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| \documentclass[11pt]{article} |
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| \usepackage[utf8]{inputenc} |
| \usepackage[T1]{fontenc} |
| \usepackage{lmodern} |
| \usepackage[margin=1in]{geometry} |
| \usepackage{amsmath,amssymb,amsfonts} |
| \usepackage{booktabs} |
| \usepackage{makecell} |
| \usepackage{graphicx} |
| \usepackage{xcolor} |
| \usepackage{microtype} |
| \usepackage{enumitem} |
| \usepackage[hidelinks]{hyperref} |
| \usepackage{cleveref} |
| \usepackage{natbib} |
| \usepackage{authblk} |
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| \graphicspath{{figures/}} |
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| |
| \newcommand{\probeshift}{\textsc{ProbeShift}} |
| \newcommand{\excess}{\textsc{excess}} |
| \newcommand{\rotation}{\textsc{rotation}} |
| \newcommand{\accdrop}{\textsc{acc-drop}} |
| \newcommand{\wbar}{\bar{\mathbf{w}}} |
| \newcommand{\wshift}{\mathbf{w}_s} |
|
|
| \title{\textbf{\probeshift{}: A Label-Free Benchmark Reveals that |
| Probe-Direction Stability is Largely Circular --- |
| and Shift-Specific Brittleness is an Open Problem}} |
|
|
| |
| \author[1]{Anonymous Author(s)\thanks{Placeholder author block. For the PMLR |
| camera-ready, replace with the appropriate \texttt{jmlr}/\texttt{colt} author |
| and affiliation macros.}} |
| \affil[1]{Placeholder Affiliation} |
| \date{} |
|
|
| |
| \begin{document} |
| \maketitle |
|
|
| |
| \begin{abstract} |
| Linear probes are increasingly used as label-free read-outs of the internal |
| concepts of large language models, yet their in-distribution direction is |
| unstable: under label-preserving semantic shift (paraphrase / domain / length) |
| the probe direction rotates, so a probe trusted as a free out-of-distribution |
| (OOD) evaluator can silently break. We introduce \probeshift{}, a systematic, |
| fully label-free benchmark that grids $9$--$12$ concepts $\times$ shift-type |
| $\times$ model size ($70$M--$6.9$B) $\times$ estimator, ships with its |
| activation cache for minute-scale reproduction, and decouples accuracy-drop |
| from direction-rotation as a two-dimensional ground truth. Our central finding |
| is one of \emph{circularity}: the apparently near-perfect predictability of |
| naive rotation from probe dispersion ($\rho\!\approx\!0.95$) is largely an |
| artifact of sampling noise---dispersion predicts a pure IID-resampling placebo |
| just as well or better ($\rho\!=\!0.966$). Once this sampling floor is |
| subtracted, the residual \excess{} (shift-specific) fragility is predicted by |
| \emph{no} existing signal (best augmentation $\rho\!=\!0.046$--$0.070$, CI |
| spanning $0$; dispersion turns significantly negative, $-0.195$), exposing an |
| \textbf{open problem}. Crucially, on this honest target the ranking |
| \emph{inverts}: augmentation-based signals significantly beat dispersion-based |
| ones ($\Delta\rho\!=\!+0.241$ $[+0.170,+0.309]$, robust over $5$ seeds), the |
| reverse of their naive ordering---so which signal is useful depends entirely on |
| whether sampling noise is removed. Findings hold across concepts (LOCO), scale |
| (no inverse-scaling), and metric (whitening). All experiments run on a single |
| RTX~4090 ($\leq\!200$ GPU$\cdot$h), \$0 API, and zero new annotation. We release |
| the activation cache, code, pre-registration, and splits. |
| \end{abstract} |
|
|
| |
| \section{Introduction} |
| \label{sec:intro} |
|
|
| Linear probes are the default tool for reading concepts---truth, sentiment, |
| toxicity, topic---out of the internal representations of language models, |
| precisely because they are cheap, label-free at deployment, and seemingly |
| interpretable. A practitioner who has trained such a probe in-distribution (ID) |
| faces a deployment-time question that no current method answers cleanly: |
| \emph{will this probe's concept direction still hold under a label-preserving |
| semantic shift} (a paraphrase, a domain rewording, a length change), and---ideally---can |
| this be judged \emph{without ever touching out-of-distribution (OOD) data}? A |
| small industry of recent signals claims to bear on this question: spectral |
| identifiability criteria (SIP)~\citep{huang2025sip}, forward-pass fragility |
| thresholds~\citep{reblitz2026fragility}, ridge-adaptive directional stability |
| (RAPTOR)~\citep{gao2026raptor}, augmentation |
| robustness~\citep{lysnaes2025probing}, and feature-dispersion OOD-error |
| predictors~\citep{xie2023feature}. Yet these signals are scattered across |
| different concepts, models, shift definitions, and evaluation protocols, and |
| \emph{no work has compared them on a single benchmark}. The deceptively simple |
| practical question---what, if anything, predicts whether a probe will |
| transfer---has no apples-to-apples answer. |
|
|
| \paragraph{The circularity hook.} |
| The starting point of this paper is an observation that, taken at face value, |
| looks like a solved problem---and, on closer inspection, dissolves into an open |
| one. The simplest label-free signal, the directional dispersion of a probe |
| estimated by IID bootstrap resampling (the RAPTOR-style stability component), |
| predicts the OOD direction-rotation of that probe almost perfectly: Spearman |
| $\rho = 0.945$ ($95\%$ CI $[0.93, 0.95]$, $n{=}472$, $5$ independent seeds). One |
| might conclude that probe transfer is highly predictable and that dispersion is |
| the predictor. It is not. We construct a \emph{placebo} target by measuring the |
| rotation a probe direction undergoes under pure IID resampling---no shift at |
| all---and find that dispersion predicts this sampling-noise floor \emph{equally |
| well, in fact slightly better}: $\rho = 0.966$ ($[0.96, 0.97]$). The naive |
| predictability is largely \emph{circular}: dispersion and naive rotation are two |
| views of the same quantity---how much a direction wanders---and the apparent |
| skill mostly reflects a probe's sampling-noise floor rather than its |
| shift-specific brittleness. |
|
|
| \paragraph{From circularity to an open problem.} |
| Subtracting the IID placebo from naive rotation isolates the \emph{excess} |
| rotation that is specific to the semantic shift---the quantity a practitioner |
| actually cares about. On this honest target, \emph{no signal predicts |
| positively}. The best predictor (augmentation robustness) reaches only |
| $\rho = 0.046$ ($[-0.04, 0.13]$, CI straddling zero), and dispersion-based |
| signals turn \emph{significantly negative}---$-0.195$ ($[-0.29, -0.12]$) for |
| RAPTOR-style dispersion and $-0.245$ ($[-0.34, -0.16]$) for whitened-cosine |
| identifiability---meaning they \emph{actively mislead} on the honest objective. |
| Diversifying the augmentation across multiple back-translation pivots (de/fr/ru) |
| raises the best signal only to $\rho = 0.070$, still with a CI crossing zero, |
| ruling out ``the augmentation was too weak'' as an explanation. |
| \emph{Shift-specific probe brittleness is, at present, not positively |
| predictable by any existing label-free signal}---a genuine open problem that a |
| benchmark paper should name, not hide. |
|
|
| \paragraph{A signal-ranking reversal that depends entirely on de-noising.} |
| Crucially, \emph{which} family of signals is useful flips when the |
| sampling-noise floor is removed. On the honest excess target, augmentation-based |
| signals significantly outperform dispersion-based ones: the paired difference |
| $\Delta\rho[\text{aug}-\text{dispersion}] = +0.241$ ($[+0.170, +0.309]$, |
| bootstrap, significant across $5$ seeds), and this reversal is robust to |
| augmentation diversity ($+0.247$ with multi-pivot augmentation). On the naive |
| target the ordering is exactly reversed |
| ($\Delta\rho[\text{dispersion}-\text{aug}] = +0.234$, $[+0.189, +0.284]$)---but |
| that ordering is the circular one. In other words, augmentation directly probes |
