Classical \(k\)-gram Language Models in R

Step 1: Loading the training corpus

kgrams offers two options for reading the text corpora used in its computations, which are basically in-memory and out-of-memory solutions:

• in-memory. The corpus is simply loaded in the R session as a character vector.
• out-of-memory. The text is read in batches of fixed size from a connection. This solution includes, for instance, reading text from a file, from an URL, or from the standard input.

The out-of-memory solution can be useful for training over large corpora without the need of storing the entire text into the RAM.

In this vignette, for illustration, we will use the example dataset kgrams::much_ado (William Shakespeare’s “Much Ado About Nothing”).

Step 2: preprocessing and tokenizing sentences

One can (and usually should) apply some transformations to the raw training corpus before feeding it as input to the \(k\)-gram counting algorithm. In particular, the algorithm considers as a sentence each entry of its pre-processed input, and pads each sentence with Begin-Of-Sentence (BOS) and End-Of-Sentence (EOS) tokens. It considers as a word any substring of a sentence delimited by (one or more) space characters.

For the moment, we only need to define the functions used for preprocessing and sentence tokenization. We will use the following functions, which leverage on the basic utilities kgrams::preprocess() and kgrams::tknz_sent(), and perform some additional steps, since we will be reading raw HTML lines from the URL connection created above.

.preprocess <- function(x) {
        # Remove speaker name and locations (boldfaced in original html)
        x <- gsub("<b>[A-z]+</b>", "", x)
        # Remove other html tags
        x <- gsub("<[^>]+>||<[^>]+$||^[^>]+>$", "", x)
        # Apply standard preprocessing including lower-case
        x <- kgrams::preprocess(x)
        # Collapse to a single string to avoid splitting into more sentences at the end of lines
        x <- paste(x, collapse = " ")
        return(x)
}

.tknz_sent <- function(x) {
        # Tokenize sentences
        x <- kgrams::tknz_sent(x, keep_first = TRUE)
        # Remove empty sentences
        x <- x[x != ""]
        return(x)
}

Step 3: get \(k\)-gram frequency counts

We can now obtain \(k\)-gram frequency counts from Shakespeare with a single command, using the function kgram_freqs(). The following stores \(k\)-gram counts for \(k\)-grams of order less than or equal to \(N = 5\):

freqs <- kgram_freqs(much_ado, # Read Shakespeare's text from connection
                     N = 5, # Store k-gram counts for k <= 5
                     .preprocess = .preprocess,  # preprocess text
                     .tknz_sent = .tknz_sent, # tokenize sentences
                     verbose = FALSE
                     )
freqs
#> A k-gram frequency table.

The object freqs is an object of class kgram_freqs, i.e. a \(k\)-gram frequency table. We can obtain a first informative summary of what this object contains by calling summary():

summary(freqs)
#> A k-gram frequency table.
#> 
#> Parameters:
#> * N: 5
#> * V: 3046
#> 
#> Number of words in training corpus:
#> * W: 26123
#> 
#> Number of distinct k-grams with positive counts:
#> * 1-grams:3048
#> * 2-grams:14373
#> * 3-grams:21201
#> * 4-grams:22705
#> * 5-grams:22974

The parameter V is the size of the dictionary, which was created behind the scenes by kgram_freqs(), using all words encountered in the text. In alternative, we could have used a pre-specified dictionary through the argument dict, and specify whether new words (not present in the original dictionary) should be added to it, or be replaced by an Unknown-Word (UNK) token, by the argument open_dict; see ?kgram_freqs for further details. The number of distinct unigrams is greater than the size of the dictionary, because the former also includes the special BOS and EOS tokens.

Notice that the functions .preprocess() and .tknz_sent() we defined above are passed as arguments of kgram_freqs()³. These are also saved in the final kgram_freqs object, and are by default applied also to inputs at prediction time.

