Understanding Recurrent Neural Architectures by Analyzing and Synthesizing Long Distance Dependencies in Benchmark Sequential Datasets

Understanding Recurrent Neural Architectures by Analyzing and Synthesizing Long Distance Dependencies in Benchmark Sequential Datasets

Abhijit Mahalunkar 0000-0001-5795-8728Technological University DublinDublinIreland abhijit.mahalunkar@mydit.ie  and  John D. Kelleher 0000-0001-6462-3248Technological University DublinDublinIreland john.d.kelleher@dit.ie
Abstract.

In order to build efficient deep recurrent neural architectures, it is essential to analyze the complexity of long distance dependencies (LDDs) of the dataset being modeled. In this context, in this paper, we present detailed analysis of the complexity and the degree of LDDs (or LDD characteristics) exhibited by various sequential benchmark datasets. We observe that the datasets sampled from a similar process or task (e.g. natural language, or sequential MNIST, etc) display similar LDD characteristics. Upon analysing the LDD characteristics, we were able to analyze the factors influencing them; such as (i) number of unique symbols in a dataset, (ii) size of the dataset, (iii) number of interacting symbols within a given LDD, and (iv) the distance between the interacting symbols. We demonstrate that analysing LDD characteristics can inform the selection of optimal hyper-parameters for SOTA deep recurrent neural architectures. This analysis can directly contribute to the development of more accurate and efficient sequential models. We also introduce the use of Strictly k-Piecewise languages as a process to generate synthesized datasets for language modelling. The advantage of these synthesized datasets is that they enable targeted testing of deep recurrent neural architectures in terms of their ability to model LDDs with different characteristics. Moreover, using a variety of Strictly k-Piecewise languages we generate a number of new benchmarking datasets, and analyse the performance of a number of SOTA recurrent architectures on these new benchmarks.

Sequential Models, Recurrent Neural Networks
copyright: nonedoi: isbn: conference: ACM SIGIR Conference on Research and Development in Information Retrieval; July 2019; Paris, Francejournalyear: 2019ccs: Mathematics of computing Information theoryccs: Computing methodologies Information extractionccs: Computing methodologies Neural networksccs: Theory of computation Regular languagesccs: Computing methodologies Supervised learning by classification

1. Introduction

Recurrent Neural Networks (RNN) laid the foundation of sequential data modeling (Elman, 1990). However, recurrent neural architectures trained using backpropagation through time (BPTT) suffer from exploding or vanishing gradients (Hochreiter, 1991; Hochreiter et al., 2001; Bengio et al., 1994). This problem presents a specific challenge in modeling sequential datasets which exhibit long distance dependencies (LDDs). LDDs describe an interaction between two (or more) elements in a sequence that are separated by an arbitrary number of positions. LDDs are related to the rate of decay of statistical dependence of two points with increasing time interval or spatial distance between them. For example, in English there is a requirement for subjects and verbs to agree, compare: “The dog in that house is aggressive” with “The dogs in that house are aggressive”. This dependence can be computed using information theoretic measure i.e. Mutual Information (Cover and Thomas, 1991; Paninski, 2003; Bouma, 2009; Lin and Tegmark, 2017).

One of the early attempts at addressing this issue was by El Hihi and Bengio (1995) who proposed a Hierarchical Recurrent Neural Network which introduced several levels of state variables, working at different time scales. Various other architectures were developed based on these principles (Chang et al., 2017; Campos et al., 2018). LSTM introduced by Hochreiter and Schmidhuber (1997) attempted to bridge minimal time lags in excess of 1000 discrete time steps by enforcing constant error flow through constant error carousels within special units. More recently, attention and memory augmented networks have delivered good performance in modeling LDDs (Merity et al., 2016; Graves et al., 2014; Salton et al., 2017). The issue of vanishing gradients can also be alleviated by maintaining spectral norm of weight matrix close to unity, thus enforcing orthogonality (Vorontsov et al., 2017).

A fundamental task for modelling sequential data is Language Modeling. A language model accepts a sequence of symbols and predicts the next symbol in the sequence. The accuracy of a language model is dependent on the capacity of the model to capture the LDDs in the data on which it is evaluated because an inability to model LDDs in the input sequence will result in erroneous predictions. In this paper, we will use the language modeling task, on a range of datasets, to evaluate the ability of RNNs to model LDDs. The standard evaluation metric for language models is perplexity. Perplexity is the measurement of how well a language model predicts the next symbol, and the lower the perplexity of a model the better the performance of the model.

