Wasserstein GAN

Martin Arjovsky, Soumith Chintala, Léon Bottou

2017 · paper · AI and machine learning

Examines ways of measuring distance between a generated and a real data distribution for training generative adversarial networks, which are known to be unstable. It argues for a different distance, the Wasserstein distance, as the training objective.

Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.

10 findings

Substance The paper defines "∥µ∥TV = sup_{A⊆X} |µ(A)|" (p.19) and δ = sup_A |Pr(A) − Pg(A)| (p.3), but then uses µ(P) − µ(Q) = ∥µ∥TV (p.27) and |µ|(X) = ∥µ∥TV = δ (p.28). That is off by a factor 2, so Eq. (7) "= −δ(Pr, Pθ)" and Theorem 4 "LG(D∗, gθ) = m/2 δ(Pr, Pθ)" do not hold under the paper's own definition.

p. 19 Appendix A, definition
∥µ∥T V = sup A⊆X |µ(A)|
p. 27 Proof of eq. (7)
= −(µ(P) −µ(Q)) = −∥µ∥T V
p. 28 End of Appendix D proof
= |µ|(X) = ∥µ∥T V = δ(Pr, Pθ)
p. 3 Definition of total variation
δ(Pr, Pg) = sup

Only the authors can say what is right here.

Judge's note: Reworked the step myself. For µ = Pr − Pθ we have µ(X) = 0, so with Hahn decomposition (P, Q), µ(P) = sup_A µ(A) = δ and µ(Q) = −δ. Hence µ(P) − µ(Q) = |µ|(X) = 2δ. Counterexample: Pr = point mass at 0, Pθ = point mass at 1. Then δ = 1 (consistent with Example 1), but inf over −1 ≤ f ≤ 1 of f(0) − f(1) = −2, not −1. Carried through the p.27 chain, this gives LD = m − mδ and LG = mδ, not (m/2)δ. The isometry claim on p.19 and the p.14 statement that bounded functions give dF = δ have the same factor issue. The qualitative proportionality survives. Theorem 4's constant is wrong as printed, hence S1 under the derivation rules.

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Substance Page 4 defines "JS(Pr, Pg) = KL(Pr∥Pm) + KL(Pg∥Pm)" without the 1/2 factors. Everywhere else the paper uses the 1/2-weighted divergence: Example 1 gives JS = log 2, p.11 has the bound "2JS(Pr, Pθ) − 2 log 2", p.12 says "log 2 ≈ 0.69 which is the highest value taken by the JS distance", and p.23 writes "JS(Pn, P) = 1/2 KL(Pn∥Pm) + 1/2 KL(P∥Pm)".

p. 4 Section 2, JS definition
JS(Pr, Pg) = KL(Pr∥Pm) + KL(Pg∥Pm)
p. 23 Proof of Theorem 2
JS(Pn, P) = 1/2KL(Pn∥Pm) + 1/2KL(P∥Pm) →0
p. 12 Section 4.2
log 2 ≈0.69 which is the highest value taken by the JS distance
p. 4 Example 1, JS calculation
log 2

Judge's note: Checked the page images. The p.4 definition plainly has no 1/2 factors. For disjoint supports, as in Example 1 with θ≠0, dP0/dPm = 2 on the support of P0, so each KL term equals log 2. The printed definition then gives 2 log 2 ≈ 1.386. Example 1 (p.4) prints log 2, p.12 calls log 2 the maximum, and the p.11 bound 2JS − 2 log 2 with the plotted quantity ½L + log 2 only works for the 1/2-weighted JS. The proof of Theorem 2 on p.23 writes the 1/2 factors explicitly. The definition is the outlier; adding 1/2 to each KL term fixes it.