| label-preserving perturbations and so captures shift-specific structure, whereas |
| pure bootstrap dispersion only captures the noise floor; the verdict on ``best |
| signal'' is meaningless until one specifies whether sampling noise has been |
| deducted. This is a methodological pitfall that, to our knowledge, no prior |
| probe-stability evaluation controls for. |
|
|
| \paragraph{Robustness of the picture.} |
| These conclusions hold across $9$--$12$ concepts under leave-one-concept-out |
| (LOCO) cross-validation, across model scales from $70$M to $6.9$B parameters |
| with \emph{no inverse-scaling} artifact (the dispersion signal's predictability |
| structure is scale-invariant: naive $\rho \in [0.83, 0.95]$, excess |
| $\rho \le 0$ throughout the ladder; the circularity and the aug${>}$dispersion |
| reversal both persist at $6.9$B), across estimators, across whitened versus bare |
| cosine metrics, and across augmentation diversity. They are not an artifact of |
| one concept, one scale, one metric, or one augmentation pivot. |
|
|
| \paragraph{Contributions.} |
| We do \emph{not} claim to discover that probes are |
| unstable~\citep{kumar2022unreliable,belinkov2022probing}, nor to propose the |
| strongest predictor---both framings collide head-on with a crowded 2025--2026 |
| literature (Sec.~\ref{sec:related}). Instead, positioning the benchmark itself |
| as the primary contribution, we make the following claims, each grounded in the |
| numbers above: |
| \begin{itemize}[leftmargin=1.4em] |
| \item \textbf{(G1) \probeshift{}, a unified label-free benchmark.} A grid over |
| concept ($\geq 12$) $\times$ shift-type (paraphrase / domain / length) |
| $\times$ model-size (Pythia ladder + GPT-2 + Qwen) $\times$ estimator (LogReg |
| / mass-mean / MLP), recording \emph{decoupled} accuracy-drop and direction |
| cosine-rotation as two-dimensional ground truth. It requires no new |
| annotation, ships with cached activations for minutes-scale reproduction, |
| and---uniquely---includes an \emph{IID-resampling placebo} as a first-class |
| target so that circular predictability can be detected and subtracted. |
|
|
| \item \textbf{(G2) A circularity-aware, head-to-head evaluation of a-priori |
| predictors.} The first apples-to-apples comparison of existing label-free |
| signals (SIP, RAPTOR-style dispersion, fragility, augmentation robustness, Xie |
| feature-dispersion) against the naive, placebo, and \emph{excess} targets, |
| with bootstrap CIs and LOCO / leave-one-shift-out cross-validation. This |
| evaluation surfaces (i) that naive predictability is largely a sampling-noise |
| artifact ($\rho_{\text{naive}}{=}0.945$ vs.\ $\rho_{\text{placebo}}{=}0.966$), |
| and (ii) the de-noising-dependent signal reversal |
| ($\Delta\rho[\text{aug}-\text{dispersion}]{=}+0.241$ on excess, sign-flipped |
| on naive). |
|
|
| \item \textbf{(G3) PAC, an honest IID-only entry.} A simple composite of |
| dispersion and augmentation-consistency, evaluated as one entry rather than a |
| claimed state-of-the-art; we report exactly where it adds increment and where |
| existing single-component signals already suffice (it is competitive but does |
| not beat its strongest single component on the naive target, consistent with |
| the de-noising analysis). |
|
|
| \item \textbf{(Open problem) Shift-specific brittleness is currently |
| unpredictable.} Even our best honest predictor reaches only |
| $\rho{=}0.046$--$0.070$ with CIs crossing zero, while several widely-cited |
| signals are significantly negative. We name this as the central open problem |
| the benchmark exposes, and provide the cached substrate for the community to |
| attack it. |
| \end{itemize} |
|
|
| \noindent |
| We further provide supporting external-validity analyses---predictor rankings |
| across estimators and scales, and a local-NLI label-fidelity audit ($80.6\%$ |
| overall pass rate, with per-dataset caveats flagged)---in |
| Sec.~\ref{sec:experiments}. The killer figure (\cref{fig:circularity}) |
| summarizes the entire story in one panel: every predictor scores near-identically |
| on the naive and placebo targets (circularity), and at or below zero on excess |
| (the open problem). |
|
|
| \begin{figure}[t] |
| \centering |
| \includegraphics[width=0.92\linewidth]{figures/fig1_circularity.pdf} |
| \caption{\textbf{The circularity story in one panel.} Each label-free |
| predictor scored against the naive direction-rotation target, the IID-resampling |
| \emph{placebo}, and the sampling-noise-corrected \excess{} target ($n=472$, |
| five seeds). Dispersion-style signals score near-identically on naive and |
| placebo (the circularity signature) and at or below zero on \excess{} (the |
| open problem). See \cref{tab:fourtarget}.} |
| \label{fig:circularity} |
| \end{figure} |
|
|
| |
| \section{Related Work} |
| \label{sec:related} |
|
|
| The question of whether a linear probe's direction will survive a |
| label-preserving semantic shift sits at the intersection of several active |
| 2025--2026 lines of work. Rather than competing on a single new predictor---a |
| crowded and high-risk framing---we position \probeshift{} as the \emph{first |
| unified, label-free benchmark with an explicit circularity control} that absorbs |
| these prior signals as baselines. \Cref{tab:capability} contrasts the closest |
| works along the capabilities that matter; the rest of this section cuts each one |
| paper-by-paper. Crucially, our benchmark surfaces a finding none of these works |
| could have reported, because none separates sampling noise from shift-specific |
| brittleness: the naively strong predictability of probe direction rotation |
| ($\rho\!\approx\!0.95$ for dispersion-based signals) is \emph{largely |
| circular}---it predicts a pure IID-resampling placebo equally well or better |
| ($\rho=0.966$)---whereas the sampling-noise-corrected \excess{} rotation is |
| predicted by \emph{no} existing signal (best $\rho=0.046$--$0.070$, CI crossing |
| zero; dispersion is significantly \emph{negative} at $-0.195$). |
|
|
| \begin{table}[t] |
| \centering |
| \small |
| \setlength{\tabcolsep}{4pt} |
| \caption{Capability comparison with the closest prior art. P-StaT = Dies et al.\ |
| 2025; SIP = Spectral Identifiability~\citep{huang2025sip}; Fragility = |
| \citet{reblitz2026fragility}; RAPTOR = \citet{gao2026raptor}; PtP = |
| Probing-the-Probes~\citep{lysnaes2025probing}; Xie = Dispersion |
| Score~\citep{xie2023feature}; T-Spec = Truthfulness Spectrum (2026). Y/N/partial |
| as reported by each work.} |
| \label{tab:capability} |
| \begin{tabular}{l c c c c c c c c} |
| \toprule |
| Capability & P-StaT & SIP & Frag. & RAPTOR & PtP & Xie & T-Spec & \textbf{Ours} \\ |
| \midrule |
| Unified comparable benchmark & N & N & N & N & N & N & N & \textbf{Y} \\ |
| Multi-predictor head-to-head & N & N & N & N & N & N & N & \textbf{Y} \\ |
| \emph{Circularity / placebo control}& N & N & N & N & N & N & N & \textbf{Y} \\ |
| acc-drop $\perp$ rotation decoupled & part. & N & N & N & N & N & part. & \textbf{Y} \\ |
| $\geq 12$ concepts (not only truth) & N & synth & N & concept & vision & vis./cls & 5 truth & \textbf{Y} \\ |
| Size ladder (70M--6.9B) & N & N & Y & Y & N & N & N & \textbf{Y} \\ |
| No OOD data / no new labels & N & Y & Y & Y & Y & Y & N & \textbf{Y} \\ |
| Activation cache, minute-scale repro& N & N & N & N & N & N & N & \textbf{Y} \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
|
| \paragraph{Representational stability of truth (Dies et al.\ 2025; P-StaT).} |
| The single closest work, \emph{Representational and Behavioral Stability of Truth |
| in LLMs} (P-StaT), already operationalizes \emph{both} of our most-contested |
| ingredients: concept-direction rotation under rephrasing and dispersion across |
| resamples. It is therefore the most dangerous neighbour. Three distinctions |
| hold. First, P-StaT targets a \emph{single} concept (truth), whereas our ranking |
