projected_grpo — vGROUT

vGROUT (vector gradient routing): route the GRPO gradient against an extracted reward-hacking direction (in the SVD-of-W basis) to reduce the reward-hack rate without tanking pass rate. A representation-routing variant of gradient routing (Cloud et al.; Shilov et al.), where the routing is gated by an extracted direction rather than a per-example data label.

Built on Ariahw, Engels & Nanda's rl-rewardhacking LeetCode benchmark. Method differs from concurrent work (Wu & Tang 2026, "Advantage Modification") by intervening at the gradient level rather than the advantage level.

See docs/spec.md, docs/brainstorm/extracted_prefs.md, and docs/papers/.

We cannot cheat (the load-bearing constraint)

The point is an alignment tool a lab would actually use, where at deployment there are known hacks and unknown hacks. So the detector is allowed to be weak: it may catch hack type A and miss type B. We then use the gradient from A to try to stop the model learning B. If that works, it mimics the generalisation to unknown hacks we'd need at deployment. A detector that already sees every hack proves nothing.

Concretely, the boundary is: using detector flags (E/C/D) to select which rollouts become contrastive pairs is fine, because that is the "weak detector for hack A" we're allowed to have. What is cheating is gating the live projection on the ground-truth grader (gt_pass) or running the full detector suite over the student's rollouts during training. The whole result is uninteresting if we let the oracle in at train time.

How it works

We're trying to ablate the "hack direction" from the training gradient on every update. The model learns by descending the gradient; if we strip out the component pointing toward reward-hacking before the optimizer step, it can't move in that direction even when the reward says it should.

To get the direction, we pair examples by hand: for each problem, one completion that solves it honestly and one that uses the kind of trick the model would learn to exploit. Then for each pair we compute the exact GRPO gradient you would get if the hack rollout had advantage +1 and the clean rollout had advantage -1: that's -grad logp(hack) + grad logp(clean) per pair. Stack these vectors over our ~10 pairs and SVD the result; the top right singular vectors are our hack-direction basis. (Mechanically this is identical to a twin-NLL extraction because GRPO with adv=+/-1 reduces algebraically to the NLL difference, but the GRPO framing is the one we mean: extraction produces a sample of the gradient GRPO itself would emit if it ever saw a perfectly-labeled pair.)

The hope is that this sample of the labeled-pair GRPO gradient covers enough of the same subspace as the actual unlabeled GRPO gradient during training that ablating along the extracted directions also ablates the relevant component of the live gradient. Not a theorem; we check it empirically by watching whether cin_t > cin_s (the v_hack basis lights up more on cached teacher rollouts than on student ones).

Everything happens in the SVD-of-W basis. Each Linear gets rotated into singular-value coordinates and we train a small per-module knob delta_S in that basis (AntiPaSTO). So the extracted directions, the live gradient, and the projection all live in delta_S space, which is low-rank per module (~500 to 2560).

Noise floor at load. SVD gives us up to K directions per module sorted by singular value, and the lower ones are mostly noise (with 10 pairs you can only fit rank-10 of real signal). We collect every singular value across every module, take a global quantile, and drop any (module, axis) whose S_i is below it. Default cut: bottom 25%. Modules whose every axis lands below get filtered out entirely. Global rather than per-module because a noisy module shouldn't be protected by having its own "top direction".

At training time: GRPO gives us a gradient on each delta_S; we subtract the component along the kept hack directions; the optimizer steps on what's left. We log cin (cosine of the live gradient with the subspace before projection) and cout (after). On a working extraction, cout should be near zero on no_gate runs (we removed the alignment), and cin_t > cin_s should hold throughout (v_hack discriminates hack from clean gradients).

What we compare

The environment has four loophole modes (run_tests, sentinel, stdout_marker, file_marker), each a different grading flaw with a truthful hint disclosing the mechanism. Problems are partitioned evenly and non-overlappingly, so a vanilla student can learn all four independently. Full prompt+hint, hack, and clean traces per mode are in the blog appendix (docs/blog/20260529_...md).

Four arms, identical model and teacher pool, differing only in the gradient policy:

• vanilla -- no intervention; the emergence reference.
• erase -- subtract the v_hack component from the live delta_S gradient (one-sided).
• route -- quarantine the v_hack component into a throwaway delta_S_hack knob, deleted at deploy. Gradient routing (Cloud et al. 2024) in the SVD basis. (v1: shared basis, relu gate on the kept-axis coords, same as erase but routed not erased.)
• route2 -- current routing arm. Per-rollout gate cos(g_rollout, v_grad) > tau (tau calibrated each step from the hack-vs-clean cosine gap) decides whether a rollout's whole gradient routes into a scale-matched, distinct-basis delta_S_hack. An exploration floor (rollout_ablate_frac) generates a fraction of rollouts knob-off so the deployed knob always sees solve signal even if the full policy saturates on hacking.
• route, weak detector -- the generalisation test: build v_hack from only 2 of the 4 modes (the "known" hacks the weak detector can flag), route on that subspace, and measure whether the 2 held-out modes are also suppressed. This is the load-bearing no-cheat check.

Deploy hack/solve is measured the same way for every arm now: knob-off forward (ablate_quarantine for route/route2, the trained model itself for vanilla/erase), n=64 prompts*group, sampled at T=0.7. So the per-arm deploy numbers are apples-to-apples.

The frozen-vs-refresh distinction is orthogonal: any arm can re-extract v_hack every N steps on the current adapter (for route, with the quarantine ablated during extraction, see the blog).

Quick start

uv sync
just smoke               # tiny-random model, projected pathway, ~1-2 min
just smoke-vanilla       # tiny-random model, vanilla pathway, ~1-2 min
just download-model      # warm Qwen3-4B cache (full preset peaks ~73GB on 96GB)
just queue-full          # queue extract + 3-seed vanilla + 3-seed projected sweep

See RESEARCH_JOURNAL.md for session-by-session findings, including the 2026-05-23 grader-bug discovery that invalidated all prior gt=0 measurements and the move from Qwen3.5-2B to Qwen3-4B (reference substrate).

Results and write-up

The paper draft is the source of truth for current numbers, figures, and the preregistered hypotheses: docs/writeup/main.tex (keynote table + figure, ablations, generalisation). Session-by-session findings and per-step log audits live in RESEARCH_JOURNAL.md.