There are a number of benchmark datasets used to train and evaluate language models: PennTree Banks (PTB) (Marcus et al., 1994), WikiText 2 (Wiki-2), WikiText 103 (Wiki-103) (Merity et al., 2016) and Hutter-Text (Text8 and Enwik8). We reviewed the SOTA language models to check their performance on these datasets. Table 1 lists the perplexity scores for test and valid sets for PTB, Wiki-2 and Wiki-103. There is a general trend in that model evaluations on Wiki-103 tend to result in lower perplexity scores followed by Wiki-2 and then PTB.

Model PTB Wiki-2 Wiki-103
FRAGE + AWD-LSTM-MoS + dynamic eval; Gong et al. (2018) -
AWD-LSTM-DOC x5; Takase et al. (2018) -
AWD-LSTM-MoS + dynamic eval; Yang et al. (2017)* -
AWD-LSTM + dynamic eval; Krause et al. (2018)* -
AWD-LSTM + continuous cache pointer; Merity et al. (2017)* - -
AWD-LSTM-DOC; Takase et al. (2018) -
AWD-LSTM-MoS + finetune; Yang et al. (2017) - -
AWD-LSTM-MoS; Yang et al. (2017) -
AWD-LSTM; Merity et al. (2017), 2017) -
Transformer with tied adaptive embeddings; Baevski and Auli (2019) - -
LSTM + Hebbian + Cache + MbPA; Rae et al. (2018) - -
Table 1. Perplexity scores (test and valid) of SOTA word-based language models

The similarity of scores for different models on these different benchmark datasets indicate that word-based dataset exhibit similar dependency structures; e.g., they exhibit similar LDD characteristics. Furthermore, our review of the language model SOTA revealed that most research on developing language models fails to explicitly analyze the characteristics of the LDDs within the datasets used to train and evaluate the models. Motivated by these two observations, this paper makes a number of research contributions.

First, we argue that a key step in modeling sequential data is to understand the characteristics of the LDDs within the data. Second, we present a method to compute and analyze the LDD characteristics for any sequential dataset, and demonstrate this method on a number of datasets that are frequently used to benchmark the state-of-the-art in sequential models. Third, based on the analysis of the LDD characteristics, we observe that LDDs are far more complex than previously assumed, and depend on at least four factors: (i) the number of unique symbols in a dataset, (ii) the size of the dataset, (iii) the number of interacting symbols within an LDD, and (iv) the distance between the interacting symbols.

Fourth, we demonstrate how understanding LDD characteristics can inform better hyperparameter selection for current state-of-the-art RNN architectures, and also aid in understanding them. We demonstrate this by using Strictly k-Piecewise (SPk) languages as a benchmarking task for sequential models. The motivation for using the SPk language modelling task, is that the standard sequential benchmarking datasets provide little to no control over the factors which directly contribute to LDD characteristics. By contrast, we can generate benchmark datasets with varying degrees of LDD complexity by modifying the grammar of the SPk language (Rogers et al., 2010; Fu et al., 2011; Avcu et al., 2017). Using these new benchmark datasets we perform evaluation experiments that specifically test the ability of different RNN architectures to model LDDs of different types. These experiments demonstrates how understanding the characteristics of the LDDs exhibited in a dataset informs the selection of appropriate hyperparameters for current state-of-the-art RNNs.

Paper Organization: Sections 2.1 presents our algorithm to compute the LDD characteristics of dataset and 2.2 introduces Strictly k-Piecewise languages. Section 3 presents our Experiments. In section 3.1 we compute and analyse the LDD characteristics of benchmarking datasets. In section 3.2 we analyse SPk languages and discuss the factors which influencing LDD characteristics. We also present a case for the use SPk languages as benchmarking datasets. In section 3.3, we demonstrate the impact of LDD characteristics on DilatedRNNs. We then argue that understanding LDD characteristics is vital in improving state-of-the-art recurrent neural architecture performance and discuss how this analysis could aid in development of better architectures. The paper concludes with discussions in section 4 and related work in section 5.

2. Preliminaries

2.1. LDD Characteristics

The experiments in Section 3.1 and 3.2 analyze the LDD characteristics of sequential datasets. This section describes the algorithm we have developed to calculate the LDD characteristic of a dataset.

Mutual information measures dependence between random variables and . These random variables have marginal distributions and and are jointly distributed as (Cover and Thomas, 1991). Mutual information, is defined as;

(1)

If and are not correlated, in other words if they are independent to each other, then and . However, if and are fully dependent on each other, then which results in the maximum value of .