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Substance Assumption 1 only asks for pointwise local Lipschitz constants with "Ez∼p[L(θ, z)] < +∞", whose neighbourhoods may shrink with z. Yet the proofs bound "≤ 2L(θ0, z)" for dominated convergence (p.25) and integrate the local bound over z (p.20) as if one neighbourhood served all z. Theorem 3's gradient formula, "when both terms are well-defined", can then fail.

p. 20 Assumption 1
Ez∼p[L(θ, z)] < +∞
p. 20 Proof of Theorem 1, integration of local Lipschitz bound
Ez[∥gθ(z) − gθ′(z)∥] ≤ ∥θ − θ′∥Ez[L(θ, z)]
p. 25 Proof of Theorem 3, domination bound
≤ 2L(θ0, z)
p. 7 Theorem 3
when both terms are well-defined.

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Judge's note: Read pp.20, 24, 25 and the statement on p.7. The p.25 domination only holds for θ in a neighbourhood of θ0 that depends on z, so dominated convergence does not apply. I checked a counterexample against the stated assumptions: Z ~ Exp(1), X = [0,1], Pr = δ0, gθ(z) = 1/2 + sgn(θ)h(|θ|e^z) with h(u) = (u−1)(2−u) on [1,2] and 0 elsewhere. g is locally Lipschitz, constant near every (0,z), and has integrable local constants, with L(0,z) = 0. Substituting u = te^z gives E gθ = 1/2 + (3 log 2 − 2)θ for |θ| < 1. With the optimal critic f(x) = −x, W = E gθ, so dW/dθ at 0 is 3 log 2 − 2 ≈ 0.0794. Since ∂θ gθ(z) = 0 at θ = 0 for every z, the right-hand side is 0 for any optimal f. Both terms are defined and they differ, so Theorem 3 fails as printed under Assumption 1. A simpler example, gθ(z) = clamp(θe^z − 1), shows the "in particular when the right hand side is well defined" claim fails: the LHS does not exist at 0. A uniform-neighbourhood integrable bound would repair this.

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Substance Section 5 MMD bullet (p.14): "{f ∈ Cb(X), ∥f∥∞ ≤ 1} will be contained in F, and therefore dF(Pr, Pθ) ≤ δ(Pr, Pθ)". The containment gives a lower bound (dF ≥ δ, or more precisely dF ≥ |Pr−Pθ|(X)), not the printed upper bound.

p. 14 Section 5, MMD bullet
will be contained in F, and therefore dF(Pr, Pθ) ≤δ(Pr, Pθ)
p. 14 Section 5, MMD bullet
will be contained in F, and therefore dF(Pr, Pθ) ≤δ(Pr, Pθ) so the regularity

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Judge's note: Checked the page 14 image: the symbol is clearly ≤. My own working: dF is a supremum over F. If G = {f ∈ Cb, ∥f∥∞ ≤ 1} ⊆ F, then dF ≥ dG. So the 'therefore' runs the wrong way. As printed, the inequality is also false. Counterexample: X=[0,1], Pr=δ0, Pθ=δ1, so δ=1 (the paper's own definition; Example 1 also gives δ=1). f(x)=1−2x lies in G, so dF ≥ 2 > 1. The sentence's own conclusion, that regularity is 'at least as bad as' TV, needs the lower bound. The likely intent is ≥, but only the authors can confirm.

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Substance On p.14 the MMD function class is printed as "F = {f ∈ H : ∥f∥∞ ≤ 1}", with the sup norm. MMD is the IPM over the unit ball of the RKHS norm, ∥f∥_H ≤ 1, and the same page later refers to maximizing equation (4) "for the ball of a RKHS".

p. 14 Section 5, MMD bullet
F = {f ∈ H : ∥f∥∞ ≤ 1}

Judge's note: Checked the p.14 image: the subscript is ∞. The paper itself points to the RKHS-norm ball: later on p.14 it speaks of maximizing (4) "for the ball of a RKHS". Its containment argument (taking H = L2(X,m) with normalized Lebesgue measure, so that the C_b sup-norm ball lies in F) only needs ∥f∥_2 ≤ ∥f∥∞, i.e. F as the H-norm ball; under the printed ∞-norm it would be trivial. A two-point check with the identity kernel gives sup over the RKHS ball of 2^{1/2}, but 2 over the printed sup-norm ball, so the two classes define different quantities.

