The next architecture we are going to present using Theano is the single-hidden-layer Multi-Layer Perceptron (MLP). An MLP can be viewed as a logistic regression classifier where the input is first transformed using a learnt non-linear transformation . This transformation projects the input data into a space where it becomes linearly separable. This intermediate layer is referred to as a This tutorial will again tackle the problem of MNIST digit classification. ## The ModelAn MLP (or Artificial Neural Network - ANN) with a single hidden layer can be represented graphically as follows: Formally, a one-hidden-layer MLP is a function , where is the size of input vector and is the size of the output vector , such that, in matrix notation: with bias vectors , ; weight matrices , and activation functions and . The vector constitutes the hidden layer. is the weight matrix connecting the input vector to the hidden layer. Each column represents the weights from the input units to the i-th hidden unit. Typical choices for include , with , or the logistic function, with . We will be using in this tutorial because it typically yields to faster training (and sometimes also to better local minima). Both the and are scalar-to-scalar functions but their natural extension to vectors and tensors consists in applying them element-wise (e.g. separately on each element of the vector, yielding a same-size vector). The output vector is then obtained as: . The reader should recognize the form we already used for To train an MLP, we learn ## Going from logistic regression to MLPThis tutorial will focus on a single-hidden-layer MLP. We start off by implementing a class that will represent a hidden layer. To construct the MLP we will then only need to throw a logistic regression layer on top. The initial values for the weights of a hidden layer should be uniformly sampled from a symmetric interval that depends on the activation function. For activation function results obtained in [Xavier10]show that the interval should be , where is the number of units in the -th layer, and is the number of units in the -th layer. For the sigmoid function the interval is . This initialization ensures that, early in training, each neuron operates in a regime of its activation function where information can easily be propagated both upward (activations flowing from inputs to outputs) and backward (gradients flowing from outputs to inputs). Note that we used a given non-linear function as the activation function of the hidden layer. By default this is If you look into theory this class implements the graph that computes the hidden layer value . If you give this graph as input to the In this tutorial we will also use L1 and L2 regularization (see As before, we train this model using stochastic gradient descent with mini-batches. The difference is that we modify the cost function to include the regularization term. We then update the parameters of the model using the gradient. This code is almost identical to the one for logistic regression. Only the number of parameters differ. To get around this ( and write code that could work for any number of parameters) we will use the list of parameters that we created with the model ## Putting it All TogetherHaving covered the basic concepts, writing an MLP class becomes quite easy. The code below shows how this can be done, in a way which is analogous to our previous logistic regression implementation. The user can then run the code by calling : The output one should expect is of the form : On an Intel(R) Core(TM) i7-2600K CPU @ 3.40GHz the code runs with approximately 10.3 epoch/minute and it took 828 epochs to reach a test error of 1.65%. To put this into perspective, we refer the reader to the results section of this page. ## Tips and Tricks for training MLPsThere are several hyper-parameters in the above code, which are not (and, generally speaking, cannot be) optimized by gradient descent. Strictly speaking, finding an optimal set of values for these hyper-parameters is not a feasible problem. First, we can’t simply optimize each of them independently. Second, we cannot readily apply gradient techniques that we described previously (partly because some parameters are discrete values and others are real-valued). Third, the optimization problem is not convex and finding a (local) minimum would involve a non-trivial amount of work. The good news is that over the last 25 years, researchers have devised various rules of thumb for choosing hyper-parameters in a neural network. A very good overview of these tricks can be found in Efficient BackProp by Yann LeCun, Leon Bottou, Genevieve Orr, and Klaus-Robert Mueller. In here, we summarize the same issues, with an emphasis on the parameters and techniques that we actually used in our code. ## NonlinearityTwo of the most common ones are the and the function. For reasons explained in Section 4.4, nonlinearities that are symmetric around the origin are preferred because they tend to produce zero-mean inputs to the next layer (which is a desirable property). Empirically, we have observed that the has better convergence properties. ## Weight initializationAt initialization we want the weights to be small enough around the origin so that the activation function operates in its linear regime, where gradients are the largest. Other desirable properties, especially for deep networks, are to conserve variance of the activation as well as variance of back-propagated gradients from layer to layer. This allows information to flow well upward and downward in the network and reduces discrepancies between layers. Under some assumptions, a compromise between these two constraints leads to the following initialization: for tanh and for sigmoid. Where is the number of inputs and the number of hidden units. For mathematical considerations please refer to [Xavier10]. ## Learning rateThere is a great deal of literature on choosing a good learning rate. The simplest solution is to simply have a constant rate. Rule of thumb: try several log-spaced values () and narrow the (logarithmic) grid search to the region where you obtain the lowest validation error. Decreasing the learning rate over time is sometimes a good idea. One simple rule for doing that is where is the initial rate (chosen, perhaps, using the grid search technique explained above), is a so-called “decrease constant” which controls the rate at which the learning rate decreases (typically, a smaller positive number, and smaller) and is the epoch/stage. Section 4.7 details procedures for choosing a learning rate for each parameter (weight) in our network and for choosing them adaptively based on the error of the classifier. ## Number of hidden unitsThis hyper-parameter is very much dataset-dependent. Vaguely speaking, the more complicated the input distribution is, the more capacity the network will require to model it, and so the larger the number of hidden units that will be needed (note that the number of weights in a layer, perhaps a more direct measure of capacity, is (recall is the number of inputs and is the number of hidden units). Unless we employ some regularization scheme (early stopping or L1/L2 penalties), a typical number of hidden units vs. generalization performance graph will be U-shaped. |

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