# The Label Bias Problem

Posted on November 8, 2019

Many sequence classification models suffer from the label bias problem. Understanding the label bias problem and when a certain model suffers from it is subtle but is essential to understand the design of models like conditional random fields and graph transformer networks.

The label bias problem mostly shows up in discriminative sequence models. At its worst, label bias can cause a model to completely ignore the current observation when predicting the next label. How and why this happens is the subject of this section. How to fix it is the subject of the next section.

Suppose we have a task like predicting the parts-of-speech for each word in a sentence. For example, take the sentence “the cat sat” which consists of the tokens [the, cat, sat]. We’d like our model to output the sequence [ARTICLE, NOUN, VERB].

A classic discriminative sequence model for solving this problem is the maximum entropy Markov model (MEMM).1 The graphical model for the MEMM is shown below. Throughout, $X\!=\![x_1, \ldots, x_T]$ is the input or observation sequence and $Y\!=\![y_1, \ldots, y_T]$ is the output or label sequence. The graphical model for the maximum entropy Markov model (MEMM).

The MEMM makes two assumptions about the data. First, $y_t$ is conditionally independent of all previous observations and labels given $x_t$ and $y_{t-1}$. Second, the observations $x_t$ are independent of one another. The first assumption is more central to the model while the second sometimes varies and is easier to relax. These assumptions can be seen from the graphical model. Mathematically, the model is written as $p(Y \mid X) = \prod_{t=1}^T p(y_t \mid x_t, y_{t-1}).$ The probabilities are computed with a softmax: $p(y_t \mid x_t, y_{t-1}) = \frac{e^{s(y_t, x_t, y_{t-1})}}{\sum_{i=1}^c e^{s(y_i, x_t, y_{t-1})}}$ where $c$ is the number of output classes and $s(y_t, x_t, y_{t-1})$ is a scoring function which should give a higher score for $y_t$ which are likely to be the correct label. Since we normalize over the set of possible output labels at each time step, we say the model is locally normalized and $p(y_t \mid x_t, y_{t-1})$ are the local probabilities. The distribution $p(Y \mid X)$ is valid since summing over all possible $Y$ of length $T$ equals one, and all the values are non-negative.

Let’s return to our example of [the, cat, sat]. We can represent the inference process on this sequence using a graph. The states (or nodes) represent the set of possible labels. The transitions (or edges) represent the possible observations along with the associated probabilities $p(y_t~\!\mid~\!x_t, y_{t-1})$. An inference graph for the MEMM with the label set {ARTICLE, NOUN, VERB}. Each arc is labelled with the probability of transitioning to the corresponding state given the observation. The sum of the probabilities on arcs leaving a node for a given observation should sum to one. For simplicity not all observation, score pairs are pictured.

To compute the probability of [ARTICLE, NOUN, VERB], we just follow the observations along each arc leading to the corresponding label. In this case the probability would be $1.0\!\times\!0.9\!\times\!1.0$.

As another example, say we drop the article and just have the sequence [cat sat]. While it’s not a great sentence, the correct label should be [NOUN, VERB]. However, if we follow the probabilities we see that the score for [ARTICLE, NOUN] is $0.9\!\times\!0.3\!=\!0.27$ whereas the score for [NOUN, VERB] is $0.1\!\times\!1.0\!=\!0.1$.

The model is not used to seeing cat at the start of a sentence, so the scores leading from the start state are poorly calibrated. What we need is information about how uncertain the model is for a given observation and previous label pair. If the model has rarely seen the observation cat from the starting node <S> then we want to know that, and it should be included in the scores.

It’s actually possible that the model did at some point implicitly store this uncertainty information. However, by normalizing the outgoing scores for a given observation, we are forcing this information to be discarded. Take a look at the following unnormalized inference graph. An unnormalized inference graph for the MEMM with the label set {ARTICLE, NOUN, VERB}. Each arc is labelled with the score of transitioning to the corresponding state given the observation. For simplicity not all observation, score pairs are pictured.

