R OBUST
E STIMATION
VIA
G ENERATIVE
A DVERSARIAL
N ETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
Robust estimation under Huber’s (cid:15) -contamination model has become an important topic in statistics and theoretical computer science.
Rate-optimal procedures such as Tukey’s median and other estimators based on statistical depth functions are impractical because of their computational intractability.
In this paper, we establish an intriguing connection between f-GANs (Nowozin et al., 2016) and various depth functions through the lens of f-Learning.
Similar to the derivation of f-GAN, we show that these depth functions that lead to rate-optimal robust estimators can all be viewed as variational lower bounds of the total variation distance in the framework of f-Learning.
This connection opens the door of computing robust estimators using tools developed for training GANs.
In particular, we show that a JS-GAN (Goodfellow et al., 2014) that uses a neural network discriminator with at least one hidden layer is able to achieve the minimax rate of robust mean estimation under Huber’s (cid:15) -contamination model.
Interestingly, the hidden layers for the neural net structure in the discriminator class is shown to be necessary for robust estimation.
I NTRODUCTION
In the setting of Huber’s (cid:15) -contamination model (Huber, 1964; 1965), one has i.i.d observations
[Equation]
(1) and the goal is to estimate the model parameter θ .
Under the data generating process (1), each observation has a 1 − (cid:15) probability to be drawn from P θ and the other (cid:15) probability to be drawn from the contamination distribution Q .
The presence of an unknown contamination distribution poses both statistical and computational challenges to the problem.
For example, consider a normal mean estimation problem with P θ =
N ( θ, I p ) .
Due to the contamination of data, the sample average, which is optimal when (cid:15) = 0 , can be arbitrarily far away from the true mean if Q charges a positive probability at inﬁnity.
Moreover, even robust estimators such as coordinatewise median and geometric median are proved to be suboptimal under the setting of (1)
(Chen et al., 2018;
Diakonikolas et al., 2016a; Lai et al., 2016).
The search for both statistically optimal and computationally feasible procedures has become a fundamental problem in areas including statistics and computer science.
For the normal mean estimation problem, it has been shown in Chen et al.
(2018) that the minimax rate with respect to the squared (cid:96) 2 loss is pn ∨ (cid:15) 2 , and is achieved by Tukey’s median (Tukey, 1975).
Despite the statistical optimality of Tukey’s median, its computation is not tractable.
In fact, even an approximate algorithm takes
O ( e Cp ) in time (Amenta et al., 2000; Chan, 2004; Rousseeuw & Struyf, 1998).
Recent developments in robust statistics are focused on the search of computationally tractable algorithms for estimating θ under Huber’s (cid:15) -contamination model (1).
The success of the efforts started from two fundamental papers Diakonikolas et al.
(2016a);
Lai et al.
(2016), where two different but related computational strategies “iterative ﬁltering” and “dimension halving” are proposed to robustly estimate the normal mean.
These algorithms can provably achieve the minimax rate pn ∨ (cid:15) 2 up to a poly-logarithmic factor within polynomial time.
The main idea behind the two methods is the fact that a good robust moment estimator can be certiﬁed efﬁciently by higher moments.
This idea was later further extended (Diakonikolas et al., 2017; Du et al., 2017;
Diakonikolas et al., 2016b; 2018a;c;b; Kothari et al., 2018) to develop robust and computable procedures for various other problems.
However, many of the computationally feasible procedures for robust mean estimation in the literature rely on the knowledge of covariance matrix and sometimes the knowledge of contamination proportion.
Even though these assumptions can be relaxed, nontrivial modiﬁcations of the algorithms are required for such an extension and statistical error rates may also be affected.
Compared with these computationally feasible procedures proposed in the recent literature for robust estimation, Tukey’s median (9) and other depth-based estimators (Rousseeuw & Hubert, 1999; Mizera, 2002; Zhang, 2002; Mizera & M¨uller, 2004; Paindaveine & Van Bever, 2017) have some indispensable advantages in terms of their statistical properties.
First, the depth-based estimators have clear objective functions that can be interpreted from the perspective of projection pursuit (Mizera, 2002).
Second, the depth-based procedures are adaptive to unknown nuisance parameters in the models such as covariance structures, contamination proportion, and error distributions (Chen et al., 2018;
Gao, 2017).
Last but not least, Tukey’s depth and other depth functions are mostly designed for robust quantile estimation, while the recent advancements in the literature are all focused on robust moments estimation.
Although this is not an issue when it comes to normal mean estimation, the difference is fundamental for robust estimation under general settings such as elliptical distributions where moments do not necessarily exist.
Given the desirable statistical properties discussed above, this paper is focused on the development of computational strategies of depth-like procedures.
Our key observation is that robust estimators that are maximizers of depth functions, including halfspace depth, regression depth and covariance matrix depth, can all be derived from the framework of f-GAN (Nowozin et al., 2016).
As a result, these depth-based estimators can be viewed as minimizers of variational lower bounds of the total variation distance between the empirical measure and the model distribution.
This observation allows us to leverage the recent developments in the deep learning literature to compute these variational lower bounds through neural network approximations.
Our theoretical results give insights on how to choose appropriate neural network classes that lead to minimax optimal robust estimation under Huber’s (cid:15) -contamination model.
R OBUST E STIMATION AND F -GAN
We start with the deﬁnition of f-divergence (Csisz ´ ar, 1964; Ali & Silvey, 1966).
Given a strictly convex function f that satisﬁes f (1) = 0 , the f-divergence between two probability distributions P and Q is deﬁned by
[Equation]
Here, we use p ( · ) and q ( · ) to stand for the density functions of P and Q with respect to some common dominating measure.
For a fully rigorous deﬁnition, see Polyanskiy & Wu (2017).
Let f ∗ be the convex conjugate of f .
That is, f ∗ ( t ) =
sup u ∈ dom f
( ut − f ( u )) .
A variational lower bound of (2) is
[Equation]
[Equation]
Note that the inequality (3) holds for any T , and it becomes an equality whenever the class T contains the function f (cid:48)
( p/q ) (Nguyen et al., 2010).
For notational simplicity, we also use f (cid:48) for an arbitrary element of the subdifferential when the derivative does not exist.
With i.i.d.
observations X 1 , ..., X n ∼ P , the variational lower bound (3) naturally leads to the following learning method
[Equation]
The formula (4) is a powerful and general way to learn the distribution P from its i.i.d.
observations.
It is known as f-GAN (Nowozin et al., 2016), an extension of GAN (Goodfellow et al., 2014), which stands for generative adversarial networks .
The idea is to ﬁnd a (cid:98) P so that the best discriminator T in the class T cannot tell the difference between (cid:98) P and the empirical distribution 1 n (cid:80) ni
=
δ X i .
2. F -L EARNING : A U NIFIED F RAMEWORK
Our f-Learning framework is based on a special case of the variational lower bound (3).
That is,
[Equation] [Equation]
where (cid:101) q ( · ) stands for the density function of (cid:101) Q .
Compare (5) with (3), and it is easy to realize that (5) is a special case of (3) with
[Equation]
Moreover, the inequality (5) becomes an equality as long as P ∈ (cid:101) Q .
The sample version of (5) leads to the following learning method
[Equation]
The learning method (7) will be referred to as f-Learning in the sequel.
It is a very general framework that covers many important learning procedures as special cases.
For example, consider the case where Q = (cid:101) Q and f ( x ) = x log x .
Direct calculations give f (cid:48)
( x ) = log x + 1 and f ∗ ( t ) =
e t − 1 .
Therefore, (7) becomes
[Equation]
which is the maximum likelihood estimator (MLE).
2. TV-L EARNING AND D EPTH -B ASED E STIMATORS
An important generator f that we will discuss here is f ( x ) =
( x − 1)+ .
