Deep learning theory and its implications:
In this study, we investigate generalization properties using loss landscape flatness, a promising predictor with recent rigorous connection to generalization gap [ 6 ] . Yet, to what extent can flatness explain generalization is still an open question in deep learning. For instance, curvature is a local measure, which means that its informativeness of robustness against global perturbation is limited. Moreover, the theoretical association between flatness and generalization is provided in the form of upper bound [ 6 , 9 ] and the bound may not always be tight, which is consistent with more variability in the generalization gap for bio-plausible learning rules but offers less predictive power. Despite observing a significant correlation between the generalization gap and a curvature-based measure in Figure 2 , the relationship appears to be messy, suggesting other factors involved in explaining generalization. Given that developing better predictors of generalization is still a work in progress [ 4 ] , we anticipate stronger theoretical tools to be applied for studying biological generalization in the future.
Our results are also consistent with existing findings linking learning rate to loss landscape curvature in deep networks [ 124 ] . In the case of bio-plausible gradient approximations, small step length along the gradient direction cannot be compensated by increasing the learning rate, as that would inadvertently increase the error vector, causing numerical issues (Eq. 9 ). Curiously, the approximation error vector \vec{e} did not seem to play a role other than restricting the learning rate. While it is well-known that stochastic gradient noise (SGN) can help with finding flat minima due to the alignment of SGN covariance and Hessian near minima [ 144 , 145 , 123 , 146 ] , that may not apply to approximation error vector \vec{e} resulting primarily from temporal truncation of the gradient (see Figure 1 A and Appendix Figure 10 ). This indicates that noise with different properties (e.g. different directions) could affect generalization differently, thereby motivating future investigations into how a broad range of biological noises — which may differ from noise in ML optimization (e.g. SGN) — can impact generalization.

Moreover, our results are closely related to a series of studies that examined the "catapulting" behavior in learning [ 147 , 148 , 149 , 150 , 151 ] . This can happen when the second order Taylor term of the loss function would dominate over the first, which would cause learning to cross the threshold for step size stability and "catapult" into a flatter region that can accommodate the step size. If the truncation noise is only aligned with the eigendirections associated with negligible eigenvalues, then it can only have limited contributions to the second-order Taylor term. On top of that, the orthogonal noise term would require a smaller learning rate to be used to avoid numerical issues, as explained earlier. Overall, these would lead to a weaker second-order Taylor term relative to the first for bio-plausible temporal credit assignment rules, which would then increase the threshold for step size stability. This increased stability is closely tied to the greater dynamical stability (for the weight update difference equation) of approximation rules predicted by Theorem 1 , due to the correspondence between loss’ Hessian matrix and the Jacobian matrix of the weight update difference equation (the correspondence is explained in Theorem 1 proof in Appendix B ).