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CRF, CTC, Word2vec NPFL114, Lecture 9 CRF, CTC, Word2Vec Milan Straka April 26, 2021 Charles University in Prague Faculty of Mathematics and Physics Institute of Formal and Applied Linguistics unless otherwise stated Structured Prediction Structured Prediction NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 2/30 Structured Prediction Consider generating a sequence of N given input . y1, … , yN ∈ Y x1, … , xN Predicting each sequence element independently models the distribution . P (yi∣ X) However, there may be dependencies among the themselves, which is difficult to capture by yi independent element classification. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 3/30 Maximum Entropy Markov Models We might model the dependencies by assuming that the output sequence is a Markov chain, and model it as P (yi∣ X, yi−1) . Each label would be predicted by a softmax from the hidden state and the previous label. The decoding can be then performed by a dynamic programming algorithm. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 4/30 Maximum Entropy Markov Models However, MEMMs suffer from a so-called label bias problem. Because the probability is factorized, each is a distribution and must sum to one. P (yi∣ X, yi−1) Imagine there was a label error during prediction. In the next step, the model might “realize” that the previous label has very low probability of being followed by any label – however, it cannot express this by setting the probability of all following labels low, it has to “conserve the mass”. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 5/30 Conditional Random Fields Let G = (V , E) be a graph such that y is indexed by vertices of G. Then (X, y) is a conditional random field, if the random variables y conditioned on X obey the Markov property with respect to the graph, i.e., P (yi∣ X, yj , i = j) = P (yi∣ X, yj ∀j : (i, j) ∈ E). By a fundamental theorem of random fields, the density of a conditional random field can be factorized over the cliques of the graph G: P (y∣X) = P (y ∣ X). ∏ C clique C of G NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 6/30 Linear-Chain Conditional Random Fields (CRF) Usually often assume that dependencies of y, conditioned on X, form a chain. Then, the cliques are nodes and edges, and we usually factorize the probability as: N N P (y∣X) ∝ exp log P (y ∣ x ) + log P (y , y ) . ( ∑ i i ∑ i i−1 ) i=1 i=2 NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 7/30 Linear-Chain Conditional Random Fields (CRF) Linear-chain Conditional Random Field, usually abbreviated only to CRF, acts as an output layer. It can be considered an extension of softmax – instead of a sequence of independent softmaxes, it is a sentence-level softmax, with additional weights for neighboring sequence elements. We start by defining a score of a label sequence y as N s(X, y; θ, A) = (Ay , y + fθ(yi∣X)) ∑i=1 i−1 i and define the probability of a label sequence y using softmax: p y X N s X z ( ∣ ) = softmaxz∈Y ( ( , ))y . For cross-entropy (and also to avoid underflow), we need a logarithm of the probability: log p(y∣X) = s(X, y) − logsumexpz∈Y N (s(X, z)), where f(x) logsumexpx (f(x)) = log(∑x e ). NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 8/30 Linear-Chain Conditional Random Fields (CRF) Computation We can compute efficiently using dynamic programming. We denote the p(y∣X) αt( k) logarithmic probability of all t-element sequences with the last label y being k. The core idea is the following: αt( k) = fθ( yt = k∣X) + logsumexpj∈Y ( αt−1( j) + Aj,k) . For efficient implementation, we use the fact that ln(a + b) = ln a + ln(1 + eln b−ln a ), so f(x)−maxx ( f(x)) logsumexpx (f(x)) = maxx (f(x)) + log(∑x e ). NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 9/30 Conditional Random Fields (CRF) Inputs: Network computing , an unnormalized probability of output sequence fθ( yt = k∣X) element probability being k at time t. Y Y Inputs: Transition matrix A ∈ R × . Inputs: Input sequence X of length N, gold labeling g ∈ Y N . Outputs: Value of log p(g∣X). Time Complexity: O(N ⋅ Y 2). For t = 1, … , N: For k = 1, … , Y : αt( k) ← fθ( yt = k∣X) If t > 1: ∣ αt(k) ← αt(k) + logsumexp (αt−1(j) + Aj,k ∣ j = 1, … , Y ) Return N N Y ∑t=1 fθ(yt = gt∣X) + ∑t=2 Agt−1, gt − logsumexpk=1(αN (k)) NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 10/30 Conditional Random Fields (CRF) Decoding We can perform decoding optimally, by using the same algorithm, only replacing logsumexp with max and tracking where the maximum was attained. Applications The CRF output layer is useful for span labeling tasks, like named entity recognition, dialog slot filling. It can be also useful for image segmentation. Figure 1 of paper "Multi-Task Cross-Lingual Sequence Tagging from Scratch", https://arxiv.org/abs/1603.06270. