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Functional Bilevel Optimization for Machine Learning
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Introduces functional implicit differentiation (FuncID) for bilevel problems whose inner objective is optimized over a function space, enabling over-parameterized neural networks as inner prediction functions via an adjoint function in L2.
Problem
Bilevel optimization with non-convex (over-parameterized neural network) inner objectives is ill-posed because inner solutions are non-unique, making the parametric implicit gradient ambiguous and unstable.
Approach
Treat the inner problem as optimization over a Hilbert function space where the inner objective is strongly convex, derive a functional implicit gradient via an adjoint function in that space, then parameterize prediction and adjoint functions with neural networks ("differentiate implicitly, then parameterize").
Key results
- On the 5K-sample dsprites IV task, FuncID outperforms DFIV (p-value=0.003, one-sided paired t-test), while AID and ITD perform notably worse.
- On CartPole with a well-specified model, FuncID reaches the maximum reward of 5 where OMD reaches 4, matching MLE.
- FuncID obtains the lowest inner-loss values among compared methods on the IV task.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use FuncID instead of DFIV | 2SLS on dsprites (5K training samples) · dsprites 2SLS (5K training samples) | test mean squared error | Helped |
| Use FuncID instead of DFIV | 2SLS on dsprites (10K training samples) · dsprites 2SLS (10K training samples) | test mean squared error | No clear effect |
| Use FuncID instead of AID or ITD | 2SLS on dsprites · dsprites 2SLS | test mean squared error | Helped |
| Use FuncID instead of gradient penalty | 2SLS on dsprites · dsprites 2SLS | test mean squared error | Helped |
| Use FuncID instead of value function penalty | 2SLS on dsprites · dsprites 2SLS | test mean squared error | Mixed |
| Use FuncID instead of OMD | Model-based RL · CartPole well-specified (32 hidden units) | average reward: 4 -> 5 | Helped |
| Use FuncID instead of MLE | Model-based RL · CartPole well-specified (32 hidden units) | average reward: 5 | No clear effect |
| Use FuncID instead of MLE | Model-based RL · CartPole misspecified (3 hidden units) | average reward | Helped |
Surprising findings
- FuncID linear, which learns only the last layer in closed form, is among the compared variants, suggesting closed-form adjoint solutions suffice in some settings.
- On CartPole, MLE (single-level) matches FuncID in the well-specified case, so the bilevel gain is only clear in the misspecified setting.
Limitations
- Convergence guarantees assume inner and adjoint problems are solved to a stated optimality error, which is hard to quantify in practice.
- Experiments are limited to L2 spaces and two applications (instrumental regression and small-scale CartPole RL).
- Requires careful hyperparameter selection like other bilevel methods.
Experimental setup
Datasets: dsprites CartPole
Benchmarks: dsprites instrumental regression CartPole
Metrics: out-of-sample mean squared error average reward inner loss outer loss
Baselines: DFIV AID ITD gradient penalty value function penalty OMD MLE
Builds on: Approximate Implicit Differentiation (AID) Iterative Differentiation (ITD) Deep Feature Instrumental Variable regression (DFIV) Optimal Model Design (OMD) adjoint sensitivity method
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