Mutual information can also be expressed using the entropy of and i.e. , and their joint entropy, as given in the equations below:

(2)
(3)

Shannon’s Entropy in Eq. 3 is known to be biased, generally underestimating true entropy from finite samples, thus, in this work, we choose the following equation to compensate for insufficient samplings (Grassberger, 2003):

(4)

where is the frequency of unique symbol i, , is the number of unique symbols, and is the logarithmic derivative of the gamma function of .

In order to measure dependence between any two symbols at a distance in a sequence, we design random variables and so that holds the subsequence of the original sequence from index till , and holds the subsequence from index till ; where represents spacing between the symbols and or is the size of the dataset. The figure below illustrates how and are defined over a sequence when .

elements

elements

Next we define a random variable that contains a sequence of paired symbols one from and one from , where the symbols in a pair have the same index in and . The figure below illustrates the definition of these pairs, each column defines one pair.

Random Variable

Random Variable

After this, we count the number of symbols that appear in and (i.e., the size of the symbol vocabulary of and ) these counts are stored in , respectively. Similarly, we count the number of unique pairs of symbols in and store this in . We then obtain the frequency of each symbol in the vocabularies of and , giving us and ; and the frequency of each of the pairs of symbols in , giving us . Using this information, and Equations 2 and 4, we calculate the mutual information at a distance in a sequence. We define the LDD characteristics of any given sequential dataset as a function of mutual information over the distance . Algorithm 1 explains the details.

for  do
     
     
      zero-matrix of size ()
     for  do
         Increment
     end for
     Compute , , for
     Compute , , for
     Compute , , for
     Compute , and using Eq. 4
     
end for
Algorithm 1 Computing LDD Characteristics

2.2. Strictly k-Piecewise Languages (SPk)

The experiments in Section 3.2 and 3.3 are based on synthetic datasets generated using SPk languages. Here we introduce SPk languages, following (Rogers et al., 2010; Fu et al., 2011; Avcu et al., 2017).

SPk languages form a subclass of regular languages. Subregular languages can be identified by mechanisms much less complicated than Finite-State Automata. Many aspects of human language such as local and non local dependencies are similar to subregular languages (Jager and Rogers, 2012). More importantly, there are certain types of long distance (non local) dependencies in human language which allow finite-state characterization (Heinz and Rogers, 2010). These type of LDDs can easily be characterizable by SPk languages and can be easily extended to other processes.

A language L, is described by a finite set of unique symbols and * (free monoid) is a set of finite sequences or strings of zero or more elements from .

Example 2.1 ().

Consider, = {1, 2, 3, 4} where 1, 2, 3, 4 are the unique symbols. A free monoid over contains all concatenations of these unique symbols. Thus, * = {, 1, 12, 13, 14, 32, 313, 2143, … }.

Definition 2.0 ().

Let, u denote a string, where u= 32. Length of a string u is denoted by which 2. A string with length zero is denoted by .

Definition 2.0 ().

A string v is a subsequence of string w, iff v = 12n and w *1*2* … *n*, where . A subsequence of length k is called a k-subsequence. Let subseqk(w) denote the set of subsequences of w up to length k.

Example 2.2 ().

Consider, = {a, b, c, d}, w = [acbd], u = [bd], v = [acd] and x = [db]. String u is a subsequence of length k = 2 or 2-subsequence of w. String v is a 3-subsequence of w. However, string x is not a subsequence of w as it does not contain [db] subsequence.

SPk languages are defined by grammar GSPk as a set of permissible k-subsequences. Here, k indicates the number of elements in a depenedency. Datasets generated to simulate 2 elements in a dependency will be generated using SP2. This is the simplest dependency structure. There are more complex chained-dependency structures which require higher k grammars.

Example 2.3 ().

Consider L, where = {a, b, c, d}. Let GSP2 be SPk grammar which is comprised of permissible 2-subsequences. Thus, GSP2 = {aa, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd}. GSP2 grammar is employed to generate SP2 language.

Definition 2.0 ().

Subsequences which are not in the grammar G are called forbidden strings111Refer section 5.2. Finding the shortest forbidden subsequences in (Fu et al., 2011) for method to compute forbidden sequences for SPk language.

Example 2.4 ().

Consider Example 2.3, although {ab} is a possible 2-subsequence, it is not part of the grammar GSP2. Hence, {ab} is a forbidden substring.

Example 2.5 ().

Consider strings u, v, w u = [bbcbdd], v = [bbdbbbcbddaa] and w = [bbabbbcbdd], where u = 6, v = 12 and w = 10. Strings u and v are valid SP2 strings because they are composed of subsequences that are in GSP2. However, w is invalid SP2 string because w contains {ab} a subsequence which is a forbidden string. These constraints apply for any string x where x.

