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Register a new userHow deep learning works --The geometry of deep learning
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Why and how that deep learning works well on different tasks remains a mystery from a theoretical perspective. In this paper we draw a geometric picture of the deep learning system by finding its analogies with two existing geometric structures, the geometry of quantum computations and the geometry of the diffeomorphic template matching. In this framework, we give the geometric structures of different deep learning systems including convolutional neural networks, residual networks, recursive neural networks, recurrent neural networks and the equilibrium prapagation framework. We can also analysis the relationship between the geometrical structures and their performance of different networks in an algorithmic level so that the geometric framework may guide the design of the structures and algorithms of deep learning systems.
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How does a 110-layer ResNet learn a high-complexity classifier using relatively few training examples and short training time? We present a theory towards explaining this in terms of Hierarchical Learning. We refer hierarchical learning as the learner learns to represent a complicated target function by decomposing it into a sequence of simpler functions to reduce sample and time complexity. We formally analyze how multi-layer neural networks can perform such hierarchical learning efficiently and automatically by applying SGD. On the conceptual side, we present, to the best of our knowledge, the FIRST theory result indicating how deep neural networks can still be sample and time efficient using SGD on certain hierarchical learning tasks, when NO KNOWN existing algorithm is efficient. We establish a new principle called backward feature correction, where training higher-level layers in the network can improve the features of lower-level ones. We believe this is the key to understand the deep learning process in multi-layer neural networks. On the technical side, we show for regression and even binary classification, for every input dimension $d>0$, there is a concept class of degree $omega(1)$ polynomials so that, using $omega(1)$-layer neural networks as learners, SGD can learn any function from this class in $mathsf{poly}(d)$ time and sample complexity to any $frac{1}{mathsf{poly}(d)}$ error, through learning to represent it as a composition of $omega(1)$ layers of quadratic functions. In contrast, we do not know any other simple algorithm (including layer-wise training or applying kernel method sequentially) that can learn this concept class in $mathsf{poly}(d)$ time even to any $d^{-0.01}$ error. As a side result, we prove $d^{omega(1)}$ lower bounds for several non-hierarchical learners, including any kernel methods, neural tangent or neural compositional kernels.
Classical linear metric learning methods have recently been extended along two distinct lines: deep metric learning methods for learning embeddings of the data using neural networks, and Bregman divergence learning approaches for extending learning Euclidean distances to more general divergence measures such as divergences over distributions. In this paper, we introduce deep Bregman divergences, which are based on learning and parameterizing functional Bregman divergences using neural networks, and which unify and extend these existing lines of work. We show in particular how deep metric learning formulations, kernel metric learning, Mahalanobis metric learning, and moment-matching functions for comparing distributions arise as special cases of these divergences in the symmetric setting. We then describe a deep learning framework for learning general functional Bregman divergences, and show in experiments that this method yields superior performance on benchmark datasets as compared to existing deep metric learning approaches. We also discuss novel applications, including a semi-supervised distributional clustering problem, and a new loss function for unsupervised data generation.
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding generalization power of overparametrized neural networks, the role of depth in deep architectures, the apparent absence of the curse of dimensionality, the surprisingly successful optimization performance despite the non-convexity of the problem, understanding what features are learned, why deep architectures perform exceptionally well in physical problems, and which fine aspects of an architecture affect the behavior of a learning task in which way. We present an overview of modern approaches that yield partial answers to these questions. For selected approaches, we describe the main ideas in more detail.
We show that a collection of Gaussian mixture models (GMMs) in $R^{n}$ can be optimally classified using $O(n)$ neurons in a neural network with two hidden layers (deep neural network), whereas in contrast, a neural network with a single hidden layer (shallow neural network) would require at least $O(exp(n))$ neurons or possibly exponentially large coefficients. Given the universality of the Gaussian distribution in the feature spaces of data, e.g., in speech, image and text, our result sheds light on the observed efficiency of deep neural networks in practical classification problems.
We propose a novel framework, called Markov-Lipschitz deep learning (MLDL), to tackle geometric deterioration caused by collapse, twisting, or crossing in vector-based neural network transformations for manifold-based representation learning and manifold data generation. A prior constraint, called locally isometric smoothness (LIS), is imposed across-layers and encoded into a Markov random field (MRF)-Gibbs distribution. This leads to the best possible solutions for local geometry preservation and robustness as measured by locally geometric distortion and locally bi-Lipschitz continuity. Consequently, the layer-wise vector transformations are enhanced into well-behaved, LIS-constrained metric homeomorphisms. Extensive experiments, comparisons, and ablation study demonstrate significant advantages of MLDL for manifold learning and manifold data generation. MLDL is general enough to enhance any vector transformation-based networks. The code is available at https://github.com/westlake-cairi/Markov-Lipschitz-Deep-Learning.
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