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Register a new userLocality defeats the curse of dimensionality in convolutional teacher-student scenarios
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Convolutional neural networks perform a local and translationally-invariant treatment of the data: quantifying which of these two aspects is central to their success remains a challenge. We study this problem within a teacher-student framework for kernel regression, using `convolutional kernels inspired by the neural tangent kernel of simple convolutional architectures of given filter size. Using heuristic methods from physics, we find in the ridgeless case that locality is key in determining the learning curve exponent $beta$ (that relates the test error $epsilon_tsim P^{-beta}$ to the size of the training set $P$), whereas translational invariance is not. In particular, if the filter size of the teacher $t$ is smaller than that of the student $s$, $beta$ is a function of $s$ only and does not depend on the input dimension. We confirm our predictions on $beta$ empirically. Theoretically, in some cases (including when teacher and student are equal) it can be shown that this prediction is an upper bound on performance. We conclude by proving, using a natural universality assumption, that performing kernel regression with a ridge that decreases with the size of the training set leads to similar learning curve exponents to those we obtain in the ridgeless case.
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Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: First, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones - such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.
In humans and animals, curriculum learning -- presenting data in a curated order - is critical to rapid learning and effective pedagogy. Yet in machine learning, curricula are not widely used and empirically often yield only moderate benefits. This stark difference in the importance of curriculum raises a fundamental theoretical question: when and why does curriculum learning help? In this work, we analyse a prototypical neural network model of curriculum learning in the high-dimensional limit, employing statistical physics methods. Curricula could in principle change both the learning speed and asymptotic performance of a model. To study the former, we provide an exact description of the online learning setting, confirming the long-standing experimental observation that curricula can modestly speed up learning. To study the latter, we derive performance in a batch learning setting, in which a network trains to convergence in successive phases of learning on dataset slices of varying difficulty. With standard training losses, curriculum does not provide generalisation benefit, in line with empirical observations. However, we show that by connecting different learning phases through simple Gaussian priors, curriculum can yield a large improvement in test performance. Taken together, our reduced analytical descriptions help reconcile apparently conflicting empirical results and trace regimes where curriculum learning yields the largest gains. More broadly, our results suggest that fully exploiting a curriculum may require explicit changes to the loss function at curriculum boundaries.
When a computational system continuously learns from an ever-changing environment, it rapidly forgets its past experiences. This phenomenon is called catastrophic forgetting. While a line of studies has been proposed with respect to avoiding catastrophic forgetting, most of the methods are based on intuitive insights into the phenomenon, and their performances have been evaluated by numerical experiments using benchmark datasets. Therefore, in this study, we provide the theoretical framework for analyzing catastrophic forgetting by using teacher-student learning. Teacher-student learning is a framework in which we introduce two neural networks: one neural network is a target function in supervised learning, and the other is a learning neural network. To analyze continual learning in the teacher-student framework, we introduce the similarity of the input distribution and the input-output relationship of the target functions as the similarity of tasks. In this theoretical framework, we also provide a qualitative understanding of how a single-layer linear learning neural network forgets tasks. Based on the analysis, we find that the network can avoid catastrophic forgetting when the similarity among input distributions is small and that of the input-output relationship of the target functions is large. The analysis also suggests that a system often exhibits a characteristic phenomenon called overshoot, which means that even if the learning network has once undergone catastrophic forgetting, it is possible that the network may perform reasonably well after further learning of the current task.
A unique cognitive capability of humans consists in their ability to acquire new knowledge and skills from a sequence of experiences. Meanwhile, artificial intelligence systems are good at learning only the last given task without being able to remember the databases learnt in the past. We propose a novel lifelong learning methodology by employing a Teacher-Student network framework. While the Student module is trained with a new given database, the Teacher module would remind the Student about the information learnt in the past. The Teacher, implemented by a Generative Adversarial Network (GAN), is trained to preserve and replay past knowledge corresponding to the probabilistic representations of previously learn databases. Meanwhile, the Student module is implemented by a Variational Autoencoder (VAE) which infers its latent variable representation from both the output of the Teacher module as well as from the newly available database. Moreover, the Student module is trained to capture both continuous and discrete underlying data representations across different domains. The proposed lifelong learning framework is applied in supervised, semi-supervised and unsupervised training. The code is available~: url{https://github.com/dtuzi123/Lifelong-Teacher-Student-Network-Learning}
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation of Laplacian. Then, we cover the cuts of graph and spectral clustering which applies clustering in a subspace of data. Different optimization variants of Laplacian eigenmap and its out-of-sample extension are explained. Thereafter, we introduce the locality preserving projection and its kernel variant as linear special cases of Laplacian eigenmap. Versions of graph embedding are then explained which are generaliz
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