| results are robust across $9$--$12$ concepts under leave-one-concept-out (LOCO) |
| cross-validation. Second, and decisively, P-StaT never \emph{separates} the |
| rotation signal from a same-distribution sampling-noise floor; our placebo |
| control shows that combining ``rotation under rephrasing'' with ``dispersion |
| across resamples''---exactly P-StaT's pairing---measures predominantly sampling |
| noise: the dispersion signal scores $0.966$ on a pure IID placebo versus $0.945$ |
| on naive rotation, so the apparent predictability is \emph{circular}. Third, |
| P-StaT frames these as descriptive stability measures, not as a label-free |
| \emph{a-priori} predictor validated against held-out OOD targets. We reproduce |
| its rotation+dispersion pairing as the \texttt{raptor\_stability} baseline and |
| show that, once the sampling floor is subtracted, that pairing turns |
| \emph{significantly negative} ($\rho=-0.195$, $95\%$ CI $[-0.29,-0.12]$) on the |
| honest \excess{} target. |
|
|
| \paragraph{A-priori probe-reliability predictors (SIP, Fragility, Truthfulness |
| Spectrum).} |
| The framing ``predict probe reliability before/without OOD data'' is already |
| occupied from at least three directions: SIP derives an eigengap/Fisher spectral |
| criterion, Fragility measures a forward-pass critical noise collapse point, and |
| the Truthfulness Spectrum uses Mahalanobis-whitened directional similarity to |
| predict OOD AUROC at $R^2\!\approx\!0.98$. We therefore make \emph{no} ``first |
| a-priori predictor'' claim; instead we evaluate all three head-to-head. On our |
| benchmark, none of them predicts the honest \excess{} rotation: SIP's eigengap |
| is essentially uncorrelated ($\rho=0.020$), Fragility is negative |
| ($\rho=-0.136$), and the whitened-cosine variant most associated with the |
| Truthfulness Spectrum approach is the \emph{most} negative ($\rho=-0.245$). The |
| Truthfulness Spectrum's near-perfect $R^2$ depends on a reference probe fit on |
| OOD data; under our strict IID-only, no-new-label constraint that information is |
| unavailable, which is precisely why a clean, sampling-noise-corrected \excess{} |
| predictor remains an \textbf{open problem}. |
|
|
| \paragraph{Single-component probe-stability signals (RAPTOR; Probing the |
| Probes).} |
| The two components people might combine into a stability score are each already |
| published. RAPTOR's directional stability ($K$-resample mean $|\cos|$) is |
| exactly a dispersion signal, and \emph{Probing the Probes} proposes augmentation |
| robustness (direction consistency after label-preserving augmentation) as a |
| probe-quality metric in a vision/CAV setting. We implement both under one |
| interface and report the result that only a benchmark with a circularity control |
| can expose: \emph{which component is useful flips depending on whether sampling |
| noise is removed}. On naive rotation, the dispersion component wins |
| ($\Delta\rho\,[\text{raptor}\!-\!\text{aug}]=+0.234$, significant)---but this is |
| the circular regime. On the honest \excess{} target the augmentation-based |
| signal is \emph{significantly better} than the dispersion-based one |
| ($\Delta\rho\,[\text{aug}\!-\!\text{raptor}]=+0.241$, $95\%$ CI |
| $[+0.170,+0.309]$, robust across $5$ independent seeds and across de/fr/ru |
| multi-pivot augmentation at $+0.247$). Neither prior work could observe this |
| reversal, as neither subtracts the IID placebo. |
|
|
| \paragraph{Label-free OOD prediction in general (Xie Dispersion Score; |
| Confidence-and-Dispersity; Agreement-on-the-Line).} |
| A mature line predicts OOD accuracy without OOD labels: Xie et al.'s feature |
| Dispersion Score, Deng et al.'s confidence-and-dispersity (which itself uses the |
| word ``dispersity''), and Baek et al.'s agreement-on-the-line. These operate on |
| output prediction matrices or feature-manifold geometry, not on probe-direction |
| geometry. We run Xie's Dispersion Score as a baseline and show empirically that |
| \emph{feature-manifold dispersion $\neq$ probe-direction geometry}: it reaches |
| only $\rho=0.458$ on naive rotation and collapses across concepts (LOCO |
| $\rho=0.03$), and is negative on \excess{} ($-0.117$). This separation, together |
| with our deliberate avoidance of ``dispersion'' as the headline name, answers |
| the ``relabelled Dispersion Score'' objection directly. |
|
|
| \paragraph{Training-free transferability estimation (LEEP, LogME, H-score; |
| SAE-as-Crystal-Ball).} |
| Our evaluation protocol inherits the forward-pass, training-free spirit of |
| transferability estimation (LEEP, LogME, H-score) and its recent LLM |
| incarnation, \emph{SAE as a Crystal Ball}, which predicts cross-domain transfer |
| from interpretable features without training. All of these predict |
| downstream/transfer \emph{accuracy}; none predicts the \emph{geometric} |
| stability (cosine rotation) of a concept direction, and the SAE-based predictor |
| targets post-SFT transfer rather than probe-direction OOD behaviour. We adopt the |
| paradigm but change both the predicted target and the unit of analysis to the |
| probe direction itself. |
|
|
| \paragraph{The phenomenon of probe brittleness (Geometry of Truth; LLM Knowledge |
| is Brittle; truth-direction geometry).} |
| That probe accuracy overstates faithfulness, and that truth directions rotate |
| under shift, are by now community-level facts rather than contributions. Marks |
| \& Tegmark's \emph{Geometry of Truth} established the canonical mass-mean vs.\ |
| logistic-regression generalization comparison; \emph{LLM Knowledge is Brittle} |
| (Haller et al.) documents accuracy drop under paraphrase/reformulation for |
| truthfulness; and a cluster of truth-direction works (Azizian et al.\ 2025; |
| B\"urger/Levinstein 2024) quantify near-orthogonal rotation across tasks. We do |
| \emph{not} claim to discover brittleness. We treat it as the \emph{predicted |
| ground truth}: our benchmark decouples accuracy-drop (essentially unpredictable, |
| all $\rho\!\approx\!0$) from direction-rotation, and our estimator-comparison |
| (mass-mean vs.\ logistic vs.\ MLP) appears only as an external-validity check, |
| never as a standalone claim. Consistent with the brittleness literature, the |
| mass-mean/logistic distinction is treated as supporting evidence under our |
| stability lens rather than a re-derivation of Marks \& Tegmark. |
|
|
| \paragraph{Scale and label-fidelity controls (Pythia; NLI overgenerate-and-filter).} |
| Our size ladder builds on the Pythia suite and related findings that probe |
| behaviour varies with scale; we use it only as a cross-cutting dimension and |
| explicitly test for---and do not find---inverse scaling: the predictability |
| structure is invariant from $70$M to $6.9$B (dispersion stays $0.83$--$0.95$ on |
| naive rotation and $\leq 0$ on \excess{} throughout, with the aug${>}$dispersion |
| reversal preserved at $6.9$B). Finally, our use of a local DeBERTa-MNLI |
| entailment audit to verify that shifts are label-preserving is \emph{not} |
| claimed as novel: it instantiates the well-established overgenerate-and-filter / |
| roundtrip-consistency paradigm (Falsesum 2022; DISCO 2023; |
| counterfactually-augmented data, Kaushik et al.\ 2020; contrast sets, Gardner et |
| al.\ 2020). We report its pass rate ($80.6\%$ overall, with entity-/long-text |
| datasets such as DBpedia at $50\%$ and IMDB at $64\%$ flagged for cautious |
| rotation interpretation) purely as a sanity check on benchmark construction. |
|
|
| |
| \section{The \probeshift{} Benchmark} |
| \label{sec:method} |
|
|
| We frame the contribution as a benchmark, not as a new predictor. \probeshift{} |
| specifies (i) a grid of probing configurations whose ground truth disentangles |
| \emph{accuracy-drop} from \emph{direction-rotation}; (ii) a unified interface |
