Now let me write the summary report. Where would you like it — a markdown file in the repo, or just here in chat?
Actually, let me just do it here with all the detail, since you said to describe all setups and results. You can tell me if you want it saved somewhere.
GP Exploit Rate Modeling: Summary Report
Motivation
The binary emergence predictor classifies whether each exploit type (N=8 per run, 2 runs = 16 samples) will exceed 10% exploit rate at prefill=0 by the final checkpoint. Simple metrics (KL divergence slope, log-exploit-lower-bound) achieve perfect classification early. The GP was investigated as a principled Bayesian approach to estimate exploit rates when observed counts are zero — borrowing strength across nearby (checkpoint, prefill) cells.
GP Models Tried
┌─────┬──────────────────────────┬────────────────────────────┬───────────────────────────────────────────────────────────┬──────────────────────────────────────────┐ │ # │ Model │ Mean function │ Key idea │ Params │ ├─────┼──────────────────────────┼────────────────────────────┼───────────────────────────────────────────────────────────┼──────────────────────────────────────────┤ │ 1 │ ExploitRateGP │ Constant μ │ 2D Matérn-5/2, binomial likelihood, Laplace approximation │ var, ℓ_ckpt, ℓ_pfx, μ │ ├─────┼──────────────────────────┼────────────────────────────┼───────────────────────────────────────────────────────────┼──────────────────────────────────────────┤ │ 2 │ ExploitRateOffsetGP │ Constant μ │ 1D GP(checkpoint) + α·KL offsets │ var, ℓ, μ, α │ ├─────┼──────────────────────────┼────────────────────────────┼───────────────────────────────────────────────────────────┼──────────────────────────────────────────┤ │ 3 │ ExploitRateConstrainedGP │ Constant μ → Linear → Tanh │ 2D GP + pairwise KL constraints, two-stage fitting │ var, ℓ_ckpt, ℓ_pfx, mean params, σ_IS, α │ └─────┴──────────────────────────┴────────────────────────────┴───────────────────────────────────────────────────────────┴──────────────────────────────────────────┘
Mean Function Variants (Model 3)
┌──────────────┬───────────────────────────────────────┬───────────────┬──────────────────────────────────────────────────┐ │ Variant │ Formula │ Parameters │ Rationale │ ├──────────────┼───────────────────────────────────────┼───────────────┼──────────────────────────────────────────────────┤ │ Constant │ μ(t) = μ₀ │ 1 (μ₀) │ Baseline; contaminated by averaging across cells │ ├──────────────┼───────────────────────────────────────┼───────────────┼──────────────────────────────────────────────────┤ │ Linear │ μ(t) = β₀ + β₁·log(ckpt) │ 2 (β₀, β₁) │ Captures checkpoint-wise trend │ ├──────────────┼───────────────────────────────────────┼───────────────┼──────────────────────────────────────────────────┤ │ Tanh S-curve │ μ(t) = β₀ + β₁·tanh(γ·t_norm)/tanh(γ) │ 3 (β₀, β₁, γ) │ Handles saturation; γ=0→linear │ └──────────────┴───────────────────────────────────────┴───────────────┴──────────────────────────────────────────────────┘
Results: gp_log_rate_p0_last threshold_level AUC (extrapolation)
┌─────┬──────────────────┬──────────┬────────┬───────┐ │ idx │ Checkpoints seen │ Constant │ Linear │ Tanh │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 0 │ 1 │ 0.445 │ 0.445 │ 0.445 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 1 │ 2 │ 0.484 │ 0.203 │ 0.328 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 2 │ 3 │ 0.625 │ 0.703 │ 0.703 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 3 │ 4 │ 0.969 │ 1.000 │ 1.000 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 4 │ 5 │ 0.906 │ 0.875 │ 0.891 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 5 │ 6 │ 0.875 │ 1.000 │ 0.766 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 6 │ 7 │ 1.000 │ 0.750 │ 1.000 │ ├─────┼──────────────────┼──────────┼────────┼───────┤ │ 7 │ 8 │ 1.000 │ 0.875 │ 0.859 │ └─────┴──────────────────┴──────────┴────────┴───────┘
Results: Simple baselines (no GP)
┌─────────────────────────┬─────────────────┬───────────────────┐ │ Metric │ Model │ First perfect AUC │ ├─────────────────────────┼─────────────────┼───────────────────┤ │ mean_neg_kl │ threshold_slope │ idx 1 (2 ckpts) │ ├─────────────────────────┼─────────────────┼───────────────────┤ │ mean_neg_kl │ logistic │ idx 1 (2 ckpts) │ ├─────────────────────────┼─────────────────┼───────────────────┤ │ log_exploit_lower_bound │ threshold_slope │ idx 2 (3 ckpts) │ ├─────────────────────────┼─────────────────┼───────────────────┤ │ exploit_rate │ threshold_level │ idx 3 (4 ckpts) │ ├─────────────────────────┼─────────────────┼───────────────────┤ │ Best GP (any variant) │ any │ idx 3 (4 ckpts) │ └─────────────────────────┴─────────────────┴───────────────────┘
Conclusions
- The GP adds no value for binary emergence prediction. KL divergence slope achieves perfect AUC from 2 checkpoints. The best GP variant reaches it at 4 checkpoints.
