In the formula for a reward-weighted probability distribution, $\pi^{*}(\mathbf{y}|\mathbf{x}) = \frac{\pi_{\theta_{\text{ref}}}(\mathbf{y}|\mathbf{x}) \exp \left(\frac{1}{\beta}r(\mathbf{x}, \mathbf{y})\right)}{Z(\mathbf{x})}$ the parameter β acts as a temperature or inverse scaling factor. How does decreasing the value of β (i.e., moving it closer to 0, but remaining positive) affect the final distribution π*?

Applying a Reward Function to a Language Model's Output

Target Policy as a Reward-Weighted Distribution

In the context of a reward-weighted probability distribution, defined as $\pi^{*}(\mathbf{y}|\mathbf{x}) = \frac{\pi_{\theta_{\text{ref}}}(\mathbf{y}|\mathbf{x}) \exp \left(\frac{1}{\beta}r(\mathbf{x}, \mathbf{y})\right)}{Z(\mathbf{x})}$, consider a scenario where a specific output, $\mathbf{y}_A$, receives a very high reward, $r(\mathbf{x}, \mathbf{y}_A)$. However, the reference distribution assigns a probability to this output that is extremely close to zero, i.e., $\pi_{\theta_{\text{ref}}}(\mathbf{y}_A|\mathbf{x}) \approx 0$. What will be the approximate probability of $\mathbf{y}_A$ in the final distribution, $\pi^{*}(\mathbf{y}_A|\mathbf{x})$?