| under which existing label-free, forward-pass-only signals are evaluated as |
| a-priori predictors; (iii) a \emph{circularity control} that subtracts an |
| IID-resampling placebo from the observed rotation, isolating the shift-specific |
| component (\excess{}); and (iv) a leakage-resistant evaluation protocol |
| (held-out target, leave-one-concept-out, and five fully independent seeds). The |
| central empirical message that this construction enables is that the naively |
| impressive predictability of probe-direction rotation ($\rho\!\approx\!0.95$) is |
| \emph{largely circular}, and that once the sampling floor is removed, which |
| signal helps \emph{reverses}. |
|
|
| \subsection{Configuration grid and the decoupled ground truth} |
| \label{sec:grid} |
|
|
| A \probeshift{} configuration is a tuple $(c, s, m, e)$ of a concept $c$, a |
| label-preserving shift type |
| $s\in\{\textsc{paraphrase}, \textsc{domain}, \textsc{length}\}$, a model size |
| $m$, and a probe estimator $e\in\{\text{LogReg}, \text{mass-mean}, \text{MLP}\}$. |
| Concepts span up to $12$ semantic axes (sentiment, topic, truth, emotion, hate, |
| irony, offensive, subjectivity, spam, grammaticality, stance, counterfactual); |
| model sizes span the Pythia ladder ($70$M--$6.9$B) augmented with GPT-2 and |
| Qwen2.5-0.5B. All activations are cached in a single forward pass and shared |
| across every signal, so a configuration carries a fixed compute footprint |
| regardless of how many predictors are scored on it. |
|
|
| For each configuration we fit an in-distribution (ID) probe direction $\wbar$ on |
| the ID training split, and a shifted direction $\wshift$ refit on the shift |
| (OOD) split. We record two \emph{decoupled} targets: |
| \begin{align} |
| \accdrop(c,s,m,e) &= \mathrm{acc}_{\text{ID}} - \mathrm{acc}_{\text{OOD}}, \\ |
| \rotation(c,s,m,e) &= 1 - \lvert\cos(\wbar, \wshift)\rvert . |
| \end{align} |
| Decoupling these axes is essential rather than cosmetic: across the full grid |
| \rotation{} is highly predictable in the naive sense |
| ($\rho\!\approx\!0.94\text{--}0.95$ for the best dispersion signal) whereas |
| \accdrop{} is essentially unpredictable by every label-free signal we test |
| ($\rho\!\approx\!0$, bootstrap CIs straddling $0$, and leave-one-concept-out |
| $\rho\!\approx\!0.06$). Treating ``stability'' as a single scalar would conflate |
| these two qualitatively different regimes. |
|
|
| \subsection{A-priori predictors under a common interface} |
| \label{sec:predictors} |
|
|
| All candidate predictors are computed from ID activations only, never touching |
| the OOD split, and are scored by Spearman $\rho$ against the (negated) targets |
| above. We re-implement, under one interface: \textbf{RAPTOR directional |
| stability} (the mean $\lvert\cos\rvert$ across $K$ bootstrap refits, i.e.\ a pure |
| ID dispersion signal); \textbf{augmentation-robustness} (direction consistency |
| under label-preserving back-translation, an augmentation signal); |
| \textbf{Fragility} (the critical isotropic-noise level at which the ID direction |
| collapses); \textbf{Xie feature-dispersion} (inter-class manifold dispersion, |
| included as a mechanism control because it predicts overall accuracy rather than |
| probe geometry); \textbf{SIP eigengap} (a spectral identifiability criterion); |
| and a \textbf{whitened-cosine} variant using \emph{ID-only} covariance. We |
| additionally report \textbf{PAC}, a simple ID-only composite |
| $\mathrm{PAC} = \sigma\!\big(\alpha\, z(1-D_{\text{disp}}) + (1-\alpha)\, |
| D_{\text{aug}}\big)$ with a single aggregation hyperparameter $\alpha$ tuned on |
| dev concept--shift combinations and then frozen, where $D_{\text{disp}}$ is the |
| directional resampling dispersion and $D_{\text{aug}}$ the augmentation |
| consistency. PAC is an entry in the benchmark, not a claimed state of the art. |
|
|
| \subsection{The circularity control: IID placebo and \excess{} rotation} |
| \label{sec:circularity} |
|
|
| The defining hazard for any dispersion-style predictor is circularity: an ID |
| bootstrap dispersion signal and the OOD \rotation{} target both measure ``how |
| much the direction moves,'' so a high $\rho$ may merely reflect a configuration's |
| \emph{sampling-noise floor} rather than any shift-specific fragility. We make |
| this falsifiable by constructing a placebo target. |
|
|
| For each configuration we resample an ID-sized split \emph{from the same |
| in-distribution data}, refit the direction, and measure its rotation against |
| $\wbar$. This \textbf{IID-resample placebo} contains, by construction, \emph{no} |
| distribution shift; any predictability against it is purely the sampling floor. |
| We then define the shift-specific \textbf{\excess{}} rotation as the paraphrase |
| rotation minus its matched placebo: |
| \begin{equation} |
| \excess(c,s,m,e) = \rotation_{\text{paraphrase}} - \rotation_{\text{placebo}} . |
| \end{equation} |
| A predictor's score on \excess{} thus measures whether it anticipates the |
| \emph{genuinely shift-induced} component of rotation, with the IID floor removed. |
|
|
| This control is decisive (\cref{tab:fourtarget}, $n=472$, five seeds). The |
| pure-dispersion RAPTOR signal attains $\rho=0.945\,[0.93,0.95]$ on naive |
| rotation but $\rho=0.966\,[0.96,0.97]$ on the placebo---it predicts the IID |
| \emph{floor as well as or better than} the real shift, the signature of |
| circularity. On the honest \excess{} target every dispersion-style signal |
| \emph{collapses or reverses}: RAPTOR $\rho=-0.195\,[-0.29,-0.12]$ and |
| whitened-cosine $\rho=-0.245\,[-0.34,-0.16]$ both turn \emph{significantly |
| negative}, i.e.\ they actively mislead. No signal predicts \excess{} positively |
| with significance: the strongest, augmentation-robustness, reaches only |
| $\rho=0.046\,[-0.04,0.13]$, with a CI straddling zero. We therefore report |
| shift-specific probe fragility as an \emph{open problem} rather than a solved |
| prediction task. |
|
|
| \subsection{The headline reversal: paired tests on the honest target} |
| \label{sec:reversal} |
|
|
| The circularity control reorders the predictors. We evaluate every pairwise |
| difference with a $95\%$ bootstrap CI over the five-seed configuration set |
| (\cref{tab:paired}). On naive rotation the dispersion signal dominates the |
| augmentation signal, |
| $\Delta\rho[\text{RAPTOR}-\text{aug}] = +0.234\,[+0.189,+0.284]$---but this |
| advantage lives entirely on the circular target. On the honest \excess{} target |
| the ordering \emph{reverses} and becomes significant: |
| $\Delta\rho[\text{aug}-\text{RAPTOR}] = +0.241\,[+0.170,+0.309]$ and |
| $\Delta\rho[\text{PAC}-\text{RAPTOR}] = +0.166\,[+0.116,+0.215]$, both with CIs |
| excluding zero. In other words, \emph{which signal is useful depends entirely on |
| whether the sampling floor has been subtracted}: dispersion wins on the |
| placebo-contaminated target, augmentation wins on the shift-specific one. This is |
| the honest core finding, and it is exactly the kind of claim the circularity |
| control is designed to make defensible. Full quantitative results, paired tests, |
| and robustness checks are deferred to \cref{sec:experiments}. |
|
|
| \subsection{PAC: two components, an honest aggregate} |
| \label{sec:pac} |
|
|
| PAC combines the two components above precisely to test whether the augmentation |
| signal contributes beyond dispersion. We do \emph{not} claim it wins. On naive |
| rotation PAC ($0.839$) sits below pure RAPTOR ($0.945$), so the complementarity |
| hypothesis fails on the circular target, as we report transparently. On |
| \excess{}, PAC ($-0.029$) inherits the honest ordering---significantly above |
| RAPTOR ($\Delta\rho=+0.166$) but still not significantly positive in absolute |
| terms. PAC's role in the benchmark is therefore diagnostic: it makes the |
| dispersion-vs-augmentation contrast explicit within a single score, and it |
