Identifying Learning Rules from Representational Plasticity

Gradient-based, performance-optimizing updates of artificial neural network (ANN) representations quantitatively predict plasticity measured in primate inferior temporal (IT) cortex after object training (Sörensen, DiCarlo & Kar, 2026). But the leading biologically motivated alternatives to backpropagation, notably feedback alignment and predictive coding, approximate gradient descent in several regimes, so it is unclear whether representational-change measurements can identify which rule produced them. We study this identifiability question exactly in the two-layer linear network. We show that backpropagation (BP), feedback alignment (FA), and predictive coding (PC) all produce first-layer updates of the form ΔW1=ηME with a shared error matrix \(E\) and a rule-specific preconditioner \(M\), and that the three rules have exactly the same reachable set of single-episode updates \(\Delta : \operatorname{rank}(\Delta) \leq 0\): when the readout, task error, and step size are unobserved, no single-episode measurement at any precision can reject any of the three rules (whereas with all three observed, single episodes generically do identify). For PC we derive the exact identity \(\Delta W_1^{PC} = (I + W_2^{\top} W_2)^{-1} \Delta W_1^{BP}\), giving relative deviation at most \(s_{\max}^2/(1+s_{\max}^2)\) in the largest readout singular value \(s_{\max}\), and we formalize the resulting non-identifiability regime under bounded measurement noise. Across an isotropic task ensemble, the update covariance has eigenvalues \(d\omega^2 s_i^2\) (BP) and \(d\omega^2 s_i^2/(1+s_i^2)^2\) (PC) on the readout's right singular vectors, while FA produces a rank-\(o\) covariance on a readout-independent random subspace. Two negative results follow: the PC covariance is exactly the BP covariance of a shrunken readout, so second moments alone cannot separate PC from BP; and, as our nonlinear-network simulations confirm, ranked covariance spectra confound the rules further. A readout-referenced projection statistic separates BP from PC deterministically (given two distinct readout singular values) and FA generically (almost surely over a nondegenerate random feedback draw, at the population level; finite-sample power is quantified by simulation), and we give a heuristic sample-complexity estimate for it. In a small nonlinear (tanh) network, the PC–BP collapse is confirmed at the level of probe-set representations, but the separation of rules from representation-only second moments partially degrades: BP separates from the local-rule family by spectral slope, while PC and FA are not separated by shape, quantifying what is lost when moving from weight space to measurable representations. All lemmas, theorems, and propositions are proved, except the sample-complexity estimate, which is a labeled heuristic; all empirical findings are simulations, plus one small trained-network check (a linear network trained by each rule on an MNIST subset, CPU, seconds), reproducible from the attached code. No neural data are analyzed, and no claim about cortex is made; we close with a proposed protocol for applying these tests to models trained in the framework of Sörensen et al. (2026).