Vanishing gradient problem

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In machine learning, the vanishing gradient problem is a difficulty found in training artificial neural networks with gradient-based learning methods and backpropagation. In such methods, each of the neural network's weights receives an update proportional to the gradient of the error function with respect to the current weight in each iteration of training. Traditional activation functions such as the hyperbolic tangent function have gradients in the range $(-1, 1)$ or $[0, 1)$ , and backpropagation computes gradients by the chain rule. This has the effect of multiplying $n$ of these small numbers to compute gradients of the "front" layers in an $n$ -layer network, meaning that the gradient (error signal) decreases exponentially with $n$ and the front layers train very slowly.

With the advent of the back-propagation algorithm in the 1970s, many researchers tried to train supervised deep artificial neural networks from scratch, initially with little success. Sepp Hochreiter's diploma thesis of 1991^[1]^[2] formally identified the reason for this failure in the "vanishing gradient problem", which not only affects many-layered feedforward networks, but also recurrent neural networks. The latter are trained by unfolding them into very deep feedforward networks, where a new layer is created for each time step of an input sequence processed by the network.

When activation functions are used whose derivatives can take on larger values, one risks encountering the related exploding gradient problem.

Solutions

Multi-level hierarchy

To overcome this problem, several methods were proposed. One is Jürgen Schmidhuber's multi-level hierarchy of networks (1992) pre-trained one level at a time through unsupervised learning, fine-tuned through backpropagation.^[3] Here each level learns a compressed representation of the observations that is fed to the next level.

Long short-term memory

Main article: Long short-term memory

Another method is the long short-term memory (LSTM) network of 1997 by Hochreiter & Schmidhuber.^[4] In 2009, deep multidimensional LSTM networks demonstrated the power of deep learning with many nonlinear layers, by winning three ICDAR 2009 competitions in connected handwriting recognition, without any prior knowledge about the three different languages to be learned.^[5]^[6]

Sven Behnke relied only on the sign of the gradient (Rprop) when training his Neural Abstraction Pyramid^[7] to solve problems like image reconstruction and face localization.

Other

Other methods also use unsupervised pre-training to structure a neural network, making it first learn generally useful feature detectors. Then the network is trained further by supervised back-propagation to classify labeled data. The deep model of Hinton et al. (2006) involves learning the distribution of a high level representation using successive layers of binary or real-valued latent variables. It uses a restricted Boltzmann machine to model each new layer of higher level features. Each new layer guarantees an increase on the lower-bound of the log likelihood of the data, thus improving the model, if trained properly. Once sufficiently many layers have been learned the deep architecture may be used as a generative model by reproducing the data when sampling down the model (an "ancestral pass") from the top level feature activations.^[8] Hinton reports that his models are effective feature extractors over high-dimensional, structured data.^[9]

Faster hardware

Hardware advances have meant that from 1991 to 2015, computer power (especially as delivered by GPUs) has increased around a million-fold, making standard backpropagation feasible for networks several layers deeper than when the vanishing gradient problem was recognized. Schmidhuber notes that this "is basically what is winning many of the image recognition competitions now", but that it "does not really overcome the problem in a fundamental way".^[10]

References

↑ S. Hochreiter. Untersuchungen zu dynamischen neuronalen Netzen. Diploma thesis, Institut f. Informatik, Technische Univ. Munich, 1991. Advisor: J. Schmidhuber.
↑ S. Hochreiter, Y. Bengio, P. Frasconi, and J. Schmidhuber. Gradient flow in recurrent nets: the difficulty of learning long-term dependencies. In S. C. Kremer and J. F. Kolen, editors, A Field Guide to Dynamical Recurrent Neural Networks. IEEE Press, 2001.
↑ J. Schmidhuber., "Learning complex, extended sequences using the principle of history compression," Neural Computation, 4, pp. 234–242, 1992.
↑ Hochreiter, Sepp; Schmidhuber, Jürgen (1997). "Long Short-Term Memory". Neural Computation. 9 (8): 1735–1780. doi:10.1162/neco.1997.9.8.1735. PMID 9377276.
↑ Graves, Alex; and Schmidhuber, Jürgen; Offline Handwriting Recognition with Multidimensional Recurrent Neural Networks, in Bengio, Yoshua; Schuurmans, Dale; Lafferty, John; Williams, Chris K. I.; and Culotta, Aron (eds.), Advances in Neural Information Processing Systems 22 (NIPS'22), December 7th–10th, 2009, Vancouver, BC, Neural Information Processing Systems (NIPS) Foundation, 2009, pp. 545–552
↑ Graves, A.; Liwicki, M.; Fernandez, S.; Bertolami, R.; Bunke, H.; Schmidhuber, J. (2009). "A Novel Connectionist System for Improved Unconstrained Handwriting Recognition". IEEE Transactions on Pattern Analysis and Machine Intelligence. 31 (5).
↑ Sven Behnke (2003). Hierarchical Neural Networks for Image Interpretation. (PDF). Lecture Notes in Computer Science. 2766. Springer.
↑ Hinton, G. E.; Osindero, S.; Teh, Y. (2006). "A fast learning algorithm for deep belief nets" (PDF). Neural Computation. 18 (7): 1527–1554. doi:10.1162/neco.2006.18.7.1527. PMID 16764513.
↑ Hinton, G. (2009). "Deep belief networks". Scholarpedia. 4 (5): 5947. doi:10.4249/scholarpedia.5947.
↑ Schmidhuber, Jürgen (2015). "Deep learning in neural networks: An overview". Neural Networks. 61: 85–117. arXiv:1404.7828. doi:10.1016/j.neunet.2014.09.003.

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