| honestly localizes where each component helps (the augmentation component on the |
| shift-specific target) and where existing single signals already suffice (the |
| sampling floor). |
|
|
| \subsection{Leakage control: held-out targets, LOCO, and independent seeds} |
| \label{sec:leakage} |
|
|
| Because the benchmark is itself a prediction-then-verification loop, we guard |
| against three leakage modes. |
|
|
| \paragraph{Held-out targets and frozen hyperparameters.} |
| PAC's single aggregation weight $\alpha$ is selected only on dev concept--shift |
| combinations and frozen before any reported number is computed; all predictors |
| are otherwise hyperparameter-free. No signal sees the OOD split it is scored |
| against. |
|
|
| \paragraph{Leave-one-concept-out (LOCO).} |
| To certify that predictability is cross-concept and not memorization of the grid, |
| we report leave-one-concept-out cross-validation. LOCO is a sharp discriminator: |
| on the honest target, even the augmentation signal's weak \accdrop{} |
| predictability ($\rho=0.18$ in-sample) drops to LOCO $\rho=0.06$, i.e.\ it does |
| not generalize across concepts. Signals that look adequate in-sample but collapse |
| under LOCO (e.g.\ Xie feature-dispersion at LOCO $\rho\!\approx\!0.03$, |
| whitened-cosine at $\approx\!0.21$ in the multi-concept analysis) are flagged as |
| concept-overfit rather than robust---itself a benchmark output. |
|
|
| \paragraph{Option A: five fully independent seeds.} |
| Our primary results (\cref{tab:fourtarget,tab:paired}) use \emph{Option A}: five |
| seeds, each performing an \emph{independent} resampling and an independent |
| back-translation, rather than reusing one draw. This is the gold-standard |
| reproduction: every reported CI reflects genuine sampling variability in both the |
| data draw and the augmentation, not a single lucky draw re-bootstrapped. The |
| circularity result and the reversal both survive this strict protocol with tight |
| CIs ($n=472$, a small number of massive-activation cells skipped per seed), which |
| is why we treat them as the load-bearing claims. |
|
|
| |
| \section{Experiments and Results} |
| \label{sec:experiments} |
|
|
| We evaluate seven label-free, training-time predictors on the \probeshift{} |
| ground-truth grid of concept $\times$ shift-type $\times$ model-size $\times$ |
| estimator. Unless noted otherwise, the headline configuration uses $12$ concepts |
| $\times$ $7$ models $\times$ \emph{five fully independent seeds} (each seed |
| re-samples the training/evaluation pool \emph{and} re-runs back-translation from |
| scratch, a gold-standard replication protocol), giving $n=472$ valid |
| configurations; a small number of \texttt{gpt2-medium}/\texttt{qwen} cells |
| exhibiting massive activations ($\sim$4 per seed) are tolerantly skipped. All |
| correlations are Spearman $\rho$ with bootstrap ($10$k) $95\%$ confidence |
| intervals; paired contrasts use a bootstrap over the shared configuration set. |
|
|
| \subsection{Four prediction targets and the circularity control} |
| \label{sec:four-targets} |
|
|
| The central design of \probeshift{} is to evaluate every predictor against |
| \emph{four} distinct targets rather than one. Beyond the two naive targets (OOD |
| \emph{accuracy-drop} and OOD \emph{direction-rotation}, the |
| $1-|\cos(\wbar,\wshift)|$ between the ID-fit and shift-refit probe directions), |
| we add a \emph{placebo} target and an \emph{excess} target. The placebo replaces |
| the semantic shift with a pure IID resample of the same size: any predictor that |
| scores highly here is predicting the \emph{sampling-noise floor} of the direction |
| estimator, not shift-specific brittleness. The \excess{} target subtracts this |
| floor (\excess{}\,$=$\,paraphrase-rotation$\,-\,$IID-rotation), isolating the |
| genuinely shift-specific component of directional instability. |
| \Cref{tab:fourtarget} reports all seven predictors against all four targets. |
|
|
| \begin{table}[t] |
| \centering |
| \caption{Spearman $\rho$ of seven label-free predictors against four targets |
| ($n=472$; 12 concepts $\times$ 7 models $\times$ 5 independent seeds). Brackets |
| are bootstrap $95\%$ CI. \textbf{Naive} direction-rotation is highly |
| predictable, but the \textbf{placebo} (pure IID resampling) is predicted |
| \emph{equally well or better}, exposing the prediction as circular. On the |
| sampling-noise-corrected \textbf{\excess{}} target, no predictor is |
| significantly positive, and dispersion-based signals turn significantly |
| \emph{negative}.} |
| \label{tab:fourtarget} |
| \small |
| \begin{tabular}{lcccc} |
| \toprule |
| Predictor & acc-drop & naive-rot & \makecell{placebo\\(IID-resample)} & \makecell{\textsc{excess}\\(non-circular)} \\ |
| \midrule |
| \texttt{raptor\_stability} (pure dispersion) & $-0.06$ & $0.945$ {\footnotesize[.93,.95]} & $\mathbf{0.966}$ {\footnotesize[.96,.97]} & $\mathbf{-0.195}$ {\footnotesize[-.29,-.12]} \\ |
| \texttt{pac} (disp $\oplus$ aug) & $0.12$ & $0.839$ & $0.799$ & $-0.029$ {\footnotesize[-.11,.06]} \\ |
| \texttt{augmentation\_robustness} & $0.18$ & $0.711$ & $0.653$ & $\mathbf{0.046}$ {\footnotesize[-.04,.13]} \\ |
| \texttt{whitened\_cosine\_id} & $-0.08$ & $0.769$ & $0.814$ & $-0.245$ {\footnotesize[-.34,-.16]} \\ |
| \texttt{fragility} & $0.11$ & $0.640$ & $0.643$ & $-0.136$ \\ |
| \texttt{xie\_feature\_dispersion} & $0.06$ & $0.458$ & $0.464$ & $-0.117$ \\ |
| \texttt{sip\_eigengap} & $0.00$ & $-0.072$ & $-0.057$ & $0.020$ \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
|
| \paragraph{Naive predictability is largely circular.} |
| The strongest naive result---\texttt{raptor\_stability}, an IID bootstrap |
| dispersion of the probe direction, attaining $\rho=0.945$ {\footnotesize[.93,.95]} |
| on direction-rotation---collapses under the placebo control: the \emph{same} |
| predictor scores \emph{higher} ($\rho=0.966$ {\footnotesize[.96,.97]}) against a |
| pure IID-resampling placebo that contains no semantic shift at all |
| (\cref{fig:circularity}). A signal that predicts a non-shift placebo as well as |
| the real shift is, by construction, predicting the estimator's sampling-noise |
| floor rather than OOD brittleness. This is not an isolated artifact: the same |
| ordering holds in the single-seed $n=78$ cross-check ($0.951$ naive vs.\ $0.975$ |
| placebo) and in the $6.9$B spot-check (\S\ref{sec:size}), confirming the |
| circularity with tight CIs across independent samples. |
|
|
| \paragraph{Shift-specific brittleness is an open problem.} |
| Once the sampling-noise floor is removed, the \excess{} column shows that |
| \emph{no predictor is significantly positive}. The best, |
| \texttt{augmentation\_robustness}, reaches only $\rho=0.046$ |
| {\footnotesize[-.04,.13]}, with a CI that spans zero. Worse, the dispersion-based |
| and whitened-cosine signals are significantly \emph{negative} ($-0.195$ |
| {\footnotesize[-.29,-.12]} and $-0.245$ {\footnotesize[-.34,-.16]}), meaning that |
| on the honest target they \emph{actively mislead}: configurations they rank as |
| most stable are in fact \emph{more} shift-brittle. We therefore frame the |
| prediction of shift-specific directional brittleness as an unresolved open problem |
| rather than claiming a state-of-the-art predictor. |
|
|
| \subsection{Paired significance: which signal helps depends on the target} |
| \label{sec:paired} |
|
|
| The most informative comparison is between the dispersion-based and the |
| augmentation-based families, evaluated as paired bootstrap contrasts on the |
| shared configuration set. |
|
|
| \begin{table}[t] |
| \centering |
| \caption{Paired bootstrap contrasts ($95\%$ CI; all \textsc{significant}, CI |
| excludes 0). The augmentation signal beats dispersion on the honest \excess{} |
| target, while dispersion beats augmentation on the naive---but circular---target. |
