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💪 Calibrage de prompts

It is possible to counteract some of the biases LLMs exhibit via calibrating **output
distributions**^{1}.

**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 `nothing`

is 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 probabilities 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.

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.

Another solution to this is **contextual calibration**^{1}, 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.

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 $p$.

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

This equation takes the original probabilities $\hat p$ 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 $\hat p$ = [0.5, 0.5].

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

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

$b = 0$

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

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

$\hat p = [0.9, 0.1]$

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

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

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].

$b$ could also be set to $-\hat p$, and $W$ to the identity matrix. This method performs
better on generation rather than classification tasks^{1}.

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

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