A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data

Understanding the structure of real data is paramount in advancing modern deep-learning methodologies. Natural data such as images are believed to be composed of features organised in a hierarchical and combinatorial manner, which neural networks capture during learning. Recent advancements show that diffusion models can generate high-quality images, hinting at their ability to capture this underlying structure. We study this phenomenon in a hierarchical generative model of data. We find that the backward diffusion process acting after a time $t$ is governed by a phase transition at some threshold time, where the probability of reconstructing high-level features, like the class of an image, suddenly drops. Instead, the reconstruction of low-level features, such as specific details of an image, evolves smoothly across the whole diffusion process. This result implies that at times beyond the transition, the class has changed but the generated sample may still be composed of low-level elements of the initial image. We validate these theoretical insights through numerical experiments on class-unconditional ImageNet diffusion models. Our analysis characterises the relationship between time and scale in diffusion models and puts forward generative models as powerful tools to model combinatorial data properties.

On the different regimes of Stochastic Gradient Descent

Modern deep networks are trained with stochastic gradient descent (SGD) whose key parameters are the number of data considered at each step or batch size $B$, and the step size or learning rate $\eta$. For small $B$ and large $\eta$, SGD corresponds to a stochastic evolution of the parameters, whose noise amplitude is governed by the `temperature' $T\equiv \eta/B$. Yet this description is observed to break down for sufficiently large batches $B\geq B^*$, or simplifies to gradient descent (GD) when the temperature is sufficiently small. Understanding where these cross-overs take place remains a central challenge. Here we resolve these questions for a teacher-student perceptron classification model, and show empirically that our key predictions still apply to deep networks. Specifically, we obtain a phase diagram in the $B$-$\eta$ plane that separates three dynamical phases: $\textit{(i)}$ a noise-dominated SGD governed by temperature, $\textit{(ii)}$ a large-first-step-dominated SGD and $\textit{(iii)}$ GD. These different phases also corresponds to different regimes of generalization error. Remarkably, our analysis reveals that the batch size $B^*$ separating regimes $\textit{(i)}$ and $\textit{(ii)}$ scale with the size $P$ of the training set, with an exponent that characterizes the hardness of the classification problem.

Dissecting the Effects of SGD Noise in Distinct Regimes of Deep Learning

Understanding when the noise in stochastic gradient descent (SGD) affects generalization of deep neural networks remains a challenge, complicated by the fact that networks can operate in distinct training regimes. Here we study how the magnitude of this noise $T$ affects performance as the size of the training set $P$ and the scale of initialization $\alpha$ are varied. For gradient descent, $\alpha$ is a key parameter that controls if the network is `lazy' ($\alpha\gg 1$) or instead learns features ($\alpha\ll 1$). For classification of MNIST and CIFAR10 images, our central results are: (i) obtaining phase diagrams for performance in the $(\alpha,T)$ plane. They show that SGD noise can be detrimental or instead useful depending on the training regime. Moreover, although increasing $T$ or decreasing $\alpha$ both allow the net to escape the lazy regime, these changes can have opposite effects on performance. (ii) Most importantly, we find that key dynamical quantities (including the total variations of weights during training) depend on both $T$ and $P$ as power laws, and the characteristic temperature $T_c$, where the noise of SGD starts affecting performance, is a power law of $P$. These observations indicate that a key effect of SGD noise occurs late in training, by affecting the stopping process whereby all data are fitted. We argue that due to SGD noise, nets must develop a stronger `signal', i.e. larger informative weights, to fit the data, leading to a longer training time. The same effect occurs at larger training set $P$. We confirm this view in the perceptron model, where signal and noise can be precisely measured. Interestingly, exponents characterizing the effect of SGD depend on the density of data near the decision boundary, as we explain.

Failure and success of the spectral bias prediction for Kernel Ridge Regression: the case of low-dimensional data

Recently, several theories including the replica method made predictions for the generalization error of Kernel Ridge Regression. In some regimes, they predict that the method has a `spectral bias': decomposing the true function $f^*$ on the eigenbasis of the kernel, it fits well the coefficients associated with the O(P) largest eigenvalues, where $P$ is the size of the training set. This prediction works very well on benchmark data sets such as images, yet the assumptions these approaches make on the data are never satisfied in practice. To clarify when the spectral bias prediction holds, we first focus on a one-dimensional model where rigorous results are obtained and then use scaling arguments to generalize and test our findings in higher dimensions. Our predictions include the classification case $f(x)=$sign$(x_1)$ with a data distribution that vanishes at the decision boundary $p(x)\sim x_1^{\chi}$. For $\chi>0$ and a Laplace kernel, we find that (i) there exists a cross-over ridge $\lambda^*_{d,\chi}(P)\sim P^{-\frac{1}{d+\chi}}$ such that for $\lambda\gg \lambda^*_{d,\chi}(P)$, the replica method applies, but not for $\lambda\ll\lambda^*_{d,\chi}(P)$, (ii) in the ridge-less case, spectral bias predicts the correct training curve exponent only in the limit $d\rightarrow\infty$.