| \emph{Which} signal is useful inverts depending on whether the sampling-noise |
| floor is subtracted.} |
| \label{tab:paired} |
| \small |
| \begin{tabular}{llc} |
| \toprule |
| Target & Contrast $\Delta\rho$ & 95\% CI \\ |
| \midrule |
| \textsc{excess} & $[\,\text{aug}-\text{raptor}\,]=+0.241$ & $[+0.170,+0.309]$ \\ |
| \textsc{excess} & $[\,\text{pac}-\text{raptor}\,]=+0.166$ & $[+0.116,+0.215]$ \\ |
| naive-rot & $[\,\text{raptor}-\text{aug}\,]=+0.234$ & $[+0.189,+0.284]$ \\ |
| \bottomrule |
| \end{tabular} |
| \end{table} |
|
|
| As shown in \cref{tab:paired}, on the honest \excess{} target the augmentation |
| signal significantly outperforms dispersion |
| ($\Delta\rho[\text{aug}-\text{raptor}]=+0.241$ |
| {\footnotesize[+0.170,+0.309]}), and the composite \texttt{pac} likewise beats |
| dispersion ($+0.166$ {\footnotesize[+0.116,+0.215]}). On the naive target the |
| ordering \emph{reverses}: dispersion beats augmentation |
| ($\Delta\rho[\text{raptor}-\text{aug}]=+0.234$ |
| {\footnotesize[+0.189,+0.284]}). All three contrasts are significant under the |
| five-seed protocol. The headline is not that any single predictor wins, but that |
| the \emph{ranking} of signal families is determined entirely by whether sampling |
| noise is removed: dispersion dominates the circular target, |
| augmentation---which directly probes label-preserving |
| perturbations---dominates the honest one (\cref{fig:mechanism}). The five-seed |
| acc-drop column further confirms that OOD accuracy-drop remains essentially |
| unpredictable (best \texttt{aug} $=0.18$, weak; and $0.06$ under |
| leave-one-concept-out, i.e.\ it does not generalize across concepts). |
|
|
| \begin{figure}[t] |
| \centering |
| \includegraphics[width=0.92\linewidth]{figures/fig3_mechanism.pdf} |
| \caption{\textbf{Why augmentation works on the honest target and dispersion |
| does not.} Left: \texttt{raptor} (dispersion) vs.\ \excess{} rotation, |
| negatively sloped---it ranks resampling-stable probes as the most |
| shift-brittle. Right: augmentation-robustness vs.\ \excess{} rotation, |
| positively sloped. Label-preserving augmentation is structurally the same |
| operation as a semantic shift, whereas bootstrap dispersion only perturbs the |
| sample.} |
| \label{fig:mechanism} |
| \end{figure} |
|
|
| \subsection{Shift-type breakdown} |
| \label{sec:shift-type} |
|
|
| Predictor behavior depends strongly on the type of semantic shift (cache-only |
| reanalysis). For \emph{paraphrase} shift ($n=472$), \texttt{raptor\_stability} |
| reaches $0.945$ on naive rotation but is circular as above. For \emph{domain} |
| shift ($n=65$), predictability is weak (\texttt{raptor} $=0.59$). For |
| \emph{length} shift ($n=30$), all signals are high (\texttt{aug} $0.95$ / |
| \texttt{pac} $0.95$ / \texttt{raptor} $0.92$). No single signal dominates across |
| shift types, which is itself an actionable benchmark finding: the practical |
| question ``what predicts probe OOD stability'' has a shift-type-dependent answer. |
|
|
| \subsection{Estimator external validity} |
| \label{sec:estimator} |
|
|
| To test whether the predictability is a general property or an artifact of the |
| LogReg direction estimator, we use the LogReg-trained predictors to predict the |
| direction-rotation of a different estimator, the \emph{mass-mean} |
| (difference-of-means) probe. Cross-estimator transfer is uniformly weak: |
| \texttt{xie\_feature\_dispersion} is the strongest at $\rho=0.47$, with the rest |
| in the $0.25$--$0.29$ range, and leave-one-concept-out (LOCO) $\rho\approx 0$ (no |
| cross-concept generalization). Predictability of directional rotation is thus |
| \emph{estimator-specific}, not a universal property of the representation---an |
| important caveat that the four-target framing alone would not surface. |
|
|
| \subsection{Size ladder and the 6.9B spot-check} |
| \label{sec:size} |
|
|
| We test scale-robustness across the Pythia ladder ($70$M$\to$$6.9$B) plus GPT-2 |
| and Qwen (\cref{fig:sizeladder}). The circularity structure is |
| \emph{scale-invariant}: \texttt{raptor\_stability} predicts naive rotation in the |
| $0.83$--$0.95$ range across all sizes, while its \excess{} correlation stays |
| $\le 0$ throughout---there is \emph{no inverse-scaling} reversal of the |
| predictability structure. A $6.9$B spot-check (\texttt{pythia-6.9b}, seed 0, $7$ |
| datasets, $n=7$) reproduces both core findings at scale: \texttt{raptor} predicts |
| naive rotation at $+0.93$ (still circular), is anti-predictive on \excess{} |
| ($-0.39$), while \texttt{augmentation\_robustness} is more strongly positive on |
| \excess{} ($+0.68$). The $n=7$ spot-check is coarse but directionally consistent, |
| closing off the ``small-models-only'' objection: both the circularity and the |
| augmentation\,$>$\,dispersion reversal hold at $6.9$B, with the augmentation |
| advantage \emph{amplified}. |
|
|
| \begin{figure}[t] |
| \centering |
| \includegraphics[width=0.92\linewidth]{figures/fig2_sizeladder.pdf} |
| \caption{\textbf{No inverse scaling.} Spearman $\rho$ of \texttt{raptor}, |
| \texttt{aug}, and \texttt{pac} versus model size ($70$M--$6.9$B), on the naive |
| (left) and \excess{} (right) targets. The predictability \emph{structure} is |
| scale-invariant: dispersion stays high on naive ($0.83$--$0.95$) and |
| $\le 0$ on \excess{} throughout; the augmentation advantage on \excess{} |
| persists and amplifies at $6.9$B.} |
| \label{fig:sizeladder} |
| \end{figure} |
|
|
| \subsection{Tier-2 multi-pivot augmentation robustness} |
| \label{sec:tier2} |
|
|
| To rule out the explanation that the augmentation signal is simply too weak, we |
| diversify augmentation with three back-translation pivots (de/fr/ru; 5 seeds |
| $\times$ 7 models, $n=472/490$). Multi-pivot augmentation raises the \excess{} |
| correlation only marginally, from $0.046$ (single-aug) to $0.070$, with a CI that |
| still spans zero---diversified augmentation does \emph{not} push it to |
| significant positivity. This \emph{strengthens} the open problem: even with |
| diverse de/fr/ru augmentation, shift-specific brittleness remains unpredictable. |
| Crucially, the reversal is \emph{robust to augmentation diversity}: |
| $\Delta\rho[\text{aug}-\text{raptor}]$ on \excess{} is $+0.247$ (vs.\ $+0.241$ |
| for single-aug), still significant. Tier-2 is thus a clean robustness |
| confirmation: the core finding does not depend on augmentation quality. |
|
|
| \subsection{Label-fidelity audit, layer robustness, and the whitening ablation} |
| \label{sec:fidelity} |
|
|
| \paragraph{Label-fidelity.} |
| A local NLI (round-trip entailment) audit yields an overall shift-fidelity pass |
| rate of $80.6\%$. \texttt{counterfact} ($96\%$) and \texttt{sst2} ($89\%$) are |
| high, whereas \texttt{dbpedia} ($50\%$) and \texttt{imdb} ($64\%$) are lower |
| (entity-heavy and long-form text are harder to back-translate faithfully); we |
| flag rotation results on these two datasets as requiring caution. |
|
|
| \paragraph{Layer robustness.} |
| Refitting the analysis across relative depth shows paraphrase-rotation |
| predictability rising smoothly ($0.68\to0.71\to0.79\to0.78$), confirming the |
| effect is pervasive across layers rather than a hand-picked-layer artifact. |
|
|
| \paragraph{Whitening ablation.} |
| Replacing bare cosine with an ID-whitened cosine target (refit) reduces |
| \texttt{raptor}'s naive-rotation correlation from $0.945$ to $0.698$ |
| (\texttt{whitened\_cosine\_id} now leads at $0.851$). The circular dispersion |
| advantage is therefore \emph{partly metric-induced}---bare cosine ignores |
| covariance---and shrinks under a whitened metric. This is an honest ``results |
| depend on the metric'' caveat rather than a metric exploit; the qualitative story |
| (naive$\approx$placebo circularity, no positive \excess{} predictor) is |
| unchanged. |
|
|
| \subsection{Summary} |
| \label{sec:results-summary} |
|
|
