Since the publication of the original pdp books (Rumelhart et al., 1986; McClelland et al., 1986) and back-propagation algorithm, the bp framework has been developed extensively. Two of the extensions that have attracted the most attention among those interested in modeling cognition have been the Simple Recurrent Network (SRN) and the recurrent back-propagation (RBP) network. In this and the next chapter, we consider the cognitive science and cognitive neuroscience issues that have motivated each of these models, and discuss how to run them within the PDPTool framework.
The Simple Recurrent Network (SRN) was conceived and first used by Jeff Elman, and was first published in a paper entitled Finding structure in time (Elman, 1990). The paper was ground-breaking for many cognitive scientists and psycholinguists, since it was the first to completely break away from a prior commitment to specific linguistic units (e.g. phonemes or words), and to explore the vision that these units might be emergent consequences of a learning process operating over the latent structure in the speech stream. Elman had actually implemented an earlier model in which the input and output of the network was a very low-level spectrogram-like representation, trained using a spectral information extracted from a recording of his own voice saying ‘This is the voice of the neural network’. We will not discuss the details of this network, except to note that it learned to produce this utterence after repeated training, and contained no explicit feature, phoneme, syllable, morpheme, or word-level units.
In Elman’s subsequent work he stepped back a little from the raw-stimulus approach used in this initial unpublished simulation, but he retained the fundamental commitment to the notion that the real structure is not in the symbols we as researchers use but in the input stream itself. In Finding structure in time, Elman presented several simulations, one addressing the emergence of words from a stream of sub-lexical elements (he actually used the letters making up the words as the elements for this), and the other addressing the emergence of sentences from a stream of words. In both models, the input at any given time slice comes from a small fixed alphabet; interest focuses on what can be learned in a very simple network architecture, in which the task posed to the network is to predict the next item in the sting, using the item at time t, plus an internal representation of the state of a set of hidden units from the previous time step.
An SRN of the kind Elman employed is illustrated in Figure 7.1. We actually show the network used in an early follow-up study by Servan-Schreiber et al. (1991), in which a very small alphabet of elements is used (this is the particular network provided with the PDPTool software, and it will be described in more detail later).
The beauty of the SRN is its simplicity. In fact, it is really just a three-layer, feed-forward back propagation network. The only proviso is that one of the two parts of the input to the network is the pattern of activation over the network’s own hidden units at the previous time step.
In Elman’s simulations, input to the network consisted of an unbroken stream of tokens from the alphabet. The simulation starts with small random connection weights in all feed-forward projections. The first token in the stream is presented, typically setting the values of the units in the input layer to a pattern of ones and zeros. Activation is propagated forward, as in a standard feed-forward back-propagation network. The teaching input or target pattern for the output layer is simply the next item in the stream of tokens. The output is compared to the target, delta terms are calculated, and weights are updated before stepping along to the next item in the stream. Localist input and output patterns are often used in SRN’s, because this choice makes it easy from the modeler to understand the network’s output, as we shall see later. However, success at learning and discovery of useful and meaningful internal representations does not depend on the use of localist input and output representations, as Elman showed in several of his simulations.
After processing the first item in the stream, the critical step—the one that makes the SRN a type of recurrent network—occurs. The state of the hidden units is ‘copied back’ to the context units, so that it becomes available as part of the input to the network on the next time step. The arrow labeled ‘copy’ in the figure represents this step. Perhaps it is worth noting that this copy operation is really just a conceptual convenience; one can think of the context units as simply providing a convenient way of allowing the network to rely on the state of the hidden units from the previous time step.
Once the copy operation is completed, the input sequence is stepped forward one step. The input now becomes the item that was the target at the previous time step, and the target now becomes the next item in the sequence. Activation is propagated forward, the output is compared to the target, delta terms are calculated via back-propagation, and weights are updated. Note that the weights that are updated include those from the context (previous hidden state) to the hidden units (current hidden state). This allows the network to learn, not only to use the input, but also to use the context, to help it make predictions.
In most of Elman’s investigations, he simply created a very long stream of tokens from the alphabet, according to some generative scheme, and then repeatedly swept through it, wrapping around to the beginning when he reached the end of the sequence. After training with many presentations, he stopped training and tested the network.
Here we briefly discuss three of the findings from Elman (1990). The first relates to the notion of ‘word’ as linguistic unit. The concept ‘word’ is actually a complicated one, presenting considerable difficulty to anyone who feels they must decide what is a word and what is not. Consider these examples: ‘line drive’, ‘flagpole’, ‘carport’, ‘gonna’, ‘wanna’, ‘hafta’, ‘isn’t’ and ‘didn’t’ (often pronounced “dint”). How many words are involved in each case? If more than one word, where are the word boundaries? Life might be easier if we did not have to decide where the boundaries between words actually lie. Yet, we have intuitions that there are points in the stream of speech sounds that correspond to places where something ends and something else begins. One such place might be between ‘fifteen’ and ‘men’ in a sentence like ‘Fifteen men sat down at a long table’, although there is unlikely to be a clear boundary between these words in running speech.
