n}{n}\right)^{\frac{2p}{2p+d+k}}\right)\leavevmode\nobreak\ ,$ (C.31)

from the expression (C.30).

Observe that the relations

$\displaystyle\frac{1}{n}\overline{R}_{n,n}\left(\hat{\overline{\mu}}_{n,n}\right)$
$\displaystyle=\frac{1}{n}\tilde{R}_{n}\left(\hat{\overline{\mu}}_{n,n}\right)-D_{n}(\hat{\overline{\mu}}_{n,n})+O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{2p}{2p+d+k}}\right),$ (C.32)
$\displaystyle\frac{1}{n}\overline{R}_{n,n}\left(\hat{\overline{\mu}}_{n,n}\right)$
$\displaystyle\geq\frac{1}{n}\overline{R}_{n,n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right),\quad\text{and}$
$\displaystyle\frac{1}{n}\overline{R}_{n,n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right)$
$\displaystyle=\frac{1}{n}\tilde{R}_{n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right)-D_{n}(\hat{\overline{\mu}}_{n,n}^{*})+O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{2p}{2p+d+k}}\right),$ (C.33)

where $\hat{\overline{\mu}}_{n,n}$ is defined in (C.24), imply the inequality

$\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}\right)\geq\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right)+\left(D_{n}(\hat{\overline{\mu}}_{n,n})-D_{n}(\hat{\overline{\mu}}_{n,n}^{*})\right)+O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{2p}{2p+d+k}}\right)\leavevmode\nobreak\ .$ (C.34)

Moreover, the term $D_{n}(\hat{\overline{\mu}}_{n,n}^{*}))$ is bounded from
above by

$\frac{2}{n}\sum_{i=1}^{n}\|\mu\left(S_{i},X_{i}\right)-\hat{\mu}_{n,n}\left(S_{i},X_{i}\right)\|_{2}\|\overline{\mu}\left(X_{i}\right)-\overline{h}\left(X_{i}\right)\|_{2}=O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d+k}+\frac{\overline{p}}{2\overline{p}+k}}\right)\leavevmode\nobreak\
,$ (C.35)

by the Cauchy-Schwarz inequality, the rate (C.28), and the choice (C.27).
Plugging this bound into (C.34) implies

$\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}\right)\geq\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right)+D_{n}(\hat{\overline{\mu}}_{n,n})+O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d+k}+\min\big{\\{}\frac{p}{2p+d+k},\frac{\overline{p}}{2\overline{p}+k}\big{\\}}}\right)\leavevmode\nobreak\
.$ (C.36)

Thus, in order to apply Lemma C.12, must must control the term
$D_{n}(\hat{\overline{\mu}}_{n,n})$.

To this end, suppose that it were known that

$\displaystyle\|\hat{\overline{\mu}}_{n,n}-\overline{\mu}\|_{2,P}=O_{P}\left(\left(\frac{\log
n}{n}\right)^{s}\right)$ (C.37)

for some constant $0\leq s<1$. In this case, the term
$D_{n}(\hat{\overline{\mu}}_{n,n})$ is similarly bounded from above by

$\frac{2}{n}\sum_{i=1}^{n}\|\mu\left(S_{i},X_{i}\right)-\hat{\mu}_{n,n}\left(S_{i},X_{i}\right)\|_{2}\|\overline{\mu}\left(X_{i}\right)-\overline{h}\left(X_{i}\right)\|_{2}=O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d+k}+s}\right)\leavevmode\nobreak\ ,$ (C.38)

by the Cauchy-Schwarz inequality and the rate (C.28), which again implies

$\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}\right)\geq\frac{1}{n}\overline{R}_{n}\left(\hat{\overline{\mu}}_{n,n}^{*}\right)+O_{P}\left(\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d+k}+\min\big{\\{}s,\frac{p}{2p+d+k},\frac{\overline{p}}{2\overline{p}+k}\big{\\}}}\right)\leavevmode\nobreak\
.$ (C.39)

Hence, by Assumption C.2, Part (iv), Lemma C.12 implies that

$\displaystyle\|\hat{\overline{\mu}}_{n,n}-\overline{\mu}\|_{2,P}=O_{P}\left(\left(\frac{\log
n}{n}\right)^{r(s)}\right)\leavevmode\nobreak\ ,$ (C.40)

where

$\displaystyle
r(s)=\max\bigg{\\{}\frac{1}{2}\left(\frac{p}{2p+d+k}+\min\bigg{\\{}s,\frac{p}{2p+d+k},\frac{\overline{p}}{2\overline{p}+k}\bigg{\\}}\right),\frac{\overline{p}}{2\overline{p}+k}\bigg{\\}}\leavevmode\nobreak\
.$ (C.41)

Observe that we have shown $\overline{\mu}_{n,n}$ converges to
$\overline{\mu}$ in $\|\cdot\|_{2,P}$ at rate $s$, then it also converges at
the rate $r(s)$.

To conclude the proof, observe that (C.37) must be satisfied by $s=0$, as
$\hat{\overline{\mu}}_{n,n}$ and $\overline{\mu}$ are subsets of
$\Lambda^{\overline{p}}_{\overline{c}(\mathcal{X})}$. Thus, if we define the
sequence $(s_{i})_{i=0}^{\infty}$ by $s_{i}=r(s_{i-1})$, with initial
condition $s_{0}=1$, we find that

$\lim_{i\to\infty}s_{i}=\frac{p}{2p+d+k}\quad\text{if}\quad\frac{\overline{p}}{2\overline{p}+k}\leq\frac{1}{2}\frac{p}{2p+d+k}\leavevmode\nobreak\
,$

and so the rate of covergence (C.29) is obtained by induction.

####  C.3.4. Auxiliary Lemmas

###### Lemma C.6 (Newey and McFadden,, 1994, Theorem 2.7, ).

If there is a function $Q_{0}\left(\theta\right)$ defined on a convex set
$\Theta$ such that (i) $Q_{0}\left(\theta\right)$ is uniquely minimized at
$\theta_{0}$, (ii) $\theta_{0}$ is an element of the interior of $\Theta$,
(ii) the function $\hat{Q}_{n}\left(\theta\right)$ defined on $\Theta$ is
concave, and (iv) $\hat{Q}_{n}\left(\theta\right)\to Q_{0}\left(\theta\right)$
for all $\theta\in\Theta$, then

$\hat{\theta}_{n}=\arg\max_{\theta\in\Theta}\text{
}\hat{Q}_{n}\left(\theta\right)$

exists with probability one and $\hat{\theta}_{n}\overset{p}{\to}\theta_{0}$
as $n\to\infty$.

###### Lemma C.7.

Under Assumption C.1, the estimators $\hat{\mu}_{m,n}$ and
$\hat{\overline{\mu}}_{m,n}$ satisfy

$\|\hat{\mu}_{m,n}-\mu_{m}^{*}\|_{P,2}\overset{p}{\to}0\quad\text{and}\quad\|\hat{\overline{\mu}}_{m,n}-\overline{\mu}_{m}^{*}\|_{P,2}\overset{p}{\to}0$

as $n\to\infty$ and $m$ is sufficiently large, respectively.

###### Proof.

It will suffice to verify the conditions of Lemma C.6. Note first that both
$\Theta_{m}$ and $\overline{\Theta}_{m}$ are convex. The population criteria
$R_{P}\left(\cdot\right)$ and $\overline{R}_{P}\left(\cdot\right)$ are both
uniquely minimized at $\mu_{m}^{*}$ and $\overline{\mu}_{m}^{*}$ over
$\Theta_{m}$ and $\overline{\Theta}_{m}$, respectively, by convexity. Both
$\mu_{m}^{*}$ and $\overline{\mu}_{m}^{*}$ are elements of the interiors of
$\Theta_{m}$ and $\overline{\Theta}_{m}$ for sufficiently large $m$ by Lemma
C.8. It is clear that the functions $R_{n}\left(h\right)$ and
$\overline{R}_{m,n}\left(\overline{h}\right)$ are strictly convex.

The pointwise convergence

$R_{n}\left(h\right)\overset{\text{p}}{\to}R_{p}\left(h\right)$

as $n\to\infty$ for $m$ sufficiently large follows from the weak law of large
numbers, and so

$\|\hat{\mu}_{m,n}-\mu_{m}^{*}\|_{P,2}\overset{p}{\to}0$ (C.42)

as $n\to\infty$ for $m$ sufficiently large follows from Lemma C.6.
Consequently, the pointwise convergence

$\overline{R}_{n}\left(\overline{h}\right)\overset{\text{p}}{\to}\overline{R}_{p}\left(\overline{h}\right)$

follows from (C.42) and the weak law of large numbers, again giving that
$\|\hat{\overline{\mu}}_{m,n}-\overline{\mu}_{m}^{*}\|_{P,2}\overset{p}{\to}0$
as $n\to\infty$ for $m$ sufficiently large by Lemma C.6.

