Open problem MAIS-O90 · posed in MAIS-A8 as Problem 7.3 · Status: open.
Tags: generalization · goal misgeneralization · proxy goals · training dynamics · probability · optimization. Difficulty: ★★.
Authored by: Claude Fable 5 directed by Lionel Levine · Audited by: GPT 5.6 Sol.
The coin line is the agenda's minimal model of goal misgeneralization: trained only on states with the coin at the right end of the line, does the network learn the proxy rule "move right" or the intended rule "go to the coin"? The two rules agree on every training state and disagree at an off-distribution probe state with the coin displaced to the left end. In the kernel limit the verdict is proved: the zero-diversity kernel gradient flow drives the probe logit to $+\infty$, so the infinite-width lazy network walks away from the displaced coin. The finite-width statement is open, and it is the one the selection question actually needs, since every trained network has finite width.
The setting, from the agenda: training inputs $x_{\mathrm a}(p,L)=(1,p,L)$ for $0\le p\le L-1$ — a constant bias, then the agent's and the coin's positions — all labeled "step right," and the probe $s^{\ast }=(\lceil L/2\rceil,0)$, agent mid-line and coin at the left end. The network is two-layer ReLU in the neural-tangent parameterization of Jacot, Gabriel, and Hongler [JGH18], $f_\theta(x)=m^{-1/2}\sum_j a_j\varphi(u_j\cdot x)$ with $\varphi(z)=\max(z,0)$ and standard Gaussian initialization, trained by full-batch gradient descent on the logistic loss with step $\eta$ (the agenda fixes a backpropagation convention, so each initialization determines a unique trajectory); $q^{\mathrm{ntk}}_0(k;m,L,\eta)$ is the probability over the initialization that the step-$k$ probe logit is positive. At diversity $\varepsilon>0$, an $\varepsilon$-fraction of training states place the coin elsewhere on the line, labeled by the intended action, but never at $0$, so the probe stays off-distribution for every $\varepsilon$; exact conventions are in MAIS-A8. The agenda's Proposition 5.3 settles the infinite-width flow by a positivity argument: some training coefficient must diverge, and the ReLU tangent kernel is positive between every training input and the probe.
Problem (MAIS-A8, Problem 7.3). The agenda's Proposition 5.3 settles the infinite-width $\varepsilon=0$ flow. Prove the finite-width probe statement
$$\lim_{m\to\infty}\liminf_{k\to\infty}q^{\mathrm{ntk}}_0(k;m,L,\eta)=1,$$
or identify the obstruction. The observable is the sign at this one probe; uniform-in-time convergence of the entire empirical kernel is not required and in general fails under cross-entropy training (Yu, Tian, and Chen). Separately, compute the probe sign of the kernel support-vector machine for $\varepsilon>0$ and small $L$.
The difficulty is the interchange of limits: finite-width trajectories track the kernel flow for fixed training time, but the probe verdict is a long-time statement, and the squared-loss finite-width guarantees of Arora, Du, Hu, Li, Salakhutdinov, and Wang [ADHL+19] do not cover logistic training at large times. Only one sign at one point must survive the interchange, which is why this is a project rather than a general theory. In the closing sub-problem, the kernel support-vector machine is the maximum-margin classifier in the tangent kernel's feature space, trained on the $\varepsilon>0$ data above. For the kernel-flow proof and the surrounding regime dichotomy, see MAIS-A8.
References
- [YTC25] Z. Yu, S. Tian, and G. Chen, Divergence of the empirical neural tangent kernel in classification problems, ICLR 2025 — why uniform-in-time kernel convergence fails under cross-entropy, forcing the probe-only formulation. arXiv:2504.11130
- [ADHL+19] S. Arora, S. S. Du, W. Hu, Z. Li, R. Salakhutdinov, and R. Wang, On exact computation with an infinitely wide neural net, NeurIPS 2019 — the squared-loss finite-width guarantee that does not cover logistic training at large times. arXiv:1904.11955
- [JGH18] A. Jacot, F. Gabriel, and C. Hongler, Neural tangent kernel: convergence and generalization in neural networks, NeurIPS 2018 — the tangent kernel and the infinite-width kernel limit. arXiv:1806.07572
Related: MAIS-O86 (whether the same limit instead matches the Gaussian posterior) · MAIS-O82 (the analogous conjecture in the standard parameterization) · MAIS-O83 (finite-width departures from the infinite-width verdicts).