SGD with Surrogates

Oracle-based Alg.



● Surrogates: Weaker optimality guarantee with a smaller comparator class

Equilibrium

Mixed Nash

Mixed Coarse Correlated

Framework Generalizes Prior Methodsξ-player 𝝀-player θ-player




SGD with Surrogates

Oracle-based Alg.


Best Response SGDwith Surrogates

SGD with Surrogates

- FTLwith Surrogates

SGD with Surrogates

- FTL*

with IndicatorsBest Response

with Rates

SPADE [N et al. 15]

NEMSIS [Kar et al. 16]

Frank-Wolfe [N et al. 15]

*Uses result of Abernethy & Wang ‘19

Equilibrium

Mixed Nash

Mixed CoarseCorrelated

Pure Nash

Pure Nash

Mixed Nash

Do not handle constraints

Heuristic Algorithm for Non-convex

Heuristic Alg.

Non-convex : e.g. sums or differences of ratios

ξ-player 𝝀-player θ-player




SGD with Surrogates

SGD SGDwith Indicators

SGD with Surrogates

Equilibrium

Mixed Nash

Mixed CoarseCorrelated

-

Oracle-based Alg.


Experiment

Experiment

Law School(black / others)

Trades-off objective for constraints

Unconstrained Baselines

Poster: East Exhibition Hall B + C #51

Open-source Library: TensorFlow Constrained Optimization (TFCO)https://github.com/google-research/tensorflow_constrained_optimization

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