The following shows how to query \(k\)-gram counts from the frequency table created above ⁴:

# Query some simple unigrams and bigrams
query(freqs, c("leonato", "enter leonato", "thy", "smartphones"))
#> [1] 38  6 52  0
# Query k-grams at the beginning or end of a sentence
query(freqs, c(BOS() %+% BOS() %+% "i", "love" %+% EOS()))
#> [1] 206   0
# Total number of words processed
query(freqs, "") 
#> [1] 26123
# Total number of sentences processed
query(freqs, EOS()) 
#> [1] 2219

The most important use of kgram_freqs objects is to create language models, as we illustrate in the next step.

Step 4. Build the final language model

kgrams provides support for creating language models using several classical smoothing techniques. The list of smoothers currently supported by kgrams can be retrieved through:

smoothers()
#> [1] "ml"    "add_k" "abs"   "kn"    "mkn"   "sbo"   "wb"

The documentation page ?smoothers provides a list of original references for the various smoothers. We will use Interpolated Kneser-Ney smoothing (Kneser and Ney 1995; see also Chen and Goodman 1999), which goes under the code "kn". We can get some usage help for this method through the command:

info("kn")
#> Interpolated Kneser-Ney
#>  * code: 'kn'
#>  * parameters: D
#>  * constraints: 0 <= D <= 1

As shown above, Kneser-Ney has one parameter \(D\), which is the discount applied to bare \(k\)-gram frequency counts or continuation counts. We will initialize the model with \(D = 0.75\), and later tune this parameter through a test corpus.

To train a language model with the \(k\)-gram counts stored in freqs, use:

kn <- language_model(freqs, "kn", D = 0.75)
kn
#> A k-gram language model.

This will create a language_model object, which can be used to obtain word continuation and sentence probabilities. Let us first get a summary of our final model:

summary(kn)
#> A k-gram language model.
#> 
#> Smoother:
#> * 'kn'.
#> 
#> Parameters:
#> * N: 5
#> * V: 3046
#> * D: 0.75
#> 
#> Number of words in training corpus:
#> * W: 26123
#> 
#> Number of distinct k-grams with positive counts:
#> * 1-grams:3048
#> * 2-grams:14373
#> * 3-grams:21201
#> * 4-grams:22705
#> * 5-grams:22974

The parameter D can be accessed and modified through the functions parameters() and param(), which have a similar interface to the base R function attributes() and attr():

parameters(kn)
#> $N
#> [1] 5
#> 
#> $V
#> [1] 3046
#> 
#> $D
#> [1] 0.75
param(kn, "D")
#> [1] 0.75
param(kn, "D") <- 0.6
param(kn, "D")
#> [1] 0.6
param(kn, "D") <- 0.75

We can also modify the order of the \(k\)-gram model, by choosing any number less than or equal to \(N = 5\) (since we stored up to \(5\)-gram counts):

param(kn, "N") <- 4 # 'kn' uses only 1:4-grams
param(kn, "N")
#> [1] 4
param(kn, "N") <- 5 # 'kn' uses also 5-grams

In the next section we show how to use this language model for basic tasks such as predicting word and sentence probabilities, and for more complex tasks such as computing perplexities and generating random text.

Word continuation and sentence probabilities

We can obtain both sentence probabilities and word continuation probabilities through the function probability(). This is generic on the first argument, which can be a character for sentence probabilities, or a word_context expression for continuation probabilities.

Sentence probabilities can be obtained as follows (the first two are sentences from the training corpus):

probability(c("Did he break out into tears?",
              "I see, lady, the gentleman is not in your books.",
              "We are predicting sentence probabilities."
              ),
            model = kn
            )
#> [1] 2.781389e-05 8.821088e-07 9.482178e-19

Behind the scenes, the same .preprocess() and .tknz_sent() functions used during training are being applied to the input. We can turn off this behaviour by explicitly specifying the .preprocess and .tknz_sent arguments of probability().

