Shuofeng Zhang

Shuofeng Zhang, Ard Louis

For overparameterized linear regression with isotropic Gaussian design and minimum-$\ell_p$ interpolator $p\in(1,2]$, we give a unified, high-probability characterization for the scaling of the family of parameter norms $ \\{ \lVert \widehat{w_p} \rVert_r \\}_{r \in [1,p]} $ with sample size. We solve this basic, but unresolved question through a simple dual-ray analysis, which reveals a competition between a signal *spike* and a *bulk* of null coordinates in $X^\top Y$, yielding closed-form predictions for (i) a data-dependent transition $n_\star$ (the "elbow"), and (ii) a universal threshold $r_\star=2(p-1)$ that separates $\lVert \widehat{w_p} \rVert_r$'s which plateau from those that continue to grow with an explicit exponent. This unified solution resolves the scaling of *all* $\ell_r$ norms within the family $r\in [1,p]$ under $\ell_p$-biased interpolation, and explains in one picture which norms saturate and which increase as $n$ grows. We then study diagonal linear networks (DLNs) trained by gradient descent. By calibrating the initialization scale $α$ to an effective $p_{\mathrm{eff}}(α)$ via the DLN separable potential, we show empirically that DLNs inherit the same elbow/threshold laws, providing a predictive bridge between explicit and implicit bias. Given that many generalization proxies depend on $\lVert \widehat {w_p} \rVert_r$, our results suggest that their predictive power will depend sensitively on which $l_r$ norm is used.

Shuofeng Zhang, Ard Louis

A wide variety of generalization measures have been applied to deep neural networks (DNNs). Although obtaining tight bounds remains challenging, such measures are often assumed to reproduce qualitative generalization trends. In this position paper, we argue that many post-mortem generalization measures -- those computed on trained networks -- are \textbf{fragile}: small training modifications that barely affect the underlying DNN can substantially change a measure's value, trend, or scaling behavior. For example, minor hyperparameter changes, such as learning rate adjustments or switching between SGD variants can reverse the slope of a learning curve in widely used generalization measures like the path norm. We also identify subtler forms of fragility. For instance, the PAC-Bayes origin measure is regarded as one of the most reliable, and is indeed less sensitive to hyperparameter tweaks than many other measures. However, it completely fails to capture differences in data complexity across learning curves. This data fragility contrasts with the function-based marginal-likelihood PAC-Bayes bound, which does capture differences in data-complexity, including scaling behavior, in learning curves, but which is not a post-mortem measure. Beyond demonstrating that many bounds -- such as path, spectral and Frobenius norms, flatness proxies, and deterministic PAC-Bayes surrogates -- are fragile, this position paper also argues that developers of new measures should explicitly audit them for fragility.

Shuofeng Zhang, Isaac Reid, Guillermo Valle Pérez et al.

The intuition that local flatness of the loss landscape is correlated with better generalization for deep neural networks (DNNs) has been explored for decades, spawning many different flatness measures. Recently, this link with generalization has been called into question by a demonstration that many measures of flatness are vulnerable to parameter re-scaling which arbitrarily changes their value without changing neural network outputs. Here we show that, in addition, some popular variants of SGD such as Adam and Entropy-SGD, can also break the flatness-generalization correlation. As an alternative to flatness measures, we use a function based picture and propose using the log of Bayesian prior upon initialization, $\log P(f)$, as a predictor of the generalization when a DNN converges on function $f$ after training to zero error. The prior is directly proportional to the Bayesian posterior for functions that give zero error on a test set. For the case of image classification, we show that $\log P(f)$ is a significantly more robust predictor of generalization than flatness measures are. Whilst local flatness measures fail under parameter re-scaling, the prior/posterior, which is global quantity, remains invariant under re-scaling. Moreover, the correlation with generalization as a function of data complexity remains good for different variants of SGD.