The unnormalized inference graph corresponds exactly to the normalized inference graph when the scores are exponentiated and normalized. However, we see something interesting here. The outgoing scores for cat from <S> are small. Recall lower scores are worse. This means the model is much less confident about the observation cat from the start state than the observation the which has a score of 100. This information is completely erased when we normalize. That’s one symptom of the label bias problem.

Notice in the unnormalized graph, the score for [ARTICLE, NOUN] is $5 + 21 = 26$ while the score for [NOUN, VERB] is $3 + 100 = 103$. The right answer gets a better score in the unnormalized graph! Note, we are adding scores here instead of multiplying them because the unnormalized scores are in log-space. In other words $p(y_t \mid x_t, y_{t-1}) \propto e^{s(y_t, x_t, y_{t-1})}$.

We can see the label bias problem quantitatively. This observation is due to Denker and Burges.2 Suppose our scoring function $s(y_t, x_t, y_{t-1})$ factorizes into the sum of two functions $f(y_t, x_t, y_{t-1})$ and $g(x_t, y_{t-1})$. Suppose further that $f(\cdot)$ mostly cares about how good the predicted label $y_t$ is given the observation $x_t$, whereas $g(\cdot)$ mostly cares about how good the observation $x_t$ is given the previous label $y_{t-1}$. If we compute the local probabilities using this factorization, we get: $p(y_t \mid x_t, y_{t-1}) = \frac{e^{f(y_t, x_t, y_{t-1}) + g(x_t, y_{t-1})}}{ \sum_{i=1}^c e^{f(y_i, x_t, y_{t-1}) + g(x_t, y_{t-1})}} = \frac{e^{f(y_t, x_t, y_{t-1})}}{\sum_{i=1}^c e^{f(y_i, x_t, y_{t-1})}}.$ The contribution of $g(\cdot)$ in the numerator and denominator cancels. This causes all the information about how likely the observation is given the previous state to be erased.

### Conservation of Score Mass

The label bias problem results from a “conservation of score mass” (Bottou, 1991).3 Conservation of score mass just says that the outgoing scores from a state for a given observation are normalized. This means that all of the incoming probability to a state must leave that state. An observation can only dictate how much of the incoming probability to send where. It cannot change the total amount of probability leaving the state. The net result is any inference procedure will bias towards states with fewer outgoing transitions.

Suppose we have three states, A, B and C, shown below. State A has four outgoing (nonzero) transitions, state B only has two and state C has just one. Suppose all three states distribute probability mass equally among their successor states: $p(y_t \mid x_t, y_{t-1})$ is uniform. An example of three states, A, B and C, which have uniform outgoing transition distributions. Label bias will cause the inference procedure to favor paths which go through state C.

Neither state A, B nor C are doing anything useful here, so we shouldn’t prefer one over the other. But the inference procedure will bias towards paths which go through state C over B and A. Paths which go through A will be the least preferred. To understand this, suppose that the same amount of probability arrives at the three states. State A will decrease the probability mass for any path by a factor of four, whereas state B will only decrease a given path’s score by a factor of two and state C won’t penalize any path at all. In every case the observation is ignored, but the state with the fewest outgoing transitions is preferred.

Even if outgoing transitions from states A and B did not ignore their observations, they would still reduce a paths score since the probabilities aren’t likely to be one. This would cause state C to be preferred even though it always ignores it’s observation.

### Entropy Bias

In a less contrived setting where the distribution $p(y_t \mid x_t, y_{t-1})$ is not the same for every observation, our model will bias towards states which have a low entropy distribution over next states given the previous state. Note this is distinct from the distribution $p(y_t \mid x_t, y_{t-1})$ which can have low entropy without directly causing label bias. However, if the conditional distribution $p(y_t \mid y_{t-1})$ has low entropy then we are potentially in trouble. For example, in the figure above, $p(y \mid \texttt{B})$ has lower entropy than $p(y \mid \texttt{A})$.