Word continuation probabilities are the conditional probabilities of words following some given context. For instance, the probability that the words "tears" or "pieces" will follow the context "Did he break out into" are computed as follows:

probability("tears" %|% "Did he break out into", model = kn)
#> [1] 0.5813744
probability("pieces" %|% "Did he break out into", model = kn)
#> [1] 1.000375e-05

The operator %|% takes as input a character vector on its left-hand side, i.e. the list of candidate words, and a length one character vector on its right-hand side, i.e. the given context. If the context has more than \(N - 1\) words (where \(N\) is the order of the language model, five in our case), only the last \(N - 1\) words are used for prediction.

Generating random text

We can sample sentences from the probability distribution defined by our language model using sample_sentences(). For instance:

set.seed(840)
sample_sentences(model = kn, 
                 n = 10,
                 max_length = 10
                 )
#>  [1] "i have studied officers ; <EOS>"                                              
#>  [2] "truly by in your company thing that you ask for [...] (truncated output)"     
#>  [3] "i protest i love the gentleman is wise ; <EOS>"                               
#>  [4] "for it . <EOS>"                                                               
#>  [5] "the best befits can i for your own hobbyhorses hence [...] (truncated output)"
#>  [6] "but by this travail fit the length july cham's beard [...] (truncated output)"
#>  [7] "don pedro she doth well as being some attires and [...] (truncated output)"   
#>  [8] "exeunt all ladies only spots of grey all the wealth [...] (truncated output)" 
#>  [9] "heighho ! <EOS>"                                                              
#> [10] "exit margaret ursula friar . <EOS>"

The sampling is performed word by word, and the output is truncated if no EOS token is found after sampling max_length words.

We can also sample with a temperature different from one. The temperature transformation of a probability distribution \(p(i)\) is defined by:

\[p_t(i) = \dfrac{\exp(\log{p(i)} / t)} {Z(t)},\] where \(Z(t)\) is the partition function, defined in such a way that \(\sum _i p_t(i) \equiv 1\). Intuitively, higher and lower temperatures make the original probability distribution smoother and rougher, respectively. By making a physical analogy, we can think of less probable words as states with higher energies, and the effect of higher (lower) temperatures is to make more (less) likely to excite these high energy states.

We can test the effects of temperature on our Shakespeare-inspired language model, by changing the parameter t of sample_sentences() (notice that the default t = 1 corresponds to the original distribution):

sample_sentences(model = kn, 
                 n = 10,
                 max_length = 10, 
                 t = 0.1 # low temperature
                 )
#>  [1] "i will not have to do with you . <EOS>"                                    
#>  [2] "i will go before and show him their examination . [...] (truncated output)"
#>  [3] "i will not be sworn but love may transform me [...] (truncated output)"    
#>  [4] "i will go get her picture . <EOS>"                                         
#>  [5] "i will go on the slightest errand now to the [...] (truncated output)"     
#>  [6] "i will not be sworn but love may transform me [...] (truncated output)"    
#>  [7] "i will not be sworn but love may transform me [...] (truncated output)"    
#>  [8] "and i will like a true drunkard utter all to [...] (truncated output)"     
#>  [9] "i will not be sworn but love may transform me [...] (truncated output)"    
#> [10] "i will not be sworn but love may transform me [...] (truncated output)"
sample_sentences(model = kn, 
                 n = 10,
                 max_length = 10, 
                 t = 10 # high temperature
                 )
#>  [1] "minds wants valuing peace speech libertines after offered being braggarts [...] (truncated output)"         
#>  [2] "smell fiveandthirty from possible knowest sickness tonight panders agony show'd [...] (truncated output)"   
#>  [3] "where's give intelligence princes finer tire scab brought rearward deserved [...] (truncated output)"       
#>  [4] "heart's evening virtues holds c hadst persuasion can finer churchbench [...] (truncated output)"            
#>  [5] "modesty thinks noncome remorse epitaphs consented mortifying whom hath expectation [...] (truncated output)"
#>  [6] "impossible yielded deceive wedding mouth unclasp absentand qualify twelve giddily [...] (truncated output)" 
#>  [7] "certainly nightraven prized grief laugh claw invincible tyrant blessed run [...] (truncated output)"        
#>  [8] "senseless beat time denies 'hundred ten forth hire' reenter toothache [...] (truncated output)"             
#>  [9] "laughed civil kill'd mean hero's yea foundation deformed appetite hour [...] (truncated output)"            
#> [10] "studied figure nine leonato enough ever herself confess authority sanctuary [...] (truncated output)"

