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Fast and unified path gradient estimators for normalizing flows
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Proposes a fast, unified path gradient estimator for normalizing flows that computes the score recursively during sampling and extends path gradients to forward KL (maximum likelihood) training by rewriting it as a reverse KL in base space.
Problem
Path gradient estimators for normalizing flows reduce variance but are several times more expensive than standard gradients, and were not scalably applicable to forward KL / maximum likelihood training, which relies on reweighting.
Approach
Derive a recursion for the derivative of the flow's log density computed alongside sampling (linear cost for coupling flows, no inversion needed for implicit flows), and show forward KL training equals reverse KL in base space via the pullback of the target, so the same fast path gradient applies to maximum likelihood without reweighting.
Key results
- Path gradients improve reverse-KL ESS on phi^4 from 85.6 to 96.0 and GMM from 92.2 to 97.4.
- Fast path gradient for explicitly invertible flows costs about 1.4-1.6x the standard gradient, versus about 2.1x for the prior method.
- For implicitly invertible U(1) flows, the implicit-differentiation fast estimator costs 2.0-2.3x the standard gradient, versus 8.2-17.5x for the baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use path gradients instead of standard gradients | GMM reverse KL training · GMM, reverse KL | ESS_p: 92.2 ± 0.0 -> 97.4 ± 0.0 | Helped |
| use path gradients instead of standard gradients | GMM reverse KL training · GMM, reverse KL | ESS_q: 93.0 ± 0.0 -> 97.4 ± 0.0 | Helped |
| use path gradients instead of standard gradients | GMM forward KL training · GMM, forward KL | ESS_p: 79.1 ± 0.0 -> 91.8 ± 0.0 | Helped |
| use path gradients instead of standard gradients | GMM forward KL training · GMM, forward KL | ESS_q: 84.1 ± 0.0 -> 91.8 ± 0.0 | Helped |
| use path gradients instead of standard gradients | phi^4 reverse KL training · phi^4, reverse KL | ESS_p: 85.6 ± 0.1 -> 96.0 ± 0.1 | Helped |
| use path gradients instead of standard gradients | phi^4 reverse KL training · phi^4, reverse KL | ESS_q: 85.6 ± 0.1 -> 96.0 ± 0.1 | Helped |
| use path gradients instead of standard gradients | phi^4 forward KL training · phi^4, forward KL | ESS_p: 85.1 ± 0.1 -> 95.6 ± 0.0 | Helped |
| use path gradients instead of standard gradients | phi^4 forward KL training · phi^4, forward KL | ESS_q: 85.1 ± 0.1 -> 95.6 ± 0.0 | Helped |
| use path gradients instead of standard gradients | U(1) reverse KL training · U(1), reverse KL | ESS_q: 40.1 ± 0.0 -> 41.1 ± 0.0 | Helped |
| use path gradients instead of standard gradients | U(1) reverse KL training · U(1), reverse KL | ELBO: 1346.42 ± .01 -> 1346.43 ± .00 | Helped |
| use the fast recursive path-gradient estimator instead of the previous path-gradient algorithm | explicitly invertible flows · phi^4, explicitly invertible flows, batch sizes 64, 1024, and 8192 | runtime factor: 1.6 ± 0.1 vs 2.1 ± 0.1; 1.4 ± 0.1 vs 2.2 ± 0.1; 1.4 ± 0.0 vs 2.1 ± 0.0 | Helped |
| use the fast recursive path-gradient estimator instead of path gradients with implicit differentiation | implicitly invertible flows · U(1), implicitly invertible flows, batch sizes 64, 1024, and 8192 | runtime factor: 2.2 ± 0.0 vs 17.5± 0.2; 2.0 ± 0.1 vs 11.0± 0.1; 2.3 ± 0.0 vs 8.2 ± 0.0 | Helped |
Limitations
- Forward KL path gradient requires a known unnormalized target energy, so it cannot be used for e.g. image generation.
- Forward KL training with path gradients relies on target samples that may be few or biased.
Experimental setup
Datasets: Gaussian mixture model (d=6, 64 modes) phi^4 lattice field theory (16x8 lattice) U(1) lattice gauge theory
Benchmarks: Gaussian mixture model phi^4 lattice field theory U(1) gauge theory
Metrics: effective sample size (ESS) ELBO runtime factor relative to standard gradient
Baselines: standard gradient estimator path gradient of Roeder et al. (2017) path gradient with implicit inversion (Kim et al.) reweighted forward KL path gradients continuous normalizing flow fast path gradient
Builds on: path gradients (Roeder et al.) doubly reparameterized gradients GDReG implicit differentiation for implicitly invertible flows
Closest papers by meaning
- Path-Gradient Estimators for Continuous Normalizing Flows (ICML 2022 · similarity 0.92)
- Normalizing Trajectory Models (NeurIPS 2026 · similarity 0.86)
- Stream-level Flow Matching with Gaussian Processes (ICML 2025 · similarity 0.86)
- Normalizing Flows are Capable Generative Models (ICML 2025 · similarity 0.86)
- Kernelised Normalising Flows (ICLR 2024 · similarity 0.85)
- Liouville Flow Importance Sampler (ICML 2024 · similarity 0.85)
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