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Substance Proof of Theorem 1 (p.20) integrates the pointwise local bound over z, although the neighbourhood U (and so Uθ = {θ′|(θ′, z) ∈ U}) depends on z. It then concludes "|W(Pr, Pθ) − W(Pr, Pθ′)| ≤ W(Pθ, Pθ′) ≤ L(θ)∥θ − θ′∥" for all θ′ ∈ Uθ, which does not follow and can fail. So local Lipschitzness, and with it the a.e. differentiability in Theorem 1.2, is not established by this argument.

p. 20 Proof of Theorem 1, second part
whenever (θ′, z) ∈U. Therefore, we can deﬁne Uθ = {θ′|(θ′, z) ∈U}.
p. 20 Proof of Theorem 1, last display
|W(Pr, Pθ) −W(Pr, Pθ′)| ≤W(Pθ, Pθ′) ≤L(θ)∥θ −θ′∥

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Judge's note: I read the step on p.20 and reworked it. For each fixed z the bound holds only for θ′ in a z-dependent set Uθ(z). Taking E_z needs the bound for all z at the same θ′, i.e. θ′ in the intersection over z, which need not be open or contain more than θ. Assumption 1 (p.20) asks only for local constants L(θ,z) with finite expectation, not uniform neighbourhoods. I checked the reviewer's counterexample against the printed assumptions. Z ~ Exp(1), X = [0,1] (compact), Pr = δ0, g(θ,z) = min(1, max(0, θe^z − 1)) is locally Lipschitz. At θ = 0, U = {θ′e^{z′} < 1} is open, contains every (0,z), and g ≡ 0 on it, so L(0,z) = 0 is valid and E L(0,Z) = 0. For θ ≠ 0, L(θ,z) is nonzero only on a bounded z-interval, so E L < ∞. For 0 < θ ≤ 1, W(Pr,Pθ) = E g = ∫_0^1 θ/(1+s) ds = θ ln 2. With z = 0, Uθ = (−∞,1) contains 0.5, and the printed inequality would give W(Pr,P0.5) ≤ 0, but the value is about 0.347. Theorem 1.2 itself is not contradicted (W here is locally Lipschitz), but the printed proof does not establish the local Lipschitz property used for a.e. differentiability. That leaves the a.e.-differentiability part of the stated theorem without a valid proof, hence S1, author_query.

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Wording The proof of Corollary 1 (p.21) uses "∥f1:k−1(z)∥ ≤ ∥z∥L^{k−1} ∏_{i=1}^{k−1} Wi". This bound fails for the affine layers and the nonlinearities with σ(0)≠0 (sigmoid, softplus) that footnote 4 explicitly allows.

p. 5 Corollary 1, footnote 4
such as the sigmoid, tanh, elu, softplus, etc
p. 21 Proof of Corollary 1, bound on intermediate layers
∥f1:k−1(z)∥
p. 21 Proof of Corollary 1, final expectation bound
C1(θ) + C2(θ)Ez∼p(z)[∥z∥]
p. 21 Proof of Corollary 1
∥f1:k−1(z)∥≤∥z∥Lk−1 Qk−1 i=1 Wi

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Judge's note: Read p.21 and footnote 4 on p.5. A Lipschitz σ only gives |σ(x)| ≤ L|x| + |σ(0)|. My own check uses a bias-free scalar network gθ(z) = σ(w2 σ(w1 z)) with sigmoid σ, at z = w1 = w2 = 0. There f_{1:1}(0) = σ(0) = 1/2, while the printed bound is 0. Also ∂g/∂w2 = σ'(0)σ(0) = 1/8 > 0, but C1(θ) contains the product of the zero weights and z = 0, so the final inequality would read 1/8 ≤ 0. The step is false as printed. The corollary itself still holds once a θ-dependent additive constant is carried through, since E∥z∥ < ∞. Hence S3.

