Contrastive Divergence

Contrastive Divergence (CD) is a training algorithm for energy-based models that approximates the gradient of the log-likelihood — using short-run MCMC (typically just 1 step of Gibbs sampling or Langevin dynamics) instead of running the chain to equilibrium, making EBM training practical.

How CD Works

• Positive Phase: Compute the gradient of the energy at data points (easy: just backprop through $E_\theta(x_{data})$).
• Negative Phase: Run $k$ steps of MCMC from the data to get approximate model samples.
• Gradient: $\nabla_\theta \log p \approx -\nabla_\theta E(x_{data}) + \nabla_\theta E(x_{MCMC})$ (push down data energy, push up sample energy).
• CD-k: $k$ is the number of MCMC steps (CD-1 is most common — just 1 step).

Why It Matters

• Practical Training: CD makes EBM training feasible by avoiding the need for converged MCMC chains.
• RBMs: CD was the breakthrough that made training Restricted Boltzmann Machines practical (Hinton, 2002).
• Bias: CD introduces bias (unconverged MCMC), but works well in practice for many EBMs.

Contrastive Divergence is the shortcut for EBM training — using a few MCMC steps instead of full equilibration to approximate the intractable gradient.