[Paper Review] A Connection between Generative Adversarial Networks, Inverse Reinforcement Learning, and Energy-Based Models

Chelsea Finn, Paul F. Christiano+2 more|arXiv (Cornell University)|Nov 11, 2016

Generative Adversarial Networks and Image Synthesis25 references214 citations

TL;DR

The paper shows that certain GANs with an evaluable generator density are mathematically equivalent to maximum entropy inverse reinforcement learning, and that this links GANs to energy-based models, enabling transferable ideas across GANs, IRL, and EBMs.

ABSTRACT

Generative adversarial networks (GANs) are a recently proposed class of generative models in which a generator is trained to optimize a cost function that is being simultaneously learned by a discriminator. While the idea of learning cost functions is relatively new to the field of generative modeling, learning costs has long been studied in control and reinforcement learning (RL) domains, typically for imitation learning from demonstrations. In these fields, learning cost function underlying observed behavior is known as inverse reinforcement learning (IRL) or inverse optimal control. While at first the connection between cost learning in RL and cost learning in generative modeling may appear to be a superficial one, we show in this paper that certain IRL methods are in fact mathematically equivalent to GANs. In particular, we demonstrate an equivalence between a sample-based algorithm for maximum entropy IRL and a GAN in which the generator's density can be evaluated and is provided as an additional input to the discriminator. Interestingly, maximum entropy IRL is a special case of an energy-based model. We discuss the interpretation of GANs as an algorithm for training energy-based models, and relate this interpretation to other recent work that seeks to connect GANs and EBMs. By formally highlighting the connection between GANs, IRL, and EBMs, we hope that researchers in all three communities can better identify and apply transferable ideas from one domain to another, particularly for developing more stable and scalable algorithms: a major challenge in all three domains.

Motivation & Objective

• Motivate and formalize connections between GANs, IRL, and EBMs.
• Show that a specialized GAN discriminator aligns with MaxEnt IRL when generator density is evaluable.
• Demonstrate that MaxEnt IRL is a special case of an energy-based model.
• Derive how GANs can train EBMs via unbiased partition function estimation.
• Discuss implications for stability and transfer of ideas across the three domains.

Proposed method

• Define a special form of the GAN discriminator that uses the generator density q(τ) and a Boltzmann energy cθ(τ).
• Show the discriminator Dτ = (1/Z exp(-cθ(τ)))/(1/Z exp(-cθ(τ)) + q(τ)) which makes the discriminator independent of the generator.
• Prove that the GAN objective matches the MaxEnt IRL objective when Z is treated as an importance-sampling estimator for the partition function.
• Derive that the generator loss equals log Z plus the sampler (entropy-regularized) objective from MaxEnt IRL.
• Extend the connection to training EBMs by deriving an unbiased partition function estimator when the generator density is available.
• Discuss implications for using density-enabled generators (e.g., autoregressive models) for stable training.

Experimental results

Research questions

• RQ1 Can a GAN with an evaluable generator density replicate the MaxEnt IRL objective?
• RQ2 What discriminator form yields equivalence between GAN training and MaxEnt IRL?
• RQ3 How do GANs relate to training energy-based models when the generator density is known?
• RQ4 Can these connections inform more stable and scalable training across GANs, IRL, and EBMs?

Key findings

• GANs with a density-evaluable generator are equivalent to a sample-based MaxEnt IRL algorithm.
• The optimal discriminator can be written to incorporate generator density, leading to independence from the generator at optimum.
• MaxEnt IRL is a special case of an energy-based model with the cost function as the energy; GANs can train EBMs when generator density is available.
• The partition function Z is estimated via importance sampling, and its gradient aligns with the IRL objective.
• The generator loss equals the MaxEnt IRL sampler loss, linking adversarial training to entropy-regularized policy optimization.
• This formalizes a pathway to transfer stability and scalability ideas across GANs, IRL, and EBMs.