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Wording Algorithm 1 says "All experiments in the paper used the default values α = 0.00005, c = 0.01, m = 64, ncritic = 5", yet the Figure 3 caption describes the lower-half run as MLPs "with substantially high learning rates (so training failed)", and Section 4.2 reports runs with Adam/high learning rates.

p. 8 Algorithm 1 caption
All experiments in the paper used the default values α = 0.00005
p. 9 Section 4.1
We use the hyper-parameters specified in Algorithm 1 for all of our experiments.
p. 10 Figure 3 caption
MLPs with substantially high learning rates (so training failed)
p. 8 Algorithm 1 caption
All experiments in the paper used the default values α = 0.00005, c = 0.01, m = 64, ncritic = 5.

Only the authors can say what is right here.

Judge's note: Checked the p.8 Algorithm 1 caption, the p.10 Figure 3 caption, and the Section 4.2 text (text-layer lines 482-483: 'when one uses a momentum based optimizer such as Adam ... or when one uses high learning rates'). The blanket statement that all experiments used the defaults is contradicted by these reported runs. The learning rate actually used is not given, so only the authors can say.

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Wording The proof of Theorem 4 (p.26) concludes from LD(D′,gθ) ≤ LD(D,gθ) that "a function D∗ : X → [0,+∞) is optimal if and only if D∗′ is". The "only if D∗′ ⇒ D∗" direction does not follow and is false.

p. 26 Proof of Theorem 4, clipping argument
optimal if and only if D∗′ is.

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Judge's note: Checked on p.26. Clipping gives E_Pr[D′] ≤ E_Pr[D] and leaves [m−D]^+ unchanged, so LD(D′) ≤ LD(D). That proves only that clipping preserves optimality. Counterexample: X = {0}, Pr = Pθ = δ0, m = 1, D(0) = 2. Then LD(D) = 2 + 0 = 2, LD(D′) = 1 + 0 = 1, and min over d ≥ 0 of d + max(0, 1−d) = 1. So D′ is optimal and D is not. The existence argument only needs the valid direction, so Theorem 4's existence claim is unaffected. Hence S3.

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Wording In the calculation of LD(D∗,gθ) on p.27, the second line reads "m + m/2 Ex∼Pr[D∗(x)] − Ex∼Pθ[f∗(x)]". Substituting D∗ = (m/2)f∗ + m/2 gives m + m/2(Ex∼Pr[f∗(x)] − Ex∼Pθ[f∗(x)]).

p. 27 Proof of Theorem 4, definition of D∗
D∗(x) = m/2 f∗(x) + m/2
p. 27 Displayed calculation of LD(D∗,gθ), second line
m + m/2 Ex∼Pr[D∗(x)] − Ex∼Pθ[f∗(x)]

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Judge's note: Checked the p.27 image: the second line keeps D∗ in the first expectation, and the second expectation is not inside the m/2 factor. Substitution cancels the constant m/2 terms, leaving m + (m/2)(E_Pr f∗ − E_Pθ f∗). Check with m = 1, Pr = Pθ = δ0, f∗ ≡ 1, D∗ ≡ 1: the first line gives 1 + 1 − 1 = 1, but the printed second line gives 1 + 0.5 − 1 = 0.5. This is a typographical slip in an intermediate line; the next line follows from the correct substitution. It is separate from the factor-of-2 issue in C029.

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42 minor (typos and the like; most checked by the judge only)

Wording Algorithm 1 line 2: "for t = 0, ..., ncritic do" taken literally runs ncritic+1 critic iterations (6 with the default value), while ncritic is defined as "the number of iterations of the critic per generator iteration".

p. 8 Algorithm 1, line 2
for t = 0, ..., ncritic do

Only the authors can say what is right here.

Judge's note: The p.8 image confirms the loop bound and the definition. The inclusive range 0..ncritic contains ncritic+1 values. This is likely a pseudocode off-by-one, but only the authors can confirm how many updates were run.

