optimizing generalized rate metrics with three players · optimizing generalized rate metrics with...
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Optimizing Generalized Rate Metrics with Three Players
Harikrishna Narasimhan, Andrew Cotter, Maya Gupta
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Constrained Learning ProblemsGeneral evaluation metric
Complexpolicy / fairness goal
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Constrained Learning Problems
Example: Fair Hiring
G1
G2
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Constrained Learning ProblemsF-measure
Example: Fair Hiring
G1
G2
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Constrained Learning ProblemsF-measure
Equal opportunity
Example: Fair Hiring
G1
G2
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Constrained Learning ProblemsF-measure
Example: Fair Hiring
G1
G2
Equal precision
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Constrained Learning ProblemsF-measure
Example: Fair Hiring
G1
G2
Equal precision
G-mean
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Constrained Learning ProblemsF-measure
Example: Fair Hiring
G1
G2
Equal precision
Match distribution
G-mean
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Constrained Learning ProblemsF-measure
Example: Fair Hiring
G1
G2
Equal precision
Match Distribution
G-mean
How does one design learning algorithms to handle general performance metrics and constraints?
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Problem Setup
where there are K prediction rates
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Problem Setup
where there are K prediction rates:
for some Expectations of counts E.g. false positive rate, coverage
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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
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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
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Prior Methods ● Solution 1: Use Surrogates‡
○ Relax indicators with continuous surrogates
○ 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]
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Prior Methods ● Solution 1: Use Surrogates‡
○ Relax indicators with continuous surrogates
○ Relaxing constraints with surrogates may make the problem infeasible
○ Surrogates may output values outside the range of
● 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
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Prior Methods ● Solution 1: Use Surrogates‡
○ Relax indicators with continuous surrogates
○ Relaxing constraints with surrogates may make the problem infeasible
○ Surrogates may output values outside the range of
● 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
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Min-max Game Formulation
Convex, monotonic
Unconstrained problem; same ideas apply with constraints
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Min-max Game Formulation
Slack variables to decouple rates
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Min-max Game Formulation
Lagrangian min-max problem
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Three-player Viewpoint
● 𝝀-player: Linear
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Three-player Viewpoint
● 𝝀-player: Linear● ξ-player: Convex
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Three-player Viewpoint
● 𝝀-player: Linear● ξ-player: Convex● θ-player: Non-continuous due to indicators
Expectation of Indicator
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Choosing Player Strategiesξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Oracle-based Alg.
(analytical solution)
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Choosing Player Strategiesξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Cost-weighted minimization oracle
Oracle-based Alg.
(analytical solution)
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Choosing Player Strategiesξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
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
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Players Find Equilibriumξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
Oracle-based Alg.
Surrogate-based Alg.
● Iterative approach returns a stochastic classifier
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Players Find Equilibriumξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
Oracle-based Alg.
Surrogate-based Alg.
● Iterative approach returns a stochastic classifier○ Near-optimality & near-feasibility guarantees (convex )
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Players Find Equilibriumξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
Oracle-based Alg.
Surrogate-based Alg.
● Iterative approach returns a stochastic classifier○ Near-optimality & near-feasibility guarantees (convex )
Equilibrium
Mixed Nash
Mixed Coarse Correlated
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Players Find Equilibriumξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
Oracle-based Alg.
Surrogate-based Alg.
● Iterative approach returns a stochastic classifier○ Near-optimality & near-feasibility guarantees (convex )
● Surrogates: Weaker optimality guarantee with a smaller comparator class
Equilibrium
Mixed Nash
Mixed Coarse Correlated
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Framework Generalizes Prior Methodsξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
Oracle-based Alg.
Surrogate-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
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Heuristic Algorithm for Non-convex
Heuristic Alg.
Non-convex : e.g. sums or differences of ratios
ξ-player 𝝀-player θ-player
Best Response SGDwith Indicators
Best Response with Indicators
Best Response SGDwith Indicators
SGD with Surrogates
SGD SGDwith Indicators
SGD with Surrogates
Equilibrium
Mixed Nash
Mixed CoarseCorrelated
-
Oracle-based Alg.
Surrogate-based Alg.
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Experiment
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Experiment
Law School(black / others)
Trades-off objective for constraints
Unconstrained Baselines
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Poster: East Exhibition Hall B + C #51
Open-source Library: TensorFlow Constrained Optimization (TFCO)https://github.com/google-research/tensorflow_constrained_optimization