Consider the three cases below. In each case there are two possible observations a and b and two possible successor states. We’d like to know which one will introduce the most label bias into the model. To answer that question, we need to make an assumption about the prior probability over observations. Suppose that the prior, $p(x_t)$, is uniform (e.g. $p(a) = p(b) = 0.5$). An example of three states, A, B and C, each with two possible outgoing transitions. Each transition is labelled with the observation and probability pair for two observations, a and b.

We can calculate $p(y_t \mid y_{t-1})$ for any state since $p(y_t \mid y_{t-1}) = \sum_i p(y_t \mid x_i, y_{t-1}) p(x_i) = \frac{1}{n} \sum_i p(y_t \mid x_i, y_{t-1})$ where we used the fact that $p(x_t)$ is uniform and there are $n$ possible observations. In our case $n\!~=\!~2$. The following figure shows $p(y_t \mid y_{t-1})$ on the corresponding arc for each example. An example of three states, A, B and C, each with two possible outgoing transitions. Each transition is labelled with the probability $p(y_t \mid y_{t-1})$ which are computed from the probabilities in the figure above.

Case (a) has the lowest entropy transition distribution whereas case (b) and (c) are equivalent. Intuitively, we expect case (a) to be worse than case (b) since it biases towards the upper path, whereas (b) does not bias towards either. However, case (c) is interesting. In case (c), the observation can have a large effect on the outcome. While this effect might be wrong if the probabilities are poorly calibrated, this case doesn’t cause label bias more than case (b) under the assumption of a uniform prior.

### Revising Past Mistakes

Another description of the label bias problem is that it does not allow a model to easily recover from past mistakes. This follows from the conservation of score mass perspective. If the outgoing score mass of a path is conserved, then at each transition the mass can only decrease. In the future, if a path encounters new evidence which makes it very likely to be correct, it cannot increase the path’s score. The most this new evidence can do is to not decrease the path’s score by preserving all of the incoming mass for that path. So the model’s ability to promote a path given new evidence is limited even if we are certain that the new observation makes this path the correct one.

## Overcoming the Label Bias Problem

As we observed in the [cat, sat] example above, if we don’t normalize scores at each step, then the path scores can retain useful information. This implies that we should avoid locally normalizing probabilities.

One option is that we don’t normalize at all. Instead, when training, we simply tune the model parameters to increase the score $s(X, Y) = \sum_{t=1}^T s(y_t, x_t, y_{t-1})$ for every $(X, Y)$ pair in our training set. The problem with this is there is no competition between possible outputs. This can result in the model maximizing $s(Y, X)$ while completely ignoring its inputs. So we need some kind of competition between outputs. If the score of one output goes up, then others should feel pressure to go down.

This can be achieved with global normalization. We compute the probability as $\gdef\Y{\mathcal{Y}}$ $p(Y \mid X) = \frac{e^{\sum_{t=1}^T s(y_t, x_t, y_{t-1})}}{ \sum_{Y^\prime \in \Y_T} e^{\sum_{t=1}^T s(y_t^\prime, x_t, y^\prime_{t-1})}}$ where $\Y_T$ is the set of all possible $Y$ of length $T$. This is exactly a linear chain conditional random field (CRF).4 The graphical model and hence dependency structure is the same as the MEMM in the previous section. The only difference is how we normalize the scores for a given input.

With this normalization scheme, the label bias problem is no longer an issue. In fact when performing inference, we need not normalize at all. The normalization term is constant for a given $X$ and hence the ordering between the possible $Y$ will be preserved without normalizing. What this means is that the inference procedure operates on an unnormalized graph just like the one we saw for the part-of-speech tagging example in the previous section.

Because transitions have unnormalized scores, they are free to affect the overall path score anyway they please. If cat is very unlikely to follow <s> the model can retain that information by keeping the scores for all transitions out of <s> low. Then whenever we see cat following the state <s>, the path score won’t be affected much since the model is uncertain of what the correct next label is.