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Wording Section 5 (p.14): "When F is the set of all measurable functions bounded between -1 and 1 ... we retrieve dF(Pr, Pθ) = δ(Pr, Pθ) the total variation distance". Under the paper's definition δ = sup_A |Pr(A) − Pθ(A)|, this IPM equals 2δ.

p. 14 Section 5, second bullet
we retrieve dF(Pr, Pθ) = δ(Pr, Pθ) the total variation distance

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Judge's note: Definitions checked: δ on p.3 and ∥µ∥TV = sup_A|µ(A)| on p.19. Example 1 gives δ=1 for disjoint point masses. Counterexample: Pr=δ0, Pθ=δ1 on [0,1], so δ=1, but f=1−2x (continuous, values in [−1,1]) gives f(0)−f(1)=2. The equality is off by a factor of 2. The topological point drawn from it still holds. The root cause is the same as in C028 (TV normalisation), but this is a separate statement in the main text.

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Wording Appendix A (p.19): "Φ is an isometric immersion" / "the total variation over Prob(X) is exactly the norm distance over Cb(X)∗". With the dual norm and ∥µ∥TV = sup_A|µ(A)| as defined on the same page, ∥Φ(Pr) − Φ(Pθ)∥ = 2δ.

p. 19 Appendix A
tells us that Φ is an isometric immersion
p. 19 Appendix A
the total variation over Prob(X) is exactly the norm distance over Cb(X)∗

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Typo p.2: "easier for a sequence of distribution to converge" should read "sequence of distributions".

p. 2 Introduction, paragraph 3
easier for a sequence of distribution to converge

Judge's note: The page image shows the phrase as quoted, in the last line of Introduction paragraph 3. It is a number-agreement typo.

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Typo Footnote 2 says "admits a density p_r(x)" (lowercase) but then writes "P_r(A) = ∫_A P_r(x)dµ(x)" (capital P_r for the density).

p. 3 Footnote 2
admits a density pr(x) with respect to µ

Judge's note: The crop shows lowercase italic p_r(x) in the text and capital italic P_r(x) in the integral. The KL definition also uses capital P_r(x) for the density. This is a minor notation inconsistency.

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Typo Footnote 2: "if and only it is absolutely continuous" is missing "if".

p. 3 Footnote 2
if and only it is absolutely continuous

Judge's note: Verified in the footnote 2 crop on p.3.

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Typo p.3: "famously assymetric" is a misspelling of "asymmetric".

p. 3 KL definition
famously assymetric
p. 3
famously assymetric and possibly

Judge's note: Visible in the page 3 crop.

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Typo p.3: "Plotting these learning curves is not only useful ... but also correlate remarkably well" has a subject-verb agreement error; it should be "correlates".

p. 3 Introduction, contributions bullet 3
but also correlate remarkably well with the observed sample quality

Judge's note: The image confirms the singular subject "Plotting ..." with the verb "is", followed by the plural verb "correlate".

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Typo p.6: after eq. (2), which is written for W(Pr, Pθ), the text says "then we end up with K·W(Pr, Pg)", switching from Pθ to Pg.

p. 6 Section 3 after eq. (2)
then we end up with K ·W(Pr, Pg)

Judge's note: The image confirms Pθ in eq. (2) and Pg in the following sentence. Pg is the generic notation from Section 2, so the meaning is clear. This is a minor notation inconsistency.

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Typo Proof of Theorem 3 (p.24) has "Kantorovich-Rubenstein duality [22]", while pp.6 and 16 spell it "Kantorovich-Rubinstein".

p. 24 Proof of Theorem 3
Kantorovich-Rubenstein duality [22]
p. 6 Section 3
Kantorovich-Rubinstein duality [22]

Judge's note: The p.24 image shows 'Rubenstein'. The text layer shows 'Rubinstein' at lines 264 and 540, so the paper is inconsistent with its own spelling.

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Typo Eq. (3): "E_{z∼p(z)}[f_w(g_θ(z)]" is missing a closing parenthesis; it should read f_w(g_θ(z)).

p. 7 Equation (3)
Ez∼p(z)[fw(gθ(z)]

Judge's note: The image of eq. (3) on p.7 shows the unbalanced parenthesis. The later expression on the same page, ∇θ f_w(g_θ(z)), is balanced.

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Typo Figure 3 caption: "The loss decreases constistently" is a misspelling of "consistently".

p. 10 Figure 3 caption
The loss decreases constistently
p. 10 Figure 3 caption
The loss decreases constistently as
p. 10 Figure 3 caption
decreases constistently

Judge's note: Visible in the Figure 3 caption image on p.10.