- The mean function matters, but inconsistently. Linear and tanh both improve mid-range extrapolation (idx 3) but introduce regressions elsewhere. No variant dominates.
- The fundamental issue is not rate estimation quality. The binary classification task has such high signal-to-noise that even crude rate estimates suffice. The GP's sophistication is solving a problem that doesn't exist here.
- Where GPs might help: trajectory prediction (continuous rate curves), multi-run generalization, or settings with more exploit types and noisier signal where simple thresholds break down.
✻ Churned for 8m 45s
❯ Can you explain again how the constrained GP works?
● Sure. The core problem it solves:
At early checkpoints with prefill=0, we observe 0 exploits out of N trials. The GP has no signal — it falls back to its prior mean. But at higher prefill levels for the same checkpoint, we sometimes do see exploits. The constrained GP propagates that signal down to
prefill=0.
The IS relationship
From importance sampling theory, in logit space (and approximately for small rates where logit ≈ log):
logit(rate_observed_at_pfx) ≈ logit(rate_true_at_pfx0) + KL(pfx)
Prefilling makes exploits easier by exactly the amount that the prefilled distribution diverges from the unprefilled one. So if we see a 5% exploit rate at prefill=20 and KL=3.0 nats, we can infer that the true unprefilled rate is roughly expit(logit(0.05) - 3.0) ≈ 0.24%.
The model
The GP models the latent logit rate f(ckpt, pfx) at every (checkpoint, prefill) cell:
g ~ GP(0, K) # GP prior on deviations f = g + μ(ckpt) # latent = deviation + mean function y_i ~ Binomial(n_i, σ(f_i)) # binomial observations
g(ckpt, 0) - g(ckpt, pfx) ~ N(-α·KL, σ²_IS) for each (ckpt, pfx>0) pair
The constraints are soft Gaussian penalties. α scales the KL (because the IS relationship is approximate in logit space — it's exact in log space). σ_IS absorbs the approximation error and IS inequality slack.
What the constraints do to inference
Without constraints, the posterior mode is found via standard Laplace approximation (R&W Algorithm 3.1), which exploits the fact that the binomial Hessian is diagonal (each observation only involves one latent value).
With pairwise constraints, the Hessian becomes non-diagonal — each constraint couples two latent values (the pfx=0 cell and the pfx>0 cell at the same checkpoint). So the code uses direct Newton iteration instead:
grad = grad_binomial + grad_constraints + grad_prior H = diag(W) + P + K⁻¹ step = H⁻¹ @ grad
where P is the constraint precision matrix (sparse — only links paired cells) and W is the binomial observation precision (diagonal).
Two-stage fitting
Joint optimization of kernel params + constraint params tends to fail because the optimizer weakens the kernel to accommodate constraints (e.g., inflating the prefill lengthscale to ignore the prefill dimension entirely). So instead:
- Stage 1: Fit an unconstrained ExploitRateGP to learn kernel hyperparameters (variance, lengthscales) and mean function (β₀, β₁, γ)
- Stage 2: Fix those kernel params, find the posterior mode under both binomial likelihood AND pairwise constraints. Optionally optimize just α and σ_IS (cheap 2D search).
Concrete effect
At checkpoint 1 with prefill=0, say we observe 0/90 exploits. The unconstrained GP predicts whatever the prior says (~0.14%). The constrained GP looks at checkpoint 1 with prefill=20 where we observed 3/90 exploits and KL=2.5, and infers a tighter estimate (~0.04%) for prefill=0. The constraint pulls the pfx=0 estimate toward what the pfx>0 data implies via the IS relationship.
