# How is inequality 31 derived from equality 30 in lemma 2 of the "Trust Region Policy Optimization" paper?

In the Trust Region Policy Optimization paper, in Lemma 2 of Appendix A (p. 11), I didn't quite understand how inequality (31) is derived from equality (30), which is:

$$\bar{A}(s) = P(a \neq \tilde{a} | s) \mathbb{E}_{(a, \tilde{a}) \sim (\pi, \tilde{\pi})|a \neq \tilde{a}} \left[ A_{\pi}(s, \tilde{a}) - A_{\pi}(s,a) \right]$$ $$|\bar{A}(s)| \le \alpha. 2 \max_{s,a} |A_{\pi}(s,a)|$$

Would you mind let me know how the inequality is derived?

$$\bar{A}(s) = P(a \neq \tilde{a}) \mathbb{E}_{(a,\tilde{a})\sim(\pi,\tilde{\pi}|a\neq\tilde{a})} [A_\pi(s, \tilde{a}) - A_\pi(s, a)]$$

Taking the absolute value of both sides, the equality remains true. We can pull the probability term out of the absolute value since it is guaranteed to be nonnegative.

$$|\bar{A}(s)| = P(a \neq \tilde{a}) |\mathbb{E}_{(a,\tilde{a})\sim(\pi,\tilde{\pi}|a\neq\tilde{a})} [A_\pi(s, \tilde{a}) - A_\pi(s, a)]|$$

By Definition 1, $$P(a \neq \tilde{a}) \leq \alpha$$. Substituting this definition in, we get:

$$|\bar{A}(s)| \leq \alpha \cdot |\mathbb{E}_{(a,\tilde{a})\sim(\pi,\tilde{\pi}|a\neq\tilde{a})} [A_\pi(s, \tilde{a}) - A_\pi(s, a)]|$$

By Jensen's Inequality, we can take the absolute value inside the expectation.

$$|\bar{A}(s)| \leq \alpha \cdot \mathbb{E}_{(a,\tilde{a})\sim(\pi,\tilde{\pi}|a\neq\tilde{a})} [|A_\pi(s, \tilde{a}) - A_\pi(s, a)|]$$

The expectation of a random variable is always upper bounded by the max value of that variable.

$$|\bar{A}(s)| \leq \alpha \cdot \max_{a,\tilde{a}|a\neq\tilde{a}} |A_\pi(s, \tilde{a}) - A_\pi(s, a)|$$

This part is a little strange, and I'm not sure if this is the logic that the authors followed, but it is still true. For any $$a,b$$, we have that $$|a - b| \leq |a| + |b|$$.

$$|\bar{A}(s)| \leq \alpha \cdot \max_{a,\tilde{a}|a\neq\tilde{a}} (|A_\pi(s, \tilde{a})| + |A_\pi(s, a)|)$$

For $$a \neq \tilde{a}$$, it must be that either $$|A_\pi(s, \tilde{a})| \geq |A_\pi(s, a)|$$ or vice versa. We can use this to replace the pair of advantage functions with 2 times the max of the two.

$$|\bar{A}(s)| \leq \alpha \cdot 2 \max_{a} |A_\pi(s, a)|$$

Now, if we take the max over $$a$$ and $$s'$$, our inequality still holds, since we are taking the maximum over a set that contains $$s$$. Making this substitution gives us (31).

$$|\bar{A}(s)| \leq \alpha \cdot 2 \max_{a,s'} |A_\pi(s', a)|$$