This freedom from the label bias problem comes at a cost. Computing the normalization term exactly is more expensive with a CRF than with an MEMM. With an MEMM we normalize locally. Each local normalization costs $\mathcal{O}(c)$ where $c$ is the number of classes, and we have to compute $T$ of them, so the total cost is $\mathcal{O}(cT)$. With a linear chain CRF, on the other-hand, the total cost using an efficient dynamic programming algorithm called the forward-backward algorithm, is $\mathcal{O}(c^2T)$.5 If $c$ is large this can be a major hit to training time. For more complex structures where there can be longer-range dependencies between the outputs, beam search is usually the only option to approximate the normalization term.6

## A Brief History of the Label Bias Problem

The first recorded observation of the label bias problem was in Léon Bottou’s PhD thesis.3 The term “label bias” was coined in the seminal work of Lafferty, McCallum and Pereira introducing conditional random fields.4 Solving the label bias problem was one of the motivations for developing the CRF. The CRF was one of earliest discriminative sequence models to give a principled solution to the label bias problem.

An even earlier sequence model which overcame the label bias problem was the check reading system proposed by Denker and Burges2, though they did not use the term label bias. This work motivated the graph transformer networks of Bottou, LeCun, Bengio and others.7 For more references on graph transformer networks visit Léon Bottou’s page on structure learning systems.

## A Few Examples

In this section we’ll look at a few examples of models, some of which suffer from label bias and some of which do not.

### Hidden Markov Model

The hidden Markov model (HMM) is a generative model which makes two assumptions about the data generating distribution. First, it assumes that the observation $x_t$ is conditionally independent of all other $y$ and $x$ given the hidden state (i.e. label) at time $t$, $y_t$. Second, the HMM makes the usual Markov independence assumption that $y_t$ is conditionally independent of all previous $y$ given $y_{t-1}$. In equations $p(X, Y) = p(y_0) \prod_{t=1}^T p(x_t \mid y_t) p(y_t \mid y_{t-1}).$ This is a very different model from the MEMM. It’s generative, not discriminative, so we estimate $p(X, Y)$ and not $p(Y \mid X)$. Interestingly, the only difference between the graphical model for an HMM and the MEMM is the direction of the arrows between $x_t$ and $y_t$. The graphical model for the hidden Markov model (HMM).

As a simple rule of thumb, generative models do not suffer from label bias. One way to see this for the HMM specifically is to look at the corresponding inference graph. An inference graph for the HMM with the label set {ARTICLE, NOUN, VERB}. The scores for the given observation on each arc are not shown.

The scores on each edge associated with an observation are given by $p(x_t \mid y_t) p(y_t \mid y_{t-1})$. In general the sum of these scores over all possible next states is not required to be one: $\sum_{i=1}^c p(x_t \mid y_i) p(y_i \mid y_{t-1}) \ne 1.$ More importantly, the sum is not a constant, but can change depending on the observation and the previous state. This implies that we do not have conservation of score mass and hence label bias is not an issue.

### Sequence-to-sequence Models

Sequence-to-sequence models with attention are very commonly used to label sequences. These models are discriminative and compute the probability of an output given an input using the chain rule $p(Y \mid X) = \prod_{t=1}^T p(y_t \mid y_{<t}, X) \quad \text{where} \quad p(y_t \mid y_{<t}, X) \propto e^{s(y_t, y_{<t}, X)}.$ The score function $s(\cdot)$ is computed using a multi-layer neural network with attention.

These models are locally normalized, $\sum_{i=1}^c p(y_i \mid y_{<t}, X) = 1,$ hence they can suffer from label bias. Whether or not this is an issue in practice remains to be seen. Some attempts have been made to design globally normalized versions of these models, though none are yet commonly used.8

### Connectionist Temporal Classification

Connectionist Temporal Classification (CTC) is a discriminative sequence model designed to solve problems where the correspondence between the input and output sequence is unknown.9 This includes problems like speech and handwriting recognition among others. (See my Distill article for an in-depth tutorial.10)

$\gdef\A{\mathcal{A}}$

For a given input-output pair $(X, Y)$, CTC allows a set of alignments $\A_{X,Y}$. We let $A\!=\![a_1, \ldots, a_T] \in \A_{X,Y}$ be one such alignment. Note, $A$ has the same length as $X$, namely $T$. The probability of a sequence $Y$ given an input $X$ can then be computed as $p(Y \mid X) = \sum_{A \in \A_{X,Y}} \prod_{t=1}^T p(a_t \mid X) \quad \text{where} \quad p(a_t \mid X) \propto e^{s(a_t, X)}.$ Like sequence-to-sequence models with attention, the score function $s(\cdot)$ is usually computed with a multi-layer neural network. Notice that CTC assumes the outputs $a_t$ are conditionally independent of one another given the input $X$.