###### Lemma C.8.

Under Assumption C.1, the unique minimizers $\mu_{m}^{*}$ and
$\overline{\mu}_{n}^{*}$ of $R_{P}\left(h\right)$ and
$\overline{R}_{P}\left(h\right)$ over $\Theta_{m}$ and $\overline{\Theta}_{m}$
satisfy

$\|\mu_{m}^{*}-\mu\|_{P,2}\to
0\quad\text{and}\quad\|\overline{\mu}_{n}^{*}-\overline{\mu}\|_{P,2}\to 0$

as $m\to\infty$, respectively.

###### Proof.

We prove only $\|\mu_{m}^{*}-\mu\|_{P,2}\to 0$ as $m\to\infty$, as the
argument for $\|\overline{\mu}_{n}^{*}-\overline{\mu}\|_{P,2}\to 0$ as
$m\to\infty$ is identical. Assume, by the way of contradiction, that there
exists some $\delta>0$ such that $\|\mu_{m}^{*}-\mu\|_{P,2}>\delta$ for all
$m$ greater than some integer $m_{1}$. As $R_{P}\left(\cdot\right)$ is
strictly convex on $\Theta$, we have that

$R_{P}\left(\mu_{m}^{*}\right)>R_{P}\left(\mu\right)+\varepsilon$ (C.43)

for all $m\geq\max\left\\{m_{0},m_{1}\right\\}$ for some $\varepsilon>0$. By
the continuity of $R_{P}\left(\cdot\right)$, there exists some
$\delta^{\prime}>0$ such that $\|\mu^{\prime}-\mu\|_{P,2}<\delta^{\prime}$
implies

$R_{P}\left(\mu^{\prime}\right)<R_{P}\left(\mu\right)+\varepsilon.$ (C.44)

By Assumption C.1, (ii), there exists some integer $m_{1}$ such that
$\|\Pi_{m}\left(\mu\right)-\mu\|_{P,2}<\delta^{\prime}$ for all $m\geq m_{1}$.
Thus, by (C.44) and (C.43), we have that

$R_{P}\left(\Pi_{m}\left(\mu\right)\right)<R_{P}\left(\mu\right)+\varepsilon<R_{P}\left(\mu_{m}^{*}\right),$

for all $m\geq\max\left\\{m_{0},m_{1}\right\\}$, giving a contradiction, as
$\mu_{m}^{*}$ minimizes $R_{P}\left(\cdot\right)$ over $\Theta_{m}$.

###### Lemma C.9 (Chen,, 2007, Theorem 3.2,).

Let $\left\\{X_{i}\right\\}_{i=1}^{n}$ be an independent sequence of random
variables with an identical distribution $P$. Fix a sieve
$\Theta_{1}\subseteq\Theta_{2}\subseteq\cdots\subseteq\Theta$. For some loss
function $\ell\left(x,\theta\right)$, let $\theta^{*}\in\Theta$ be the
population risk minimizer

$\theta^{*}=\underset{\theta\in\Theta}{\arg\min}\text{
}\mathbb{E}_{P}\left[\ell\left(\theta,X_{i}\right)\right]$

and let $\|\cdot\|$ be some norm on $\Theta$. Define the function class

$\mathcal{F}_{n}=\left\\{\ell\left(\theta,\cdot\right)-\ell\left(\theta^{*},\cdot\right):\|\theta-\theta^{*}\|\leq\delta,\theta\in\Theta_{n}\right\\}.$

For some constant $b>0$, let

$\delta_{n}=\underset{\delta\in\left(0,1\right)}{\inf}\frac{1}{\sqrt{n}\delta^{2}}\int_{b\delta^{2}}^{\delta}\sqrt{H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|\right)}\text{d}w\leq
1,$

where $H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|\right)$ is the
$L^{r}\left(P\right)$ metric entropy with bracketing of the class
$\mathcal{F}_{n}$. Assume that the following conditions hold:

(i) There exists constants $c_{1},c_{2}>0$ such that

$c_{1}\mathbb{E}\left[\ell\left(\theta,X_{i}\right)-\ell\left(\theta^{*},X_{i}\right)\right]\leq\|\theta-\theta^{*}\|^{2}\leq
c_{2}\mathbb{E}\left[\ell\left(\theta,X_{i}\right)-\ell\left(\theta^{*},X_{i}\right)\right]$

for any $\theta\in\Theta$.

(ii) There is a constant $C_{1}>0$ such that for sufficiently small $\eta>0$,

$\sup_{\theta\in\Theta_{n}:\|\theta-\theta^{*}\|\leq\eta}\operatorname{Var}\left(\ell\left(\theta,X_{i}\right)-\ell\left(\theta^{*},X_{i}\right)\right)\leq
C_{1}\eta^{2}.$

(iii) For any $\eta>0$, there exists a constant $s\in\left(0,2\right)$ such
that

$\sup_{\theta\in\Theta_{n}:\|\theta-\theta^{*}\|\leq\eta}|\ell\left(\theta,X_{i}\right)-\ell\left(\theta^{*},X_{i}\right)|\leq\eta^{s}U\left(X_{i}\right)$

with $\mathbb{E}\left[U\left(Z_{i}\right)^{\gamma}\right]\leq C_{2}$ for some
$\gamma\geq 2$.

If $\hat{\theta}_{n}$ is a sequence of estimators satisfying

$\displaystyle\|\hat{\theta}_{n}-\theta^{*}\|$
$\displaystyle=o_{P}\left(1\right),$ (C.45)
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\ell\left(\hat{\theta}_{n},X_{i}\right)$
$\displaystyle\geq\sup_{\theta\in\Theta_{n}}\frac{1}{n}\sum\ell\left(\theta,X_{i}\right)+O_{P}\left(\varepsilon_{n}^{2}\right),\quad
and$ (C.46)
$\displaystyle\max\left\\{\delta_{n},\inf_{\theta\in\Theta_{n}}\|\theta^{*}-\theta\|\right\\}$
$\displaystyle=O\left(\varepsilon_{n}\right),$ (C.47)

as $n\to\infty$ for some sequence $\varepsilon_{n}$, then
$\|\hat{\theta}_{n}-\theta^{*}\|=O_{P}\left(\varepsilon_{n}\right)$ as
$n\to\infty$.

###### Lemma C.10.

Suppose that Assumption C.2 holds.

(i) There exists a function $K\left(Y,Z\right)$ and a constant $C$ such that
such that

$|L\left(h,a\right)-L\left(\mu,a\right)|\leq
K\left(y,z\right)|h\left(z\right)-\mu\left(z\right)|,$

for each
$a=\left(y,z,g\right)\in\mathbb{R}\times\mathcal{S}\times\left\\{0,1\right\\}$,
where

$\sup_{z\in\mathcal{Z}}\text{ }\mathbb{E}_{P}\left[K\left(Y,Z\right)^{2}\mid
Z=z\right]\leq C.$

(ii) There exists a function $\overline{K}\left(S,Z\right)$ and a constant
$\overline{C}$ such that such that

$|\overline{L}\left(\overline{h},a\right)-\overline{L}\left(\overline{\mu},a\right)|\leq\overline{K}\left(s,z\right)|\overline{h}\left(x\right)-\overline{\mu}\left(x\right)|,$

for each
$a=\left(y,z,g\right)\in\mathbb{R}\times\mathcal{S}\times\left\\{0,1\right\\}$,
where

$\sup_{x\in\mathcal{X}}\text{
}\mathbb{E}_{P}\left[\overline{K}\left(S,X\right)^{2}\mid
X=x\right]\leq\overline{C}.$

###### Proof.

We provide the details of the verification of the first claim, as verification
of the second claim follows by an identical argument. For any
$h\in\Lambda_{c}^{p}\left(\mathcal{Z}\right)$ and $z\in\mathcal{Z}$, we have
that

$\displaystyle|L\left(h,a\right)-L\left(\mu,a\right)|$
$\displaystyle=g|\left(y-h\left(z\right)\right)^{2}-\left(y-\mu\left(z\right)\right)^{2}|$
$\displaystyle=g\cdot|2\left(y-\lambda\left(z\right)h\left(z\right)+\left(1-\lambda\left(z\right)\right)\mu\left(z\right)\right)|\cdot|h\left(z\right)-\mu\left(z\right)|$

for some function $\lambda\left(z\right)\in\left[0,1\right]$, by the Mean
Value Theorem. Thus, we can set

$K\left(y,z\right)=g\cdot|2\left(y-\lambda\left(z\right)h\left(z\right)+\left(1-\lambda\left(z\right)\right)\mu\left(z\right)\right)|.$

We can verify that, for some constant $C$, we have that

$\displaystyle\mathbb{E}_{P}\left[K\left(Y,Z\right)^{2}\mid Z=z\right]$
$\displaystyle=4\mathbb{E}_{P}\left[G\left(Y-\lambda\left(Z\right)h\left(Z\right)+\left(1-\lambda\left(Z\right)\right)\mu\left(Z\right)\right)^{2}\mid
Z=z\right]$
$\displaystyle=\mathbb{E}_{P}\left[G\left(\left(Y-\mu\left(Z\right)\right)-\lambda\left(Z\right)\left(h\left(Z\right)-\mu\left(Z\right)\right)\right)^{2}\mid
Z=z\right]\leq C$

for all $z\in\mathcal{Z}$, by the bound (C.26) in Assumption C.2 and the fact
that both $h\left(z\right)$ and $\mu\left(z\right)$ are in
$\Lambda_{c}^{p}\left(\mathcal{Z}\right)$, and are therefore bounded on
$\mathcal{Z}$.

