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pith:CZ346JSL

pith:2026:CZ346JSLYI4GRFJN5WL7J3VUT7

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$f$-Trajectory Balance: A Loss Family for Tuning GFlowNets, Generative Models, and LLMs with Off- and On-Policy Data

Jake Fawkes, Jason Hartford

A family of losses lets generative models match any f-divergence on-policy while keeping the same global minimizer off-policy.

arxiv:2605.15417 v1 · 2026-05-14 · cs.LG · cs.AI

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Claims

C1strongest claim

This construction can be extended to the whole family of f-divergences, leading to a family of losses whose on-policy gradients are that of the corresponding f-divergence, but retain the same global minimizer off-policy. Specifically, the on-policy gradients lead to a one to one correspondence between translation invariant loss functions on the target and model log probabilities, and f-divergences.

C2weakest assumption

The paper assumes that the surrogate losses remain valid and share the same global minimizer when evaluated off-policy, which rests on the translation-invariance property of the loss functions on log probabilities; if this invariance does not hold for the chosen f-divergence or if the correspondence is not one-to-one, the off-policy guarantee fails.

C3one line summary

The work defines a family of surrogate losses for generative models and LLMs whose on-policy gradients match those of any f-divergence while sharing the same off-policy minimizer as the original MSE loss.

References

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[1] PMLR, 2020. D. Go, T. Korbak, G. Kruszewski, J. Rozen, N. Ryu, and M. Dymetman. Aligning language models with prefer- ences through f-divergence minimization.arXiv preprint arXiv:2302.08215, 2023. J. 2020 · doi:10.64434/tml

[2] If µ=p θ, the expected auto-differentiated gradients match the f-divergence gradient: ∇θDf(pθ∥p⋆) = Epθ[∇θLf(∆θ(y))]. A.1.1. PROOF OFPART1: CONVEXITY ANDGLOBALMINIMIZER Let the scalar loss function wi

[3] Gradient of the Surrogate Loss:The gradient of the loss Lf with respect to the backward parameters ϕ, estimated on-policy, is: ∇ϕJon =E τ∼π F [∇ϕLf(logu)] =E τ∼π F [(f ′(u)−f ′(1))∇ϕ(−logπ B)] =E τ∼π

[4] Gradient of the Candidate Divergence:Consider the generic divergence Dg(πF ∥πB) = R πB(τ)g πF (τ) πB(τ) dτ. Differentiating with respect toϕ: ∇ϕDg = Z (∇ϕπB ·g(u) +π Bg′(u)∇ϕu)dτ Using the identity∇ ϕ

[5] •Gradient Weight: wDG i = ∆i −E B[∆(y)]

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Receipt and verification

First computed	2026-05-20T00:00:57.505206Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1677cf264bc23868952ded97f4eeb49feb0af8ad34fe8caf3495e4d0b799a852

Aliases

arxiv: 2605.15417 · arxiv_version: 2605.15417v1 · doi: 10.48550/arxiv.2605.15417 · pith_short_12: CZ346JSLYI4G · pith_short_16: CZ346JSLYI4GRFJN · pith_short_8: CZ346JSL

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/CZ346JSLYI4GRFJN5WL7J3VUT7 \
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Canonical record JSON

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  "metadata": {
    "abstract_canon_sha256": "4078c284f01852446425428635bb59b3cb2f88ba3d6ac510c0ffc3b13bfa024b",
    "cross_cats_sorted": [
      "cs.AI"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T21:02:07Z",
    "title_canon_sha256": "a4ee575b2dcc713ff1fa49a4956eb5aa306519dc23e85057571a540450c0f516"
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  "source": {
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