Elman’s approach to these issues, as previously mentioned, was to break utternances down into a sequence of elements, and present them to an SRN. In his letter-in-word simulation, he actually used a stream of sentences generated from a vocabulary of 15 words. The words were converted into a stream of elements corresponding to the letters that spelled each of the words, with no spaces. Thus, the network was trained on an unbroken stream of letters. After the network had looped repeatedly through a stream of about 5,000 elements, he tested its predictions for the first 50 or so elements of the training sequence. The results are seen in Figure 7.2.
What we see is that the network tends to have relatively high prediction error for the first letter of each word. The error tends to drop throughout the word, and then suddenly to shoot up again at the first letter of the next word. This is not always true – sometimes, afer a few downward steps, there is an uptick within a word, but such uptics generally correspond to places where there might be the end of what we ordinarily call a word. Thus, the network has learned something that corresponds at least in part with our intuitive notion of ‘word’, without building in the concept of word or ever making a categorical decision about the locations of word boundaries.
The other two findings come from a different simulation, in which the elements of the sequences used corresponded to whole words, strung together again to form simple sentences. The set of words Elman used corresponded to several familiar nouns and verbs. Each sentence involved a verb, and at least one noun as subject, with an optional subsequent noun as direct object. Verbs and nouns fell into different sub-types, – there were, for example, verbs of perception (which require an animate subject but can take any noun as object) and verbs of consumption, which require something consumable, and verbs of descruction, each of which had different restrictions on the nouns that could occur with it as subject and object. Crucially, the input patterns representing the nouns and verbs were randomly assigned, and thus did not capture in any way the coocurrence structure of the domain. Over the course of learning, however, the network came to assign each input its own internal representation. In fact, the hidden layer reflected both the input and the context; as a result, the patterns the network learned to assign provided a highly context-sensitive form of lexical representation.
The next two figures illustrate findings from this simulation. The first of these (Figure 7.3) shows a cluster analysis based on the average pattern over the hidden layer assigned to each of the different words in the corpus. What we see is that the learned average internal representations indicate that the network has been able to learn the category structure and even the sub-category structure of the “lexicon” of this simple artificial language. The reason for this is largely that the predictive consequences of each word correspond closely to the syntactic category and sub-category structure of the language. One may note, in fact, that the category structure encompasses distinctions that are usually treated as syntactic (noun or verb, and within verbs, transitive vs intransitive) as well as distinctions that are usually treated as semantic (fragile-object, food item), and at least one distinction that is clearly semantic (animate vs. inanimate) but is also often treated as a syntactically relevant “subcategorization feature” in linguistics. The second figure (Figure 7.4) shows a cluster analysis of the patterns assigned two of the words (BOY and GIRL) in each of many different contexts. The analysis establishes that the overall distinction between BOY and GIRL separates the set of context-sensitive patterns into two highly similar subtrees, indicating that the way context shades the representation of BOY is similar to the way in which it shades the representation of GIRL.
Overall, these three simulations from Elman (1990) show how both segmentation of a stream into larger units and assignment of units into a hierarchical similarity structure can occur, without there actually being any enumerated list of units or explicit assignment to syntactic or semantic categories.
Elman continued his work on SRN’s through a series of additional important and interesting papers. The first of these (Elman, 1991) explored the learning of sentences with embedded clauses, illustrating that this was possible in this simple network architecture, even though the network did not rely on the computational machinery (an explicitly recursive computational structure, including the ability to ‘call’ a computational process from within itself) usually thought to be required to deal with imbeddings. A subsequent and highly influential paper (Elman, 1993) reported that success in learning complex embedded structures depended on starting small – either starting with simple sentences, and gradually increasing the number of complex ones, or limiting the network’s ability to exploit context over long sequences by clearing the context layer after every third element of the training sequence. However, this later finding was later revisited by Rohde and Plaut (1999). They found in a very extensive series of investigations that starting small actually hurt eventual performance rather than helped it, except under very limited circumstances. A number of other very interesting investigations of SRN’s have also been carried our by Tabor and collaborators, among other things using SRN’s to make predictions about participants reading times as they read word-by-word through sentences (Tabor et al., 1997).
The simple recurrent network introduced by Elman (1990) also spawed investigations by Servan-Schreiber et al. (1991). We describe this work in greater detail than some of the other work following up on Elman (1990) because the exercise we provide is based on the Servan-Schreiber et al. (1991) investigations.