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 11/30 Connectionist Temporal Classification Let us again consider generating a sequence of given input , but this y1, … , yM x1, … , xN time M ≤ N and there is no explicit alignment of x and y in the gold data. Figure 7.1 of the dissertation "Supervised Sequence Labelling with Recurrent Neural Networks" by Alex Graves. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 12/30 Connectionist Temporal Classification We enlarge the set of output labels by a – (blank) and perform a classification for every input element to produce an extended labeling. We then post-process it by the following rules (denoted as B): 1. We collapse multiple neighboring occurrences of the same symbol into one. 2. We remove the blank –. Because the explicit alignment of inputs and labels is not known, we consider all possible alignments. Denoting the probability of label at time as t, we define l t pl t t def t′ α s p ( ) = . ∑ ∏ πt′ extended t′ =1 labelings π: B(π1:t ) =y1:s NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 13/30 CRF and CTC Comparison In CRF, we normalize the whole sentences, therefore we need to compute unnormalized probabilities for all the (exponentially many) sentences. Decoding can be performed optimally. In CTC, we normalize per each label. However, because we do not have explicit alignment, we compute probability of a labeling by summing probabilities of (generally exponentially many) extended labelings. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 14/30 Connectionist Temporal Classification Computation When aligning an extended labeling to a regular one, we need to consider whether the extended labeling ends by a blank or not. We therefore define t t def t′ α s p ( ) = − ∑ ∏ πt′ extended t′ =1 labelings π: B(π1:t ) =y1:s , πt= − t t def t′ α s p ( ) = ∗ ∑ ∏ πt′ extended t′ =1 labelings π: B(π1:t ) =y1:s , πt= − and compute t as t t . α (s) α− ( s) + α∗( s) NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 15/30 Connectionist Temporal Classification Computation – Initialization We initialize αs as follows: 1 1 α− ( 0) ← p− 1 1 α p ∗(1) ← y1 Computation – Induction Step We then proceed recurrently according to: Figure 7.3 of the dissertation "Supervised Sequence Labelling with t t t−1 t−1 Recurrent Neural Networks" by Alex Graves. α− ( s) ← p−( α∗ ( s) + α− ( s)) t t−1 t−1 t−1 p α s α s α s y y t ys ( ∗ ( ) + − ( − 1) + ∗ ( − 1)), if s = s−1 α s ∗( ) ← t t−1 t−1 p α s α s y y { ys ( ∗ ( ) + − ( − 1)), if s = s−1 NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 16/30 CTC Decoding Unlike CRF, we cannot perform the decoding optimally. The key observation is that while an optimal extended labeling can be extended into an optimal labeling of a larger length, the same does not apply to regular (non-extended) labeling. The problem is that regular labeling corresponds to many extended labelings, which are modified each in a different way during an extension of the regular labeling. Figure 7.5 of the dissertation "Supervised Sequence Labelling with Recurrent Neural Networks" by Alex Graves. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 17/30 CTC Decoding Beam Search To perform beam search, we keep k best regular (non-extended) labelings. Specifically, for each regular labeling we keep both t and t , which are probabilities of all (modulo beam y α− ( y) α∗( y) search) extended labelings of length t which produce the regular labeling y; we therefore keep k regular labelings with highest t t . α− ( y) + α∗( y) To compute the best regular labelings for longer prefix of extended labelings, for each regular labeling in the beam we consider the following cases: adding a blank symbol, i.e., contributing to t+1 both from t and t ; α− ( y) α− ( y) α∗( y) adding a non-blank symbol, i.e., contributing to t+1 from t and to possibly α∗ ( ⋅) α− ( y) different t+1 from t . α∗ ( ⋅) α∗( y) Finally, we merge the resulting candidates according to their regular labeling and keep only the k best. NPFL114, Lecture 9 CRF CTC CTCDecoding Word2Vec CLEs Subword Embeddings 18/30 Unsupervised Word Embeddings The embeddings can be trained for each task separately.
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