✻ Brewed for 35s
❯ True rate is observed rate plus KL. It feels a bit unintuitive to me. Like it seems like that should only hold if the proposal distribution is optimized or something to maximize the rate. I don't know. I mean, there's a sort of, there's a duration by which I came
to it. I mean, I can go through that again, but I don't know if you can explain the intuition there as well.
● You're right to find it unintuitive — it's actually an inequality, not an equality, and the derivation makes the intuition clearer.
The derivation
We want to know P(exploit) (the true, unprefilled rate). We only observe samples from Q (the prefilled model). By change of measure:
P(exploit) = Σ_x P(x) · 1(exploit)
= Σ_x Q(x) · [P(x)/Q(x)] · 1(exploit)
= Q(exploit) · E_{Q|exploit}[P(x)/Q(x)]
Take logs:
log P(exploit) = log Q(exploit) + log E_{Q|exploit}[P(x)/Q(x)]
Apply Jensen's inequality (log is concave):
log E_{Q|exploit}[P(x)/Q(x)] ≥ E_{Q|exploit}[log P(x) - log Q(x)]
That last term — the expected log-likelihood ratio of exploit completions under Q — if exploit completions are typical sequences under Q, is approximately -KL(Q||P) by the AEP. So:
log P(exploit) ≥ log Q(exploit) - KL(Q||P)
Or rearranging: the observed rate upper-bounds the true rate, with KL as the gap.
Why approximate equality?
Jensen's bound is tight when P(x)/Q(x) is approximately constant across exploit completions. That happens when exploit completions are "typical" under Q — i.e., when the prefill successfully makes exploit reasoning the natural continuation mode, so all exploit completions have similar log-likelihood ratios.
Your intuition about the proposal needing to be "optimized" is exactly right. If the prefill barely nudges the distribution (low KL, exploits still rare under Q), the exploit completions are atypical under Q, the likelihood ratios have high variance, and Jensen's gap is large. If the prefill makes exploits the dominant continuation mode (exploits are typical under Q), the bound is tighter.
Why α ≈ 0.15, not 1.0
Three things conspire to make the effective coefficient much less than 1:
- Jensen's gap — the inequality is not equality
- logit ≈ log breaks down — at higher rates the logit diverges from log, weakening the relationship
- KL is measured per-token then summed — but the relevant KL for the IS bound is over complete sequences, which may not decompose cleanly
So α absorbs all the slack. The fact that it's 0.15 rather than ~1.0 tells you the bound is fairly loose in practice — the IS relationship gives you directional information but not a precise correction factor.
The honest summary
It's a theoretically-motivated soft constraint that says "if KL is high at this (checkpoint, prefill) pair, the pfx=0 rate should be proportionally lower." The direction is right, the magnitude is approximate, and σ_IS absorbs the rest. It's better than ignoring the high-prefill data entirely, but it's not a tight enough relationship to do precise inference — which is part of why the GP doesn't outperform simple KL slope thresholds.
✻ Churned for 1m 48s
❯ Yeah, I mean, I wonder if there's an assumption here which is like going from Prefiel 0 to Prefiel 10 has the same sort of impact as going from Prefiel 90 to Prefiel 100, which it almost certainly doesn't, right? That's sort of assuming that the first 10 and the
last 10 tokens get you about the same KL cost.
● Exactly right. The constraint uses total KL at that prefill level and applies a single α to all of them:
f(ckpt, 0) - f(ckpt, pfx=10) ~ N(-α·KL₁₀, σ²)
f(ckpt, 0) - f(ckpt, pfx=100) ~ N(-α·KL₁₀₀, σ²)
Same α, but the marginal value of prefill tokens is almost certainly non-uniform. The first few tokens that shift from "normal coding" to "let me think about exploiting..." are probably doing most of the work — high KL per token and high marginal impact on exploit
rate. By token 90-100, the model is already deep in exploit mode, each additional token contributes some KL but barely moves the exploit rate further.
So the mapping from total KL to log rate inflation is probably concave — the first nats of KL buy more rate inflation than later nats. A linear model (single α) overpredicts the gap at high prefill, which means the high-prefill constraints pull the pfx=0 estimate too low.
The code has a sigma_slope parameter that gives more slack to high-KL constraints (σ²_c = σ²_base + (KL·slope)²), and a kl_cap that truncates KL values. Both are band-aids for this nonlinearity.
The principled fix would be something like α(KL) — a diminishing-returns function instead of a constant. But at that point you're fitting yet another functional form to a relationship you don't have much data to calibrate, which is the same trap the mean function work fell into.