These authors were especially interested in exploring the relationship between SRNs an classical automata, including Finite State Transition Networks, and were also interested in the possibility that SRNs might provide useful models of implicit sequence learning in humans (Cleeremans and McClelland, 1991). Servan-Schreiber et al. (1991) investigated the learning of sequences that could be generated by a simple finite-state transition network grammar of the kind used to generate stimuli used in human implicit learning experiments by Rebur (1976), and illustrated in Figure 7.5. Sequence generation occurs as follows. Each sequence begins with the symbol ‘B’, and enters the state associated with the node in the diagram marked ‘#0’. From there, a random choice is made to follow one of the two links leading out of node 0. After the random choice, the symbol written beside the link is added as the second element of the sequence, and the process transitions to the state at the end of the chosen link. The choice could, for example, be the upper link leaving node 0, producing a T and shifting to node 1. The process iterates as before, choosing one of two links out of the current node, adding the symbol written next to the link as the next element of the seqeuence, and shifting to the node at the end of the link. When node 5 is reached, there is only one remaining possibility; the symbol E is written, and the sequence then ends.
In their simulation of learning sequences generated by the Finite State Transition Network, Servan-Schreiber et al. (1991) assumed that the learner is trained with a series of sequences, which may be generated at random by the sequence-generating finite-state transition network. At the beginning of each sequence, the context representation is reset of a neutral initial state, and the initial element B is presented on the input. The network then tries to predict the next element of the sequence, which then is presented as the target. Small adjustments to the weights are made, and then the process steps forward through the sequence. Now the second element of the sequence (the one after the B) is presented at the the input, and the third item in the sequence becomes the target. The process of propagating activation forward, adjusting the the connections, and stepping forward to the next element continues, until the last element of the sequence (the E symbol) is reached. At that point, processing of the sequence ends. The process then begins again with the next sequence.
In their paper, Servan-Schreiber et al. (1991) demonstrated that a network like the one shown in Figure 7.1 could essentially learn to become the transition network grammar through a completely gradual and continuous learning process. That is, as a result of training, the patterns at the hidden layer came to correspond closely to the nodes of the finite state transition network, in the sense that there was essentially one hidden pattern corresponding to each of the nodes in the network, regardless of the sequence of elements leading to that node. However, the network could also learn to ‘shade’ its representation of each node to a certain extent with the details of the prior context, so that it would eventually learn to capture subtle idiosyncratic constraints when trained repeatedly on a fixed corpus of legal example sentences. The authors also went on to show that the network could learn to use such subtle shading to carry information forward over an embedded sequence, but this will only happen in the SRN if the material needed at the end of the embedding is also of some relevance within the sequence.
The reader is referred to the paper by Servan-Schreiber et al. (1991) for further details of these investigations. Here we concentrate on describing the simulation model that allows the reader to explore the SRN model, using the same network and one of the specific training sets used by Servan-Schreiber et al. (1991).
The srn is a specific type of back-propagation network. It assumes a feed-forward architecture, with units in input, hidden, and output pools. It also allows for a special type of hidden layer called a “context” layer. A context layer is a hidden layer that receives a single special type of projection from another layer containing the same number of units, typically a hidden layer. This special type of projection (called a ‘copy-back projection’) allows the pattern of activation left over on the sending units from the last input pattern processed to be copied onto the receiving units for use as context in processing the next input pattern.
The srn network type also allows provides a construct called the “sequence”, which consists of one or more input-output pairs to be presented in a fixed order.
The idea is that one might experience a series of sequences, such that each sequence has a fixed structure, but the order in which the sequences appear can be random (permuted) within each epoch. In the example provided, a sequence is a sequence of characters beginning with a B and ending with an E.
Sequences can be defined in the pattern file in two different ways:
This line specifies that the following entries will be used to construct actual input output pattern pairs, as follows. The character strings following SeqLocal are treated as labels for both the input and output units, with the first label being used for the first input unit and the first output unit etc. Single characters are used in the example but strings are supported.
Specific sequences are then specified by lines like the following:
Here each line begins with a pname, followed by a series of instances of the different labels previously defined. The case shown above generates four input-output pattern pairs; the first input contains a 1 in the first position and 0’s elsewhere and the first target contains a 1 in the second position and 0’s elsewhere. The second input is the same as the first target, etc. Thus, in this model the user specifies the actual symbolic training sequences and these are translated into actual input and output patterns for the user.
Using the last target from the previous sequence as the first element if the input for the next sequence. If the first label in the string of labels is the special label ‘#’, the last target pattern from the previous sequence is used as the first input pattern for the current sequence. In this case, we also copy the state of the hidden units from the network’s prediction of the last target to the context layer, instead of resetting the context to the clearval.