###### Lemma C.11 (Chen and Shen,, 1998, Lemma 2, ).

If $\mathcal{X}$ is a compact subset of $\mathbb{R}^{d}$ and
$h\in\Lambda_{c}^{p}\left(\mathcal{X}\right)$, then

$\|h\|_{\infty}\leq 2\|h\|_{2}^{\frac{2p}{2p+d}}c^{1-\frac{2p}{2p+d}},$

where $\|\cdot\|_{\infty}$ denotes the $L_{\infty}$ norm and $\|\cdot\|_{2}$
denotes the $L_{2}$ norm under the Lebesgue measure.

###### Lemma C.12.

Continue the notation of Lemma C.9. Assume that the conditions (i)-(iii) of
Lemma C.9 are satisfied. Suppose that the function class
$\Theta=\Lambda_{c}^{p}\left(\mathcal{X}\right)$, where $\mathcal{X}$ is some
compact subset of $\mathbb{R}^{d}$, that the sieve spaces $\Theta_{n}$ are
linear and satisfy

$\mathsf{dim}\left(\Theta_{n}\right)\asymp\left(\frac{n}{\log
n}\right)^{\frac{d}{2p+d}}\quad\text{and}\quad\inf_{h\in\Theta_{n}}\|h-\mu\|\asymp\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d}}.$ (C.48)

If $\hat{\theta}_{n}$ is a sequence of estimators satisfying the consistency
condition (C.45) and optimization condition (C.46) for the sequence
$\varepsilon_{n}$, then

$\|\hat{\theta}_{n}-\theta^{*}\|_{P,2}=O_{P}\left(\max\left\\{\varepsilon_{n},\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d}}\right\\}\right)$ (C.49)

as $n\to\infty$.

###### Proof.

As the conditions of Lemma C.9 are satisfied, we have that

$\displaystyle\|\hat{\theta}_{n}-\theta^{*}\|_{P,2}$
$\displaystyle=O_{P}\left(\max\left\\{\varepsilon_{n},\delta_{n},\inf_{h\in\Theta_{n}}\|h-\mu\|\right\\}\right)$
$\displaystyle=O_{P}\left(\max\left\\{\varepsilon_{n},\delta_{n},\left(\frac{\log
n}{n}\right)^{\frac{p}{2p+d}}\|\right\\}\right),$ (C.50)

where

$\delta_{n}=\underset{\delta\in\left(0,1\right)}{\inf}\frac{1}{\sqrt{n}\delta^{2}}\int_{b\delta^{2}}^{\delta}\sqrt{H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|_{2}\right)}\text{d}w\leq
1$

and $H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|_{2}\right)$ is the
$L^{2}\left(P\right)$ metric entropy with bracketing of the function class

$\mathcal{F}_{n}=\left\\{L\left(h,\cdot\right)-L\left(\mu,\cdot\right):\|h-\mu\|_{P,2}\leq\delta,h\in\Theta_{n}\right\\}.$

Now, Condition (iii) of Lemma C.9 implies that

$H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|_{2}\right)\leq\log
N\left(w^{1+\frac{d}{2p}},\Theta_{n},\|\cdot\|_{P,2}\right),$

where $N\left(w,\mathcal{F}_{n},\|\cdot\|_{2}\right)$ is the covering number
of $\mathcal{F}_{n}$. Moreover, we have that

$\log
N\left(w^{1+\frac{d}{2p}},\mathcal{F}_{n},\|\cdot\|_{2}\right)\lesssim\mathsf{dim}\left(\Theta_{n}\right)\log\left(\frac{1}{w}\right)$

as $\Theta_{n}$ is an element of a finite dimensional linear sieve (see e.g.,
Section 3.2 of Chen,, 2007). Thus, we have that

$\displaystyle\frac{1}{\sqrt{n}\delta^{2}}\int_{b\delta^{2}}^{\delta}\sqrt{H_{[]}\left(w,\mathcal{F}_{n},\|\cdot\|_{2}\right)}\text{d}w$
$\displaystyle\leq\frac{1}{\sqrt{n}\delta^{2}}\int_{b\delta^{2}}^{\delta}\sqrt{\log
N\left(w^{1+\frac{d}{2p}},\Theta_{n},\|\cdot\|_{P,2}\right)}\text{d}w$
$\displaystyle\lesssim\frac{1}{\delta}\sqrt{\frac{\mathsf{dim}\left(\Theta_{n}\right)}{n}\log\frac{1}{\delta}}$

and that therefore

$\delta_{n}=O\left(\sqrt{\frac{\mathsf{dim}\left(\Theta_{n}\right)\log
n}{n}}\right)=O\left(\left(\frac{\log n}{n}\right)^{\frac{p}{2p+d}}\right)$
(C.51)

by Assumption C.2 and the choice (C.48). The proof is then complete by
plugging the rate (C.51) into the bound (C.50).

##  Appendix D Simulation

###  D.1. Data

The data from Banerjee et al., (2015) were acquired from
https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/NHIXNT
on September 10, 2021. We restrict attention to data from the program
evaluation in Pakistan. Prior to cleaning, there are $1299$ households in this
sample. We omit data from $399$ households who attrited from the sample prior
to either of the post-treatment outcome measurements. We also omit data from
46 households who were missing measurements of all consumption variables for
either of the post-treatment outcome measurements. Consequently, we obtain a
cleaned sample of 854 households. Of these households, 446 were randomly
assigned to participate in the program. There was perfect compliance with
treatment assignment.

Some of the individuals are missing measurements for some of the variables.
There are at most $14$ individuals missing a measurement of a given variable.
For most continuously valued variables, we impute missing values with the mean
value of that outcome. Five of the continuously valued variables are heavily
censored, meaning that more than 50% of the observations are equal to
zero.242424These variables are total informal loans outstanding, total formal
loans outstanding, savings deposit amount, income from agriculture, income
from business, and revenue from animals We impute missing values with zero for
these variables. For categorical or binary valued variables, we impute missing
values with the modal outcome of that variable.

We include variables belonging in five categories: consumption, food security,
assets, finance, and income and revenue. The consumption variables are the
total monthly consumption and the total monthly consumptions on food, non-
food, and durable commodities. The food-security variables are an index for
overall food-security and five binary variables indicating different aspects
of food security (e.g., did a child in the household skip a meal). The assets
variables are value of assets, the value of productive assets, and value of
the household’s assets, each aggregated in two ways. The financial variables
are the total amount of formal and informal loans outstanding and the total
value of the households savings. The income and revenue variables are the
income from agriculture, the income from business, the income from paid labor,
the revenue from animals, and the self-assessed perception of economic status.
We do not include the total income from business as a baseline covariate, as
it is equal to zero for all individuals in the sample. We refer the reader to
the appendix of Banerjee et al., (2015) for further information on the
construction of these variables.

###  D.2. Simulation Calibration

We calibrate our simulation to the Banerjee et al., (2015) data with GANs. A
GAN is composed of a pair of competing neural networks (Goodfellow et al.,,
2014). The objective of one neural network is to generate data similar to the
training data. The objective of the other neural network is to distinguish
between the true and generated data. These networks iteratively compete until
the true and generated data are difficult to distinguish.

There are three GANs that contribute to generating a data point $A_{i}$. The
first GAN is fit to the marginal distribution of the pre-treatment covariates
(i.e., $X_{i}$). The second GAN is fit to the distribution of the short-term
outcomes conditioned on pre-treatment covariates and treatment (i.e.,
$S_{i}|X_{i},W_{i}$). The third GAN takes two forms, depending on whether we
are generating data designed to satisfy the assumptions in the Latent
Unconfounded Treatment or Statistical Surrogacy Models. For the Latent
Unconfounded Treatment Model, the third GAN is fit to the distribution of the
long-term outcome conditioned on the short-term outcomes, the pre-treatment
covariates, and treatment (i.e., $Y_{i}|S_{i},X_{i},W_{i}$). For the
Statistical Surrogacy Model, the analogous distribution does not condition on
treatment. In the Statistical Surrogacy Model treatment is assumed to be
independent of $Y_{i}$, conditional on $S_{i}$ and $X_{i}$, by Assumption 2.5.
By contrast, in the Latent Unconfounded Treatment Model, treatment is only
independent of the potential outcomes $Y_{i}(w)$, conditional on $S_{i}$ and
$X_{i}$, by Assumption 2.4.

