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Abstract:High-dimensional data must be highly structured to be learnable. Although the compositional and hierarchical nature of data is often put forward to explain learnability, quantitative measurements establishing these properties are scarce. Likewise, accessing the latent variables underlying such a data structure remains a challenge. In this work, we show that forward-backward experiments in diffusion-based models, where data is noised and then denoised to generate new samples, are a promising tool to probe the latent structure of data. We predict in simple hierarchical models that, in this process, changes in data occur by correlated chunks, with a length scale that diverges at a noise level where a phase transition is known to take place. Remarkably, we confirm this prediction in both text and image datasets using state-of-the-art diffusion models. Our results show how latent variable changes manifest in the data and establish how to measure these effects in real data using diffusion models.

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Authors:Beatriz Borges, Negar Foroutan, Deniz Bayazit, Anna Sotnikova, Syrielle Montariol, Tanya Nazaretzky, Mohammadreza Banaei, Alireza Sakhaeirad, Philippe Servant, Seyed Parsa Neshaei(+80 more)

Abstract:AI assistants are being increasingly used by students enrolled in higher education institutions. While these tools provide opportunities for improved teaching and education, they also pose significant challenges for assessment and learning outcomes. We conceptualize these challenges through the lens of vulnerability, the potential for university assessments and learning outcomes to be impacted by student use of generative AI. We investigate the potential scale of this vulnerability by measuring the degree to which AI assistants can complete assessment questions in standard university-level STEM courses. Specifically, we compile a novel dataset of textual assessment questions from 50 courses at EPFL and evaluate whether two AI assistants, GPT-3.5 and GPT-4 can adequately answer these questions. We use eight prompting strategies to produce responses and find that GPT-4 answers an average of 65.8% of questions correctly, and can even produce the correct answer across at least one prompting strategy for 85.1% of questions. When grouping courses in our dataset by degree program, these systems already pass non-project assessments of large numbers of core courses in various degree programs, posing risks to higher education accreditation that will be amplified as these models improve. Our results call for revising program-level assessment design in higher education in light of advances in generative AI.

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Abstract:How much data is required to learn the structure of a language via next-token prediction? We study this question for synthetic datasets generated via a Probabilistic Context-Free Grammar (PCFG) -- a hierarchical generative model that captures the tree-like structure of natural languages. We determine token-token correlations analytically in our model and show that they can be used to build a representation of the grammar's hidden variables, the longer the range the deeper the variable. In addition, a finite training set limits the resolution of correlations to an effective range, whose size grows with that of the training set. As a result, a Language Model trained with increasingly many examples can build a deeper representation of the grammar's structure, thus reaching good performance despite the high dimensionality of the problem. We conjecture that the relationship between training set size and effective range of correlations holds beyond our synthetic datasets. In particular, our conjecture predicts how the scaling law for the test loss behaviour with training set size depends on the length of the context window, which we confirm empirically for a collection of lines from Shakespeare's plays.

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Abstract:Understanding what makes high-dimensional data learnable is a fundamental question in machine learning. On the one hand, it is believed that the success of deep learning lies in its ability to build a hierarchy of representations that become increasingly more abstract with depth, going from simple features like edges to more complex concepts. On the other hand, learning to be insensitive to invariances of the task, such as smooth transformations for image datasets, has been argued to be important for deep networks and it strongly correlates with their performance. In this work, we aim to explain this correlation and unify these two viewpoints. We show that by introducing sparsity to generative hierarchical models of data, the task acquires insensitivity to spatial transformations that are discrete versions of smooth transformations. In particular, we introduce the Sparse Random Hierarchy Model (SRHM), where we observe and rationalize that a hierarchical representation mirroring the hierarchical model is learnt precisely when such insensitivity is learnt, thereby explaining the strong correlation between the latter and performance. Moreover, we quantify how the sample complexity of CNNs learning the SRHM depends on both the sparsity and hierarchical structure of the task.

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Abstract: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.

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Abstract: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.

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Authors:Leonardo Petrini, Francesco Cagnetta, Umberto M. Tomasini, Alessandro Favero, Matthieu Wyart

Abstract:Learning generic high-dimensional tasks is notably hard, as it requires a number of training data exponential in the dimension. Yet, deep convolutional neural networks (CNNs) have shown remarkable success in overcoming this challenge. A popular hypothesis is that learnable tasks are highly structured and that CNNs leverage this structure to build a low-dimensional representation of the data. However, little is known about how much training data they require, and how this number depends on the data structure. This paper answers this question for a simple classification task that seeks to capture relevant aspects of real data: the Random Hierarchy Model. In this model, each of the $n_c$ classes corresponds to $m$ synonymic compositions of high-level features, which are in turn composed of sub-features through an iterative process repeated $L$ times. We find that the number of training data $P^*$ required by deep CNNs to learn this task (i) grows asymptotically as $n_c m^L$, which is only polynomial in the input dimensionality; (ii) coincides with the training set size such that the representation of a trained network becomes invariant to exchanges of synonyms; (iii) corresponds to the number of data at which the correlations between low-level features and classes become detectable. Overall, our results indicate how deep CNNs can overcome the curse of dimensionality by building invariant representations, and provide an estimate of the number of data required to learn a task based on its hierarchically compositional structure.

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Abstract: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.

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Abstract:A central question of machine learning is how deep nets manage to learn tasks in high dimensions. An appealing hypothesis is that they achieve this feat by building a representation of the data where information irrelevant to the task is lost. For image datasets, this view is supported by the observation that after (and not before) training, the neural representation becomes less and less sensitive to diffeomorphisms acting on images as the signal propagates through the net. This loss of sensitivity correlates with performance, and surprisingly correlates with a gain of sensitivity to white noise acquired during training. These facts are unexplained, and as we demonstrate still hold when white noise is added to the images of the training set. Here, we (i) show empirically for various architectures that stability to image diffeomorphisms is achieved by spatial pooling in the first half of the net, and by channel pooling in the second half, (ii) introduce a scale-detection task for a simple model of data where pooling is learned during training, which captures all empirical observations above and (iii) compute in this model how stability to diffeomorphisms and noise scale with depth. The scalings are found to depend on the presence of strides in the net architecture. We find that the increased sensitivity to noise is due to the perturbing noise piling up during pooling, after being rectified by ReLU units.

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Abstract:Despite their success, understanding how convolutional neural networks (CNNs) can efficiently learn high-dimensional functions remains a fundamental challenge. A popular belief is that these models harness the compositional and hierarchical structure of natural data such as images. Yet, we lack a quantitative understanding of how such structure affects performances, e.g. the rate of decay of the generalisation error with the number of training samples. In this paper we study deep CNNs in the kernel regime: i) we show that the spectrum of the corresponding kernel and its asymptotics inherit the hierarchical structure of the network; ii) we use generalisation bounds to prove that deep CNNs adapt to the spatial scale of the target function; iii) we illustrate this result by computing the rate of decay of the error in a teacher-student setting, where a deep CNN is trained on the output of another deep CNN with randomly-initialised parameters. We find that if the teacher function depends on certain low-dimensional subsets of the input variables, then the rate is controlled by the effective dimensionality of these subsets. Conversely, if the teacher function depends on the full set of input variables, then the error rate is inversely proportional to the input dimension. Interestingly, this implies that despite their hierarchical structure, the functions generated by deep CNNs are too rich to be efficiently learnable in high dimension.

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