This leads to the total variation distance
D f ( P (cid:107) Q ) =
12 (cid:82) | p − q | .
With f (cid:48) ( x ) =
I { x ≥ 1 } and f ∗ ( t ) =
t I { 0 ≤ t ≤ 1 } , the TV-Learning is given by
[Equation]
Proposition 2.1.
The TV-Learning (8) includes the following special cases:
Take Q =
{ N ( η, I p ) : η ∈ R p } and (cid:101) Q =
{ N ( (cid:101)
η, I p ) : (cid:107) (cid:101)
η − η (cid:107) ≤ r } .
As r → 0 , (8) becomes (cid:98) θ = argmax η ∈ R p inf (cid:107) u
(cid:107)
[Equation]
(cid:88)
[Equation] [Equation]
Take Q = (cid:8) P y,X = P y | X P X : P y | X
=
N ( X T η, 1) , η ∈ R p (cid:9) , and (cid:101) Q = (cid:8) P y,X
=
P y
| X P X : P y | X
= N ( X T (cid:101) η, 1) , (cid:107) (cid:101)
η − η (cid:107) ≤ r (cid:9) .
As r → 0 , (8) becomes
[Equation]
Take Q =
{ N (0 , Γ) : Γ ∈ E p } , where E p is the class of all p
×
p covariance matrices, and (cid:101)
[Equation]
=
[Equation] [Equation]
(cid:101)
[Equation]
As r → 0 ,
8) becomes
[Equation]
The formula (9) is recognized as Tukey’s median, the maximizer of Tukey’s halfspace depth.
A traditional understanding of Tukey’s median is that (9) maximizes the halfspace depth (Donoho et al., 1992) so that (cid:98) θ is close to the center of all one-dimensional projections of the data.
In the f-Learning framework, N ( (cid:98) θ, I p ) is understood to be the minimizer of a variational lower bound of the total variation distance.
The formula (10) gives the estimator that maximizes the regression depth proposed by Rousseeuw & Hubert (1999).
It is worth noting that the derivation of (10) does not depend on the marginal distribution P X in the linear regression model.
Finally, (11) is related to the covariance matrix depth (Zhang, 2002; Chen et al., 2018; Paindaveine & Van Bever, 2017).
All of the estimators (9), (10) and (11) are proved to achieve the minimax rate for the corresponding problems under Huber’s (cid:15) -contamination model (Chen et al., 2018; Gao, 2017).
2. F ROM F -L EARNING TO F -GAN
The connection to various depth functions shows the importance of TV-Learning in robust estimation.
However, it is well-known that depth-based estimators are very hard to compute (Amenta et al., 2000; van Kreveld et al., 1999; Rousseeuw & Struyf, 1998), which limits their applications only for very low-dimensional problems.
On the other hand, the general f-GAN framework (4) has been successfully applied to learn complex distributions and images in practice (Goodfellow et al., 2014; Radford et al., 2015; Salimans et al., 2016).
The major difference that gives the computational advantage to f-GAN is its ﬂexibility in terms of designing the discriminator class T compared with the pre-speciﬁed choice (6) in f-Learning.
While f-Learning provides a uniﬁed perspective in understanding various depth-based procedures in robust estimation, we can step back into the more general f-GAN for its computational advantages, and to design efﬁcient computational strategies.
R OBUST M EAN E STIMATION VIA GAN
In this section, we focus on the problem of robust mean estimation under Huber’s (cid:15) -contamination model.
Our goal is to reveal how the choice of the class of discriminators affects robustness and statistical optimality under the simplest possible setting.
That is, we have i.i.d.
observations X 1 , ..., X n ∼ (1 − (cid:15) )
N ( θ, I p )
+ (cid:15)Q , and we need to estimate the unknown location θ ∈ R p with the contaminated data.
Our goal is to achieve the minimax rate pn ∨ (cid:15) 2 with respect to the squared (cid:96) 2 loss uniformly over all θ ∈ R p and all Q .
3. R ESULTS FOR TV-GAN
We start with the total variation GAN (TV-GAN) with f ( x ) =
( x − 1)
+ in (4).
For the Gaussian location family, (4) can be written as
[Equation]
with D ( x ) =
T ( x ) in (4).
Now we need to specify the class of discriminators D to solve the classiﬁcation problem between N ( η, I p ) and the empirical distribution 1 n (cid:80) ni =
δ X i .
One of the simplest discriminator classes is the logistic regression,
[Equation]
With D ( x ) = sigmoid ( w T x + b ) in (13), the procedure (12) can be viewed as a smoothed version of TVLearning (8).
To be speciﬁc, the sigmoid function sigmoid ( w T x + b ) tends to an indicator function as (cid:107) w (cid:107) → ∞ , which leads to a procedure very similar to (9).
In fact, the class (13) is richer than the one used in (9), and thus (12) can be understood as the minimizer of a sharper variational lower bound than that of (9).
Theorem 3.1.
Assume p n + (cid:15) 2 ≤ c for some sufﬁciently small constant c > 0 .
With i.i.d.
observations
X 1 , ..., X n ∼ (1 − (cid:15) )
N ( θ, I p )
+ (cid:15)Q , the estimator (cid:98) θ deﬁned by (12) satisﬁes
[Equation]
with probability at least 1 − e − C (cid:48)
( p + n(cid:15) 2 ) uniformly over all θ ∈ R p and all Q .
The constants C, C (cid:48) > 0 are universal.
Though TV-GAN can achieve the minimax rate pn ∨ (cid:15) 2 under Huber’s contamination model, it may suffer from optimization difﬁculties especially when the distributions Q and N ( θ, I p ) are far away from each other, as shown in Figure 1.
3. R ESULTS FOR JS-GAN
Given the intractable optimization property of TV-GAN, we next turn to Jensen-Shannon GAN (JS-GAN) with f ( x ) = x log x − ( x + 1) log x +12 .
The estimator is deﬁned by
[Equation]
with D ( x ) = log T ( x ) in (4).
This is exactly the original GAN (Goodfellow et al., 2014) specialized to the normal mean estimation problem.
The advantages of JS-GAN over other forms of GAN are studied extensively in the literature (Lucic et al., 2017; Kurach et al., 2018).
Unlike TV-GAN, our experiment results show that (14) with the logistic regression discriminator class (13) is not robust to contamination.
However, if we replace (13) by a neural network class with one or more hidden layers, the estimator will be robust and will also work very well numerically.
To understand why and how the class of the discriminators affects the robustness property of JS-GAN, we introduce a new concept called sigmoid Jensen-Shannon divergence.
Let g : R p
→
R d be a function that maps a multivariate observation to a d -dimensional feature space.
The sigmoid Jensen-Shannon divergence between two probability distributions P and Q with respect to the feature g is deﬁned as
[Equation]
In other words, P and Q are distinguished by a logistic regression classiﬁer using the feature g ( X ) .
It is easy to see that JS g ( P, Q ) is a variational lower bound of the original Jensen-Shannon divergence.
The key property of JS g ( P, Q ) is given by the following proposition.
Proposition 3.1.
Assume W is a convex set that contains an open neighborhood of 0 .
Then, JS g ( P, Q ) =
if and only if E P g ( X ) =
E Q g ( X ) .
The proposition asserts that JS g ( · , · ) cannot distinguish P and Q if the feature g ( X ) has the same expected value under the two distributions.
This generalized moment matching effect has also been studied by Liu et al.
(2017) for general f-GANs.
However, the linear discriminator class considered in Liu et al.
(2017) is parameterized in a different way compared with the discriminator class here.
When we apply Proposition 3.1 to robust mean estimation, the JS-GAN is trying to match the values ofn (cid:80) ni
=1 g ( X i ) and E N ( η,I η ) g ( X ) for the feature g ( X ) used in the logistic regression classiﬁer.
