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Register a new userAn Analytical Theory of Curriculum Learning in Teacher-Student Networks
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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.
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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}
Recently, consistency-based methods have achieved state-of-the-art results in semi-supervised learning (SSL). These methods always involve two roles, an explicit or implicit teacher model and a student model, and penalize predictions under different perturbations by a consistency constraint. However, the weights of these two roles are tightly coupled since the teacher is essentially an exponential moving average (EMA) of the student. In this work, we show that the coupled EMA teacher causes a performance bottleneck. To address this problem, we introduce Dual Student, which replaces the teacher with another student. We also define a novel concept, stable sample, following which a stabilization constraint is designed for our structure to be trainable. Further, we discuss two variants of our method, which produce even higher performance. Extensive experiments show that our method improves the classification performance significantly on several main SSL benchmarks. Specifically, it reduces the error rate of the 13-layer CNN from 16.84% to 12.39% on CIFAR-10 with 1k labels and from 34.10% to 31.56% on CIFAR-100 with 10k labels. In addition, our method also achieves a clear improvement in domain adaptation.
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.
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.
Learning in Deep Neural Networks (DNN) takes place by minimizing a non-convex high-dimensional loss function, typically by a stochastic gradient descent (SGD) strategy. The learning process is observed to be able to find good minimizers without getting stuck in local critical points, and that such minimizers are often satisfactory at avoiding overfitting. How these two features can be kept under control in nonlinear devices composed of millions of tunable connections is a profound and far reaching open question. In this paper we study basic non-convex one- and two-layer neural network models which learn random patterns, and derive a number of basic geometrical and algorithmic features which suggest some answers. We first show that the error loss function presents few extremely wide flat minima (WFM) which coexist with narrower minima and critical points. We then show that the minimizers of the cross-entropy loss function overlap with the WFM of the error loss. We also show examples of learning devices for which WFM do not exist. From the algorithmic perspective we derive entropy driven greedy and message passing algorithms which focus their search on wide flat regions of minimizers. In the case of SGD and cross-entropy loss, we show that a slow reduction of the norm of the weights along the learning process also leads to WFM. We corroborate the results by a numerical study of the correlations between the volumes of the minimizers, their Hessian and their generalization performance on real data.
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