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CF-OPT: Counterfactual Explanations for Structured Prediction
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Proposes CF-OPT, a first-order augmented-Lagrangian method that generates plausible counterfactual explanations for deep-learning-plus-linear-optimization pipelines by searching in a cost-aware VAE latent space with a hypersphere plausibility regularizer.
Problem
Structured learning pipelines (a differentiable prediction model followed by a black-box linear optimization layer) are opaque, and naive counterfactual search in feature space yields adversarial rather than plausible explanations.
Approach
Formulate relative, absolute, and new epsilon-explanations as constrained problems, solve them with the modified differential method of multipliers in a VAE latent space, add a latent hypersphere plausibility regularizer, and train the VAE with a cost-aware loss penalizing differences in predicted costs after reconstruction.
Key results
- Cost-aware VAEs with alpha >= 0.5 yield significantly better counterfactual loss, and alpha=2 was selected for experiments.
- Latent hypersphere regularization yields more plausible explanations on both reconstruction-error and decision-focused reconstruction-error metrics than unregularized search, and unlike log-likelihood regularization does not sharply increase loss as beta grows.
- Tabular experiments show iteration counts for relative and epsilon-explanations are nearly independent of model depth and scale well with decision dimension, while absolute explanations scale worse.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train the VAE with cost-aware regularization instead of the cost-agnostic objective (ablation) | CF-OPT on Warcraft Maps · Warcraft Maps test set | relative regret after VAE reconstruction | Helped |
| Train the VAE with cost-aware regularization, with α in 0 < α ≤ 2 (ablation) | CF-OPT on Warcraft Maps · Warcraft Maps test set | squared feature reconstruction error | Helped |
| Use cost-aware VAE training with α≥0.5 rather than cost-agnostic training (ablation) | CF-OPT absolute explanations · Warcraft Maps | explanation loss | Helped |
| Add latent hypersphere plausibility regularization (ablation) | CF-OPT · Warcraft Maps absolute explanations | reconstruction error | Helped |
| Add latent hypersphere plausibility regularization (ablation) | CF-OPT · Warcraft Maps absolute explanations | decision-focused reconstruction error | Helped |
| Replace latent hypersphere regularization with log-likelihood regularization (ablation) | CF-OPT · Warcraft Maps absolute explanations | explanation loss | Hurt |
| Use latent hypersphere regularization instead of log-likelihood regularization (ablation) | CF-OPT · Warcraft Maps absolute explanations | explanation loss | No clear effect |
| Use a latent hypersphere instead of a centered ball for the plausibility region (ablation) | CF-OPT · Counterfactual explanations | performance | Helped |
| Increase ε for ε-explanations (ablation) | CF-OPT · Tabular shortest paths and knapsack | squared distance to initial context | Hurt |
| Use a cost-aware VAE rather than an unregularized VAE for the plausibility experiment (ablation) | CF-OPT · Warcraft Maps absolute explanations | decision-focused reconstruction error | Mixed |
Surprising findings
- Naive counterfactual search in high-dimensional feature space produces adversarial examples that vanish under VAE reconstruction, so plain reconstruction error fails to detect them, requiring a decision-focused metric.
Limitations
- Covariates must be continuous; categorical or discrete features are not handled.
- Only local optimality guaranteed since the MDMM is applied to a highly non-convex problem.
- Absolute explanations become costly for large decision spaces due to combinatorial explosion.
Experimental setup
Datasets: Warcraft shortest path maps (10,000 training examples) synthetic shortest path on grid contextual multi-dimensional knapsack
Benchmarks: Warcraft shortest path maps
Metrics: relative regret squared L2 distance to initial context reconstruction error decision-focused reconstruction error number of iterations explanation loss
Baselines: naive feature-space counterfactual search log-likelihood latent regularization cost-agnostic VAE
Builds on: modified differential method of multipliers variational autoencoder C-CHVAE SPO+ loss
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