This explains what we observe in our numerical experiments.
A neural net without any hidden layer is equivalent to a logistic regression with a linear feature g ( X ) =
( X T , 1) T ∈ R p +1 .
Therefore, whenever η = 1 n (cid:80) ni
=1 X i , we have
JS g (cid:0) 1 n (cid:80) ni
=
δ X i , N ( η, I p ) (cid:1)
= 0 , which implies that the sample mean is a global maximizer of (14).
On the other hand, a neural net with at least one hidden layers involves a nonlinear feature function g ( X ) , which is the key that leads to the robustness of (14).
We will show rigorously that a neural net with one hidden layer is sufﬁcient to make (14) robust and optimal.
Consider the following class of discriminators,
D
[Equation]
The class (15) consists of two-layer neural network functions.
While the dimension of the input layer is p , the dimension of the hidden layer can be arbitrary, as long as the weights have a bounded (cid:96) 1 norm.
The nonlinear activation function σ ( · ) is allowed to take 1) indicator: σ ( x )
=
I { x ≥ 1 } , 2) sigmoid: σ ( x ) = 1 1+ e − x , 3) ramp: σ ( x ) = max(min
( x + 1 / 2 , 1) , 0) .
Other bounded activation functions are also possible, but we do not exclusively list them.
The rectiﬁed linear unit (ReLU) will be studied in Appendix A.1.
Theorem 3.2.
Consider the estimator (cid:98) θ deﬁned by (14) with D speciﬁed by (15).
Assume pn + (cid:15) 2 ≤ c for some sufﬁciently small constant c > 0 , and set κ = O (cid:0)(cid:112) pn + (cid:15) (cid:1) .
With i.i.d.
observations X 1 , ..., X n ∼ (1 − (cid:15) )
N ( θ, I p )
+ (cid:15)Q , we have
[Equation]
, with probability at least 1 − e − C (cid:48)
( p + n(cid:15) 2 ) uniformly over all θ ∈ R p and all Q .
The constants C, C (cid:48) > 0 are universal.
E LLIPTICAL D ISTRIBUTIONS
An advantage of Tukey’s median (9) is that it leads to optimal robust location estimation under general elliptical distributions.
In this section, we show that JS-GAN shares the same property.
A random vector X ∈ R p follows an elliptical distribution if it admits a representation
[Equation]
where U is uniformly distributed on the unit sphere { u ∈ R p : (cid:107) u (cid:107) = 1 } and ξ ≥ 0 is a random variable independent of U that determines the shape of the elliptical distribution (Fang, 2017).
The center and the scatter matrix is θ and Σ = AA T .
For a unit vector v , let the density function of ξv T U be h .
Note that h is independent of v because of the symmetry of U .
Then, there is a one-to-one relation between the distribution of ξ and h , and thus the triplet ( θ, Σ , h ) fully parametrizes an elliptical distribution.
Note that h and Σ
= AA T are not identiﬁable, because ξA = ( cξ )( c − 1 A ) for any c > 0 .
Therefore, without loss of generality, we can restrict h to be a member of the following class
[Equation]
This makes the parametrization ( θ, Σ , h ) of an elliptical distribution fully identiﬁable, and we use EC ( θ, Σ , h ) to denote an elliptical distribution parametrized in this way.
The JS-GAN estimator is deﬁned as
[Equation]
where E p ( M ) is the set of all positive semi-deﬁnite matrix with spectral norm bounded by M .
Theorem 4.1.
Consider the estimator (cid:98) θ deﬁned above with D speciﬁed by (15).
Assume M
=
O
(1) , pn + (cid:15) 2 ≤ c for some sufﬁciently small constant c > 0 , and set κ = O (cid:0)(cid:112) pn + (cid:15) (cid:1) .
With i.i.d.
observations X 1 , ..., X n ∼ (1 − (cid:15) )
EC ( θ, Σ , h ) + (cid:15)Q , we have
[Equation]
, with probability at least 1 − e − C (cid:48)
( p + n(cid:15) 2 ) uniformly over all θ ∈ R p , Σ ∈ E p ( M ) (for some M = O (1) ) and all Q .
The constants C, C (cid:48) > 0 are universal.
Remark 4.1.
The result of Theorem 4.1 also holds (and is proved) under the strong contamination model (Diakonikolas et al., 2016a).
That is, we have i.i.d.
observations X 1 , ..., X n ∼ P for some P satisfying
TV ( P, EC ( θ, Σ , h )) ≤ (cid:15) .
Note that Theorem 4.1 guarantees the same convergence rate as in the Gaussian case for all elliptical distributions.
This even includes multivariate Cauchy where mean does not exist.
Therefore, the location estimator (16) is fundamentally different from Diakonikolas et al.
(2016a);
Lai et al.
(2016), which is only designed for robust mean estimation.
We will show such a difference in our numerical results.
To achieve rate-optimality for robust location estimation under general elliptical distributions, the estimator (16) is different from (14) only in the generator class.
They share the same discriminator class (15).
This underlines an important principle for designing GAN estimators: the overall statistical complexity of the estimator is only determined by the discriminator class.
The estimator (16) also outputs ( (cid:98) Σ , (cid:98) h ) , but since the discriminator class is speciﬁcally designed for location estimation, there is no theoretical guarantee for ( (cid:98) Σ , (cid:98) h ) .
This will be systematically studied in a future project.
N UMERICAL E XPERIMENTS
In this section, we give extensive numerical studies of robust mean estimation via GAN.
After introducing the implementation details in Section 5.1, we verify our theoretical results on minimax estimation with both
TV-GAN and JS-GAN in Section 5.2.
Comparison with other methods on robust mean estimation in the literature is given in Section 5.3.
The effects of various network structures are studied in Section 5.4.
Finally, adaptation to unknown covariance is studied in Section 5.5.
We assume i.i.d.
observations are drawn from (1 − (cid:15) )
N (0 p , I p ) + (cid:15)Q with (cid:15) and Q to be speciﬁed.
5. I MPLEMENTATIONS
We adopt the standard algorithmic framework of f-GANs (Nowozin et al., 2016) for the implementation of JSGAN and TV-GAN for robust mean estimation.
In particular, the generator for mean estimation is G η ( Z ) =
Z + η with Z ∼ N (0 p , I p ) ; the discriminator D is a multilayer perceptron (MLP), where each layer consisting of a linear map and a sigmoid activation function and the number of nodes will vary in different experiments to be speciﬁed below.
Details related to algorithms, tuning, critical hyper-parameters, structures of discriminator networks and other training tricks for stabilization and acceleration are discussed in Appendix B.1.
The source code to reproduce the results will be released upon request.
5. N UMERICAL S UPPORT FOR THE M INIMAX R ATES
We verify the minimax rates achieved by TV-GAN (Theorem 3.1) and JS-GAN (Theorem 3.2) via numerical experiments.
Two main scenarios we consider here are (cid:112) p/n < (cid:15) and (cid:112) p/n > (cid:15) , where in both cases, various types of contamination distributions Q are considered.
Speciﬁcally, the choice of contamination distributions
Q includes N ( µ ∗ 1 p , I p ) with µ ranges in { 0 . 2 , 0 . 5 , 1 , 5 } , N (0 .
5 ∗ 1 p , Σ) and Cauchy ( τ ∗ 1 p ) .
Details of the construction of the covariance matrix Σ is given in Appendix B.2.
The distribution Cauchy ( τ ∗ 1 p ) is obtained by combining p independent one-dimensional standard Cauchy with location parameter τ j = 0 .
The main experimental results are summarized in Figure 2, where the (cid:96) 2 error we present is the maximum error among all choices of Q , and detailed numerical results can be founded in Tables 7, 8 and 9 in Appendix.