Example 2.6 ().

Let GSP3 = {aaa, aab, abb, baa, bab, bba, bbb, …} and forbidden string = {aba} be SP3 grammar which is comprised of permissible -subsequences. Thus, u = [aaaaaaab], where u = 8 is a valid SP3 string and v = [aaaaabaab], where v = 9 is an invalid SP3 string as defined by the grammar GSP3.

The extent of LDD exhibited by a certain SPk language is almost equal to the length of the strings generated which abides by the grammar. However, as per definition 2.2, the strings generated using this method will also exhibit dependencies of shorter lengths. It should be noted that the length of the LDD is not the same as k. The length of the LDD is the maximum distance between two elements in a dependency, whereas k specifies the number of elements in the dependency (as defined in the the SPk grammar).

Example 2.7 ().

As per Example 2.5, v = [bbdbbbcbddaa], consider b in the first position, subsequence {ba} exhibits dependency of 10 and 11. Similarly, subsequence {bd} exhibits dependency of 3, 9 and 10.

Figure 1. The automaton for GSP2 where nl=6

Figure 1 depicts a finite-state diagram of GSP2, which generates strings of synthetic data. Consider a string x from this data, it is x = 6, generated strings x, generated using grammar GSP2. The forbidden string for this grammar is {ab}. Since {ab} is a forbidden string, the state diagram has no path (from state 0 to state 11) because such a path would permit the generation of strings with {ab} as a subsequence, e.g. {abcccc} Traversing the state diagram generates valid strings e.g. {accdda, caaaaa}.

Various GSPk could be used to define an SPk depending on the set of forbidden strings chosen. Thus, we can construct rich datasets with different properties for any SPk language. Forbidden strings allow for the elimination of certain logically possible sequences while simulating a real world dataset where the probability of occurrence of that particular sequence is highly unlikely. Every SPk grammar is defined with at least one forbidden string.

3. Experiments

3.1. LDD Characteristics of Natural Datasets

In Section 1 we introduced the task of language modelling, and reviewed the SOTA results across the standard benchmark natural language datasets: PennTree Banks (PTB) (Marcus et al., 1994), WikiText 2 (Wiki-2), WikiText 103 (Wiki-103) (Merity et al., 2016) and Hutter-Text (Text8 and Enwik8). When applied to natural language datasets, language modelling can be framed as word-based language modeling (unique symbols are words) or character-based language modeling (unique symbols are characters). The perplexity scores reported in Section 1 were all word-based language modeling results. However, all of these natural language benchmark datasets can be used to train both variants of language models.

We are interested in understanding how the characteristics of a dataset (in particular characteristics of the LDDs in the data) affect the performance of RNN models. With this goal in mind we analysed the attributes of the standard datasets at both the character and word level. Table 2 lists aggregate statistics for these datasets at both these levels. In order to examine the characteristics of the LDDs within each of these datasets we plotted a curve of the mutual information within the dataset at different distances. For each dataset two of these curves were created, one at the word level and one at the character level.

To plot a curve for a dataset we first applied the algorithm from Section 2.1 in an iterative manner for different sizes of (ranging from 1 to the length of the dataset). Then these results were plotted on a log-log axis, with the x-axis denoting the distance between two symbols (either characters or words, and with a range from 1 to the length of the dataset) and the y-axis denotes the mutual information (in nats) between these two random variables. We refer to these plots as the LDD characteristics of a dataset.

LDD characteristics at a character level were computed for PTB, Wiki-2, Wiki-103, Text8 and Enwik8 and are displayed in figure 2(a). Word-level LDD characteristics were computed for PTB, Wiki-2, Wiki-103 and Text8 and are displayed in figure 1(a).

Word-based Character-based
Dataset Words Length Characters Length
Enwik8 NA NA 6062 98620454
Text8 253855 17005208 27 100000000
PTB 10000 1085779 48 5639751
Wiki2 33278 2551843 282 12755448
Wiki103 267735 103690236 1249 536016456
Table 2. Features of natural language datasets
(a)
(b)
Figure 2. (a) LDD characteristics of word-based language dataset; (b) Broken power law fit for the word-based datasets. Green Line is the first power law fit; Red Line is the second power law fit.
(a)
(b)
(c)
Figure 3. LDD characteristics of (a) character-based language dataset; (b) unpermuted sequential MNIST & permuted sequential MNIST using multiple seeds; (c) Geolife dataset (human mobility)

