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# Chapter 7 PPO

### Page 176, Section 7.2 Proximal Policy Optimization (PPO)

*Thanks to Jérémie Clair Coté for suggesting we clarify this and for the discussion, and HyeAnn Lee for correction.*

Page 176, the last sentence of the 1st paragraph and the first two sentences of the 2nd paragraph read:

$$
\text{To see why this is the case, consider when } r\_t(\theta)A\_t \text{ would assume large positive} \newline \text{values, which is either } A\_t > 0, r\_t(\theta) > 0, \text{ or } A\_t < 0, r\_t(\theta) < 0.
$$

$$
\text{When } A\_t > 0, r\_t(\theta) > 0, \text{ if } r\_t(\theta) \text{ becomes much larger than 1, the upper clip term} \newline 1 - \epsilon \text{ applies to upper-bound } r\_t(\theta) \le 1 + \epsilon, \text{ hence } J^{CLIP} \le (1 + \epsilon)A\_t. \text{ On the} \newline \text{ other hand, when } A\_t < 0, r\_t(\theta) < 0, \text{ if } r\_t(\theta) \text{ becomes much smaller than 1, the} \newline \text{lower clip term } 1 - \epsilon \text{ applies to again upper-bound } J^{CLIP} \le (1 - \epsilon)A\_t.
$$

This is confusing because **(1)** $$r\_t(\theta)$$ cannot be < 0 because it is a ratio of two probabilities and **(2)** there is a typo when referring to the upper clip term. The sentences should be replaced with:

$$
\text{To see why this is the case, let's consider when } A\_t \text{ is either } \gt 0 \text{ or } \lt 0. \text{ Note} \newline \text{ that } r\_t(\theta) \text{ is always} \ge 0 \text{ because it is a ratio of two probabilites}.
$$

$$
\text{When } A\_t > 0, \text{ if } r\_t(\theta) > 1 + \epsilon, \text{ the upper clip term } 1 + \epsilon \text{ applies to upper-bound } \newline r\_t(\theta) \le 1 + \epsilon, \text{ hence } J^{CLIP} \le (1 + \epsilon)A\_t. \text{ On the other hand, when } A\_t < 0, \text{ if } \newline r\_t(\theta) < 1 - \epsilon \text{ the lower clip term } 1 - \epsilon \text{ applies to again upper-bound } \newline J^{CLIP} \le (1 - \epsilon)A\_t.
$$

### Page 178, Section 7.3 PPO Algorithm, Algorithm 7.2

*Thanks to Jérémie Clair Coté for this correction.*

Algorithm 7.2 PPO with clipping, line 35:

$$
\theta\_C = \theta\_C + \alpha\_C \nabla\_{\theta\_C} L\_{val}(\theta\_C)
$$

contains a typo. The second term on the right hand side of the equation should be subtracted not added since the loss is being minimized. It should read:

$$
\theta\_C = \theta\_C - \alpha\_C \nabla\_{\theta\_C} L\_{val}(\theta\_C)
$$

Note that the actor parameter update on line 33 of algorithm 7.2 is correct because the policy "loss" for PPO is formulated as an objective to be maximized (see equation 7.39 on page 177).