We estimate the GANs needed to calibrate our simulation with the “wgan” python
package associated with Athey et al., (2021). This package is available at
https://github.com/gsbDBI/ds-wgan. Table 1 displays the choices of tuning
parameters used for this calibration. All other tuning parameter choices are
set to the wgan defaults.

Tuning Parameter |  | Distribution: | $X_{i}$ | $S_{i}\mid X_{i},W_{i}$ | $Y_{i}\mid S_{i},X_{i}$ | $Y_{i}\mid S_{i},X_{i},W_{i}$
---|---|---|---|---|---|---
Batch Size |  |  | 256 | 256 | 256 | 256
Epochs |  |  | 30000 | 30000 | 5000 | 5000
Learning Rate | Critic |  | $10^{-4}$ | $10^{-4}$ | $10^{-4}$ | $10^{-4}$
Generator |  | $10^{-4}$ | $10^{-4}$ | $10^{-4}$ | $10^{-4}$
Critic Dropout | Critic |  | 0 | 0.1 | 0.1 | 0.1
Generator |  | 0.1 | 0.1 | 0.1 | 0.1
Critic Gradient Penalty |  |  | 20 | 20 | 20 | 20

Table 1. Tuning Parameter Choices in GAN Estimation

Notes: Table 1 displays tuning parameter choices for the GAN estimation using
the wgan package associated with Athey et al., (2021). Any tuning parameter
choices not listed are set to the wgan defaults.

Figure 3 compares the moments of the data from Banerjee et al., (2015) to the
GAN generated simulation data. The first two rows compare the means and
variances of these data sets, and are displayed in log-scale. The second two
rows compare correlations within and across variable types. For example, the
third row compares the correlations of the pre-treatment covariates in the
true and generated data, and the fourth row compares the correlations of the
pre-treatment covariates and short-term outcomes. Note that in this comparison
the long-term outcomes $Y_{i}$ includes both total assets, chosen as the long-
term outcome in the main text, and total consumption, chosen as the long-term
outcome in Section D.5.

Figure D.4. Validation

---

Notes: Figure D.4 displays scatterplots comparing the moments of the data from
Banerjee et al., (2015) to the GAN generated simulation data. Columns
differentiate between different types of variables. In the first three rows,
the columns correspond to the pre-treatment covariates, the short-term
outcomes, and the long-term outcomes, respectively. The columns of the fourth
row correspond to the three pairs of these three variable types. Rows
differentiate between different types of moments. The first two rows compare
the means and variances of the the true and generated data. The second two
rows compare correlations of the true and generated data, within and across
variable types, respectively. The x-axis of each sub-panel measures the
moments of the true data. The y-axis of each sub-panel measures the moments of
the generated data. The x and y axes in the first two rows are displayed in
log-scale. A forty-five degree line is displayed in all sub-panels. Triangular
and square dots correspond to data generated the Latent Unconfounded Treatment
and Statistical Surrogacy Models, respectively. Circular dots correspond to
data shared by both models. Blue and green dots denote moments conditioned on
treatment being set to one and zero, respectively. Black dots denote
unconditioned moments.

The joint distributions of the true and generated data match remarkably
closely. This match holds both for data generated for the contexts in which
treatment is and is not observed in the observational sample, and continues to
hold if moments are conditioned on the treatment assignment being set to one
and zero, respectively.

Figure D.5 displays histograms comparing the true and generated distributions
for the long-term outcomes, conditioned on treatment assignment. These
distributions again align remarkably closely. In each case, the distributions
have similarly shaped supports and right tails.

Figure D.5. Comparison of True and Generated Long-Term Outcome Distributions

Panel A: Total Consumption
---
Panel B: Total Assets

Notes: Figure D.5 displays histograms comparing the true and generated long
term outcome distributions, conditioned by treatment assignment. Panel A
compares the distributions for total consumption. Panel B compares
distributions for total assets. Histograms for the generated and true data are
displayed in green and blue, respectively. Solid and dotted lines border the
histograms for the data generated for Latent Unconfounded Treatment and
Statistical Surrogacy Models, respectively.

###  D.3. Design

To generate an observation $A_{i}$, we first generate a value of the pre-
treatment covariates $X_{i}$ from the first GAN. This generated value of
$X_{i}$ is then input into the second GAN to generate values of the treated
and untreated short-term potential outcomes, i.e., generating $S_{i}(1)$ and
$S_{i}(0)$ by taking draws from the GAN approximating the distributions of
$S_{i}|X_{i},W_{i}=1$ and $S_{i}|X_{i},W_{i}=0$. We repeat this step, using
the generated values $(X_{i},S(1))$ and $(X_{i},S_{i}(0))$ as inputs into the
third GAN to generate the long-term treated and untreated potential outcomes
$Y_{i}(1)$ and $Y_{i}(0)$, respectively. Each observation is assigned to being
either “experimental” or “observational” uniformly at random, i.e., we draw
each $G_{i}$ from a Bernoulli distribution with parameter $1/2$. Treatment is
always assigned uniformly at random for experimental observations.

We consider two procedures for assigning treatment for the observational
units. First, as a baseline, we assign treatment uniformly at random, leaving
treatment assignment unconfounded. Second, we induce confounding.
Specifically, we assign treatment with probability

$\frac{1}{1+\exp(-(\tilde{\tau}_{c,i}+\tilde{\tau}_{a,i})\cdot\phi)}$

where $\tilde{\tau}_{c,i}$ and $\tilde{\tau}_{a,i}$ denote the true treatment
effects on short-term total consumption and total assets for unit $i$,
normalized across observational units, and $\phi$ is a parameter indexing the
degree of confounding. Written differently, the probability of being assigned
treatment increases logistically with the true treatment effects on short-term
outcomes. Large values of $\phi$ induce more confounding.

By repeating this process, we generate ten million simulation draws. We take
the empirical distribution of this sample as the population distribution. In
this sample, we observe both treated and untreated potential outcomes, and,
thereby, the true treatment effects for each individual. It is worth noting
the significant computational scale of the experiment reported in this
section. The simulation requires parallelization on a large-scale computing
cluster and uses roughly 60 years of cpu time.

Panel A of Figure D.6 plots the probability that an observational unit is
assigned to treatment as a function of $\tilde{\tau}_{c,i}$ and
$\tilde{\tau}_{a,i}$, the true treatment effects on short-term total
consumption and total assets for unit $i$, normalized across observational
units. This curve is displayed for $\phi$ equal to $0.50$ and $0.66$, which
are the parameter values that we consider in out simulation analysis. Panels B
and C display histograms of the treatment probabilities for all of the
simulation draws for $\phi$ equal to $0.50$ and $0.66$. Observe that
increasing $\phi$ increases the confounding in the data, in the sense that
there are more simulation draws with propensity scores very different from one
half.

Figure D.6. Visualization of Confounding Parameterization

Panel A: Parameterization of Confounding
---
Panel B: Histogram of Propensities, $\phi=0.5$ | Panel C: Histogram of Propensities, $\phi=0.66$
|

Notes: Panel A of Figure D.6 displays curves giving the probability of a
simulation draw being assigned to treatment as a function of
$\tilde{\tau}_{c,i}$ and $\tilde{\tau}_{a,i}$, the true treatment effects on
short-term total consumption and total assets for unit $i$, normalized across
observational units. Panels B and C display histograms of the treatment
probabilities for all simulation draws for $\phi$ equal to $0.50$ and $0.66$,
respectively.

###  D.4. Estimation Details

##### Cross-fitting:

Following the choice made in Chernozhukov et al., 2018a , we set $k$ equal to
five for the $k$-fold cross-fitting construction. We expect that alternative
choices for this hyper-parameter will not impact the results of the simulation
substantially, and so do not report results for different choices of $k$.

##### Propensity Score Estimation:

For the sake of numerical stability across simulations, we threshold all
estimates of propensity scores to lie in $[0.05,0.95]$, which amounts to
enforcing a value of $\varepsilon$ chosen for the strict overlap Assumption
2.2 in the estimation of propensity scores. The choice of this hyper-parameter
may substantively impact results in cases, e.g., in which the covariate
distributions of the experimental and observational distributions have limited
overlap. We do not report results for alternative choices of these
hyperparameters, but view the adaptation of the methods developed in Section 4
to setting in which there is limited overlap along one or more direction as a
useful direction for further research.

##### Linear Regression:

We use ordinary linear regression and a generalized linear model with a
binomial response type when estimating outcome mean and propensity score type
nuisance functions, respectively.

##### Lasso:

We implement cross-validated Lasso (Tibshirani,, 1996) with the R package
“glmnet’ (Hastie and Qian,, 2014). In each cross-fitting fold, we use $5$-fold
cross-validation to choose the regularization parameter that gives the minimum
mean cross-validation error. We use the binomial response type when estimating
propensity scores.