LDD characteristics of character-based and word-based tasks follow expected trends (Ebeling and Poeschel, 2002; Montemurro and Pury, 2002). It is seen that mutual information decay follows a power law (Lin and Tegmark, 2017). For character-based datasets, strong dependence (higher power law decay) is observed between characters at a distance up to ; beyond which the curve exhibits a long flat tail indicating lower dependence. This point of inflection is of much interest. For word-based datasets, strong dependence is observed between words at a distance up to 10 across various datasets. This inflection point indicates the presence of a broken power law. A Broken power law is a piecewise function, consisting of two or more power laws, combined with a threshold (inflection point) (Jóhannesson et al., 2006). For e.g. with two power laws:

(5)

In figure 1(b), we fit broken power laws to word-based datasets to study the features of the LDD characteristics. For a given dataset, we observe that . The higher value of is due to a faster rate of reduction in the frequency of contextually correlated words in a sequence, as the spacing between them increases. This signifies the presence of a strong grammar. Beyond the point of inflection, it is understood that the pairs are not contextually correlated which results in a flatter curve or lower value of . This analysis enables us to approximate the contextual boundary of the natural language data. Also, the absolute of value of mutual information is an indicator of the degree of the short and long distance dependencies present in a dataset.

The fact that our above analysis of the English datasets found a very large value for indicates that a dataset with good distribution of English text will exhibit a high value of mutual information at lower values of followed by a steep decay of mutual information. Recall from Section 1 that we noted a trend in the results reported across the standard benchmark datasets where Wiki-103 tended to deliver the best perplexity score followed by Wiki-2 and PTB. Our analysis of the LDD characteristics provides an explanation for this trend. Language models have very good performance on Wiki-2 due to the fact that they can take advantage of large and very low mutual information in the flat region. Furthermore, language models marginally outperform on Wiki-2 as compared to the PTB due to higher mutual information at lower values of .

The natural language datasets analysed above are not the only datasets used to evaluate language models. The sequential MNIST dataset is also widely used as a benchmarking dataset. The dataset contains 240,000 training images and 40,000 test images which are 28x28 pixel wide. In order to use them in a sequential task, the images are converted into a single vector of 784 pixels by concatenating all the rows of a single image. There are 256 unique values and total length of the data is 54880000. We generated an LDD characteristic plot for the entire dataset by concatenating all the sequential data of the images and then applying algorithm 1 on this data. We also designed permuted sequential MNIST datasets with various seeds and computed plotted the LDD characteristics of these datasets. The LDD characteristics are plotted in figure 2(b). The MNIST LDD curve shows us that the unpermuted Sequential MNIST (blue line) exhibits a peculiar decay. High mutual information is observed between pixels spaced at small distances. We also observe mutual information peaks at multiples of 28, where 28 is the width of the MNIST images. This is in accordance with the properties of the image, i.e. nearby pixels (along the rows and columns) tend to be similar due to spatial dependencies. And when we convert the image into a sequential vector, the dependent pixels get spaced out by a factor of 28 (width of the image in this case), hence we observe peaks at multiples of 28. Also, when we analyze the decay trend, we observe exponential decay indicating lack of LDDs for (Lin and Tegmark, 2017). By introducing permutations in the sequential MNIST data, we loose the spatial dependencies within the image. Permutations result in chaotic data, thereby increasing the joint entropy of the two random variables, resulting in significantly lower mutual information. The LDD curve stays almost flat for . Beyond which the mutual information decay is exponential also indicating lack of LDDs for .

RNN models always perform better on unpermuted sequential MNIST compared with permuted sequential MNIST. Our analysis of the LDD characteristics of these datasets provides an explanation for why this is the case. Unpermuted sequential MNIST has LDDs of less than 300, due to the pixel dependencies and exponential decay. Datasets possessing such short-range dependency can be easily modeled using simple models e.g. Hidden Markov Models (HMMs) as they don’t require long memory. In the case of permuted sequential MNIST, we again observe LDDs of not more than (due to exponential decay beyond that). However, the flat curve with very low mutual information is usually a result of noisy data. It is this noisy data (rather than complex LDDs) that is responsible for reducing the performance of the SOTA sequential models on permuted sequential MNIST. Overall, however we would argue that both of these datasets are inadequate at benchmarking sequential models due to their limitation in generating complex LDDs.

We also plotted the LDD characteristics of GPS trajectory dataset collected in Geolife project (Microsoft Research Asia) by 178 users in a period of over four years (from April 2007 to October 2011), see figure 2(c). A GPS trajectory of this dataset is represented by a sequence of time-stamped points, each of which contains the information of latitude, longitude and altitude which was converted to a unique location number. These trajectories were recorded by different GPS loggers and GPS phone (Zheng et al., 2011). Upon analyzing the plot of the LDD characteristics in this data, it’s evident that human mobility also has power law decay suggesting the presence of LDDs.