We separately explore the relation between the error and one of (cid:15) , √ p and 1 / √ n with the other two parameters ﬁxed.
The study of the relation between the (cid:96) 2 error and (cid:15) is in the regime (cid:112) p/n < (cid:15) so that (cid:15) dominates the minimax rate.
The scenario (cid:112) p/n > (cid:15) is considered in the study of the effects of √ p and 1 / √ n .
As is shown in Figure 2, the errors are approximately linear against the corresponding parameters in all cases, which empirically veriﬁes the conclusions of Theorem 3.1 and Theorem 3.2.
5. C OMPARISON WITH O THER M ETHODS
We perform additional experiments to compare with other methods including dimension halving (Lai et al., 2016) and iterative ﬁltering (Diakonikolas et al., 2017) under various settings.
We emphasize that our method does not require any knowledge about the nuisance parameters such as the contamination proportion (cid:15) .
Tuning
GAN is only a matter of optimization and one can tune parameters based on the objective function only.
[Table]
Table 1 shows the performances of JS-GAN, TV-GAN, dimension halving, and iterative ﬁltering.
The network structure, for both JS-GAN and TV-GAN, has one hidden layer with 20 hidden units when the sample size is 50,000 and 2 hidden units when sample size is 5,000.
The critical hyper-parameters we apply is given in
Appendix and it turns out that the choice of the hyper-parameter is robust against different models when the net structures are the same.
To summarize, our method outperforms other algorithms in most cases.
TV-GAN is good at cases when Q and N (0 p , I p ) are non-separable but fails when Q is far away from N (0 p , I p ) due to optimization issues discussed in Section 3.1 (Figure 1).
On the other hand, JS-GAN stably achieves the lowest error in separable cases and also shows competitive performances for non-separable ones.
5. N ETWORK S TRUCTURES
We further study the performances of TV-GAN and JS-GAN with various structures of neural networks.
The main observation is tuning networks with one-hidden layer becomes tough as the dimension grows (e.g. p ≥ 200 ), while a deeper network can signiﬁcantly reﬁne the situation perhaps by improving the landscape.
Some experiment results are given in Table 2.
On the other hand, one-hidden layer performs not worse than deeper networks when dimension is not very large (e.g. p ≤ 100 ).
More experiments are given in Appendix B.4.
Additional theoretical results for deep neural nets are given in Appendix A.1.
[Table]
5. A DAPTATION TO U NKNOWN C OVARIANCE
The robust mean estimator constructed through JS-GAN can be easily made adaptive to unknown covariance structure.
We deﬁne
[Equation]
Compared with (14), (17) optimizes over both the mean and the covariance matrix.
The estimator (cid:98) θ , as a result, is rate-optimal even when the true covariance matrix is not necessarily identity and is unknown (see Theorem
A.2 in Appendix A.2).
Below, we demonstrate some numerical evidence of the optimality of the estimator (17) in Table 3.
[Table]
A DAPTATION TO E LLIPTICAL D ISTRIBUTIONS
We consider the estimation of the location parameter θ in elliptical distribution EC ( θ, Σ , h ) by the JS-GAN deﬁned in (16).
In particular, we study the case with i.i.d.
observations X 1 , ..., X n ∼ (1 − (cid:15) )
Cauchy ( θ, I p )
+ (cid:15)Q .
The density function of Cauchy ( θ, Σ) is given by p ( x ; θ, Σ) ∝ | Σ | − 1 / 2 (cid:0) 1 + ( x − θ ) T Σ − 1 ( x − θ ) (cid:1) − (1+ p ) / 2 .
Compared with Algorithm (1), the difference lies in the choice of the generator.
We consider the generator
G 1 ( ξ, U )
=
g ω ( ξ ) U + θ , where g ω ( ξ ) is a non-negative neural network parametrized by ω and some random variable ξ .
The random vector U is sampled from the uniform distribution on { u ∈ R p : (cid:107) u (cid:107) = 1 } .
If the scatter matrix is unknown, we will use the generator G 2 ( ξ, U ) =
g ω ( ξ ) AU
+ θ , with AA T modeling the scatter matrix.
Table 4 shows the comparison with other methods.
Our method still works well under Cauchy distribution, while the performance of other methods that rely on moment conditions are affected in this setting.
[Table]
Cauchy (0 .
5 ∗ 1 p , I p )
[Table]
Cauchy (5 .
∗ 1 p , I p )
[Table] [Table]
Cauchy (0 .
5 ∗ 1 p , 5 ∗ I p )
[Table]
A A DDITIONAL T HEORETICAL R ESULTS A. R ESULTS FOR D EEP R E LU N ETWORKS
In this section, we investigate the performance of discriminator classes of deep neural nets with the ReLU activation function.
Since our goal is to learn a p -dimensional mean vector, a deep neural network discriminator without any regularization will certainly lead to overﬁtting.
Therefore, it is crucial to design a network class with some appropriate regularizations.
Inspired by the work of Bartlett (1997); Bartlett & Mendelson (2002), we consider a network class with (cid:96) 1 regularizations on all layers except for the second top layer with an (cid:96)regularization.
With G H 1 ( B ) =
(cid:8) g ( x ) = ReLU ( v T x ) : (cid:107) v (cid:107) 1 ≤ B (cid:9) , a neural network class with l + 1 layers is deﬁned as
[Equation]
Combining with the last sigmoid layer, we obtain the following discriminator class,
[Equation]
Note that all the activation functions are ReLU ( · ) except that we use sigmoid ( · ) in the top two layers.
A theoretical guarantees of the class deﬁned above is given by the following theorem.
Theorem A.1.
Assume p log p n ∨ (cid:15) 2 ≤ c for some sufﬁciently small constant c > 0 .
Consider i.i.d.
observations
X 1 , ..., X n ∼ (1 − (cid:15) )
N ( θ, I p )
+ (cid:15)Q and the estimator (cid:98) θ deﬁned by (14) with D = F HL ( κ, τ, B ) with H ≥ 2 p , 2 ≤ L = O (1) , 2 ≤ B = O (1) , and τ = √ p log p .
We set κ = O (cid:18)(cid:113) p log p n + (cid:15) (cid:19) .
Then, for the estimator (cid:98) θ deﬁned by (14) with D = F HL ( κ, τ, B ) , we have
[Equation]
with probability at least 1 − e − C (cid:48)
( p log p + n(cid:15) 2 ) uniformly over all θ ∈ R p such that (cid:107) θ (cid:107) ∞ ≤ √ log p and all Q .
The theorem shows that JS-GAN with a deep ReLU network can achieve the error rate p log p n ∨ (cid:15) 2 with respect to the squared (cid:96) 2 loss.
The condition (cid:107) θ (cid:107) ∞ ≤ √ log p for the ReLU network can be easily satisﬁed with a simple preprocessing step.
We split the data into two halves, whose sizes are log n and n − log n , respectively.
Then, we calculate the coordinatewise median (cid:101) θ using the small half.
It is easy to show that (cid:107) (cid:101) θ − θ (cid:107) ∞ ≤ (cid:113) log p log n ∨ (cid:15) with high probability.
Then, for each X i from the second half, the conditional distribution of X i − (cid:101) θ given the ﬁrst half is (1 − (cid:15) )
N ( θ − (cid:101) θ, I p )+ (cid:15) (cid:101) Q .
Since (cid:113) log p log n ∨ (cid:15) ≤ √ log p , the condition (cid:107) θ − (cid:101) θ (cid:107) ∞ ≤ √ log p is satisﬁed, and thus we can apply the estimator (14) using the shifted data X i − (cid:101) θ from the second half.
The theoretical guarantee of Theorem A.1 will be
[Equation]
with high probability.
Hence, we can use (cid:98) θ
+ (cid:101) θ as the ﬁnal estimator to achieve the same rate in Theorem A.1.
