generating text with recurrent neural networks
Introduction
The goal of this paper is to introduce a new type of recurrent neural network for character-level language modelling that allows the input character at a given timestep to multiplicatively gate the connections that make up the hidden-to-hidden layer weight matrix. The paper also introduces a solution to the problem of vanishing and exploding gradients by applying a technique called Hessian-Free optimization to effectively train a recurrent network that, when unrolled in time, has approximately 500 layers. At the date of publication, this network was arguably the deepest neural network ever trained successfully.
The problem solved by language modelling involves predicting the next character or word in a sequence given some number of preceding characters or words. Recurrent neural networks are naturally applicable to this problem, since they make predictions based on a current input and a hidden state whose value is determined by some number of previous inputs. Alternative methods that the authors compare their results to include a hierarchical Bayesian model called a 'sequence memoizer', and a mixture of context models referred to as PAQ, which actually includes word-level information (rather strictly character-level information). The multiplicative RNN introduced in this paper improves on the state-of-the-art for solely character-level language modelling, but is somewhat worse than the state-of-the-art for text compression.
To give a brief review, an ordinary recurrent neural network is parameterized by three weight matrices, [math]\displaystyle{ \ W_{hi} }[/math], [math]\displaystyle{ \ W_{hh} }[/math], and [math]\displaystyle{ \ W_{oh} }[/math], and functions to map a sequence of [math]\displaystyle{ N }[/math] input states [math]\displaystyle{ \ [i_1, ... , i_N] }[/math] to a sequence of hidden states [math]\displaystyle{ \ [h_1, ... , h_N] }[/math] and a sequence of output states [math]\displaystyle{ \ [o_1, ... , o_N] }[/math]. The matrix [math]\displaystyle{ \ W_{hi} }[/math] parameterizes the mapping from the current input state to the current hidden state, while the matrix [math]\displaystyle{ \ W_{hh} }[/math] parameterizes the mapping from the previous hidden state to current hidden state, such that the current hidden state is function of the previous hidden state and the current input state. Finally, the matrix [math]\displaystyle{ \ W_{oh} }[/math] parameterizes the mapping from the current hidden state to the current output state. So, at a given timestep [math]\displaystyle{ t }[/math], the values of the hidden state and output state are as follows:
- [math]\displaystyle{ \ h_t = tanh(W_{hi}i_t + W_{hh}h_{t-1} + b_h) }[/math]
- [math]\displaystyle{ \ o_t = W_{oh}h_t + b_o }[/math]
Typically, the output state is converted into a probability distribution over characters or words using the softmax function. The network can then be treated as a generative model of text by sampling from this distribution and providing the sampled output as the input to the network at the next timestep.
Recurrent networks are known to be very difficult to train due to the existence a highly unstable relationship between a network's parameter and the gradient of its cost function. Intuitively, the surface of the cost function is intermittently punctuated by abrupt changes (which can lead to exploding gradients) and nearly flat plateaus (which lead to vanishing gradients) that can effectively become poor local minima when a network is trained through gradient descent. Techniques for improving training include the use of Long Short-Term Memory networks <ref> Hochreiter, Sepp, and Jürgen Schmidhuber. "Long short-term memory." Neural computation 9.8 (1997): 1735-1780. </ref>, in which memory units are used to selectively preserve information from previous states, and the use of Echo State networks, <ref> Jaeger, H. and H. Haas. "Harnassing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication." Science, 204.5667 (2004): 78-80. </ref> which learns only the output weights on a network with recurrent connections that implement a wide range of time-varying patterns.
Hessian-Free Optimization
While this optimization technique is described elsewhere in Martens (2010), its use is essential to obtaining the successful results reported in this paper. In brief, the technique involves computing
Multiplicative Recurrent Neural Networks
The authors report that using a standard neural network trained via Hessian-free optimization produces only mediocre results. As such, they introduce a new architecture called a multiplicative recurrent neural network (MRNN). The motivating intuition behind this architecture is the idea the input at a given time step should both additively contribute to the hidden state (though the mapping performed by the input-to-hidden weights) and additionally determine the weights on the recurrent connections to the hidden state. In other words, the idea is to define a unique weight matrix [math]\displaystyle{ \ W_{hh} }[/math] for each possible input. The reason this design is hypothesized to the improve the predictive adequacy of the model is due to the idea that the conjunction of the input at one time step and the hidden state at the previous time step is important. Capturing this conjunction requires the input to influence the contribution of the previous hidden state to the current hidden state. Otherwise, the previous hidden state and the current input will make entirely independent contributions to the calculation of the current hidden state. Formally, this changes the calculation of the hidden state at a given time step as follows:
- [math]\displaystyle{ \ h_t = tanh(W_{hi}i_t + W^{i_t}_{hh}h_{t-1} + b_h) }[/math]
As a first approach to implementing this MRNN, the authors suggest using a tensor of rank 3 to store the hidden-to-hidden weights. The idea is that the tensor stores one weight matrix per possible input; when the input is provided as a one-hot vector, tensor contraction (i.e. a generalization of matrix multiplication) can be used to extract the 'slice' of the tensor that contains the appropriate set of weights. One problem with this approach is that it quickly becomes impractical to store the hidden-to-hidden weights as a tensor if the dimensionality of the hidden state has a large number of dimensions. For instance, if a network's hidden layer encodes a vector with 1000 dimensions, then the number of parameters in the tensor that need to be learned will be equal to [math]\displaystyle{ \ 1000^2 * N }[/math], where [math]\displaystyle{ \ N }[/math] is the vocabulary size. In short, this method will add many millions of parameters to a model for a non-trivially sized vocabulary.
To fix this problem, the tensor is factored using a technique described in Taylor & Hinton (2009) <ref>Taylor, G. and G. Hinton. "Factored Conditional Restricted Boltzmann Machines for Modeling Motion Style" ICML (2009) </ref>. The idea is to
Quantitative Experiments
Qualitative Experiments
Discussion
Bibliography
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