The .net file specifies the specific architecture of the model as in other PDPtool programs. To have a pool of units treated as a ‘context’ layer, it should be of type ‘hidden’ and should receive a single projection from some other layer (typically a hidden layer) which has its constraint_type field set to ‘copyback’. When the constraint type is ‘copyback’, there are no actual weights, the state of the ‘sending’ units is simply copied to the ‘receiving’ units at the same time that the next target input is applied (except at the beginning of a new sequence, where the state of each of the context units is set to the clearval, which by default is set to .5).
The srn program’s inner loops have been converted to fast mex code (c code that interfaces with MATLAB). To turn on the fast mode for network training, click the ’fast’ option in the train window. This option can also be set by typing the following command on the MATLAB command prompt:
There are some limitations in executing fast mode of training.In this mode, network output logs can only be written at epoch level. Any smaller output frequency if specified, will be ignored.When running with fast code in gui mode, network viewer will only be updated at epoch level.Smaller granularity of display update is disabled to speed up processing.
There is a known issue with running this on linux OS.MATLAB processes running fast version appear to be consuming a lot of system memory. This memory management issue has not been observed on windows. We are currently investigating this and will be updating the documentation as soon as we find a fix.
The exercise is to replicate the simulation discussed in Sections 3 and 4 of Servan-Schreiber et al. (1991). The training set you will use is described in more detail in the paper, but is presented here in Figure 7.6. This particular set contains 21 patterns varying in length from 3 to 8 symbols (plus a B at the beginning and an E at the end of each one).
To run the exercise, download the latest version of pdptool, set your path to include pdptool and all of its children, and change to the pdptool/srn directory. Type ‘gsm’ (standing for “Graded State Machine”) at the matlab prompt. After everything loads you will see a display showing (at the right of the screen) the input, hidden and output units, and a vector representing the target for the output units (see Figure 7.7). To the left of the input is the context. Within the input, output, and target layers the units are layed out according to Figure 7.1. Your exercise will be to test the network after 10, 50, 150, 500, and 1000 epochs of training. The parameters of the simulation are a little different from the parameters used in the published article (it is not clear what values were actually used in the published article in some cases), and the time course of learning is a little extended relative to the results reported in the paper, but the same basic pattern appears.
One thing to be clear about at the outset is that the training and testing is organized at the sequence level. Each sequence corresponds to a string that could be generated by the stochastic finite state automaton shown in Figure 7.5. The ptrain option (which is the one used for this exercise) permutes the order of presentation of sequences but presents the elements of the sequence in its cannonical sequential order. Each sequence begins with a B and ends with an E, and consists of a variable number N of elements. As describe above, the sequence is broken down into N-1 input-target pairs, the first of which has a B as input and the successor of B in the sequence as its target, and the last of which has the next to last symbol as its input and E as its target. When the B symbol is presented the context is reset to .5’s. It makes sense to update the display during testing at the pattern level, so that you can step through the patterns within each sequence. During training I update the display at the epoch level or after 10 epochs.
Before you begin, consider:
OK, now run through a test, before training. You should find that the network produces near uniform output over the output units at each step of testing.
NOTE: For testing, set update to occur after 1 pattern in the test window. Use the single step mode and, in this case, quickly step through, noticing how little the output changes as a function of the input. The weights are initialized in a narrow range, making the initial variation in output unit activation rather tiny. You can examine the activations of the output units at a given point by typing the following to the matlab console:
When you get tired or stepping through, hit run in the test window. The program will then quickly finish up the test.
The basic goal of this exercise is to allow you to watch the network proceed to learn about the 0th, 1st, and higher-order structure of the training set. You have already examined the 0th and 1st order structure; the higher-order structure is the structure that depends on knowing something about what happened before the current input. For example, consider the character V . What can occur after a V , where the V is preceded by a T? What happens when the V is preceeded by an X? By a P? By another V ? Similar questions can be asked about other letters.
Set nepochs in the training options panel to ten, and run ten epochs of training, then test again.
What would you say has been learned at this point? Explain your answer by referring to the pattern of activation across the output units for different inputs and for the the same input at different points in the sequence.
Continue training, testing after a total of 50, 150, 500, and 1000 epochs. Answer the same question as above, for each test point point.
Summarize the time course of learning, drawing on your results for specific examples as well as the text of section 4 of the paper. How do the changes in the representations at the hidden and context layers contribute to this process?
Write about 1 page about the concept of the SRN as a graded state machine and its relation to various types of discrete-state automata, based on your reading of the entire article (including especially the section on spanning embedded sequences).
Try to be succinct in each of your answers. You may want to run the whole training sequence twice to get a good overall sense of the changes as a function of experience.