##### GRF:

We implement generalized random forests (Athey et al.,, 2019) with the R
package “grf”. When $h\in\\{1,3\\}$, we set the “honesty” hyper-parameter
equal to false. In each cross-fitting fold, we choose the hyper-parameters
“sample.fraction”, “mtry”, and “min.node.size” by setting the grf hyper-
parameter “tune.parameters” equal to “all”. This option chooses $1000$ hyper-
parameter values randomly, then chooses the hyper-parameter values with the
minimum 5-fold cross-validation error. When $h=10$, we set “sample.fraction”
equal to 0.1 in order to optimize memory usage, but tuned the other parameters
in the same manner. We use regression forests to predict both long-term
outcome means and propensity scores.

##### XGBoost:

We implement XGboost (Chen and Guestrin,, 2016) with the R package “xgboost”
(Chen et al.,, 2015). We set the learning rate $\eta$ and max depth equal to
$0.1$ and $2$, respectively. We choose the number of rounds with 5-fold cross
validation, with the maximum number of rounds set to $100$. We use the root
mean squared error and log-loss as the evaluation metrics for the estimation
of long-term outcome means and propensity scores, respectively.

###  D.5. Additional Results

Figures D.7 and D.8 displays results analogous to Figures 2 and 3 in the main
text, when total household consumption is chosen as the long-term outcome of
interest. The results are very similar to the those reported in the main text.

Figure D.7. Comparison of Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure D.7 compares measurements of the absolute bias and root mean
squared error for the estimators formulated in Section 4, in addition to the
difference in means estimator defined in (5.1). The y-axes are displayed in
logs, base 10. The long-term outcome is total household consumption. Panels A
and B display results for the Latent Unconfounded Treatment and Statistical
Surrogacy Models defined in 2.1 and 2.2, respectively. The columns of each
panel vary the sample size multiplier $h$. Each sub-panel displays results for
the baseline, unconfounded, case, as well as for the cases that the
confounding parameter $\phi$ has been set to $1/2$ and $2/3$. Results for each
estimator are displayed with dots of different colors. Results for different
nuisance parameter estimators are displayed with dots of different shapes.

Figure D.8. Finite-Sample Performance with Different Nuisance Parameter
Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure D.8 displays measurements of the quality of the estimators
formulated in Section 4.1 implemented with several alternative choices of
nuisance parameter estimators. The long-term outcome is total household
consumption. Panels A and B display results for the estimators defined in 4.1
for the Latent Unconfounded Treatment and Statistical Surrogacy Models defined
in 2.1 and 2.2, respectively. The columns of each panel vary the sample size
multiplier $h$. The rows of each panel display the absolute value of the bias,
one minus the coverage probability, and the root mean squared error of each
estimator, from top to bottom, respectively. A dotted line denoting one minus
the nominal coverage probability, 0.05, is displayed in each sub-panel in the
third row. Each sub-panel displays a bar graph comparing measurements of the
performance of the estimator defined in 4.1, constructed with three types of
nuisance parameter estimators, with the difference in means estimator (5.1).
Each sub-panel displays results for the baseline, unconfounded, case, as well
as for the cases that the confounding parameter $\phi$ has been set to $1/2$
and $2/3$.

Figure D.9. Comparison of Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure D.9 compares measurements of variance for the estimators
formulated in Section 4, in addition to the difference in means estimator
defined in (5.1). The y-axes are displayed in logs, base 10. The long-term
outcome is total household assets. All other formatting is analogous to Figure
2 in the main text.

Figure D.10. Comparison of Estimators

Panel A: Latent Unconfounded Treatment
---
Panel B: Statistical Surrogacy

Notes: Figure D.10 compares measurements of variance for the estimators
formulated in Section 4, in addition to the difference in means estimator
defined in (5.1). The y-axes are displayed in logs, base 10. The long-term
outcome is total household consumption. All other formatting is analogous to
Figure D.7

Tables 2 and 3 present estimates of coverage probabilities of confidence
intervals constructed around the estimators formulated in 4.1 with several
alternative choices of nuisance parameter estimators when total assets and
total consumption are chosen as the long-term outcome of interest,
respectively. Wald intervals constructed around the difference in means
estimator severely undercover when there is confounding in the observational
data set. The Confidence intervals (A.41) constructed around the estimators
formulated in 4.1 tend to undercover, although the under-coverage is not as
severe. Broadly, coverage is worse under the Statistical Surrogacy Model. In
this case, coverage becomes very poor when generalized random forests are used
to estimate nuisance parameters. By contrast, in the same situation, the
coverage probabilities are much closer to the nominal level when nuisance
parameters are estimated with the Lasso or XGBoost. As an exception, these
intervals over cover if nuisance parameters are estimated with linear
regression.

|  | Treatment Observed |  | Treatment Not Observed
---|---|---|---|---
Confounding: | Baseline | 0.5 | 0.66 |  | Baseline | 0.5 | 0.66
Sample Size: | (1) | (2) | (3) |  | (4) | (5) | (6)
Difference in Means | $\times 1$ | 0.953 | 0.803 | 0.712 |  | 0.953 | 0.811 | 0.735
$\times 3$ | 0.956 | 0.545 | 0.353 |  | 0.955 | 0.564 | 0.396
$\times 10$ | 0.953 | 0.095 | 0.011 |  | 0.954 | 0.101 | 0.021
Linear | $\times 1$ | 0.992 | 0.995 | 0.994 |  | 0.976 | 0.979 | 0.977
$\times 3$ | 0.999 | 0.999 | 1.000 |  | 0.974 | 0.981 | 0.983
$\times 10$ | 1.000 | 1.000 | 1.000 |  | 0.946 | 0.983 | 0.985
Lasso | $\times 1$ | 0.946 | 0.908 | 0.901 |  | 0.904 | 0.862 | 0.843
$\times 3$ | 0.945 | 0.922 | 0.911 |  | 0.823 | 0.877 | 0.884
$\times 10$ | 0.945 | 0.930 | 0.929 |  | 0.522 | 0.742 | 0.784
GRF | $\times 1$ | 0.947 | 0.901 | 0.868 |  | 0.791 | 0.438 | 0.334
$\times 3$ | 0.947 | 0.900 | 0.877 |  | 0.882 | 0.520 | 0.405
$\times 10$ | 0.946 | 0.842 | 0.743 |  | 0.905 | 0.583 | 0.467
XGBoost | $\times 1$ | 0.947 | 0.908 | 0.891 |  | 0.868 | 0.784 | 0.746
$\times 3$ | 0.952 | 0.909 | 0.900 |  | 0.916 | 0.834 | 0.807
$\times 10$ | 0.949 | 0.925 | 0.916 |  | 0.917 | 0.849 | 0.839

Table 2. Coverage Probability Estimates: Total Assets

Notes: Table 2 compares estimates of the coverage probabilities of confidence
intervals constructed around each estimator. The long-term outcome of interest
is taken to be total assets. The nominal coverage probability is set to $0.95$
in each case. Standard Wald intervals are constructed around the difference in
means estimator (5.1). Confidence intervals are constructed with (A.41) for
the estimators formulated in 4.1.

|  | Treatment Observed |  | Treatment Not Observed
---|---|---|---|---
Confounding: | Baseline | 0.5 | 0.66 |  | Baseline | 0.5 | 0.66
Sample Size: | (1) | (2) | (3) |  | (4) | (5) | (6)
Difference in Means | $\times 1$ | 0.953 | 0.910 | 0.887 |  | 0.948 | 0.914 | 0.895
$\times 3$ | 0.948 | 0.820 | 0.763 |  | 0.957 | 0.834 | 0.788
$\times 10$ | 0.950 | 0.554 | 0.374 |  | 0.949 | 0.590 | 0.436
Linear | $\times 1$ | 0.994 | 0.997 | 0.996 |  | 0.985 | 0.983 | 0.985
$\times 3$ | 0.999 | 1.000 | 0.999 |  | 0.985 | 0.984 | 0.985
$\times 10$ | 1.000 | 1.000 | 1.000 |  | 0.987 | 0.986 | 0.983
Lasso | $\times 1$ | 0.945 | 0.935 | 0.929 |  | 0.926 | 0.909 | 0.910
$\times 3$ | 0.941 | 0.938 | 0.931 |  | 0.930 | 0.924 | 0.926
$\times 10$ | 0.941 | 0.940 | 0.937 |  | 0.910 | 0.910 | 0.909
GRF | $\times 1$ | 0.948 | 0.937 | 0.926 |  | 0.852 | 0.705 | 0.654
$\times 3$ | 0.945 | 0.932 | 0.928 |  | 0.898 | 0.753 | 0.692
$\times 10$ | 0.943 | 0.925 | 0.908 |  | 0.918 | 0.747 | 0.680
XGBoost | $\times 1$ | 0.950 | 0.936 | 0.929 |  | 0.918 | 0.884 | 0.877
$\times 3$ | 0.951 | 0.934 | 0.929 |  | 0.931 | 0.905 | 0.899
$\times 10$ | 0.946 | 0.938 | 0.935 |  | 0.937 | 0.903 | 0.898

Table 3. Coverage Probability Estimates: Total Consumption

Notes: Table 3 compares estimates of the coverage probabilities of confidence
intervals constructed around each estimator. The long-term outcome of interest
is taken to be total consumption. The nominal coverage probability is set to
$0.95$ in each case. Standard Wald intervals are constructed around the
difference in means estimator (5.1). Confidence intervals are constructed with
(A.41) for the estimators formulated in 4.1.