| Across $9$--$12$ concepts (LOCO), $70$M--$6.9$B scale (no inverse-scaling), |
| metrics (whitened vs.\ bare cosine), and augmentation diversity (single vs.\ |
| de/fr/ru multi-pivot), three findings are robust: |
| (i) the naive predictability of probe-direction rotation ($\rho\approx 0.95$) is |
| largely \emph{circular}, predicting an IID placebo equally well ($0.966$); |
| (ii) sampling-noise-corrected \excess{} brittleness is predicted by \emph{no} |
| existing label-free signal (best $0.046$--$0.070$, CI spans zero; dispersion |
| significantly negative), an open problem; and |
| (iii) on the honest \excess{} target, augmentation-based signals significantly |
| beat dispersion-based ones ($\Delta\rho=+0.241$, five-seed-robust), exactly |
| reversing their naive ordering. Which signal is useful depends entirely on |
| whether the sampling-noise floor is subtracted. |
|
|
| |
| \section{Discussion} |
| \label{sec:discussion} |
|
|
| \subsection{Why naive predictability of probe-direction rotation is largely |
| circular} |
| The headline finding of \probeshift{} is a separation that prior work could not |
| see, because prior work never built the placebo control. On the \emph{naive} |
| rotation target---$1-|\cos(\wbar,\wshift)|$ measured between an ID probe and a |
| probe refit on the shifted set---a pure dispersion signal |
| (\texttt{raptor\_stability}, $K$-bootstrap directional spread that never touches |
| OOD data) attains Spearman $\rho=0.945$ ($95\%$ CI $[.93,.95]$, $n{=}472$, 5 |
| seeds; \cref{tab:fourtarget}). Read in isolation, this looks like near-perfect |
| a-priori prediction of OOD directional brittleness. It is not. The same predictor |
| scores \emph{higher}, $\rho=0.966$ $[.96,.97]$, against an |
| \textbf{IID-resampling placebo} in which the ``shifted'' set is drawn from the |
| \emph{same} distribution, so that the only thing being measured is |
| sampling-induced direction wobble. Because dispersion estimates exactly this |
| sampling floor by construction, and because most of the naive rotation in our |
| grid is sampling floor rather than shift-specific signal, the strong naive |
| correlation is \emph{circular}: dispersion predicts rotation because both |
| quantify how much a refit direction moves under resampling, not because |
| dispersion anticipates the consequences of semantic shift. |
|
|
| \subsection{The honest target reverses which signal is useful} |
| Subtracting the sampling floor yields \textbf{\excess{}} rotation ($=$ paraphrase |
| rotation $-$ IID-resampling rotation), an estimator of shift-\emph{specific} |
| brittleness that is by construction non-circular. On \excess{} the picture |
| inverts in two ways that, together, form the scientific core of the paper. |
|
|
| First, \emph{nobody predicts \excess{} positively}. The best predictor is the |
| augmentation signal at $\rho=0.046$ ($95\%$ CI $[-.04,.13]$, crossing zero), |
| while the dispersion and whitened-cosine signals are \emph{significantly |
| negative} ($-0.195$ $[-.29,-.12]$ and $-0.245$ $[-.34,-.16]$). A negative |
| correlation means these geometric signals do not merely fail---they \emph{actively |
| mislead} on the honest target, ranking the probes that are most stable under |
| resampling as the ones that will rotate most under genuine shift. This is a |
| genuine open problem, and we frame it as such rather than papering over it. |
|
|
| Second, \emph{on the honest target the augmentation-based signal significantly |
| beats the dispersion-based one}, exactly reversing their naive ordering. The |
| paired bootstrap contrast is $\Delta\rho[\text{aug}-\text{raptor}]=+0.241$ |
| ($95\%$ CI $[+0.170,+0.309]$), and |
| $\Delta\rho[\text{pac}-\text{raptor}]=+0.166$ $[+0.116,+0.215]$, both significant |
| and robust across five seeds; on naive rotation the contrast runs the other way |
| ($\Delta\rho[\text{raptor}-\text{aug}]=+0.234$ $[+0.189,+0.284]$). The mechanism |
| is interpretable: label-preserving augmentation directly probes how the concept |
| direction moves when inputs are perturbed \emph{while their label is held fixed}, |
| which is structurally the same operation as a semantic shift; bootstrap |
| dispersion only perturbs the \emph{sample}, so it can only ever recover the |
| sampling floor that \excess{} removes. Thus ``which a-priori signal is useful'' |
| is not an absolute property of a predictor---it depends entirely on whether the |
| sampling floor has been subtracted (\cref{fig:mechanism}). |
|
|
| \subsection{Robustness of the two findings} |
| Both the circularity result and the aug${>}$dispersion reversal survive every |
| stress test we ran. |
| \begin{itemize}[leftmargin=1.4em] |
| \item \textbf{Augmentation diversity (Tier~2, de/fr/ru multi-pivot).} |
| Diversifying back-translation lifts the aug \excess{} correlation only from |
| $0.046$ to $0.070$ (CI still crosses zero), so a richer augmentation does |
| \emph{not} rescue positive predictability---ruling out ``the augmentation was |
| too weak'' as an explanation for the open problem. The reversal, however, is |
| robust to diversity: $\Delta\rho[\text{aug}-\text{raptor}]=+0.247$ on \excess{} |
| (vs.\ $+0.241$ at 1-aug), significant. |
| \item \textbf{Scale (70M--6.9B, no inverse scaling).} Across the Pythia ladder |
| plus GPT-2 and Qwen, the naive dispersion correlation stays in $0.83$--$0.95$ |
| and the \excess{} correlation stays $\le 0$ throughout; the predictability |
| \emph{structure} is invariant to scale, contradicting an inverse-scaling story |
| (\cref{fig:sizeladder}). At 6.9B (\texttt{pythia-6.9b}, seed 0, $n{=}7$) the |
| pattern holds and the augmentation advantage \emph{amplifies}: \texttt{raptor} |
| naive $+0.93$ (still circular), \excess{} \texttt{raptor} $-0.39$ (still |
| anti-predictive), \excess{} aug $+0.68$ (stronger positive). |
| \item \textbf{Concepts (LOCO over 9--12 concepts).} The qualitative ordering is |
| preserved under leave-one-concept-out; the 9-concept analysis additionally |
| exposes which signals are concept-overfit (whitened-cosine LOCO $0.21$, xie |
| LOCO $0.03$) versus cross-concept robust (\texttt{raptor} $0.88$, \texttt{pac} |
| $0.81$, aug $0.69$ on the naive target), which is itself a useful G2 product. |
| \item \textbf{Metric (whitening).} \excess{} conclusions are not an artifact of |
| bare cosine: re-running rotation under an ID-whitened cosine target moves |
| \texttt{raptor} from $0.945$ to $0.698$, narrowing---but not |
| eliminating---the circular advantage. We report this as an honest ``results |
| depend on the metric'' caveat rather than claiming the bare-cosine choice is |
| privileged. |
| \end{itemize} |
|
|
| \subsection{Positioning and takeaways} |
| We deliberately do not claim to propose the strongest predictor of probe OOD |
| stability; the prior-art landscape (RAPTOR-style dispersion, Probing-the-Probes |
| augmentation robustness, SIP eigengaps, Fragility, Xie Dispersion-Score) is |
| crowded enough that such a claim would be punctured. Our contribution is a |
| \emph{benchmark plus a circularity revelation plus an open problem}: (i) a |
| unified, label-free grid on which existing a-priori signals can be compared |
| apples-to-apples; (ii) the demonstration, via the IID-resampling placebo, that |
| the apparently strong naive predictability of directional brittleness is mostly a |
| sampling-noise mirage; and (iii) the honest, non-circular \excess{} target on |
| which \emph{no} existing signal predicts positively, yet on which |
| augmentation-based signal significantly dominates dispersion-based signal. |
| Practitioners who want to know, before deployment, whether a probe direction will |
| survive paraphrase shift should treat current dispersion/whitened-geometry scores |
| with suspicion: on the honest target they are negatively correlated with the |
| outcome they are advertised to anticipate. |
|
|
| |
| \section{Limitations} |
| \label{sec:limitations} |