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Typo p.11: "which is is a lower bound of 2JS(Pr, Pθ)−2 log 2" has a duplicated "is".

p. 11 Section 4.2
which is is a lower bound
p. 11 Section 4.2
which is is a lower bound of
p. 11
which is is a lower bound of 2JS(Pr, Pθ)−2 log 2

Judge's note: Visible in the p.11 image.

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Typo p.11: "This quantity clearly correlates poorly the sample quality" is missing "with".

p. 11 Section 4.2 last line
correlates poorly the sample quality
p. 11 Section 4.2, last line
This quantity clearly correlates poorly the sample quality.

Judge's note: Visible in the p.11 image.

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Typo p.11: "we have succesfully used the loss metric" is a misspelling of "successfully".

p. 11 Section 4.2
we have succesfully used
p. 11 Section 4.2
we have succesfully used the loss metric
p. 11 Section 4.2
succesfully used the loss metric

Judge's note: Visible in the p.11 image.

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Typo "yeald" is misspelled for "yield" twice: p.13 "can yeald to radically different metrics" and p.14 "This will yeald the same behaviour".

p. 13 Section 5
can yeald to radically diﬀerent metrics
p. 14 EBGAN bullet
This will yeald the same behaviour
p. 13 Section 5
of functions can yeald to radically diﬀerent metrics

Judge's note: Both occurrences are visible in the p.13 and p.14 images. In the p.13 occurrence, the word "to" after "yield" is also unidiomatic ("yield" or "lead to").

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Typo EBGAN bullet (p.14): "the cost for the generator will aproximate the total variation distance". 'aproximate' should be 'approximate'.

p. 14 EBGAN bullet
will aproximate the total variation distance
p. 14 Section 5, EBGAN bullet
the cost for the generator will aproximate the total variation distance
p. 14 EBGAN bullet
aproximate the total variation

Judge's note: The misspelling is visible in the page 14 image.

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Typo EBGAN bullet (p.14): "This will yeald the same behaviour". 'yeald' should be 'yield'.

p. 14
This will yeald the same behaviour

Judge's note: Visible in the page 14 image.

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Typo Section 5 (p.15): "there is more in commmon than not". 'commmon' should be 'common'.

p. 15 Section 5
more in commmon than not
p. 15 Section 5
there is more in commmon than not

Judge's note: Visible in the page 15 crop.

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Typo "as in it’s power as a statistical test" (p.15) and "We will denote gθ(z) it’s evaluation" (p.20). The possessive should be 'its'.

p. 15 GMMN bullet
as in it’s power as a statistical test
p. 20 Assumption 1
We will denote gθ(z) it’s evaluation
p. 15 Section 5, GMMN bullet
(as in it’s power as a statistical test approaching 1)

Judge's note: Both instances are visible in the crops. Other uses of 'it's' in the paper (e.g. 'it's easier') are correct contractions.

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Typo Reference [13] (p.17): "In Proceedings of the 33nd International Conference on Machine Learning". The ordinal should be '33rd'.

p. 17 Reference [13]
Proceedings of the 33nd

Judge's note: Visible in the page 17 image. This is an ordinal-suffix error.

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Typo Reference [17] (p.17): "f-gan: Training generative neural samplers using variational divergence minimization. pages 271–279, 2016." Page numbers are given but no venue.

p. 17 Reference [17]
variational divergence minimization. pages 271–279, 2016.
p. 17 Reference [17]
variational divergence minimization. pages 271–279,

Judge's note: Page 17 image checked: the entry has no proceedings or journal name. The entry is incomplete.

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Typo References use "Corr" ([11], [19], [24], [25]) alongside "CoRR" ([8], [9], [18] and others).

p. 17 References [11], [19]
Corr, abs/1611.02163, 2016.
p. 17 Reference [8]
CoRR, abs/1412.6980, 2014.
p. 17 refs [11],[19]
Corr, abs/1611.02163

Judge's note: Page 17 image shows 'Corr' in [11] and [19] and 'CoRR' in [8], [9] and [18]. The text layer shows 'Corr' in [24] and [25] as well. This is a cosmetic inconsistency.