3.2. LDD characteristics of SPk datasets

Natural datasets present little to no control over the factors the affect LDDs. This, limits our ability to understand LDDs in more detail. SPk languages exhibit some types of LDDs occurring in natural datasets. Moreover, by modifying the SPk grammar we can control the LDD characteristics within a dataset generated by the grammar. To understand and validate the interaction between an SPk grammar and the characteristics of the data it generates we used a number of SPk grammars to generate datasets and analysed the properties of these datasets.

We used SP2, SP4 and SP16 grammars to generate these datasets. Using grammars with , enabled us to generate datasets with different dependency structures ( and ) and, hence, to analyze the impact of dependency structure on LDD characteristics.

In-order to analyse the impact of vocabulary size on LDD characteristics, we generated SP2 grammars where (size of vocabulary = ) and (size of vocabulary = ). We generated strings of maximum length of and using SP2 grammar. As explained in Example 2.6, by increasing the length of the generated strings, the distance between dependent elements is also increased, resulting in longer LDDs. We can then simulate LDD lengths as and . We also choose two sets of forbidden strings for SP2 grammar, {} and {}. We also generate two sizes of the same SP2 grammar to study the impact of the size of the data on the LDD characteristics, where one dataset was twice the size of the other. The datasets were generated using foma (Hulden, 2009) and python (Mahalunkar and Kelleher, 2018). Figure 4 shows plots of the LDD characteristics of these datasets.

(a)
(b)
(c)
(d)
(e)
(f)
Figure 4. LDD characteristics of datasets of (a) SP2 grammar exhibiting LDDs of length 20, 100, 200 and 500. (b) SP2, SP4 and SP16 grammar exhibiting LDD of length 100. (c) SP2 grammar with vocabulary of 4 and 26. (d) SP2 grammar with varying forbidden strings (e) SP2 grammar with varying size of the datasets (f) All the SPk grammars in a plot

Figure 3(a) plots LDD characteristics of SP2 languages with maximum string length of . The point where mutual information decay is faster, the inflection point, lies around the same point on x-axis as the maximum length of the LDD. This confirms that SPk can generate datasets with varying lengths of LDDs.

Figure 3(b) plots the LDD characteristics of SP2, SP4 and SP16 grammars. The strings in all the grammars are up to 100. Hence, we can observe the mutual information decay beyond . k defines the number of correlated or dependent elements in a dependency rule. As k increases the grammar becomes more complex and there is an overall reduction in frequency of the dependent elements in a given sequence (due to lower probability of these elements occurring in a given sequence). Hence, the mutual information is lower. This can be seen with dataset of SP16 vs SP2 and SP4. It is worth noting that datasets with lower mutual information curves tend to present more difficulty during modeling (Mahalunkar and Kelleher, 2018)

The impact of vocabulary size can be seen in figure 3(c) where the LDD characteristics of SP2 datasets with vocabulary size () and are plotted. Both these datasets contain strings of maximum length 20. Hence the mutual information decays at 20. Both curves have identical decay indicating a similar grammar. However the overall mutual information of the dataset with is much lower then the mutual information of the dataset with . This is because a smaller vocabulary results in an increase in the probability of the occurrence of each elements.

Figure 3(d) plots the LDD characteristics of SP2 grammar with two set of forbidden strings as {} and {}. It is seen that the dataset with more forbidden strings exhibited less steeper mutual information decay than the one with less number of forbidden strings. This can be attributed to the fact that datasets with more complex forbidden strings tend to exhibit more complex grammar as explained in section 2.2. By introducing more number of forbidden strings, it is possible to synthesize more complex LDDs as seen in the plot. In figure 3(e) we can observe the impact of the size of the dataset sampled from the same grammar. It can be seen that datasets sampled from the same grammar are less likely to be affected by the size of the dataset.

These grammars allow for the generation of rich datasets by setting the parameter k, the maximum length of the strings generated, size of vocabulary and by choosing appropriate forbidden substrings. This presents a compelling case to use these grammars to benchmark state-of-the-art sequential models.

3.3. Experiments with Dilated Recurrent Neural Networks

DilatedRNNs use multi-resolution dilated recurrent skip connections to extend the range of temporal dependencies in every layer and upon stacking multiple such layers are able to learn temporal dependencies at different scales (Chang et al., 2017). This stacking of multi-resolution layers helps in passing contextual information over long distances which otherwise would have vanished via a single layer. Thus, the size of the dilations should, ideally, be tailored to match the LDD characteristics present in the dataset, and, in particular, the max dilation should match the max significant LDDs present in the dataset being modeled. The dilations per layer, and the number of layers, within a DilatedRNN are controlled by hyper-parameters (Chang et al., 2017).