# Audio Defect Detection in Music with Deep Networks

###### Abstract

With increasing amounts of music being digitally transferred from production
to distribution, automatic means of determining media quality are needed.
Protection mechanisms in digital audio processing tools have not eliminated
the need of production entities located downstream the distribution chain to
assess audio quality and detect defects inserted further upstream. Such
analysis often relies on the received audio and scarce meta-data alone.
Deliberate use of artefacts such as clicks in popular music as well as more
recent defects stemming from corruption in modern audio encodings call for
data-centric and context-sensitive solutions for detection. We present a
convolutional network architecture following end-to-end encoder-decoder
configuration to develop detectors for two exemplary audio defects. A click
detector is trained and compared to a traditional signal processing method,
with a discussion on context sensitivity. Additional post-processing is used
for data augmentation and workflow simulation. The ability of our models to
capture variance is explored in a detector for artefacts from decompression of
corrupted MP3 compressed audio. For both tasks we describe the synthetic
generation of artefacts for controlled detector training and evaluation. We
evaluate our detectors on the large open-source Free Music Archive (FMA) and
genre-specific datasets.

##  1 Introduction

In recent decades, digital means of media delivery have become increasingly
popular, not least with the advent of high-speed internet and the ubiquity of
digital playback and recording devices ranging from studio digital signal
processing hardware to MP3 players and mobile phones. The greater availability
of technology required to produce digital media now allows for small-scale
studios to create high quality content and instantly transfer it to
distributors such as music labels.

Digital media files can suffer from various degradations that occur during
transport and processing of the media. Automatic means exist for detection and
correction of certain data errors in uncompressed audio, but many complex
defects remain untackled. In this paper, we explore the efficacy of applying a
data-driven machine learning approach using Deep Neural Networks to two audio
defects, which music labels are confronted with in quality assurance of
incoming media.

For our first scenario, we define clicks as discontinuities affecting a few
signal samples, resulting in very short broadband impulses. A prominent source
of similar artefacts are buffer under-runs, where, due to synchronisation
issues during digital audio processing, a few samples of an old or zeroed
signal are transmitted instead of the current signal. Although existing
mastering software offers methods to remedy similar artefacts such as clicks,
crackle and clipping, restoration often requires manual selection of noise
profile, target segments or thresholds. This is due to ambiguities introduced
by e.g. signal quality or, depending on the genre, degradations voluntarily
applied to audio as effects, rendering current methods costly in large-scale
application.

Our hypothesis is that, using a data-driven approach, our network can
distinguish deliberate clicks (such as electronic snare drums) from defects,
and thereby enable automatic processing of large electronic music corpora.
This is an essential challenge in our scenario, where manual inspection may
not be feasible because of operational constraints.

In our second scenario, we aim to detect corruption of binary MP3 111See
standards https://www.iso.org/standard/19180.html and
https://www.iso.org/standard/31537.html data via the artefacts audible after
decoding such files. The MP3 audio encoding family uses a psychoacoustic model
to guide data reduction. Due to the transformations applied to data during MP3
decoding, a large variation of effects is possible, ranging from added
whistling noise over various missing frequency bands to broadband noise. In
contrast to noise normally added during the process of lossy encoding itself,
the artefacts stemming from unnoticed corruption have not been approached for
detection yet.

Such MP3 corruption may happen within a production chain where manual
transcoding is performed by different production agents, as illustrated with
the following use case: A distributor receives a sound file in lossless format
(e.g. pulse-code modulation format .wav) as a studio quality delivery from a
production entity. A degradation is detected at this point by listening to the
audio. During discussion of this defect, it is found that the production
entity sent a transcoded file they prepared for checking an earlier, defect
MP3 delivery. The decoded MP3 was transmitted rather than the original .wav
file intended for the delivery. Having no dependable meta-data on intermediary
processing, the distributor can only rely on the audio itself for quality
assurance. We found the above case to have practical relevance in the music
industry, and designed our method to aid error detection in similar
circumstances.

Our two detection scenarios are chosen to benefit from our data-driven
approach: while the click detection may make use of signal context to
determine the “musicality” of a click candidate, the MP3 glitch detector can
benefit from the network’s ability to capture the variation of artefacts
during training. In contrast to the click artefacts, the artistic use of the
MP3 glitch artefacts is currently limited, reducing the chance to confuse such
intentional use as degradation. The resulting models are designed for robust
scanning of large media libraries in an unsupervised batch processing
scenario.

In the following we present: an adaptation of the Wave-U-Net deep architecture
to the detection and localisation of audio defects, two separate detectors
built upon this to respectively detect clicks and MP3 glitches, methods for
simulation of these artefacts assuring significance of inserted defects, and a
large-scale evaluation against several music datasets.

##  2 Related Work

Traditional methods for detecting clicks and similar non-stationary noise in
audio aim at detecting discontinuities, using autoregressive modelling in the
raw audio or sparse optimisation in the spectral domain[1, 2]. For restoration
of analog media, where issues like clicks, overload and high frequency noise
are common, wavelet-based implementations are used in commercial noise removal
software [3, 4]. Recently, Deep Neural Networks have been used for noise and
obstruction removal mostly in images but also for audio and other time-based
signals[5, 6, 7]. Removal of obstructions from images is a task particularly
close to our task as it deals with less stationary noise [8, 9]. Matsui et
al.[10] use a convolutional neural network similar to our architecture to
remove fences from images. Restoration methods differ from our detection task
in that explicit detection reporting and evaluation of false positives are not
needed when applied as an audio de-noising effect.

For lossy audio encodings such as MP3, the artefacts arising during encoding,
mainly as a trade-off between quality and bitrate, vary with methods and
encoders and have been extensively discussed for the MP3 format [11].
Nevertheless, corruption of files introduces new, different artefacts that may
remain unnoticed during decoding and thus are the subject of our detector.

Research in the related field of Computational Auditory Scene Analysis (CASA)
concerns itself with the detection and labelling of events in an audio stream.
Deep Neural Networks are now being increasingly used in this field, often with
a spectral feature extraction pre-processing step. Mesaros et al.[12] report
the detection of “clicks” as one of 61 classes in a detection task on their
private dataset, with a recognition rate of around 65 percent. It is not clear
how these clicks compare to the clicks stemming from digitally signal failures
which we tackle in this paper. The focus in CASA is to detect recorded sounds
while being robust to noise in the recording, thus clicks to be detected would
relate to physical events (see "mouse click" in the CASA dataset [13]). From a
CASA perspective, intentional clicks, as frequently found in electronic music,
would not necessarily be distinguished from those stemming from defects.

This difficulty of ambiguity in audio defect annotation is noted by Alonso-
Jiménez et al.[14] who describe their implementation and evaluation of
established audio-defect detection algorithms. Their optimised algorithms
detect a significant number of audio defects in a database from a commercial
streaming service, noting that there may be a long tail of likely, but
undetected degradations. In our experiments (Section 6.1) we evaluate their
click detector implementation after Vaseghi[15], complementing their results
with accuracy metrics on a large dataset containing synthesized defects.

Research on error concealing scenarios, where defects are already identified,
e.g. due to missing packets at the network layer during transport of an audio
stream[16], can inform us of the complexity and artefacts resulting from such
failures. Deep convolutional networks have been recently introduced into the
field of audio inpainting - filling a gap in audio in such a way that the
error is concealed - to great effect[17, 18]. Their ability to encode and
model variation in large datasets results in more intelligible speech
reconstruction, when compared to conventional concealment approaches.

We aim to exploit gains from training with large datasets for our detectors.
Although large datasets of music, such as the FMA dataset, are available, we
are not aware of any open datasets with audio defect labels. This may be due
to the fact that defects are mainly corrected in production, and are - in
terms of playback time - very rare. The task of audio anomaly detection deals
with this issue using unsupervised learning to model usual/common signals on
unlabeled data. Autoregressive networks[19] and autoencoders[20] have recently
been used to detect unusual acoustic events via their high reconstruction loss
after such training. Our architecture is similar to the above, in that it
shares an information bottleneck to learn representations, but we use a
supervised learning approach with synthetic examples similar to the work in
[21] to better control the type of artefacts detected, avoiding the detection
of e.g. new instruments or audio samples as anomalies.