On the other hand, our experiments show that this preprocessing step is not needed.
We believe that the assumption (cid:107) θ (cid:107) ∞ ≤ √ log p is a technical artifact in the analysis of the Rademacher complexity.
It can probably be dropped by a more careful analysis.
A. R ESULTS WITH U NKNOWN C OVARIANCE
The robust mean estimator constructed through JS-GAN can be easily made adaptive to unknown covariance structure.
We deﬁne
[Equation]
where E p ( M ) = { Γ ∈ E p : (cid:107) Γ (cid:107) op ≤ M } .
Compared with (14), (18) optimizes over both the mean and the covariance matrix.
The estimator (cid:98) θ , as a result, is rate-optimal even when the true covariance matrix is not necessarily identity and is unknown.
Theorem A.2.
Consider the estimator (cid:98) θ deﬁned by (18) with D speciﬁed by (15).
Assume M
=
O
(1) , pn + (cid:15) 2 ≤ c for some sufﬁciently small constant c > 0 , and set κ = O (cid:0)(cid:112) pn + (cid:15) (cid:1) .
With i.i.d.
observations
X 1 , ..., X n ∼ (1 − (cid:15) )
N ( θ, Σ) + (cid:15)Q , we have
[Equation]
with probability at least 1 − e − C (cid:48)
( p + n(cid:15) 2 ) uniformly over all θ ∈ R p , Σ ∈ E p ( M ) and all Q .
The constants
C, C (cid:48) > 0 are universal.
Theorem A.2 enjoys the same error rate as in Theorem 3.2, even if the covariance matrix is not known.
This is because the complexity of (18) is determined by the discriminator class through
[Equation]
regardless of the generator class.
Similar results of adaptation also hold for D being deep neural network classes, and will be omitted in this paper.
B D ETAILS OF E XPERIMENTS B. T RAINING D ETAILS
The implementation for JS-GAN is given in Algorithm 1, and a simple modiﬁcation of the objective function leads to that of TV-GAN.
Several important implementation details are discussed below.
• How to tune parameters?
The choice of learning rates is crucial to the convergence rate, but the minimax game is hard to evaluate.
We propose a simple strategy to tune hyper-parameters including the learning rates.
Suppose we have estimators (cid:98) θ 1 , . . .
, (cid:98) θ M with corresponding discriminator networks D (cid:98) w 1 ,..., D (cid:98) w M .
Fixing η = (cid:98) θ , we further apply gradient descent to D w with a few more epochs (but not many in order to prevent overﬁtting, for example 10 epochs) and select the (cid:98) θ with the smallest value of the objective function (14) (JS-GAN) or (12) (TV-GAN).
We note that training discriminator and generator alternatively usually will not suffer from overﬁtting since the objective function for either the discriminator or the generator is always changing.
However, we must be careful about the overﬁtting issue when training the discriminator alone with a ﬁxed η , and that is why we apply an early stopping strategy here.
Fortunately, the experiments show if the structures of networks are same (then of course, the dimensions of the inputs are same), the choices of hyper-parameters are robust to different models and we present the critical parameters in Table
to reproduce the experiment results in Table 1 and Table 2.
[Table]
[Table] [Table]
( G η ( Z ))
[Table]
Input : Observation set S =
{ X 1 , . . .
, X n } ∈ R p , discriminator network D w ( x ) , generator network G η ( z ) =
z + η , learning rates γ d and γ g for the discriminator and the generator, batch size m , discriminator steps in each iteration K , total epochs T , average epochs T 0 .
Initialization : Initialize η with coordinatewise median of S .
Initialize w with N (0 , .
05) independently on each element or Xavier (Glorot & Bengio, 2010).
1: for t = 1 , . . .
, T do 2: for k = 1 , . . .
, K do 3:
Sample mini-batch { X 1 , . . .
X m } from S .
Sample { Z 1 , . . .
Z m } from N (0 , I p ) 4: g w ← ∇ w [ 1 m Σ mi =1 log D w ( X i ) + 1 m Σ mi =1 log(1 − D w
( G η ( Z i )))] 5: w ← w + γ d g w 6: end for 7:
Sample { Z 1 , . . .
Z m } from N (0 , I p ) 8: g η ← ∇ η [ 1 m Σ mi =1 log(1 − D w ( G η ( Z i )))
] 9: η ← η − γ g g η 10: end for
Return : The average estimate η over the last T 0 epochs.
[Table]
• When to stop training?
Judging convergence is a difﬁcult task in GAN trainings, since sometimes oscillation may occur.
In computer vision, people often use a task related measure and stop training once the requirement based on the measure is achieved.
In our experiments below, we simply use a sufﬁciently large T which works well, but it is still interesting to explore an efﬁcient early stopping rule in the future work.
• How to design the network structure?
Although Theorem 3.1 and Theorem 3.2 guarantee the minimax rates of TV-GAN without hidden layer and JS-GAN with one hidden layer, one may wonder whether deeper network structures will perform better.
From our preliminary experiments, TV-GAN with one hidden layer is signiﬁcantly better than TV-GAN without any hidden layer.
Moreover, JS-GAN with deep network structures can signiﬁcantly improve over shallow networks especially when the dimension is large (e.g. p ≥ 200 ).
For a network with one hidden layer, the choice of width may depend on the sample size.
If we only have 5,000 samples of 100 dimensions, two hidden units performs better than ﬁve hidden units, which performs better than twenty hidden units.
If we have 50,000 samples, networks with twenty hidden units perform the best.
• How to stabilize and accelerate TV-GAN?
As we have discussed in Section 3.1, TV-GAN has a bad landscape when N ( θ, I p ) and the contamination distribution Q are linearly
separable (see Figure 1).
An outlier removal step before training TV-GAN may be helpful.
Besides, spectral normalization
(Miyato et al., 2018) is also worth trying since it can prevent the weight from going to inﬁnity and thus can increase the chance to escape from bad saddle points.
To accelerate the optimization of TVGAN, in all the numerical experiments below, we adopt a regularized version of TV-GAN inspired by Proposition 3.1.
Since a good feature extractor should match nonlinear moments of P = (1 − (cid:15) )
N ( θ, I p )
+ (cid:15)Q and N ( η, I p ) , we use an additional regularization term that can accelerate training and sometimes even leads to better performances.
Speciﬁcally, let D ( x ) = sigmoid ( w T Φ( x )) be the discriminator network with w being the weights of the output layer and Φ D ( x ) be the corresponding network after removing the output layer from D ( x ) .
The quantity Φ D ( x ) is usually viewed as a feature extractor, which naturally leads to the following regularization term (Salimans et al., 2016;
Mroueh et al., 2017), deﬁned as
[Equation]
where T (Φ , P ) =
E P
Φ
( X ) (moment matching) or T (Φ , P ) =
Median X ∼ P Φ D ( X ) (median matching).
B. S ETTINGS OF C ONTAMINATION Q
We introduce the contamination distributions Q used in the experiments.
We ﬁrst consider Q = N ( µ, I p ) with µ ranges in { 0 . 2 , 0 . 5 , 1 , 5 } .
Note that the total variation distance between N (0 p , I p ) and N ( µ, I p ) is of order (cid:107)
p − µ (cid:107) = (cid:107) µ (cid:107) .
We hope to use different levels of (cid:107) µ (cid:107) to test the algorithm and verify the error rate in the worst case.
Second, we consider Q = N (1 . 5 ∗ 1 p , Σ) to be a Gaussian distribution with a non-trivial covariance matrix Σ .
The covariance matrix is generated according to the following steps.
First generate a sparse precision matrix Γ = ( γ ij ) with each entry γ ij = z ij ∗ τ ij , i ≤ j , where z ij and τ ij are independently generated from Uniform (0 . 4 , 0 .