The CTC model is a special case in that it is both locally normalized and globally normalized. Because of the conditional independence assumption, the two are equivalent. At the level of an individual alignment, we have $p(A \mid X) = \prod_{t=1}^T \frac{e^{s_t(a_t, X)}}{\sum_{i=1}^c e^{s_t(a_i, X)}} = \frac{\prod_{t=1}^T e^{s_t(a_t, X)}}{\prod_{t=1}^T \sum_{i=1}^c e^{s_t(a_i, X)}}.$ We can rewrite the denominator using the fact that $\prod_{j=1}^m \sum_{i_j=1}^n a_{i_j} = \left(\sum_{i_1=1}^n a_{i_1}\right) \ldots \left(\sum_{i_m=1}^n a_{i_m}\right) = \sum_{i_1=1}^m \ldots \sum_{i_m=1}^n \prod_{j=1}^m a_{i_j}$ to get $p(A \mid X) = \frac{e^{\sum_{t=1}^T s_t(a_t, X)}}{\sum_{A^\prime} e^{\sum_{t=1}^T s_t(a^\prime_t, X)}}.$

Used on its own, CTC does not suffer from label bias. There are a couple of ways to see this. First, as we described, CTC is globally normalized at the level of an alignment and label bias results from local normalization.

Second, the conditional independence assumption made by CTC removes label bias. If the next state prediction does not depend on any previous state, then there is no label bias. The model acts as if the transition probabilities $p(y_t \mid y_{t-1})$ are uniform and the same for all $y_{t-1}$. This means the entropies of these distributions are all the same and maximal.

The model does have conservation of score mass in the sense that $\sum_{i=1}^c p(y_i \mid X) = 1$. However, new evidence can arbitrarily influence the plausibility of a given path. The model can favor paths which have a certain label at a given time step by giving the corresponding $p(y_t \mid X)$ a value close to one. This will in turn make all paths which do not predict $y_t$ have scores very close to zero. However, the expressiveness of the model is also limited since it cannot select for paths based on previously predicted labels.

1. Andrew McCallum, Dayne Freitag, and Fernando C.N. Pereira. Maximum Entropy Markov Models for Information Extraction and Segmentation. ICML 2000. (pdf)

2. John Denker and Christopher Burges. Image Segmentation and Recognition, 1994. (link)  2

3. Léon Bottou. Une Approche théorique de l’Apprentissage Connexionniste: Applications à la Reconnaissance de la Parole, Ph.D. thesis, Université de Paris XI, Orsay, France, 1991. (link)  2

4. John Lafferty, Andrew McCallum, and Fernando C.N. Pereira. Conditional Random Fields: Probabilistic Models for Segmenting and Labeling Sequence Data, 2001. (link)  2

5. Charles Sutton, and Andrew McCallum. An introduction to conditional random fields. Foundations and Trends® in Machine Learning 4.4 2012: 267-373. (link)

6. Ronan Collobert, Awni Hannun, and Gabriel Synnaeve. A fully differentiable beam search decoder. ICML 2019. (link)

7. Léon Bottou, Yoshua Bengio, and Yann LeCun. Global training of document processing systems using graph transformer networks. Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition. IEEE, 1997. (link)

8. Sam Wiseman and Alexander M. Rush. Sequence-to-sequence learning as beam-search optimization. 2016. (link)

9. Alex Graves, Santiago Fernández, Faustino Gomez, and Jürgen Schmidhuber. Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks. ICML, 2006. (link)

10. Awni Hannun. Sequence modeling with CTC. Distill (2017). (link)