Houjun Liu

SU-CS224N MAY022024

Zero-Shot Learning

GPT-2 is able to do many tasks with not examples + no gradient updates.

Instruction Fine-Tuning

Language models, by default, are not aligned with user intent.

  1. collect paired examples of instruction + output across many tasks
  2. then, evaluate on unseen tasks

~3 million examples << n billion examples

dataset: MMLU

You can generate an Instruction Fine-Tuning dataset by asking a larger model for it (see Alpaca).

Pros + Cons

  • simple and straightforward + generalize to unseen tasks
  • but, its EXPENSIVE to collect ground truth data
    • ground truths maybe wrong
    • creative tasks may not have a correct answer
    • LMs penalizes all token-level mistakes equally, but some mistakes are worse than others
    • humans may generate suboptimal answers

Human Preference Modeling

Imagine if we have some input \(x\), and two output trajectories, \(y_{1}\) and \(y_{2}\).

Suppose we have \(R(x, y)\).

We desire:

\begin{equation} \mathbb{E}_{\hat{y} \sim p_{\theta}(y | x)} R(x, y) \end{equation}

RLHF, in broad strokes

  1. do Instruction Fine-Tuning
  2. estimate a reward model \(R(x,y)\)
  3. maximize that reward model

Model Preferences as an NLP Problem


\begin{equation} RM_{\phi}(x, y) \end{equation}

which models a human preference scores.

Get preference data

To get the preference data actually, ask the humans to RANK the rankings.

Bradley-Terry Preference Model

Suppose a human chose \(y^{w}\) over \(y^{l}\). Then, Bradley-Terry Preference Model tells us that a good reward model \(R\) will minimize:

\begin{equation} J(\phi) = -\mathbb{E}_{(x, y^{w}, y^{l})} \qty[\log \sigma \qty(R_{\phi}(x, y^{w}) - R_{\phi}(x, y^{l}))] \end{equation}


Then, we optimize this:

\begin{equation} \max_{\theta} \mathbb{E}\qty[ RM_{\phi}(x, \hat{y}) - \beta \log \qty( \frac{P_{\theta}^{RL} (\hat{y} | x)}{P_{\theta}^{orig} (\hat{y} | x)})] \end{equation}

we have a penalty term to prevent large drifts.


What if there is a way to write \(R_{\phi}(x,y)\) directly in terms of \(p_{\theta}^{RL}(\hat{y}|x)\)?

Our goal is to solve this problem:

\begin{equation} \max_{\theta} \mathbb{E}\qty[ RM(x, \hat{y}) - \beta \log \qty( \frac{P_{\theta}^{RL} (\hat{y} | x)}{P_{\theta}^{orig} (\hat{y} | x)})] \end{equation}

There’s actually a closed-form solution to this!

\begin{equation} p^{*} (\hat{y} | x) = \frac{1}{Z(x)} p^{orig} (\hat{y}|x) \exp \qty(\frac{1}{\beta} RM(x| \hat{y})) \end{equation}

where \(Z(x) = \sum_{\hat{y} \in y}^{} p^{orig}(\hat{y}|x)\).

Notice! Computing the normalization term \(Z\) is intractable! But first, we rearrange this equation to get:

\begin{equation} RM(x, \hat{y}) = \beta \log \frac{p^{*}(\hat{y}|x)}{p^{PT}(\hat{y}|x)} + \beta \log Z(x) \end{equation}

Now, we want to solve for a \(p^{*}\) given reward signal, so let’s parametrize it:

\begin{equation} RM_{\theta}(x, \hat{y}) = \beta \log \frac{p_{\theta}(\hat{y}|x)}{p^{PT}(\hat{y}|x)} + \beta \log Z(x) \end{equation}

(issue: wait, but in the beginning \(\theta = PT\), so \(\log (1) = 0\), and this whole thing is \(0\)…. we will get to that, also \(Z\) is still intractable.)

Now! Recall Bradley-Terry Preference Model: a good \(RM_{\theta}\) should

\begin{equation} \min_{\phi} -\mathbb{E}_{(x, y^{w}, y^{l})} \qty[\log \sigma \qty(R_{\phi}(x, y^{w}) - R_{\phi}(x, y^{l}))] \end{equation}

Plugging our expression for \(RM_{\theta}\) from an equation ago into here, you’ll notice the \(Z\) CANCELLS OUT! And this gives:

\begin{equation} \min_{\theta} -\mathbb{E}_{(x, y^{w}, y^{l})} \qty[\log \sigma \qty(\beta \log \frac{p_{\theta}(y^{w}|x)}{p_{PT}(y^{w}|x)} - \beta \log \frac{p_{\theta}(y^{l}|x)}{p_{PT}(y^{l}|x)})] \end{equation}