In Chang et al. (2017) it is reported that for PTB and sequential MNIST (unpermuted and permuted) the best performance is achieved with max dilations of 64 and 256 respectively. However, Chang et al. (2017) provide no explanation for why these dilations are the optimal settings for these datasets. Given this context, it is interesting that our analysis of the LDD characteristics of PTB and sequential MNIST (unpermuted and permuted) in figures 2(a) (red line) and 2(b) respectively, found the mutual information inflection point for each dataset has similar value as the max dilations reported in Chang et al. (2017). For the PTB dataset, the inflection point is between 40 to 60 (x-axis) and for permuted sequential MNIST datasets, the inflection point is between 200-300 (x-axis). Based on this, we formulated the following hypothesis:

Hypothesis 1 ().

For DilatedRNNs, the hyperparameter value of maximum dilation which lies in the same region as the inflection point of the LDD characteristics, delivers the best performance.

We trained DilatedRNNs on the permuted sequential MNIST dataset (Chang et al., 2017) for three sets of hyper-parameters to confirm that the max dilation and the inflection point have similar value. The settings are mentioned in the table 3. The first 3 settings were used for this experiment. To prevent any influence of other hyper-parameters on training, the original code was kept unchanged except for the setting of dilations. The testing accuracy for each task is plotted in figure 5 for all the 3 tasks. As expected, the set of hyper-parameters with max dilation of 256, delivered the best performance. After analyzing the LDD characteristics, the choice of dilations (powers of 2) make sense as they provide maximum LDD coverage with the least number of stacked layers and hyper-parameters.

   Task No    Task Name   # of layers    Size of Dilations
1 Dilations upto 128 8 1, 2, 4, 8, 16, 32, 64, 128
2 Dilations upto 256 9 1, 2, 4, 8, 16, 32, 64, 128, 256
3 Dilations upto 512 10 1, 2, 4, 8, 16, 32, 64, 128, 256, 512
4 Custom Dilations upto 280 10 1, 2, 4, 8, 16, 32, 64, 128, 256, 280
Table 3. Tasks for training DilatedRNNs

Our analysis of the LDD characteristics of the sequential MNIST task, section 3.1, indicated that this dataset does not exactly follow power law decay. This presents an interesting opportunity to deviate the set of dilations from the powers of 2 in-order to deliver better performance. We trained a DilatedRNN network with max dilation of 280, as we observed 280 as the point of inflection for permuted sequential MNIST. The settings for this experiment is mentioned task no. 4 in table 3. The performance curve is plotted in figure 5 (purple line). This new set of hyper-parameters delivered better performance as compared to max dilation of 256. Both these experiments and results recorded in (Chang et al., 2017) confirmed our hypothesis 1.

Hence, by observing the LDD characteristics, we ensured faster and optimal training of DilatedRNNs by preventing the need for a grid search for this optimal hyper-parameter. We suspect max dilations less than the 280 (such as, up-to 128) inhibit the networks ability to fully capture all LDDs in the dataset, whereas dilations greater than the inflection point (such as, 512) results in the network learning contextually uncorrelated pairs from the data, which leads to degraded testing accuracy. After analyzing the performance of DilatedRNNs it is evident that exponential decay of mutual information in sequential MNIST achieved perplexity of due to lack of LDDs.

Figure 5. Testing accuracy plot of DilatedRNNs with max dilations of 128, 256, 512 and 280.

4. Discussion

The LDD characteristics of a dataset is indicative of the presence of a certain type of grammar in the dataset. For example, our analysis of word-based and character-based datasets in section 3.1, indicate that the word-based grammar is very different from character-based grammar. Understanding the properties of the underlying grammar that produces a language (data sequence) can can aid in choosing optimal sequential model to learn on a given dataset of that language. For example, the maximum length of LDDs is much smaller in word-based datasets as compared to character-based datasets. But at the same time word-based LDD characteristics exhibit higher value of overall mutual information. This is why the sequential model that performs best on the word-based language modeling task will not necessarily be the best for the character based language modeling task.

It can also be noted that even though a specific grammar does induce similar LDD characteristics, there are subtle variations. These variations depend on a number of factors such as size of the vocabulary, size of the dataset, dependency structure (for e.g. “k” and “forbidden strings”) and presence of any other noisy data (or presence of another grammar as seen with Enwik8 dataset). Thus, if a sequential model such as recurrent neural architecture intends to model a dataset, knowing these factors would greatly benefit in selecting the best hyper-parameters of the sequential model.

