Reward Function in Terms of Policy Models and Normalization Factor

In a particular policy optimization framework, the target policy, denoted as $\pi_{\theta}(\mathbf{y}|\mathbf{x})$, is determined by the following relationship involving a reference policy $\pi_{\theta_{\text{ref}}}$, a reward function $r(\mathbf{x}, \mathbf{y})$, a positive temperature parameter $\beta$, and a normalization term $Z(\mathbf{x})$: $\pi_{\theta}(\mathbf{y}|\mathbf{x}) = \frac{\pi_{\theta_{\text{ref}}}(\mathbf{y}|\mathbf{x}) \exp(\frac{1}{\beta}r(\mathbf{x}, \mathbf{y}))}{Z(\mathbf{x})}$ Given this formula, what is the primary effect of significantly increasing the reward $r(\mathbf{x}, \mathbf{y})$ for a single, specific output $\mathbf{y}$, while keeping all other factors constant?

Consider the following equation that defines a target policy $\pi_{\theta}$ based on a reference policy $\pi_{\theta_{\text{ref}}}$, a reward function $r(\mathbf{x}, \mathbf{y})$, a positive scaling parameter $\beta$, and a normalization term $Z(\mathbf{x})$: $\pi_{\theta}(\mathbf{y}|\mathbf{x}) = \frac{\pi_{\theta_{\text{ref}}}(\mathbf{y}|\mathbf{x}) \exp(\frac{1}{\beta}r(\mathbf{x}, \mathbf{y}))}{Z(\mathbf{x})}$ True or False: If the reward function $r(\mathbf{x}, \mathbf{y})$ is equal to zero for all possible outputs $\mathbf{y}$ given an input $\mathbf{x}$, the target policy $\pi_{\theta}(\mathbf{y}|\mathbf{x})$ will be identical to the reference policy $\pi_{\theta_{\text{ref}}}(\mathbf{y}|\mathbf{x})$.

Impact of the Scaling Parameter on Policy Behavior