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Typo Appendix A (p.19): "where the supremum is taken all Borel sets in X" is missing 'over'. The same page has "With this definitions" for 'these'.

p. 19 Appendix A
where the supremum is taken all Borel sets

Judge's note: Both phrases are visible in the page 19 image.

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Typo Proof of Theorem 1 (p.20) prints "∥gθ(z) − g′θ(z′)∥ ≤ L(θ, z)(∥θ − θ′∥ + ∥z − z′∥)" with the prime on g; it should be gθ′(z′).

p. 20 Proof of Theorem 1, local Lipschitz inequality
∥gθ(z) −g′ θ(z′)∥≤L(θ, z)(∥θ −θ′∥+ ∥z −z′∥)

Judge's note: The crop and page 20 show the prime as a superscript on g with subscript θ, not on θ. The surrounding text sets up the pair (θ′, z′), and the next display uses gθ′(z), so this is a notation slip.

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Typo Proof of Theorem 1 (p.20): "By taking expectations and z′ = z we" has no main verb (e.g. 'we get').

p. 20 Proof of Theorem 1
By taking expectations and z′ = z we

Judge's note: The page image shows the sentence ending with "we" and then the display, with no verb. This is a missing word.

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Typo Proof of Corollary 1 (p.21): the first term of the second inequality is printed "L^H ∏_{i=H}^{K} ∥Wi∥", but it should be ∏_{i=1}^{H}, as in the line above and in C1(θ).

p. 21 Proof of Corollary 1, second inequality
≤LH K Y i=H ∥Wi∥

Judge's note: The page 21 image shows lower limit i=H and upper limit K. K is not defined anywhere, the line above uses ∏_{i=1}^H WiDi, and C1(θ) on the next line uses ∏_{i=1}^H ∥Wi∥. This is a typographic index slip.

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Typo Proof of Corollary 1 (p.21): "L(θ, Z) ≤ ∥∇θ,x gθ(z)∥ + ϵ" should read L(θ, z) ≤ ∥∇θ,z gθ(z)∥ + ϵ.

p. 21 Proof of Corollary 1
L(θ, Z) ≤∥∇θ,xgθ(z)∥+ϵ

Judge's note: The crop shows both a capital Z and the subscript x. g is a function of (θ,z), and the next display uses ∇θ,z. There is no variable x here, so both are notation slips.

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Typo "Radamacher's theorem" (p.21) and "Radamacher's Theorem" (p.24) misspell Rademacher.

p. 21 Proof of Theorem 1
by Radamacher’s theorem
p. 24 Proof of Theorem 3
By Radamacher’s Theorem
p. 21 Proof of Thm 1
Radamacher’s theorem
p. 22 Proof of Thm 2
Radon-Nykodim derivative

Judge's note: I saw the spelling on the p.21 and p.24 images. The other spellings in reviewer note 2 ('Radon-Nykodim', 'Kantorovich-Rubenstein') are judged separately in C040 and C043. Rests on outside knowledge: the theorem is named after Hans Rademacher.

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Typo Proof of Corollary 1 (p.21): "Dk is are the diagonal Jacobians" has a duplicated verb.

p. 21 Proof of Corollary 1
Dk is are the diagonal Jacobians
p. 21 Proof of Corollary 1
Dk is are the diagonal

Judge's note: Seen in the crop of page 21.

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Typo Proof of Theorem 2 (p.22): "Which implies Pm(Am) ≤ 1/ϵ δ(Pn, P)" should read Pm(An).

p. 22 Proof of Theorem 2
Which implies Pm(Am) ≤1 ϵ δ(Pn, P)
p. 22 Proof of Thm 2
Pm(Am) ≤1
p. 22
Which implies Pm(Am) ≤1

Judge's note: The page 22 image shows Pm(Am). Only An = {fn > 1+ϵ} is defined, and both the preceding and following displays use Pm(An). This is a subscript slip.