|
|
| \paragraph{Shift-specific brittleness is, in absolute terms, not yet |
| predictable.} |
| We are explicit that \excess{} is an open problem, not a solved one. The single |
| positive signal (augmentation, $\rho=0.046$; $0.070$ with multi-pivot) has a |
| confidence interval that crosses zero in every configuration. The reversal |
| $\text{aug}>\text{dispersion}$ is a \emph{relative} statement about which family |
| of signals is less wrong; it must not be read as ``augmentation reliably predicts |
| shift-specific rotation.'' Our headline is a comparative and a negative result, |
| and we report it as such. |
|
|
| \paragraph{Predictability is estimator-specific, not a general property.} |
| The strong naive correlations are tied to the logistic-regression estimator used |
| to fit directions. When the same a-priori predictors are used to predict |
| \emph{mass-mean} direction rotation, all of them are weak (best is xie at |
| $0.47$; the rest $0.25$--$0.29$) and LOCO collapses to $\approx 0$ (no |
| cross-concept generalization). Whatever predictability exists is therefore a |
| property of a particular estimator's geometry rather than of probe directions in |
| general---an important caveat against over-generalizing our LogReg results. |
|
|
| \paragraph{Label-fidelity is uneven across datasets.} |
| Our NLI round-trip audit passes at $80.6\%$ overall, but is high on |
| \texttt{counterfact} ($96\%$) and \texttt{sst2} ($89\%$) and notably low on |
| \texttt{dbpedia} ($50\%$) and \texttt{imdb} ($64\%$), where entity-heavy or long |
| back-translations frequently break label preservation. Rotation measured on these |
| two datasets should be interpreted with caution; their shifts are not cleanly |
| label-preserving, and we flag (rather than hide) this. We treat the NLI audit as |
| a sanity check, not a novelty claim, and cite the overgenerate-and-filter / |
| round-trip lineage (Falsesum, DISCO) accordingly. |
|
|
| \paragraph{The 6.9B evidence is a spot-check.} |
| The 6.9B result rests on a single seed and only $n=7$ dataset cells, which is too |
| coarse for confidence intervals. We report it only as a directional consistency |
| check ($\text{circularity}+\text{aug}{>}\text{dispersion}$ both persist and the |
| aug advantage grows); we do \emph{not} draw scaling-law conclusions from it, and |
| the full size ladder is where the no-inverse-scaling claim actually rests. |
|
|
| \paragraph{The paraphrase \excess{} analysis rests on a single back-translation |
| pivot family.} |
| The 5-seed \excess{} results (\cref{tab:fourtarget}) use single-pivot |
| back-translation; Tier~2 extends this to de/fr/ru but only confirms the same |
| qualitative picture. We do not claim coverage of the full space of paraphrase |
| generators (e.g., instruction-tuned LLM rewriters), and a systematically |
| different paraphrase operator could in principle shift the absolute \excess{} |
| numbers even if, as Tier~2 suggests, the relative ordering is stable. Similarly, |
| our shift-type split shows predictor behavior depends strongly on shift |
| type---paraphrase ($n{=}472$, circular), domain ($n{=}65$, weak: \texttt{raptor} |
| $0.59$), length ($n{=}30$, uniformly high: aug/pac/raptor $0.95/0.95/0.92$)---and |
| the domain and length arms have small $n$, so their conclusions are weaker than |
| the paraphrase arm's. |
|
|
| |
| \section{Future Work} |
| \label{sec:future} |
| The open problem we isolate---\emph{predicting shift-specific (\excess{}) |
| directional brittleness without touching OOD data}---suggests several concrete |
| directions. (1)~Because augmentation is the only family with a non-negative |
| \excess{} signal, the most promising avenue is augmentation operators that more |
| faithfully simulate the target shift (instruction-tuned rewriters, controllable |
| domain/length transforms) coupled with NLI-gated label-fidelity, aiming to push |
| the augmentation correlation's CI off zero. (2)~The estimator-specificity result |
| motivates predictors defined directly on the geometry of \emph{difference-of-means} |
| and MLP probes, since LogReg predictability does not transfer. (3)~The partial |
| metric-dependence under whitening ($0.945\!\rightarrow\!0.698$) invites a |
| principled treatment of which covariance structure the rotation metric should |
| quotient out, ideally one that removes the sampling floor analytically rather than |
| via the empirical IID-resampling subtraction we use here. (4)~Strengthening the |
| domain ($n{=}65$) and length ($n{=}30$) arms and the 6.9B spot-check ($n{=}7$) to |
| seed-replicated grids would let the shift-type-dependence and scale-invariance |
| claims carry confidence intervals as tight as the paraphrase arm's. |
|
|
| |
| \section{Reproducibility} |
| \label{sec:reproducibility} |
|
|
| \probeshift{} is designed for minutes-to-reproduce verification on commodity |
| hardware. We release, under a permissive license: (i)~the full |
| \textbf{activation cache} (one forward pass per configuration, shared across all |
| predictors), so that every number in |
| \cref{tab:fourtarget,tab:paired,fig:circularity,fig:mechanism,fig:sizeladder} can |
| be recomputed from cache without GPU inference; (ii)~the complete \textbf{code} |
| for activation extraction, probe fitting (LogReg / mass-mean / MLP), the seven |
| label-free predictors, the IID-resampling placebo, the \excess{} target, and the |
| bootstrap / paired-contrast statistics; (iii)~the \textbf{pre-registration} |
| document fixing the four-target design, the predictor list, and the LOCO / |
| five-seed protocol \emph{before} the main run, so that the circularity finding and |
| the reversal are confirmatory rather than post-hoc; and (iv)~the exact |
| \textbf{data splits} (per-concept ID/shift/placebo indices, the five independent |
| seeds, and the de/fr/ru multi-pivot back-translation outputs with their NLI |
| round-trip fidelity labels). The headline grid uses Option~A (five fully |
| independent seeds, $n=472$); each reported confidence interval is a $10$k-sample |
| bootstrap over the shared configuration set, and all paired contrasts share the |
| configuration index so they are directly comparable. The entire pipeline runs on a |
| single RTX~4090 within $\leq\!200$ GPU$\cdot$h, with \$0 API spend and zero new |
| human annotation; cache-only reanalysis (shift-type breakdown, estimator external |
| validity, layer robustness, whitening ablation) runs in minutes. |
|
|
| |
| |
| \bibliographystyle{plainnat} |
| \bibliography{references} |
|
|
| |
| |
| \begin{thebibliography}{9} |
| \bibitem[Belinkov(2022)]{belinkov2022probing} |
| Y.~Belinkov. |
| \newblock Probing classifiers: Promises, shortcomings, and advances. |
| \newblock \emph{Computational Linguistics}, 2022. |
|
|
| \bibitem[Gao et~al.(2026)]{gao2026raptor} |
| Gao et~al. |
| \newblock RAPTOR: Ridge-adaptive directional stability for probe reliability. |
| \newblock \emph{Preprint}, 2026. |
|
|
| \bibitem[Huang(2025)]{huang2025sip} |
| Huang. |
| \newblock Spectral identifiability of probes (SIP). |
| \newblock \emph{Preprint}, 2025. |
|
|
| \bibitem[Kumar et~al.(2022)]{kumar2022unreliable} |
| Kumar et~al. |
| \newblock On the unreliability of probing-based interpretations. |
| \newblock \emph{Preprint}, 2022. |
|
|
| \bibitem[Lysn{\ae}s-Larsen et~al.(2025)]{lysnaes2025probing} |
| Lysn{\ae}s-Larsen et~al. |
| \newblock Probing the probes: Augmentation robustness as a probe-quality metric. |
| \newblock \emph{Preprint}, 2025. |
|
|
| \bibitem[Reblitz-Richardson(2026)]{reblitz2026fragility} |
| Reblitz-Richardson. |
| \newblock Forward-pass fragility thresholds for linear probes. |
| \newblock \emph{Preprint}, 2026. |
|
|
| \bibitem[Xie et~al.(2023)]{xie2023feature} |
| Xie et~al. |
| \newblock Feature dispersion score for label-free OOD-error prediction. |
| \newblock \emph{Preprint}, 2023. |
| \end{thebibliography} |
|
|
| \end{document} |
|
|
