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Unlocking Global Optimality in Bilevel Optimization: A Pilot Study
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Gives two sufficient "benign landscape" conditions (joint and blockwise PL) for the penalty reformulation of bilevel problems, proves PBGD converges to the global optimum under them, and verifies both conditions for linear representation learning and data hyper-cleaning.
Problem
Global optimality of bilevel optimization is open because bilevel landscapes can have spurious local solutions; existing convergence guarantees are mostly to stationary points or local optima.
Approach
Analyze the penalty reformulation L_gamma = f + gamma(g - g*) rather than the nested objective; define joint and blockwise PL conditions on L_gamma, show penalty-based bilevel gradient descent (PBGD) converges globally under either (Jacobi or Gauss-Seidel update), and verify the conditions along the PBGD trajectory for two linear bilevel problems.
Key results
- PBGD converges almost linearly to the global optimum of bilevel representation learning and data hyper-cleaning, consistent with theory (iteration complexity O(log^2(1/epsilon)))
- Enlarging the penalty constant gamma reduces the target optimality gap, consistent with gamma = O(epsilon^{-0.5})
- PBGD, F2SA and BOME all converge to the global optimum in the representation learning task
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase penalty constant γ (ablation) | PBGD · Representation learning and data hyper-cleaning | optimality gap | Helped |
| Use excessively large step sizes α or β (ablation) | PBGD · Representation learning | convergence | Hurt |
| Use small step sizes α or β (ablation) | PBGD · Representation learning | convergence speed | Hurt |
| Use PBGD with proper step sizes | PBGD · Representation learning | convergence | Helped |
| Use PBGD with proper step sizes | PBGD · Data hyper-cleaning | convergence | Helped |
| Use F²SA or BOME instead of PBGD | Penalty-based first-order bilevel methods · Representation learning | convergence | No clear effect |
Surprising findings
- The nested bilevel objective F(u) can fail the PL condition and convexity even when both levels are PL or jointly convex, and the penalized objective can have saddle points even when both levels are PL
Limitations
- Global guarantees are proven only for linear (overparameterized) models with least-squares losses in two specific applications
- Joint PL and blockwise PL conditions do not imply each other and do not add up in general
- Requires assumptions such as full-rank solution existence and diagonal-type conditions for data hyper-cleaning
Experimental setup
Metrics: relative error (log scale) optimality gap
Baselines: F2SA BOME
Builds on: PBGD Polyak-Lojasiewicz condition F2SA BOME
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