8) and Bernoulli (0 . 1) .
We then deﬁne γ ij = γ ji for all i > j and ¯Γ =
Γ + ( | min eig (Γ) | + 0 .
I p to make the precision matrix symmetric and positive deﬁnite, where min eig (Γ) is the smallest eigenvalue of Γ .
The covariance matrix is Σ = ¯Γ − 1 .
Finally, we consider Q to be a Cauchy distribution with independent component, and the j th component takes a standard Cauchy distribution with location parameter τ j = 0 .
B. C OMPARISON D ETAILS
In Section 5.3, we compare GANs with the dimension halving (Lai et al., 2016) and iterative ﬁltering (Diakonikolas et al., 2017).
• Dimension Halving.
Experiments conducted are based on the code from https://github.com/ kal2000/AgnosticMeanAndCovarianceCode .
The only hyper-parameter is the threshold in the outlier removal step, and we take C = 2 as suggested in the ﬁle outRemSperical.m.
• Iterative Filtering.
Experiments conducted are based on the code from https://github.com/ hoonose/robust-filter .
We assume (cid:15) is known and take other hyper-parameters as suggested in the ﬁle ﬁlterGaussianMean.m.
B. S UPPLEMENTARY E XPERIMENTS FOR N ETWORK S TRUCTURES
The experiments are conducted with i.i.d.
observations drawn from (1 − (cid:15) )
N (0 p , I p ) + (cid:15)N (0 .
5 ∗ 1 p , I p ) with (cid:15) = 0 .
Table 6 summarizes results for p = 100 , n ∈ { 5000 , 50000 } and various network structures.
We observe that TV-GAN that uses neural nets with one hidden layer improves over the performance of that without any hidden layer.
This indicates that the landscape of TV-GAN might be improved by a more complicated network structure.
However, adding one more layer does not improve the results.
For JS-GAN, we omit the results without hidden layer because of its lack of robustness (Proposition 3.1).
Deeper networks sometimes improve over shallow networks, but this is not always true.
We also observe that the optimal choice of the width of the hidden layer depends on the sample size.
[Table]
B. T ABLES FOR TESTING THE MINIMAX RATES
Tables 7, 8 and 9 show numerical results corresponding to Figure 2.
separable, as discussed in Section 3.1 (see Figure 1).
[Table] [Table]
[Table]
C P ROOFS OF P ROPOSITION 2.1 AND P ROPOSITION 3.
In the ﬁrst example, consider
[Equation]
In other words, Q is the class of Gaussian location family, and (cid:101) Q is taken to be a subset in a local neighborhood of N ( η, I p ) .
Then, with Q = N ( η, I p ) and (cid:101) Q = N ( (cid:101) η, I p ) , the event (cid:101) q
( X ) /q
( X ) ≥ 1 is equivalent to (cid:107) X − (cid:101) η (cid:107) 2 ≤ (cid:107) X − η (cid:107) 2 .
Since (cid:107) (cid:101) η − η (cid:107) ≤ r , we can write (cid:101) η
= η + (cid:101) ru for some (cid:101) r ∈ R and u ∈ R p that satisfy 0 ≤ (cid:101) r ≤ r and (cid:107) u (cid:107)
= 1 .
Then, (8) becomes (cid:98) θ = argmin
[Equation]
p
[Equation]
Letting r → 0 , we obtain (9), the exact formula of Tukey’s median.
The next example is a linear model y |
X ∼ N
( X T θ, 1) .
Consider the following classes
[Equation]
Here, P y,X stands for the joint distribution of y and X .
The two classes Q and (cid:101) Q share the same marginal distribution P X and the conditional distributions are speciﬁed by N ( X T η, 1) and N ( X T (cid:101) η, 1) , respectively.
Follow the same derivation of Tukey’s median, let r → 0 , and we obtain the exact formula of regression depth (10).
It is worth noting that the derivation of (10) does not depend on the marginal distribution P X .
The last example is on covariance/scatter matrix estimation.
For this task, we set Q = { N (0 , Γ) : Γ ∈ E p } , where E p is the class of all p × p covariance matrices.
Inspired by the derivations of Tukey depth and regression depth, it is tempting to choose (cid:101) Q in the neighborhood of N (0 , Γ) .
However, a native choice would lead to a deﬁnition that is not even Fisher consistent.
We propose a rank-one neighborhood, given by
[Equation]
Then, a direct calculation gives
[Equation] [Equation]
Since lim (cid:101)
[Equation]
= 1 , the limiting event of (22) is either I {| u T X
| 2 ≤ u T Γ u } or I {| u T X | 2 ≥ u T Γ u } , depending on whether (cid:101) r tends to zero from left or from right.
Therefore, with the above Q and (cid:101) Q , (8) becomes (11) under the limit r → 0 .
Even though the deﬁnition of (21) is given by a rank-one neighborhood of the inverse covariance matrix, the formula (11) can also be derived with (cid:101) Γ − 1 =
Γ − 1 + (cid:101) ruu T in (21) replaced by (cid:101) Γ =
Γ + (cid:101) ruu T by applying the Sherman-Morrison formula.
A similar formula to (11) in the literature is given by
[Equation]
which is recognized as the maximizer of what is known as the matrix depth function (Zhang, 2002; Chen et al., 2018; Paindaveine & Van Bever, 2017)
The β in (23) is a scalar deﬁned through the equation P ( N (0 , 1) ≤ √ β )
= 3 / 4 .
It is proved in Chen et al.
(2018) that (cid:98) Σ achieves the minimax rate under Huber’s (cid:15) -contamination model.
While the formula (11) can be derived from TV-Learning with discriminators in the form of I (cid:110) dN (0 , (cid:101) Γ) dN (0 , Γ) ( X ) ≥ 1 (cid:111) , a special case of (6), the formula (23) can be derived directly from TVGAN with discriminators in the form of I (cid:110) dN (0 ,β (cid:101) Γ) dN (0 ,β Γ) ( X ) ≥ 1 (cid:111) by following a similar rank-one neighborhood argument.
This completes the derivation of Proposition 2.1.
To prove Proposition 3.1, we deﬁne F ( w ) =
E P log sigmoid ( w T g ( X ))
+ E Q log(1 − sigmoid ( w T g ( X )))
+ log 4 , so that JS g ( P, Q ) = max w ∈W F ( w ) .
The gradient and Hessian of F ( w ) are given by
[Equation]
Therefore, F ( w ) is concave in w , and max w ∈W F ( w ) is a convex optimization with a convex W .
Suppose
JS g ( P, Q ) = 0 .
Then max w ∈W F ( w ) =
= F (0) , which implies ∇ F (0) = 0 , and thus we have E P g
( X ) =
E Q g ( X ) .
Now suppose E P g ( X ) =
E Q g ( X ) , which is equivalent to ∇ F (0) = 0 .
Therefore, w = 0 is a stationary point of a concave function, and we have JS g ( P, Q ) = max w ∈W F ( w )
=
F (0) = 0 .
D P ROOFS OF M AIN R ESULTS
In this section, we present proofs of all main theorems in the paper.
We ﬁrst establish some useful lemmas in
Section D.1, and the the proofs of main theorems will be given in Section D.2.
D. S OME A UXILIARY L EMMAS
Lemma D.1.
Given i.i.d.
observations X 1 , ..., X n ∼ P and the function class D deﬁned in (13), we have for any δ > 0 , sup
[Equation]
n
[Equation]
, with probability at least 1 − δ for some universal constant C > 0 .
Proof.
Let f ( X 1 , ..., X n ) =
sup D ∈D (cid:12)(cid:12) 1 n (cid:80) ni
=1 D ( X i ) − E D ( X ) (cid:12)(cid:12) .