Ronneberger et al. [22] originally presented the U-Net as a deep convolutional
neural network for the task of segmenting biomedical images. The network
structure allows for efficient learning of spatial, or - in the case of audio
- temporal and frequency patterns. It has since been used for various end-to-
end audio transformation tasks [23]. Stoller et al. [24] modified this
structure to work directly on the one-dimensional audio signal as input,
resulting in the Wave-U-Net. In their source-separation task, they employ the
network on overlapping excerpts of the original signal at a low 8kHz sample
rate.

##  3 Model Architecture

In this paper we introduce the Hook-Net as a novel adaptation of the Wave-U-
Net for the task of detecting artefacts in audio signals. We apply this model
as an end-to-end approach, feeding raw waveform segments into the network to
receive a time series of classification results. After initial experiments
with a U-Net on spectrogram features, we found training to be more effective
when using raw audio input, which may be due to the extreme brevity of our
defects. The time-aligning horizontal connections of input and output promise
to help capture the context of distortions in the input waveform.

Our network takes as input segments 16384 samples of audio (at 44100Hz sample
rate) and outputs a time series of 128 output samples that are individually
quantised to the binary decision on original vs. degraded. The original Wave-
U-Net implies that input and output share the same sampling rate. To reduce
computational cost while operating at source input sample rates, the Hook-Net
introduces an imbalance: the output time resolution is reduced by a factor of
128 with regards to the input sample rate, resulting in a classification
sample rate of 344.5Hz.222This also allows comparisons with spectrogram-based
models, which did not perform as well and are omitted for brevity.

Figure 1: Architecture of the Hook-Net

This is reflected in the network structure as displayed in Figure 1. Our
contracting (left side) path consists in blocks comprising two sequences of
(zero-padded convolution – batch-norm – activation) layers followed by max-
pooling. Here, the temporal resolution is halved every block, while the number
of filters increases. An expanding block consists of an upscaling operation,
followed by concatenation of the skip connection from the contracting path and
a regular convolution layer. In our model, resolution of the axis mapped to
time in our output is only doubled every second block - resulting in a reduced
temporal resolution at the network output.

For the upscaling operation we follow the original U-Net [22], but employ
transposed convolution of stride 2 only every second layer, and stride 1
otherwise. Thereby some steadiness remains in the growth of resolution across
the expanding path, despite the reduced final output resolution. For the
horizontal skip-connections, connecting the contracting and expansive path, we
use max-pooling to adapt the time resolutions between the corresponding
layers. We furthermore add vertical skip connections on the contracting path,
bridging every block of two convolutions. This strategy is motivated by the
training benefits reported in residual networks[25].

In the following we describe the generation of click and MP3 glitch artefacts.

##  4 Click artefact generation

Within the scope of this paper, a click degradation corresponds to a fault in
the already digitised signal: we define a click as a discontinuity, where the
signal changes sharply to a random value for 1-3 samples but then continues
unchanged. With this definition we aim to cover and simulate defects from
digital signal transport, commonly resulting from buffer under-run during
playback or mixing in a DAW or timing errors during digital transport over
wire.

Clicks are inserted on-the-fly into the network input audio segments for
training, validation and test scenarios. The position of the click is
randomised, and one click is inserted with a probability of 0.1 per audio
segment, with a small variation in length. The amplitude value of the click is
calculated as a random offset of the current signal, from a uniform
distribution within $[0.3,1)$. Each initial offset sign is chosen randomly,
then signs that would create clipping are inverted.

The minimum amplitude offset ($0.3$) of the inserted clicks’ amplitudes is
introduced to assure that the signal change and resulting degradation is
significant. In absence of perceptual data on the acoustic salience of
inserted artefacts, this heuristic should create clicks that are likely to be
audible, where they are not perceptually masked by close preceding transients
or loud broad-band noise.

###  4.1 Audio post-processing via SoX

In order to simulate potential post-processing effects that may have occurred
on audio signals with previously undetected clicks, we use the
SoX333http://sox.sourceforge.net/ sound processor to slightly alter the audio
segments after the click-insertion steps. Segments are post-processed
regardless of whether a click has actually been inserted. A random combination
of reverb, two-band EQ and compression is applied, and strength and filter
parameters are chosen randomly within ranges that apply only mild changes to
the signal. In Section 6.1 we report results for click detector training and
detection with and without post-processing.

###  4.2 Click target vector

The target vector for training and testing of the detector is a 128-component
floating point vector that is set to 1 on the (resampled) position
corresponding to the location of an inserted click, and 0 otherwise. In our
experiments we simulate the problem of rare clicks that may be overlooked
during production. There is at most 1 click per input segment.

##  5 MP3 glitch artefact generation

This use-case tackles degradations that result from data corruption in the
commonly used MPEG-1/2 Audio Layer III (MP3) lossy audio compression format.
We will refer to these as _glitch_ defects. This degradation is interesting as
it can easily be “overlooked” in quality assurance. Moreover, the generation
approach described below can be generalised to other audio codecs.

Figure 2: Typical example of glitch degradation. Top: linear frequency
spectrogram of original signal. Bottom: same for signal with two degradations
at frame 0 (intermittent) and frame 80 with binary ground truth annotations
from spectrogram comparison.

The acoustic shape of the degradation varies strongly and often depends on the
content and amplitude of the surrounding audio data. Fig.2 displays a typical
glitch artefact: the degradation consists in added high-frequency content.
This effect appears when comparing the original (top) and degraded (bottom)
spectrogram contents between 80-100 frames. The same example features another
glitch effect at the start of the frame, affecting a lower part of the
spectrum. Acoustically, the effect is similar to a short whistle. An audio
example demonstrating glitch artefacts is available
online444https://osf.io/uqner/?view_only=042774933537440299dd48a4083305b1.
Note that the overall signal energy does not always change. The often well-
embedded and adaptive nature of these glitches render automatic detection
difficult.

###  5.1 Simulation of data corruption

Unlike the click degradations, which can be inserted on-the-fly as in Section
4, MP3 glitch degradations are calculated on a per-piece basis before training
due to the less easily indexable file format of MP3 which hinders exact
seeking.

In order to simulate data corruption in the compressed format, we modify MP3
encoded data during a decoding process which we survey on a frame by frame
basis: the MP3 format encodes an audio stream into a series of MP3 frames.
Each of these frames contains a header, containing format information and
parameters of the MP3 encoding process itself. An optional integrity check for
the header is often omitted to save bitrate. The
lame555https://linux.die.net/man/1/lame encoder used in the present study does
not include CRC checking. The header is followed by the encoded audio data,
the size of this block being determined by the bitrate used for the frame
during encoding. If a corrupted file remains undetected before and during
decoding, introduced errors may be audible as glitches in playback, but become
(from a data integrity point of view) undistinguishable from the original
signal. This may easily happen if a command-line decoding tool only issues a
warning in case of a data corruption which the decoding process can recover
from in successive frames.

We generate glitches as follows: First, the input audio is transcoded to
128kbps mono MP3 files. During the following decoding, frames are randomly
selected for glitch insertion given a probability of $P_{glitch}=0.05$. In
contrast to click generation we control glitch likelihood per MP3 frame. The
data in these frames is then partially overwritten with random data of a
randomised length. We found an average overwritten range of 120 bytes (an
average frame contains 418 bytes) with a standard deviation of 60 bytes to
give realistic results. No distinction was made between the header and data
sections of the MP3 frame. In the rare case that the decoding of the degraded
frame is not possible, the original MP3 frame is decoded and treated as non-
degenerated. The same method is used where the decoded data is of a different
length than the original un-degraded signal. Frames not selected for glitch
insertion are decoded inbetween such that the degraded decoding of the MP3
data results in a file with the same length as the original.

###  5.2 MP3 glitch target vector

MP3 glitches are inserted during pre-processing. To identify the actual parts
of the decoded wave signal affected by the glitch artefact after decoding and
segmentation, we employ a spectral distance measure comparing the degradedly
decoded audio to a clean decoding of the audio. Using a frame wise thresholded
difference of a power spectrogram at the frame rate of the network output (128
spectrogram frames), we determine whether significant degradation has taken
place for each of the 128 output values. Fig. 2 (bottom) shows a resulting
glitch classification target.

##  6 Experiments

We compare our detectors on the Creative-Commons licensed FMA-Large666Online
at https://github.com/mdeff/fma/. dataset, containing roughly 30-second
snippets of 106,574 tracks from 16341 artists within 161 genres, the most
frequent being "Experimental", "Electronic", and "Rock"[26]. The music is
stored in 320kbps stereo MP3 format.

Depending on the scenario, the data is either decompressed and degraded on-
the-fly, or, as in the MP3 glitch scenario, data is already pre-processed with
degradations added, and loaded as raw waveform alongside corresponding ground
truth data. The Hook-Net models take as input audio segments of 16384 samples
at a sampling rate of 44.1kHz, corresponding to 0.37 seconds.

