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Generalization Analysis for Controllable Learning
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Builds a unified theoretical framework for controllable learning and derives generalization bounds via a novel vector-contraction inequality whose dependence on the number of task targets is only logarithmic, plus bounds for embedding-based and hypernetwork-based methods.
Problem
Generalization performance of controllable learning methods, which adapt model behavior to test-time task requirements, lacks theoretical understanding, especially how bounds depend on the number of task targets.
Approach
A vector-contraction inequality exploiting l-infinity Lipschitz continuity of losses and a projection operator that decouples task targets, yielding Rademacher-based bounds with no dependence on the number of targets except logarithmic factors; modular Rademacher bounds for FNN and Transformer components are combined into bounds for embedding-based and hypernetwork-based controllable learners.
Key results
- Weighted Hamming loss bound scales as O(1/sqrt(n)) with no dependence on number of task targets c except logarithmic factors (Theorem 4.7).
- Bipartite ranking bound has logarithmic dependence on c with rate O(1/sqrt(s0)), where s0 = n min(min_j p_j, 1-min_j p_j) (Theorem 4.9).
- Transformer single-layer Rademacher bound is O(1/sqrt(n)) with no dependence on sequence length T (Theorem 5.6).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the ℓ∞-Lipschitz loss and projection function class in the vector-contraction bound | General controllable learning · General controllable learning | generalization bound dependence on number of task targets | Helped |
| Use the Transformer control function in the embedding-based method | Embedding-based controllable learning · Embedding-based controllable learning | generalization bound | Mixed |
| Use a hypernetwork to generate task-specific model parameters | Hypernetwork-based controllable learning · Hypernetwork-based controllable learning | generalization bound | Hurt |
| Increase class imbalance | General controllable learning · Multi-target bipartite ranking loss | generalization bound | Hurt |
Surprising findings
- The square-root dependence on c is inevitable for l2-Lipschitz losses, while l-infinity Lipschitz continuity removes it.
Limitations
- Depth-dependent bounds become vacuous via peeling for deep networks; analysis mainly valid for shallow models.
- Bipartite ranking bound degrades under class imbalance and with large sum of label probabilities.
- No empirical experiments validating the bounds.
Experimental setup
Metrics: Rademacher complexity generalization bound
Builds on: Maurer vector-contraction inequality Lei et al. 2019 l-infinity Lipschitz vector contraction Edelman et al. 2022 Transformer bounds Clémençon et al. 2008 U-process ranking bounds
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