It is clear that f ( X 1 , ..., X n ) satisﬁes the bounded difference condition.
By McDiarmid’s inequality (McDiarmid, 1989), we have
[Equation]
with probability at least 1 − δ .
Using a standard symmetrization technique (Pollard, 2012), we obtain the following bound that involves Rademacher complexity,
[Equation] [Equation]
where (cid:15) 1 , ..., (cid:15) n are independent Rademacher random variables.
The Rademacher complexity can be bounded by Dudley’s integral entropy bound, which gives
[Equation]
n n (cid:88) i = (cid:15) i D ( X i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) E 1 √ n (cid:90) (cid:112) log N ( δ, D , (cid:107) · (cid:107) n ) dδ, where N ( δ, D , (cid:107) · (cid:107) n ) is the δ -covering number of D with respect to the empirical (cid:96) 2 distance (cid:107)
f − g (cid:107) n = (cid:113) 1 n (cid:80) ni
=
( f ( X i ) − g
( X i )) 2 .
Since the VC-dimension of D is O ( p ) , we have N ( δ, D , (cid:107) · (cid:107) n ) (cid:46) p (16 e/δ ) O
( p ) (see Theorem 2.6.7 of Van Der Vaart & Wellner (1996)).
This leads to the bound 1 √ n (cid:82) 2 0 (cid:112) log N ( δ, D , (cid:107) · (cid:107) n ) dδ (cid:46) (cid:112) pn , which gives the desired result.
Lemma D.2.
Given i.i.d.
observations X 1 , ..., X n ∼ P , and the function class D deﬁned in (15), we have for any δ > 0 ,
[Equation]
Proof.
Let f ( X 1 , ..., X n ) =
sup
[Equation]
∈D
[Equation]
i
[Equation]
Since
[Equation]
we have
[Equation]
Therefore, by McDiarmid’s inequality (McDiarmid, 1989), we have
[Equation]
with probability at least 1 − δ .
By the same argument of (24), it is sufﬁcient to bound the Rademacher complexity E sup D ∈D (cid:12)(cid:12) 1 n (cid:80) ni
=1 (cid:15)
i log(2 D ( X i ))
(cid:12)(cid:12) .
Since the function ψ ( x ) = log(2 sigmoid ( x )) has Lipschitz constant 1 and satisﬁes ψ (0) =
0 , we have
[Equation]
which uses Theorem 12 of Bartlett & Mendelson (2002)
By H ¨ older’s inequality, we further have
[Equation]
Note that for a monotone function σ : R → [0 , 1] , the VC-dimension of the class { σ ( u T x + b ) : u ∈ R , b ∈ R } is O ( p ) .
Therefore, by using the same argument of Dudley’s integral entropy bound in the proof Lemma D.1, we have
[Equation]
which leads to the desired result.
Lemma D.3.
Given i.i.d.
observations X 1 , .., X n ∼ N ( θ, I p ) and the function class F HL ( κ, τ, B ) .
Assume (cid:107) θ (cid:107) ∞ ≤ √ log p and set τ = √ p log p .
We have for any δ > 0 ,
[Equation]
with probability at least 1 − δ for some universal constants C > 0 .
Proof.
Write f
( X 1 , ..., X n ) =
sup D ∈F HL ( κ,τ,B ) (cid:12)(cid:12) 1 n (cid:80) ni
=1 log D ( X i ) − E log D
( X ) (cid:12)(cid:12) .
Then, the inequality (25) holds with probability at least 1 − δ .
It is sufﬁcient to analyze the Rademacher complexity.
Using the fact that the function log(2 sigmoid ( x )) is Lipschitz and H¨older’s inequality, we have
[Table]
HL −
[Table]
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)n n
[Table]
=
[Table]
(cid:15)
i
[Table]
2 p
[Table]
h =
[Table]
u h g h ( X i ) + b (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ 8 √ pκ E sup g ∈G HL − 1 ( B ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12) n n (cid:88) i
= (cid:15) i g
[Table]
X
[Table]
(cid:12) + 4 κτ E
[Table]
(cid:12) n n (cid:88) i
=
(cid:15)
i
[Table]
(cid:12) .
Now we use the notation Z i =
X i − θ ∼
N (0 , I p ) for i = 1 , ..., n .
We bound E sup g ∈G HL − 1 ( B ) (cid:12)(cid:12) 1 n (cid:80) ni
=1 (cid:15)
i g ( Z i + θ ) (cid:12)(cid:12) by induction.
Since
E (cid:32) sup g ∈G H 1 ( B ) n n (cid:88)
i = (cid:15)
i g ( Z i + θ ) (cid:33) ≤ E (cid:32) sup (cid:107) v (cid:107) 1 ≤ B n n (cid:88)
i = (cid:15)
i v T ( Z i + θ ) (cid:33) ≤ B (cid:32)
E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)n n (cid:88) i = (cid:15)
i Z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ + (cid:107) θ (cid:107) ∞
E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 1 n n (cid:88) i = (cid:15) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) ≤ CB √ log p + (cid:107) θ (cid:107) ∞ √ n , and
E (cid:32) sup g ∈G
Hl +1 ( B ) n n (cid:88) i
= (cid:15)
i g ( Z i + θ ) (cid:33) ≤ E (cid:32) sup (cid:107) v (cid:107) 1 ≤ B,g
h ∈G
Hl ( B ) n n (cid:88) i = (cid:15)
i
H (cid:88) h
= v h g h
( Z i + θ ) (cid:33) ≤ B E (cid:32) sup
g ∈G Hl
( B ) (cid:12) (cid:12)(cid:12)(cid:12)(cid:12) 1 n n (cid:88)
i = (cid:15)
i g ( Z i + θ ) (cid:12) (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) ≤ 2 B E (cid:32) sup
g ∈G Hl ( B ) n n (cid:88) i
= (cid:15)
i g (
[Equation]
we have
[Equation]
Combining the above inequalities, we get
[Equation]
This leads to the desired result under the conditions on τ and (cid:107) θ (cid:107) ∞ .
D. P ROOFS OF M AIN T HEOREMS
Proof of Theorem 3.1.
We ﬁrst introduce some notations.
Deﬁne F ( P, η ) = max w,b F w,b ( P, η ) , where
F
[Equation] [Equation]
P,
[Equation]
=
E P sigmoid
[Equation]
T X + b )
[Equation]
E N ( η,I p ) sigmoid ( w T X + b ) .
With this deﬁnition, we have (cid:98) θ = argmin η F ( P n , η ) , where we use P n for the empirical distributionn (cid:80) ni
=
δ X i .
We shorthand N ( η, I p ) by P η , and then
[Equation] [Equation]
[Equation] [Equation]
[Equation] [Equation]
(cid:114) log
[Equation]
With probability at least 1 − δ , the above inequalities hold.
We will explain each inequality.
Since
[Equation]
we have
[Equation]
which implies (26) and (30).
The inequalities (27) and (29) are implied by Lemma D.1 and the fact that
[Equation]
The inequality (28) is a direct consequence of the deﬁnition of (cid:98) θ .
Finally, it is easy to see that F ( P θ , θ ) = 0 , which gives (31).
In summary, we have derived that with probability at least 1 − δ ,
[Equation]
for all w ∈ R p and b ∈ R .
For any u ∈ R p such that (cid:107) u (cid:107) = 1 , we take w = u and b = − u T θ , and we have
[Equation]
, where f ( t ) = (cid:82) 1
+ e z
+ t φ ( z ) dz , with φ ( · ) being the probability density function of N (0 , 1) .
It is not hard to see that as long as | f ( t ) − f (0) | ≤ c for some sufﬁciently small constant c > 0 , then | f ( t ) − f (0)
| ≥ c (cid:48) | t | for some constant c (cid:48) > 0 .
