Editing Openai/693c0f4f-255c-8008-92e9-0cd44c6d6226 (section)

==== Let’s sketch a few variants. ====

===== Data for each training LoRA ttt: =====
* Prompts batch PtP_tPt (like DnD)
* True LoRA weights ΔWt\Delta W_tΔWt
* True Delta Activation δt\delta_tδt

Model:
# Condition encoder: et=E(Pt)e_t = E(P_t)et=E(Pt) (e.g., text encoder + pooling)
# LoRA generator: ΔW^t=Gθ(et)\hat{\Delta W}_t = G_\theta(e_t)ΔW^t=Gθ(et)
# Delta head: Option 1 (cheaper): a small learned approximator δ^t=Hϕ(ΔW^t)\hat\delta_t = H_\phi(\hat{\Delta W}_t)δ^t=Hϕ(ΔW^t) Option 2 (more faithful): actually run base+LoRA on a fixed probe set to compute δ^t\hat\delta_tδ^t.

Loss:

L=λweights∥ΔW^t−ΔWt∥2⏟classical DnD-style loss+λδ∥δ^t−δt∥2⏟auxiliary behavioral loss\mathcal{L} = 
\lambda_\text{weights} \underbrace{\|\hat{\Delta W}_t - \Delta W_t\|^2}_{\text{classical DnD-style loss}}
+ \lambda_\delta \underbrace{\|\hat\delta_t - \delta_t\|^2}_{\text{auxiliary behavioral loss}}L=λweightsclassical DnD-style loss∥ΔW^t−ΔWt∥2+λδauxiliary behavioral loss∥δ^t−δt∥2
What does the δ-loss do?
* It pushes the generated LoRA to not only be close in parameter space, but also to induce the correct global behavior on probe prompts.
* It gives an extra gradient signal that’s much lower-dim and semantically aligned, which can stabilize training especially when weight loss is noisy / multi-modal.

Weak points:
* If you compute δ^t\hat\delta_tδ^t via true forward passes on probes, this adds heavy cost per step.
* If you approximate HϕH_\phiHϕ (LoRA → δ), you’ve added another network that might be lossy; but maybe that’s okay because δ itself is coarse.

===== To avoid directly regressing millions of weights from prompts, you can insert δ as a bottleneck: =====

Stage 1: learn a task encoder using δ
* Train EψE_\psiEψ s.t. Eψ(Pt)≈δtE_\psi(P_t) \approx \delta_tEψ(Pt)≈δt.
* That’s a simple regression in low dimension; way easier than full LoRA.

This makes δ the “task embedding space” learned from actual finetuned behaviors.

Stage 2: learn δ → LoRA map

Now train a separate generator GθG_\thetaGθ to map δ to LoRA:

ΔW^t=Gθ(δt)\hat{\Delta W}_t = G_\theta(\delta_t)ΔW^t=Gθ(δt)
with weight-level loss:

Lstage2=∥ΔW^t−ΔWt∥2\mathcal{L}_\text{stage2} = \|\hat{\Delta W}_t - \Delta W_t\|^2Lstage2=∥ΔW^t−ΔWt∥2
At inference:
* New dataset ➜ prompts PnewP_{\text{new}}Pnew
* Stage 1: δ^new=Eψ(Pnew)\hat\delta_{\text{new}} = E_\psi(P_{\text{new}})δ^new=Eψ(Pnew)
* Stage 2: ΔW^new=Gθ(δ^new)\hat{\Delta W}_{\text{new}} = G_\theta(\hat\delta_{\text{new}})ΔW^new=Gθ(δ^new)

Now δ is both:
* A learned task representation (Stage 1), supervised by behavioral deltas;
* The bottleneck that the LoRA generator uses (Stage 2).

Pros:
* You’ve decomposed a very hard mapping “prompts → millions of weights” into: - “prompts → 4k-dim δ” (reasonable), and - “δ → weights” (pure parameter-space mapping).
* You can even pretrain Stage 2 using any LoRA–δ pairs you have, independent of prompts.

Weaknesses:
* δ is intentionally lossy. Many LoRAs can share similar δ, especially if probes are generic. So δ alone might not contain enough info to reconstruct exact LoRA.
* You’re back to weight-level MSE in Stage 2; though G has an easier job because δ already encodes “what kind of task” it is.

You can combine both worlds: in Stage 2, also add a δ-consistency loss:

∥Delta(ΔW^t)−δt∥2\|\text{Delta}(\hat{\Delta W}_t) - \delta_t\|^2∥Delta(ΔW^t)−δt∥2
so that even if weights aren’t exactly matching, their induced δ is.

===== A more behaviorally principled variant: =====
# Prompts PtP_tPt → embedding ete_tet
# Generator G(et)=ΔW^tG(e_t) = \hat{\Delta W}_tG(et)=ΔW^t
# Use two behavioral losses instead of relying heavily on raw weight MSE: - Delta loss: ∥Delta(ΔW^t)−δt∥2\|\text{Delta}(\hat{\Delta W}_t) - \delta_t\|^2∥Delta(ΔW^t)−δt∥2 (behavior on a fixed probe set) - Output distillation loss: On a small set of task-specific prompts QtQ_tQt, minimize, e.g., KL(pbase+ΔW^t(⋅∣q) ∥ pbase+ΔWt(⋅∣q))\text{KL}\big(p_{\text{base}+\hat{\Delta W}_t}(\cdot|q) \,\|\, p_{\text{base}+\Delta W_t}(\cdot|q)\big)KL(pbase+ΔW^t(⋅∣q)∥pbase+ΔWt(⋅∣q)) so that generated LoRA matches the real LoRA’s outputs.

Now weight-level supervision becomes optional or just a small regularizer. Most of the signal is:
* “your LoRA should behave like this known one on probes and on some task prompts”.

This is much closer to what we actually want (function match), and δ is a cheap, global summarization of that behavior.

Weaknesses:
* Needs two model evaluations (base+true-LoRA and base+gen-LoRA) during training. Expensive but maybe ok if you restrict to small probe sets.
* Still doesn’t solve the identifiability problem perfectly: lots of LoRAs can match the same δ and distillation on small Q_t but differ elsewhere. But that might be fine in practice.