🔴 Calibrating LLMs

It is possible to counteract some of the biases LLMs exhibit via calibrating output distributions¹.

What exactly does it mean to calibrate an output distribution?

Let's walk through a quick example: Say we have a sentiment analysis task with two possible labels, Positive and Negative. Consider what happens when the LLM is prompted with Input: nothing Sentiment: . This input doesn't contain any context which the LLM can use to make a sentiment prediction, so it is called a context-free input.

Since nothingis neither a positive nor a negative concept, we would expect the LLM to output a probability of about 0.5 for both Positive and Negative. However, often (and for this example) that will not be the case.

p("Positive" | "Input: nothing Sentiment:") = 0.9

p("Negative" | "Input: nothing Sentiment:") = 0.1

Given these label probabilites for a context-free input, we know that the LLM's output distribution is likely biased towards the label Positive. This may cause the LLM to favor Positive for all inputs, even if the input is not actually positive.

If we can somehow calibrate the output distribution, such that context-free inputs are assigned a probability of 0.5 for both Positive and Negative, then we can often remove the bias towards Positive and the LLM will be more reliable on both context-free inputs and inputs with context.

Non-Technical Solution

A non-technical solution to this problem is to simply provide few shot examples where context-free exemplars are effectively assigned a probability of 0.5 for both Positive and Negative.

For example, we could provide the following few shot examples which show each context-free exemplar being classified as both Positive and Negative:

Input: I hate this movie. Sentiment: Negative
Input: I love this movie. Sentiment: Positive
Input: N/A Sentiment: Positive
Input: N/A Sentiment: Negative
Input: nothing Sentiment: Positive
Input: nothing Sentiment: Negative
Input: I like eggs. Sentiment:

To my knowledge, this solution has not been explored in the literature, and I am not sure how well it works in practice. However, it is a simple solution that demonstrates what calibration is trying to achieve.

Technical Solution

Another solution to this is contextual calibration¹, where we adjust special calibration parameters, which ensure that context-free inputs like Input: nothing Sentiment: are assigned a probability of about 0.5 for both labels. Note that in practice this method performs calibration over multiple different context free inputs (e.g. Input: N/A Sentiment: , Input: [MASK] Sentiment: ). It averages the calibration parameters that work best for each context-free input to find the best calibration parameters for the LLM.

Example

Let's go through an example of computing the calibration parameters for one context-free input. Note that this example is not reproducible with GPT-3 due to the fact that it can't be restricted to the labels Positive and Negative.

Consider again the above example where the LLM assigns the following probabilities to the labels for a context-free input:

p("Positive" | "Input: nothing Sentiment:") = 0.9

p("Negative" | "Input: nothing Sentiment:") = 0.1

We want to find some probability distribution q such that

q("Positive" | "Input: nothing Sentiment:") = 0.5

q("Negative" | "Input: nothing Sentiment:") = 0.5

We will do so by creating a linear transformation that adjusts (calibrates) the probabilities of $p$.

$\^{q} = \text{Softmax}(W\^{p} + b)$

This equation takes the original probabilities $\^{p}$ and applies the weights $W$ and bias $b$ to them. The weights $W$ and bias $b$ are the calibration parameters, which, when applied to the context-free example's probabilites, will yield $\^{q}$ = [0.5, 0.5].

Computing W and b

We need to somehow compute the weights $W$ and bias $b$. One way to do this is:

$W = \text{diag}(\^{p})^{-1}$

$b = 0$

Although the definition of $W$ may seem a bit strange at first, but it is just taking the inverse of each value in $\^{p}$ in order to find a $W$ that will transform the original probabilities $\^{p}$ into the calibrated probabilities [0.5, 0.5].

Let's verify that this works for the example above:

$\^{p} = [0.9, 0.1]$

$W = \text{diag}(\^{p})^{-1} = \text{diag}([0.9, 0.1])^{-1} = \begin{bmatrix} 0.9 & 0 \\ 0 & 0.1 \end{bmatrix}^{-1} = \begin{bmatrix} 1.11 & 0 \\ 0 & 10 \end{bmatrix}$

$\^{q} = \text{Softmax}(W\^{p} + b) = \text{Softmax}(\begin{bmatrix} 1.11 & 0 \\ 0 & 10 \end{bmatrix}*{[0.9, 0.1]} + 0) = \text{Softmax}([1, 1]) =[0.5, 0.5]$

As mentioned above, we would perform this same process for multiple different context-free inputs, and average the calibration parameters that work best for each context-free input to find the best calibration parameters for the LLM. This means that the final calibration parameters willl probably not map any of the context-free inputs to exactly [0.5, 0.5].

Another method

$b$ could also be set to $-\^{p}$, and $W$ to the identity matrix. This method performs better on generation rather than classification tasks¹.

Takeaways

LLMs are often predisposed (biased) towards certain labels. Calibration can be used to counteract this bias.

---

1. Zhao, T. Z., Wallace, E., Feng, S., Klein, D., & Singh, S. (2021). Calibrate Before Use: Improving Few-Shot Performance of Language Models. ↩