Errata and Notes for:

Reinforcement Learning: An Introduction, 2nd edition
by Richard S. Sutton and Andrew G. Barto

Errata for the second printing (2020) of the second edition:

Last updated April 25, 2022

• p1, second paragraph: "methods.of" --> "methods. That is, we adopt the perspective of"  (Samuel Pettersson)
  
• p29, second paragraph: "about 1.55" would be better as "about 1.54"  (Tetsuro Tanaka)
  
• p35, Equation 2.9 should say n>0, not n>=0  (John Senneker)
• p58, Exercise 3.11: "S_t" --> "S_t=s"  (Marlos Machado)
  
• p68, paragraph 3: this would be clearer with the word "tabular" inserted before "continuing tasks"  (Gabriel Paludo Licks)
• p80: "finite number of policies" --> "finite number of deterministic policies" (Samuel Pettersson)
• p80, in the box, at the end of the initialization line, add: "; V(terminal)=0"  (Samuel Pettersson)
• p102: The first equation in the middle group is not correct. It and the sentence leading up to it should read: We know that $\tilde v_*$ is the unique solution to the Bellman optimality equation \eqrf{bellman-v*} with altered transition probabilities: \tilde v_*=\max_a\sum_{s',r} \Bigl[ (1-\e)\p(s',r|s,a) + \sum_{a'}\frac{\varepsilon}{|\A(s)|} \p(s',r|s,a')\Bigr]\Bigl[r+\g\tV(s')\Bigr]  (Samuel Pettersson)
• p129: The convergence of Sarsa also requires the usual conditions on the step sizes (2.7).  (Samuel Pettersson)
• p164, 2nd line: "contents of the" --> "contents of the model"    (Samuel Pettersson)
• p211, bottom, or possibly top of the next page: "the length the interval" --> "the length of the interval"  (Prabhat Nagarajan)
• p253, paragraph 2, first line: "returns" --> "rewards"
  
• p259, Equation 11.6: The last importance sampling ratio should be \rho_{t+n} instead of \rho_{t+n-1}.  (Nikolay Gudkov)
• p259, Equation 11.6: Some versions of the pdf erroneously start the equation with both occurrences of w indexed at t+n. The second occurrence should be indexed by t+n-1  (Prabhat Nagarajan)
• p259, the last line of the second paragraph: G_{t:n} should be G_{t:t+n}.  (Nikolay Gudkov)
• p259, Equation 11.8: w_t-1  -->  w_t+n-1  (A Reader)
  
• p269, in the first full sentence after (11.20): "for v" --> "for v_{\bf w}" (Nikolay Gudkov)
• p269, 4th para, 2nd line: \Pi\bar\delta_{v_{\bf w}} should be \Pi\bar\delta_{\bf w}  (Nikolay Gudkov)
• p269, last sentence before (11.22): v should be v_{\bf w}  (Nikolay Gudkov)
• p275, line 5: "even given even" --> "even given" (Nikolay Gudkov)
• p278, last equation: {\bf x}_s{\bf x}_s --> {\bf x}(s){\bf x}{s) (Nikolay Gudkov)
• p282, first line after the equations: M_{t-1}=0 --> M_{-1}=0 (Nikolay Gudkov)
• p289, before (12.2): [0,1] --> [0,1)   (Jiuqi Wang)
• p289, in the caption for Figure 12.1: digram --> diagram  (Prabhat Nagarajan)
• p308: "to the extent that we not bootstrapping" --> to the extent that we are not bootstrapping" (Prabhat Nagarajan)
• p309, before  (12.23): "the truncated version of this return" --> "this final \lambda-return" (Nikolay Gudkov)
• p335, first line of Section 13.7: "actions spaces" --> "action spaces" (Prabhat Nagarajan)
• p331, second line of Section 3.5: "the only the" --> "only the" (Ansel Blume)
• p336, Exercise 13.4: "gaussian" --> "Gaussian"  (Samuel Pettersson)
• p459, Equation 17.1: This equation is a definition and should have a dot over the equal sign
• p463, 2/3rds down the page: "the the option's policy" --> "the option's policy"  (Prabhat Nagarajan)
• p469, "plateau problem" --> "mesa phenomena"  (Prabhat Nagarajan)
• p479, last line: Minh --> Mnih  (Prabhat Nagarajan)
• p495: The reference for John (1994) is missing. It should be John, G. H. (1994). When the best move isn't optimal: Q-learning with exploration. In Proceedings of the Association for the Advancement of Artificial Intelligence, p. 1464. (Michael Przystupa)
• p496, in reference for Keiflin and Janak: "Ffrom" --> "From" (Steven Bagley)
  

Errata for the first printing (2018) of the second edition:

• The title of the final section, Section 17.6, was mistakenly printed as a repeat of an earlier sections title. It should be "The Future of Artificial Intelligence." This is also wrong in the table of contents.
  
• The phrase "function approximation" was mistakenly abbrieviated to "function approx." many times in the printed book.
  