Degraded (as well as non-degraded) audio and target data are then used for
network training, using a batch size of 200 segments. During configuration of
the Hook-Net, we tested variations on general parameters such as numbers of
filters and found the models with 13 contracting/expanding blocks (see Section
3), 15 filters per contraction and 5 filters per expansion, totalling at
27,307,633 trainable parameters, performing well for our tasks at hand.
Initial experiments consistently showed reduced precision in smaller models.
Training is performed in Tensorflow, using the ADAM[27] optimiser, on single
Nvidia V100 GPU.

###  6.1 Click detection

We selected a subset of 57,928 pieces from FMA-Large for training and
evaluation of our click detector Hook-Net. Pieces were separated into non-
overlapping training (40,000 pieces), validation (8964 pieces), and test (8964
pieces) sets. For each piece, 50 consecutive audio segments (18.6s in total)
were extracted with no overlap, resulting in 2,000,000 training segments and
448,200 validation and 448,200 test segments. In the above datasets, 0.078% of
the individual target values is set to 1 (on average, in every 10th segment,
one value in the target vector is marked as containing a click), which is
reflected in the initial setting of the network outputs’ activation biases.
Evaluation is performed on the target values as smallest units, not summarised
at the segment level. The model was trained using a root mean square loss
weighted towards the “click” class with a learning rate of 0.001. We report
results from models of the epoch with best validation set accuracy.

As a baseline for the click detection task we apply a generic click detector
(SigClick) as
implemented777https://essentia.upf.edu/reference/std_ClickDetector.html in the
open source Essentia library[28] to the data segments. To apply this to our
segments of 16384 samples, we add an additional sub-segmentation step, using
sub-segments of 4096 samples length, with an overlap of 2048 samples. Click
positions returned by SigClick are transformed into a binary output vector of
128 samples.

| test data | acc_t | pr_t | rec_t | f1_t
---|---|---|---|---|---
Hook-Net | FMA | 99.9993 | 99.77 | 99.24 | 99.47
SigClick | FMA | 99.95 | 84.39 | 90.52 | 84.60
Hook-Net | $\text{FMA}_{\text{post}}$ | 99.9991 | 99.70 | 99.02 | 99.32
$\text{Hook-Net}_{\text{post}}$ | $\text{FMA}_{\text{post}}$ | 99.9995 | 99.86 | 99.11 | 99.46
SigClick | $\text{FMA}_{\text{post}}$ | 99.97 | 85.52 | 81.04 | 80.62
Table 1: Click detector performances for plain (top) and post-processed
(bottom) click-degraded data (higher is better, best per dataset in bold):
test-set accuracy (acc_t), test precision(pr_t), recall(rec_t) and f-measure
(f1_t) for click detection. In percent.

Table 1 compares the test set performance of the Hook-Net to that of the best
performing (by highest validation set accuracy) configuration (threshold 35)
of the SigClick detector between tested thresholds of 30(default), 33, 35, 40,
and 50. The Hook-Net models were selected by highest validation set accuracy,
which was achieved after 13 epochs (36 total) for the click data (Hook-Net)
and 10 epochs (14 total) for training with post-processed click data
($\text{Hook-Net}_{\text{post}}$).

Due to the bias of the dataset and target vectors towards 0 (no click), we
concentrate on precision and recall which is reported for the click class. For
the application of finding defects in large commercial databases, the
precision of click detection is of great importance. Top of Table 1 shows the
Hook-Net achieves significantly better precision and recall of clicks. The
bottom of the table confirms this using post-processed click data as test set.
Added variation from post-processing results in lower recall values,
particularly with SigClick. The difference in training with (Hook-Net
${}_{\text{post}}$) and without (Hook-Net) post-processing shows good
generalisation from the generated click artefacts to the diverse post-
processed artefacts.

These results highlight a critical difference in the data-driven paradigm
applied in network training versus the more generalistic detection in
SigClick: while the assumption for a general click detector is to detect any
existing click (given other preconditions e.g. sufficient distance to the
preceding one), the Hook-Net has been trained to detect our inserted clicks
while ignoring other click instances in the music. Note that we do only
compare the performances on our clicks simulating digital defects such as
buffer under-run and timing errors. The SigClick detector may detect a wider
range of clicks, but for this initial experiment we refrained from training
more generic models due to the lack of datasets with clicks stemming from
defects in music production, as we aim to minimise false positives.

Manual verification showed that the FMA dataset does feature many examples of
electronic “glitch” music with intentional or expected clicks. The goal of our
detector is to differentiate such clicks from the ones added due to signal
failures.

| dataset | acc_t | pr_t | rec_t | f1_t
---|---|---|---|---|---
Hook-Net | electronic | 99.9997 | 99.91 | 99.74 | 99.82
SigClick | electronic | 99.94 | 80.34 | 98.06 | 84.94
Hook-Net | rock pop | 99.99992 | 100.0 | 99.90 | 99.95
SigClick | rock pop | 99.99 | 92.08 | 94.37 | 92.05
Hook-Net | classical | 100.0 | 100.0 | 100.0 | 100.0
SigClick | classical | 99.992 | 92.47 | 99.11 | 95.34
Table 2: Click performance in control datasets. Test-set accuracy (acc_t),
test precision(pr_t), recall(rec_t) and f-measure (f1_t).

We validate this relation to musical genre in Table 2 by applying the above
model and SigClick on three smaller genre-consistent datasets of 11400
(electronic), 10000 (pop rock) and 31600 (classical) segments not included in
the former training. Following the hypothesis that the electronic genre
features more intentional click samples than the classical genre, we see the
precision of SigClick very high for the classical data but significantly
dropping in the rock/pop and electronic genres, while the Hook-Net maintains
high precision with only little difference. The values for post-processed data
not reported here due to space limitations confirm this effect.

###  6.2 MP3 glitch detection

For this task, pre-caching of degraded audio allows us to use larger subsets
of FMA for training (66476 pieces), validation (14244 pieces) and test (14244
pieces).

For each piece, 50 consecutive audio segments were extracted as above,
resulting in 3,323,800 training segments and 712,200 validation and test
segments. Segments were randomised within each of the above datasets. While a
click only results in 1 target value to be set to 1, glitches affect multiple
target values per segment due to whole MP3 frames being affected by each
corruption. This results in train and validation datasets containing 8.04% of
the target values marked as glitched (test set: 8.19%).

Training is performed using root mean square loss, with initial learning rate
of 0.001 and reduction-on-plateau (factor 0.1, patience 10 epochs) of the
learning rate. Across 40 training epochs, the best model was selected based on
its validation accuracy (acc_v) measure.

| ep. | acc_v | acc_t | pr_t | rec_t | f1_t
---|---|---|---|---|---|---
Hook-Net | 31 | 98.52 | 98.46 | 92.83 | 88.04 | 90.37
Table 3: Glitch detection performance. Epoch with greatest validation-set
accuracy, validation accuracy (acc_v), test-set accuracy (acc_t), test
precision(pr_t), recall(rec_t) and f-measure (f1_t) for glitch detection. In
percent.

Table 3 shows the performance of this best model. Given the bias towards the
non-degraded class we report f-measure and precision regarding the glitched
class due to their relevance for our application scenario. The Hook-Net glitch
model generalises well from the validation to the test set, with an f1-measure
of 0.9037.

##  7 Conclusions and Future Work

We presented the Hook-Net convolutional neural network architecture with two
novel applications to detect click and data corruption errors in digital audio
recordings. The design goal of the architecture is to capture the typical
shape and variation of artefacts in the direct audio signal, with respect to
their acoustic context. The detectors localise errors with a time resolution
of less than 10 milliseconds.

For click detection, we demonstrated an end-to-end simulation, post-processing
and detector training method. Our evaluation shows the resulting detector
outperforms a state-of-the-art baseline on the large FMA popular music
dataset, using synthetically generated defects. A genre-specific evaluation
experiment shows the practical relevance of inclusion of context and the
capability of our model to capture this: discontinuities resembling clicks in
audio may represent intentional music content depending on their context. The
achieved precision ($>99.77$%) renders the detector suitable for the testing
of large commercial music databases.

Our second proposed application is aimed to filter degradations in modern
compressed audio from persisting unnoticed in subsequent use of the defect
audio. We describe the MP3 decoding glitch as a relatively novel type of audio
degradation for which detection is difficult due to its variation. Our
evaluation shows that our detector generalises well on common glitch
artefacts. The proposed training tasks come with a bias towards non-defective
audio, which we assume to be strong in real-world applications. This is
tackled using large training and validation datasets with synthesised artefact
insertion. The simulation of data corruption with subsequential MP3 decoding
promises a stronger realism of the artefacts synthesized in this task. This
process is applicable to other audio codecs, depending on decoder robustness
and consistency checks.

In future work we plan to extend the range of lossy compression defects
simulated and apply our architecture to further and more generalised local
audio degradations. We also aim to reduce model complexity without negatively
affecting performance.

##  8 Acknowledgements

This work was performed using HPC resources from GENCI-IDRIS (Grant
2020-AD011011379).

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