This is probably another reason why the GP constraints don't help much — the IS relationship is weakest precisely at the high-prefill cells where you have the most exploit signal to propagate.
✻ Baked for 40s
❯ And so how does this relate to the Jensen gap argument previously, right? Like I'm accumulating KL, but I'm not increasing Q_exploit. So what's going on? So I'm just making the bound worse, but is it like variance that's increasing or something? I don't know.
● Yeah, that's the right question. Let me trace through what happens to the bound as prefill increases.
The exact decomposition is:
log P(exploit) = log Q(exploit) + log E_{Q|exploit}[P(x)/Q(x)]
↑ plateaus ↑ this is the actual correction
And the IS bound replaces the exact correction with:
log P(exploit) ≥ log Q(exploit) + E_{Q|exploit}[log P(x)/Q(x)] ↑ ≈ -KL
The Jensen gap is log E[w] - E[log w] where w = P(x)/Q(x). For log-normal-ish weights this is approximately Var(log w) / 2.
So as you go from prefill 10 to prefill 100:
- log Q(exploit) plateaus (model is already in exploit mode)
- KL keeps growing (each additional forced token adds to the total divergence)
- The exact correction log E_{Q|exploit}[P/Q] must be roughly constant (since P(exploit) is fixed and Q(exploit) plateaus)
- But E[log P/Q] ≈ -KL keeps getting more negative
- So the Jensen gap = (exact correction) - (-KL) grows
The gap is growing because you're accumulating KL "cost" from tokens that don't actually change the exploit outcome. Tokens 90-100 make log P(x)/Q(x) more negative for every completion, but they do so approximately uniformly — they don't differentially affect exploit vs non-exploit completions. So you're adding noise to the importance weights without adding signal.
In variance terms: the log importance weights log P(x) - log Q(x) get a large negative shift from the late prefill tokens, but the variance of those weights across exploit completions also grows (or at least the gap between log E[w] and E[log w] grows), because you're in a regime where E[w] is dominated by rare completions where P(x)/Q(x) happens to be large.
So yes — it's fundamentally a variance problem. The late prefill tokens add KL without adding discriminative power, which inflates the importance weight variance and loosens the bound.
✻ Cooked for 1m 54s
❯ Your analysis is not quite correct because the KL is the KL of the pre-fill, not the KL of the actual event. So we have like... Yeah, we're not looking at P of X where X is like, you know, exploit and pre-fill and Q of X is like exploit and pre-fill. The KL that
we're computing and comparing is just the KL of the pre-fill. So we might need to re-drive it so we've got the right formula.
● Right, you're correct — I was sloppy. Let me set up the framework properly.
Setup:
- Model P generates sequences token-by-token
- We force prefill tokens p_{1:N}, then let the model continue
- The continuation distribution IS P(· | p_{1:N}) — same model, just conditioned
So the distribution Q over full sequences is:
Q(x) = δ(x_{1:N} = p_{1:N}) · P(x_{N+1:T} | p_{1:N})
The likelihood ratio for any completion x = (p_{1:N}, continuation):
P(x) / Q(x) = P(p_{1:N}) · P(cont | p_{1:N}) / [1 · P(cont | p_{1:N})] = P(p_{1:N})
This is constant across all continuations — the P/Q ratio doesn't depend on whether the continuation exploits or not. It's just the probability of generating the prefill under the base model.
Now, IS with restricted support (Q only generates sequences starting with our prefill):
P(exploit) ≥ P(p_{1:N}) · P(exploit | p_{1:N})
This is an inequality because there are other paths to exploitation that don't start with our prefill. In log space:
log(rate_0) ≥ log(rate_N) + log P(p_{1:N}) log(rate_0) ≥ log(rate_N) - surprisal(prefill)
So the relevant quantity is the surprisal of the prefill, -log P(p_{1:N}), not a KL divergence in the usual sense. And this is a lower bound on the true rate, not an equality. The gap comes from the restricted support of Q, not from Jensen.
Now the question: is the "KL" in the data files actually this surprisal, or is it something else (like per-token divergence between two models measured at the prefill positions)?
✻ Baked for 2m 17s
❯ P(exploit) ≥ P(p_{1:N}) · P(exploit | p_{1:N})
This is an inequality because there are other paths to exploitation that don't start with our prefill <- not if we take expectation over prefills
❯ P(exploit) ≥ P(p_{1:N}) · P(exploit | p_{1:N})
This is an inequality because there are other paths to exploitation that don't start with our prefill <- not if we take expectation over prefills