• p11, 5 lines from bottom: "(see (Section 16.1))" --> "(Section 16.1)”
• p19, 8 lines from bottom: "(Section 16.2)" --> "(Section 15.9)"
• p30, Exercise 2.2: The values specified for R_1 and R_3 should have minus signs in front of them
• p35, in (2.9): n>=0 --> n>0  (Prabhat Nagarajan)
• p64, after the figure: v_pi -->  v_*
  
• p81: At the top of this page there should be a bold heading: "Example 4.2: Jack's Car Rental". Related to this, the example on page 84 should be Example 4.3. Also on page 81, there should be a paragraph break after "the policy that never moves any cars." (Sam Ritchie)
  
• p98, start of last paragraph: For Monte Carlo policy evaluation --> For Monte Carlo policy iteration
• p107, the 3rd line is cut off; it should read: "using the behavior policy that selects right and left with equal probability."
  
• p117, in 5.5: probabalistic --> probabilistic
• p143, in Exercise 7.1: "a sum TD errors" --> "a sum of TD errors"  (Prabhat Nagarajan)
• p149, in the pseudocode algorithm, in the upper limit for the \rho product: "\tau+n-1" --> "\tau+n", and in the comment: "t+n-1" --> "\tau+n". (Jyothis Vasudevan, Zixuan Jiang, Yifan Wang, and Zhiqi Pan)
  
• p153, end of first paragraph: occuring --> occurring
• p155: Equations (7.17) and (7.18) are mistakenly the same. The first equation should actually be a three-line derivation as given in the online book, here: http://www.incompleteideas.net/book/RLbook2018.pdf#page=177. The second equation should be numbered (7.17). (Xiang Gu)
• p156, line 2: (7.11) --> (7.5)      (Xiang Gu)
• p156, in the algorithm, the line "G <- 0:" should be replaced with the two lines: "If t+1 < T:" and then, indented from that: "G <- Q(S_{t+1},A_{t+1})"    (Mark Rowland and Will Dabney)
  
• p178, line 10: polices --> policies   (Kaniska Mohanty)
  
• p180, line 24: "sill" --> "still"  (James R)
• p198 2/3 down page: "s \mapsto g" --> "s \mapsto u"
• p202, after (9.7): "(S_t)" --> "(s)"   (Dhawal Gupta)
  
• p204, bottom: "approximate state-value function" --> "approximate the state-value function"
• p206, end of (9.11): "\Re^d \times \Re^d" --> "\Re^{d\times d}"  (Dhawal Gupta)
• p208, beginning of second paragraph: "the these" --> "these".  (Miguel Drummond)
  
• p212, line 1: "length the interval" --> "length of the interval" (Prabhat Nagarajan)
• p212, second to last line: The i index should start at 1, not 0. (Chris Harding)
• p220, middle of page: horizonal --> horizontal
• p229, In (9.22) and the equation above it labeled (from (9.20)), all the x's should have their time index reduced by 1: x_t --> x_{t-1} and x_{t+1} --> x_t  (Frederic Godin)
• p229, bottom of page: forgeting --> forgetting
• p241: "approximated by linear combination" --> "approximated by a linear combination" (Prabhat Nagarajan)
• p244, line 14: w_t --> w_{t-1} (Frederic Godin)
• p246, footnote 1: ",A)" in the call to tiles should be ",[A])" as this last argument to tiles must be a list (Martha Steenstrup)
  
• p248, Exercise 10.1, line 2: "or in" --> "in"
  
• p256 in 10.3: "Tsitiklis" --> "Tsitsiklis" (Prabhat Nagarajan)
• p259, above (11.6): "Expected Sarsa"  -->  "Sarsa" (Xiang Gu)
• p267, within Figure 11.3: "\overline{TDE}=0" --> \min \overline{TDE}
• p279, end of paragraph below (11.27): "naive residual-gradient" --> "residual gradient" (Abhishek Naik)
  
• p286 in 11.7: "Mahadeval" --> "Mahadevan" (Prabhat Nagarajan)
• p289, in the caption for Figure 12.1: digram --> diagram  (Prabhat Nagarajan)
• p302, middle: auxilary --> auxiliary
• p304, just before Example 12.1: a second closing parenthes is needed after “inactive (=0)” (Abhishek Naik)
• p308: "to the extent that we not bootstrapping" --> "to the extent that we are not bootstrapping" (Prabhat Nagarajan)
• p310, shortly after the three equations on top: "sum left" --> "sum on" (Abhishek Naik)
• p321, after equation (13.1): d should be d'
• p327, the first equation should not be an equality, but a proportional-to. And in the second line following that system of equations, the word "equal" should be changed to "proportional" (Mirco Musolesi)
  