As seen in LDD characteristics of sequential MNIST, it is evident that the use of standard sequential MNIST in benchmarking tasks is out of place due to the absence of long-range correlations. This presents a compelling case to analyze LDD characteristics of benchmark datasets before they are selected for this job. Also, permutations lead to more complex dependency structure. Thus by altering existing datasets (with short-range dependency) in a way to introduce long-range correlations or LDDs and then analyzing the LDD characteristics presents a more systematic way of building more rich datasets. Even in SPk languages, the choice of forbidden strings allows for the introduction of more complex dependency structure, hence introducing stronger long-range correlations in the generated dataset. This results in a systematic control of the design of benchmarking tasks. This can also be verified by computing LDD characteristics of the generated datasets.

One implication of these experiments is that having multiple benchmark datasets from a single domain does not necessarily improve the experimental testing of a models capacity to model LDDs: essentially, LDDs are fixed within a domain and sampling more datasets from that domain simply results in testing the model on LDDs with similar characteristics. Consequently, the relatively limited set of domains and tasks covered by benchmark datasets indicates that current benchmarks do not provide enough LDD variety to extensively test the capacity of state-of-the-art architectures to model LDDs.

5. Related Work

5.1. Mutual Information and LDDs

Mutual information has previously been used to compute LDDs. In (Ebeling and Poeschel, 2002), two literary texts, Moby Dick by H. Melville and Grimm’s tales were used to analyze maximum length of LDDs present in English text. Correlations were found between few hundred letters. More specifically, strong dependence was observed (large ) upto 30 characters indicating strong grammar, beyond which point the curve exhibited a long tail indicating weak dependence.

(Lin and Tegmark, 2017) analyzed LDD characteristics of enwik8. It was observed that LDDs with power-law correlations tend to be more difficult to model. They argued that LSTMs are capable of modeling sequential datasets exhibiting LDDs with power law correlations such as natural languages far more effectively than markov models; due to power-law decay of hidden state of the LSTM network controlled by the forget gate.

In another experiment, it was observed that DNA nucleotides exhibited long-range power law correlations (Peng et al., 1992; Melnik and Usatenko, 2014).

5.2. Neural Networks and Artificial Grammars

Formal Language Theory, primarily developed to study the computational basis of human language is now being used extensively to analyze any rule-governed system (Chomsky, 1956, 1959; Fitch and Friederici, 2012). Formal languages have previously been used to train RNNs and investigate their inner workings. The Reber grammar (Reber, 1967) was used to train various 1st order RNNs (Casey, 1996; Smith and Zipser, 1989). The Reber grammar was also used as a benchmarking dataset for LSTM models (Hochreiter and Schmidhuber, 1997). Regular languages, studied by Tomita (Tomita, 1982), were used to train 2nd order RNNs to learn grammatical structures of the strings (Watrous and Kuhn, 1991; Giles et al., 1992).

Regular languages are the simplest grammars (type-3 grammars) within the Chomsky hierarchy which are driven by regular expressions. Strictly k-Piecewise languages are natural and can express some of the kinds of LDDs found in natural languages (Jager and Rogers, 2012; Heinz and Rogers, 2010). This presents an opportunity of using SPk grammar to generate benchmarking datasets (Avcu et al., 2017; Mahalunkar and Kelleher, 2018). In Avcu et al. (2017), LSTM networks were trained to recognize valid strings generated using SP2, SP4, SP8 grammar. LSTM could recognize valid strings generated using SP2 and SP4 grammar but struggled to recognize strings generated using SP8 grammar, exposing the performance bottleneck of LSTM networks. It was also observed that by increasing the maximum length of the generated strings of SP2 language thereby increasing the length of LDDs, the performance of LSTM degraded (Mahalunkar and Kelleher, 2018).

6. Conclusion

The foundational contribution of this paper represent a synthesis of distinct themes of research on LDDs from multiple fields, including information theory, artificial neural networks for sequential data modeling, and formal language theory. The potential impact of this synthesis for neural networks research include: an appreciation of the multifaceted nature of LDDs; a procedure for measuring LDD characteristics within a dataset; an evaluation and critique of current benchmark datasets and tasks for LDDs; an analysis of how the use of these standard benchmarks and tasks can be misleading in terms of evaluating the capacity of a neural architectures to generalize to datasets with different forms of LDDs; and, a deeper understanding of the relationship between hyper-parameters and LDDs within language model architectures which can directly contribute to the development of more accurate and efficient sequential models.

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