Prior Contrastive Estimation (PCE)

Summary

Prior Contrastive Estimation (PCE) is the simplification of ACE that draws the contrastive samples from the prior $p(\theta )$ instead of a learned inference network $q_{\varphi }$. This removes the need to learn any variational parameters — there is no $\varphi$ — so it is cheaper and simpler to train, at the cost of only being tight as $L\to \infty$ (it loses ACE’s “good network” route to tightness). PCE is essentially the InfoNCE mutual-information bound applied to experimental design.

Overview

ACE’s inference network $q_{\varphi }(\theta \mid y)$ must be learned, adding parameters and training cost. If the prior is already a reasonable proposal for estimating the marginal $p(y\mid \xi )$, we can skip the network and use prior draws as contrasts. The bound then depends only on the design $\xi$, so optimization is a pure design problem.

Main Content

Definition: PCE lower bound (Foster 2020, Eq. 12)

With ${\theta }_0\sim p(\theta )$, $y\sim p(y\mid {\theta }_0,\xi )$, and contrastive samples ${\theta }_{1:L}\sim p(\theta )$ drawn from the prior:

$$I_{PCE}(\xi ,L):=\mathbb{E}\left[\log \frac{p(y\mid {\theta }_0,\xi )}{\frac{1}{L+1}{\sum }_{\ell =0}^{L}p(y\mid {\theta }_{\ell },\xi )}\right],$$

expectation over $p({\theta }_0)p(y\mid {\theta }_0,\xi )p({\theta }_{1:L})$. This is the $q_{\varphi }=p(\theta )$ special case of ACE, so it inherits Theorem 1: it is a valid lower bound, monotone in $L$, and tight as $L\to \infty$ (but only case-2 tightness — no “perfect network” route).

Connection to InfoNCE (Foster 2020, Eq. 13)

The InfoNCE / information-noise-contrastive-estimation bound from representation learning (van den Oord et al. 2018) is, for data $x_k$, representations $z_k$, and critic $f_{\psi }(x,z)\ge 0$:

$$\mathrm{MI}(x;z)\ge \mathbb{E}\left[\frac{1}{K}\sum _{k=1}^K\log \frac{f_{\psi }(x_k,z_k)}{\frac{1}{K}{\sum }_{\ell =1}^K{f}_{\psi }(x_{\ell },z_k)}\right].$$

Writing $\theta$ for $x$ and $y$ for $z$, PCE is the case where the optimal critic $p(z\mid x)=p(y\mid \theta ,\xi )$ is known (it is the likelihood) — so PCE is the experimental-design instance of InfoNCE with a known critic.

Unnormalized prior densities

A practical bonus: PCE (and ACE) only need the prior up to proportionality. If $p(\theta )=A\cdot \gamma (\theta )$ with $A$ independent of $(\xi ,\varphi ,y)$ and $\gamma$ an unnormalized density, then (Foster 2020, Eq. 15)

$$I(\xi )\ge \mathbb{E}\left[\log \frac{p(y\mid {\theta }_0,\xi )}{\frac{1}{L+1}{\sum }_{\ell =0}^L\frac{\gamma ({\theta }_{\ell })p(y\mid {\theta }_{\ell },\xi )}{q_{\varphi }({\theta }_{\ell }\mid y)}}\right]-\log A,$$

and the derivatives of $\log A$ vanish. This matters in iterated design, where the prior at step $t$ is the previous posterior $p(\theta \mid y_{1:t-1},{\xi }_{1:t-1})$, known only up to its normalizing constant.

When to prefer PCE vs ACE

• PCE: prior is an adequate proposal for $p(y\mid \xi )$; no variational training wanted; low-to-moderate dimension. PCE performed well in low dimensions but degraded as dimension increased (the prior becomes an inefficient proposal).
• ACE: the inference network can closely approximate the posterior, or sampling from the prior is inefficient (high dimension) — ACE/BA learn adaptive proposals and avoid the under-estimation.

Connections

• Special case of Adaptive Contrastive Estimation (ACE) (set $q_{\varphi }=p(\theta )$).
• InfoNCE / NCE lineage: ties BOED to contrastive representation learning and noise-contrastive estimation.
• In Rainforth 2023, PCE is one of the “contrastive bounds” cited as enabling consistent stochastic-gradient design optimization (their §3.3.1).

See Also

• Adaptive Contrastive Estimation (ACE) — the adaptive (learned-proposal) generalization
• Likelihood-Free ACE and Gradient Estimation — gradient estimation that applies to PCE too
• High-Dimensional Design Applications — where PCE wins (low-D) and loses (high-D)