• p327, 5 lines from bottom: "boxed" --> "boxed algorithm"  (Douglas De Rizzo Meneghetti)
• p329, bottom: \w\in\Re^m --> \w\in\Re^d
• p335, first line of Section 13.7: "actions spaces" --> "action spaces" (Prabhat Nagarajan)
• p337, line 25: critizes --> criticizes (Prabhat Nagarajan)
• p337, bottom: Schall --> Schaal
• p350, third line: "Rescoral-Wgner" --> "Rescorla-Wagner" (Kyle Simpson)
• p350, (14.6): the z indices should be t and t-1 (Abhishek Naik)
• p354, 11 lines from the bottom: "Rescoral-Wagner" --> "Rescorla-Wagner" (Kyle Simpson)
• p371, in the note on section 14.2.2, 3rd line from bottom: Schmajuk—> Schmajuk’s
• p372, in the note on section 14.3, line 2: Thorndikes —> Thorndike’s
• p398, near the bottom: "receiving action R_t+1" --> "receiving reward R_t+1"  (Caleb Bowyer)
• p399, the two eligibility-trace update equations (z) should have a gamma just before their lambdas (Abhishek Naik)
• p399, in the line after the equations, \lambda^w c --> \lambda^{\bf w} and \lambda^w a --> \lambda^{\theta} (Abhishek Naik)
• p400: The left side of (15.3) is missing a logarithm (ln) between the grad symbol (Nabla) and the policy symbol (pi)  (Jiahao Fan)
  
• p400, in (15.3) and again 7 lines down: "A_t-\pi(A_t|S_t" --> "A_t-\pi(1|S_t"
• p401, third paragraph, three times: "A_t-\pi(A_t|S_t" --> "A_t-\pi(1|S_t"
• p415, biblio section 15.8: "\pi(A_t|S_t" -->  "\pi(1|S_t"
  
• p436, eight lines from the bottom: "Tesauro and colleages" --> "Tesauro and colleagues" (Raymund Chua)
• p447, 14 lines from the bottom: "Figure 16.7, were $\theta$ is" -> "Figure 16.7, where $\theta$ is" (Kyle Simpson)
• p451, middle left: user-targed --> user-targeted
• p460, in paragraph 3, then again in paragraph 4: auxilary --> auxiliary
• p461, bottom line: \g_\omega(S_{t+1}) --> 1-\g_\omega(S_{t+1})
• p462, 2nd line: \g_\omega(S_{t+2})  -->  1-\g_\omega(S_{t+2})
• p463, 2/3rds down the page: "the the option's policy" --> "the option's policy"  (Prabhat Nagarajan)
• p463, 6th line from bottom: C_t=\g(S_t)\cdot\ind{S_t=s'}  -->  C_t=(1-\g_\omega(S_t))\ind{S_t=s'}
  
• p465, middle right: there should be no commas in the list defining tau
• p465: This definition and discussion of Markov state is insufficient. In addition to a Markov state being sufficient to make the one-step predictions (17.6), it must also be incrementally updatable as in (17.9).
  
• p504: "Pavlov, P. I." --> "Pavlov, I. P." (Brian Christian)

Notes:

• It is easy to make a URL that refers to a specific page of the online version of the book. Use a URL of the form "http://www.incompleteideas.net/book/RLbook2018.pdf#page=X" where X is the number of the page you want to refer to plus 22. For example, to refer to page number 100, you would use "http://www.incompleteideas.net/book/RLbook2018.pdf#page=122".
• Thermal soaring (Section 16.8) has since been extended to real gliders. See Reddy, G., Ng, J. W., Celani, A., Sejnowski, T. J., Vergassola, M. Glider soaring via reinforcement learning in the field. Nature 562:236-239, 2018.
• The differential Sarsa algorithm for the average-reward case, shown on page 251, only converges to the true action values up to an additive constant. That is, \hat q(s,a,w_\infty) = q_*(s,a) + Q for some scalar Q. (To see this, note that adding an arbitrary constant, say 100, to the estimated values of all actions would have no effect on any of the TD errors (10.10).) If you wanted to estimate the true differential action values, you would have to estimate Q in addition to running the given algorithm. It is not hard to see that under asymptotic on-policy conditions, the average of q_*(S_t,A_t) is zero. It follows that Q is the asymptotic average of \hat q(S_t,A_t,w_t). Thus one could estimate Q by \bar Q, updated by \bar Q_t = \bar Q_{t-1} + beta * (\hat q(S_t,A_t,w_t) - \bar Q_{t-1}). Then if at time t you want an estimate of the true value you would use q_*(s,a) \approx \hat q(s,a,w_t) - \bar Q_t.
• (first printing) In the computation of the reward baseline, described in words after (2.12) on page 37, it probably would be better if the baseline at step t included only rewards up to time t (i.e., not the reward at time t). This would be more consistent with the averages described earlier in this chapter and with the baselines used in later chapters, and is arguably necessary for the theory developed in the box on page 39. Thanks to Frederic Godin for pointing this out.
• The results in Figure 4.3 are for asynchronous value iteration---the algorithm given on page 83. Because of this, the results are dependent on the order in which the states are swept through. The given results are for sweeping through the states from Capital=1 to Capital=99, which is probably a poor order in terms of rapidly converging to the optimal value function.
• Exercise 7.3. This is a difficult exercise.   Igor Karpov made a thorough answer available, but at present only part of it is still available, from the wayback machine, at https://web.archive.org/web/20160326154910/http://www.cs.utexas.edu/~ikarpov/Classes/RL/RandomWalk/.