As explained above, sampling with low temperature gives much more weight to probable sentences, and indeed the output is very repetitive. On the contrary, high temperature makes sentence probabilities more uniform, and in fact our output above looks quite random.

Compute language model’s perplexities

Perplexity is a standard evaluation metric for the overall performance of a language model. It is given by \(P = e^H\), where \(H\) is the cross-entropy of the language model sentence probability distribution against a test corpus empirical distribution:

\[ H = - \dfrac{1}{W}\sum _s\ \ln (\text {Prob}(s)) \] Here \(W\) is total number of words in the test corpus (following Ref. (Chen and Goodman 1999), we include counts of the EOS token, but not the BOS token, in \(W\)), and the sum extends over all sentences in the test corpus. Perplexity does not give direct information on the performance of a language model in a specific end-to-end task, but is often found to correlate with the latter, which provides a practical justification for the use of this metric. Notice that better models are associated with lower perplexities, and that \(H\) is proportional to the negative log-likelihood of the corpus under the language model assumption.

Perplexities can be computed in kgrams using the function perplexity(), which can read text both from a character vector and from a connection. We will take our test corpus again from Shakespeare’s opus, specifically the play “A Midsummer Night’s Dream”, which is example data from kgrams namespace:

midsummer[840]
#> [1] " when truth kills truth o devilishholy fray !"

We can compute the perplexity of our Kneser-Ney \(5\)-gram model kn against this corpus as follows:

perplexity(midsummer, model = kn)
#> [1] 376.3205

We can use perplexity to tune our model parameter \(D\). We compute perplexity over a grid of values for D and plot the results. We do this for the \(k\)-gram models with \(k \in \{2, 3, 4, 5\}\):

D_grid <- seq(from = 0.5, to = 0.99, by = 0.01)
FUN <- function(D, N) {
        param(kn, "N") <- N
        param(kn, "D") <- D
        perplexity(midsummer, model = kn)
}
P_grid <- lapply(2:5, function(N) sapply(D_grid, FUN, N = N))
oldpar <- par(mar = c(2, 2, 1, 1))
plot(D_grid, P_grid[[1]], type = "n", xlab = "D", ylab = "Perplexity", ylim = c(300, 500))
lines(D_grid, P_grid[[1]], col = "red")
lines(D_grid, P_grid[[2]], col = "chartreuse")
lines(D_grid, P_grid[[3]], col = "blue")
lines(D_grid, P_grid[[4]], col = "black")

Perplexity as a function of the discount parameter of Interpolated Kneser-Ney 2-gram (red), 3-gram (green), 4-gram (blue) and 5-gram (black) models.

par(oldpar)

We see that the optimal choices for D are close to its maximum allowed value D = 1, for which the performance of the 2-gram model is slightly worse than the higher order models, and that the 5-gram model performs generally worse than the 3-gram and 4-gram models. Indeed, the optimized perplexities for the various \(k\)-gram orders are given by:

sapply(c("2-gram" = 1, "3-gram" = 2, "4-gram" = 3, "5-gram" = 4),
       function(N) min(P_grid[[N]])
       )
#>   2-gram   3-gram   4-gram   5-gram 
#> 324.3317 320.0537 319.4006 319.4984

which shows that the best performing model is the 4-gram one, while it seems that the 5-gram model is starting to overfit (which is not very surprising, given the ridiculously small size of our training corpus!).