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Typo Proof of Theorem 2 (p.22): "Radon-Nykodim derivative" misspells Radon-Nikodym.

p. 22 Proof of Theorem 2
Radon-Nykodim derivative

Judge's note: Seen on the page 22 image. Rests on outside knowledge: the standard spelling is Radon-Nikodym.

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Typo Proof of Theorem 2 (p.22): "for our purposes 3 will sufice" is a misspelling of 'suffice'.

p. 22 Proof of Theorem 2
3 will suﬁce

Judge's note: The page 22 image shows 'sufice' with a single f (the ligature in the text layer reflects this).

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Typo Proof of Theorem 3 (p.24): "= ∇θ[Ex∼Pr[f(x)] − Ez∼p(z)[f(gθ(z))]" opens the bracket after ∇θ and never closes it.

p. 24 Proof of Theorem 3
= ∇θ[Ex∼Pr[f(x)] −Ez∼p(z)[f(gθ(z))]

Judge's note: Counting brackets in the crop: three '[' and two ']', so the outer bracket is unclosed.

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Typo Proof of Theorem 3 (p.24): "which we knows exists" should be 'which we know exists'.

p. 24 Proof of Theorem 3
which we knows exists

Judge's note: Seen in the crop of page 24.

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Typo End of the proof of Theorem 3 (p.25): "the mere existance of the left hand side" misspells 'existence'.

p. 25 End of Theorem 3 proof
the mere existance of the left hand side
p. 25 end of proof
mere existance of the left hand side

Judge's note: Seen in the crop of page 25.

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Typo Theorem 4 statement: "Let Pr be a the real data distribution" has duplicated articles.

p. 26 Theorem 4 statement
Let Pr be a the real data distribution

Judge's note: The page 26 image shows 'be a the real data distribution'.

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Typo Appendix D: "does not depen- dent on θ" should read "does not depend on θ".

p. 26 Appendix D
does not depen- dent on θ

Judge's note: The page 26 image shows 'does not dependent on θ' (hyphenated across a line break).

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Typo Proof of Theorem 4 (p. 27): "= m + m/2 E_{x∼Pr}[D∗(x)] − E_{x∼Pθ}[f∗(x)]" should have f∗ in the first expectation.

p. 27 Proof of Theorem 4
= m + m 2 Ex∼Pr[D∗(x)] −Ex∼Pθ[f ∗(x)]

Judge's note: The image confirms D∗ inside the first expectation. Since D∗ = (m/2)f∗ + m/2, E_r[D∗] − E_θ[D∗] = (m/2)(E_r f∗ − E_θ f∗). The next line uses this form, so the intended expression is evident.

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Typo Proof of eq. (7) (p. 27): "EE_{x∼Pr}[f∗(x)]" has a stray duplicated E.

p. 27 Proof of eq. (7)
EEx∼Pr[f ∗(x)] −Ex∼Pθ[f ∗(x)]
p. 27 Hahn decomposition proof
EEx∼Pr[f ∗(x)]

Judge's note: The page 27 image shows 'EE_{x∼Pr}[f∗(x)] − ...' at the start of the Hahn decomposition display.

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Typo Proof of Theorem 4 (p. 27): the final line "= m/2 δ(Pr, Pg)" uses Pg where the theorem and the rest of the proof use Pθ.

p. 27 Proof of Theorem 4, LG line
= m 2 δ(Pr, Pg)

Judge's note: The page 27 image confirms 'δ(Pr, Pg)' in the last line of the LG derivation. Theorem 4 states the result as (m/2)δ(Pr, Pθ). This is a notational inconsistency for the same quantity.

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Typo Proof of Theorem 4 (p. 27): "We then have 0 ≤ D(x) ≤ m" should refer to the newly defined D∗(x).

p. 27 Proof of Theorem 4
We then have 0 ≤ D(x) ≤ m

Judge's note: The page 27 image shows the bound is stated right after defining D∗(x) = (m/2)f∗(x) + m/2. Because f∗ ∈ [−1,1], the bound applies to D∗. The printed 'D(x)' is a dropped asterisk.

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