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C-GAIL: Stabilizing Generative Adversarial Imitation Learning with Control Theory
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Applies control theory to GAIL training dynamics, shows the desired state is not an equilibrium, and proposes C-GAIL, a discriminator regularizer that speeds convergence and reduces oscillation.
Problem
GAIL training is unstable: losses and returns oscillate and convergence is slow, and the desired state (policy equals expert, discriminator at 1/2) is not an equilibrium of GAIL's training dynamics.
Approach
Model GAIL as a gradient-flow ODE in function space, show the entropy term prevents convergence to the desired state, and add a linear negative-feedback controller; in practice this becomes a quadratic penalty centered at D=1/2 added to the discriminator loss, with asymptotic stability proven only in a one-step setting.
Key results
- On Hopper, C-GAIL-DAC converges 5x faster than GAIL-DAC; on Ant it reduces oscillation range by around 10x.
- Toward end of training, state Wasserstein for C-GAIL-DAC is more than two times smaller than GAIL-DAC on all five MuJoCo environments.
- In the Atari table, C-GAIL's Hero return is 23912.73 vs 13942.51 for GAIL, but on Pong and Seaquest C-GAIL is not better than GAIL.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the discriminator regularizer to GAIL | GAIL · Ant | return: $2087 \pm 187$ -> $2411 \pm 21$ | Helped |
| Add the discriminator regularizer to GAIL | GAIL · Half-Cheetah | return: $3293 \pm 239$ -> $3435 \pm 50$ | Helped |
| Add the discriminator regularizer to GAIL | GAIL · Hopper | return: $2579 \pm 85$ -> $2636 \pm 8$ | Helped |
| Add the discriminator regularizer to GAIL | GAIL · Swimmer | return: $295 \pm 3$ -> $ 298 \pm 0$ | Helped |
| Add the discriminator regularizer to GAIL | GAIL · Walker2d | return: $2589 \pm 121$ -> $2633 \pm 12$ | Helped |
| Add the discriminator regularizer to GAIL-DAC | GAIL-DAC · MuJoCo tasks | convergence speed: 5x faster on Hopper; 2x faster on Half-Cheetah | Helped |
| Add the discriminator regularizer to GAIL-DAC | GAIL-DAC · MuJoCo tasks | state Wasserstein: more than two times smaller | Helped |
| Add the discriminator regularizer to GAIL-DAC | GAIL-DAC · Walker 2D | return oscillation range: more than 3 times | Helped |
| Add the discriminator regularizer to GAIL-DAC | GAIL-DAC · Ant | return oscillation range: around 10x | Helped |
| Vary controller strength k among 0.1, 1, and 10 (ablation) | C-GAIL · MuJoCo tasks | normalized return | Mixed |
| Add the discriminator regularizer to GAIL-DAC | GAIL-DAC · Reacher | iterations to reach 95% of return: 0.02M -> 0.01M | Helped |
Surprising findings
- The desired state (policy equals expert, discriminator at 1/2) is not an equilibrium of standard GAIL dynamics because of the entropy term, so GAIL cannot converge to it even with unlimited data and capacity.
- The practical C-GAIL drops the policy-side controller because it requires expert policy likelihoods, so the stability guarantee does not strictly carry over to the implemented method.
Limitations
- Theoretical stability proof holds only in a one-step simplified setting and for the discriminator-plus-policy ODE, not the full trajectory setting.
- Practical implementation applies the controller only to the discriminator; the policy-side controller requires the expert policy and has no convergence guarantee.
- Continuous-time analysis does not account for discrete updates used in practice.
Experimental setup
Datasets: MuJoCo expert demonstrations (Half-Cheetah, Ant, Hopper, Reacher, Walker2d, Swimmer) Atari expert demonstrations (BeamRider, Pong, Q*bert, Seaquest, Hero)
Benchmarks: MuJoCo Half-Cheetah MuJoCo Ant MuJoCo Hopper MuJoCo Reacher MuJoCo Walker2d MuJoCo Swimmer Atari BeamRider Atari Pong Atari Q*bert Atari Seaquest Atari Hero
Metrics: normalized return state Wasserstein distance gradient steps to reach 95% of max return return mean and std
Baselines: GAIL GAIL-DAC BC AIRL DAgger WAIL BC+GAIL DiffAIL
Builds on: GAIL DAC control theory for GANs
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