This implies
[Equation]
with probability at least 1 − δ .
The proof is complete.
Proof of Theorem 3.2.
We continue to use P η to denote N ( η, I p ) .
Deﬁne
[Equation]
where
F w,u,b
[Equation]
P,
[Equation]
with D ( x ) = sigmoid (cid:16)(cid:80)
j ≥
w j σ ( u Tj x + b j ) (cid:17) .
Then,
[Equation] [Equation]
[Equation]
(cid:32)(cid:114)
[Equation]
δ
[Equation]
n
[Equation] [Equation]
Cκ (cid:32)(cid:114) p n
[Equation]
(cid:114) log(
[Equation]
δ ) n (cid:33) (35) ≤ F
[Equation]
P
[Equation]
θ )
[Equation]
4 κ(cid:15)
[Equation]
2 Cκ (cid:32)(cid:114) p n
[Equation] [Equation]
δ
[Equation]
n (cid:33) (36) = 4 κ(cid:15)
+ 2 Cκ (cid:32)(cid:114) p n + (cid:114)
[Equation]
The inequalities (32)-(36) follow similar arguments for (26)-(30).
To be speciﬁc, (33) and (35) are implied by
Lemma D.2, and (34) is a direct consequence of the deﬁnition of (cid:98) θ .
To see (32) and (36), note that for any w such that (cid:107) w (cid:107) 1 ≤ κ , we have
[Equation]
A similar argument gives the same bound for | log(2(1 − D ( X ))) | .
This leads to
[Equation]
which further implies (32) and (36).
To summarize, we have derived that with probability at least 1 − δ ,
[Equation] [Equation]
[Equation]
for all (cid:107) w (cid:107) 1 ≤ κ , (cid:107) u j (cid:107) ≤ 1 and b j .
Take w 1 = κ , w j = 0 for all j > 1 , u 1 = u for some unit vector u and b 1 = − u T θ , and we get
[Equation]
where
[Equation]
with Z ∼ N (0 , 1) .
Direct calculations give
[Equation]
Therefore, f δ (0) = 0 , f (cid:48)
δ (0) =
12 ( E σ ( Z ) − E σ ( Z + δ )) , and f (cid:48)(cid:48) δ ( t ) ≥ − 12 .
By the inequality
[Equation]
we have κf (cid:48) δ (0) ≤ f δ ( κ ) + κ 2 / 4 .
In view of (38), we have
[Equation]
It is easy to see that for the choices of σ ( · ) , (cid:82) σ ( z ) φ ( z )
dz −
[Equation]
σ ( z + t ) φ ( z ) dz is locally linear with respect to t .
This implies that
[Equation]
Therefore, with a κ (cid:46) (cid:112) pn + (cid:15) , the proof is complete.
Proof of Theorem 4.1.
We use P θ, Σ ,h to denote the elliptical distribution EC ( θ, Σ , h ) .
Deﬁne
[Equation]
where
[Equation]
with D ( x ) = sigmoid (cid:16)(cid:80)
j ≥
w j σ ( u Tj x + b j ) (cid:17) .
Let P be the data generating process that satisﬁes
TV ( P, P θ, Σ ,h ) ≤ (cid:15) [Equation] and then there exist probability distributions Q 1 and Q 2 , such that
[Equation]
The explicit construction of Q 1 , Q 2 is given in the proof of Theorem 5.1 of Chen et al.
This implies that
[Equation]
Then [Equation] the same argument in Theorem 3.2 (with (37) replaced by (41)) leads to the fact that with probability at least 1 − δ ,
[Equation]
for all (cid:107) w (cid:107) 1 ≤ κ , (cid:107) u j (cid:107) ≤ 1 and b j .
Take w 1 = κ , w j = 0 for all j > 1 , u 1 = u/ (cid:112) u T (cid:98) Σ u for some unit vector u and b 1 = − u T θ/ (cid:112) u T (cid:98) Σ u , and we get
[Equation]
where
[Equation]
where δ = u T ( (cid:98) θ − θ ) √ u T (cid:98) Σ u and ∆ = √ u T [Equation] u √ u T
[Equation]
Σ u .
A similar argument to the proof of Theorem 3.2 gives
[Equation]
Since
[Equation]
the above bound is equivalent to
[Equation]
where H ( δ ) =
(cid:82) σ ( δ + s ) (cid:98)
h ( s ) ds .
The above bound also holds for κ 2 ( H ( δ ) − H (0)) by a symmetric argument, and therefore the same bound holds for κ 2 | H ( δ ) − H (0) | .
Since H (cid:48) (0) =
(cid:82) σ ( s )(1 − σ ( s ))
(cid:98) h
( s ) ds
= 1 , H ( δ ) is locally linear at δ = 0 , which leads to a desired bound for δ = u T ( (cid:98) θ − θ ) √ u T (cid:98) Σ u .
Finally, since u T (cid:98) Σ u ≤ M , we get the bound for u T ( (cid:98) θ − θ ) .
The proof is complete by taking supreme over all unit vector u .
Proof of Theorem A.1.
We continue to use P η to denote N ( η, I p ) .
Deﬁne
F
[Equation]
P
[Equation]
sup
[Equation]
with
[Equation]
Follow the same argument in the proof of Theorem 3.2, use Lemma D.3, and we have
[Equation]
uniformly over D ∈ F HL ( κ, τ, B ) with probability at least 1 − δ .
Choose w 1 = κ and w j = 0 for all w j > 1 .
For any unit vector (cid:101) u ∈ R p , take u 1 h = − u 1( h + p )
= (cid:101) u h for h = 1 , ..., p and b 1 = − (cid:101)
u T θ .
For h = 1 , ..., p , set g 1 h ( x ) = max( x h , 0) .
For h = p + 1 , ..., 2 p , set g 1 h ( x ) = max( − x h − p , 0) .
It is obvious that such u and b satisfy (cid:80) h u 2 1 h ≤ 2 and | b 1 | ≤ (cid:107) θ (cid:107) ≤ √ p (cid:107) θ (cid:107) ∞ ≤ √ p log p .
We need to show both the functions max( x, 0) and max( − x, 0) are elements of G HL − 1 ( B ) .
This can be proved by induction.
It is obvious
that max( x h , 0) , max( − x h , 0) ∈
G H 1 ( B ) for any h = 1 , ..., p .
Suppose we have max( x h , 0) , max( − x h , 0) ∈
G Hl ( B ) for any h
= 1 , ..., p .
Then,
[Equation]
Therefore, max( x h , 0) , max( − x h , 0) ∈
G Hl +1 ( B ) as long as B ≥ 2 .
Hence, the above construction satisﬁes
D ( x ) = sigmoid ( κ sigmoid ( (cid:101) u T ( x − θ )))
∈ F HL ( κ, τ, B ) , and we have
[Equation]
where the deﬁnition of f δ ( t ) is given by (39) with Z ∼ N (0 , 1) and σ ( · ) is taken as sigmoid ( · ) .
Apply the a similar in the proof of Theorem 3.2, we obtain the desired result.
Proof of Theorem A.2.
We use P η, Γ to denote the distribution N ( η, Γ) .
Deﬁne
[Equation]
max
[Equation]
w
[Equation]
1 ≤ κ,u,b F w,u,b ( P, ( η, Γ)) , where
[Equation]
with D ( x ) = sigmoid (cid:16)(cid:80)
j ≥
w j σ ( u Tj x + b j ) (cid:17) .
Then, following the same arguments that derive (32)-(36), we get
[Equation]
In other words, we have with probability at least 1 − δ ,
[Equation]
for all (cid:107) w (cid:107) 1 ≤ κ , (cid:107) u j (cid:107) ≤ 1 and b j .
Take w 1 = κ , w j = 0 for all j > 1 , u 1 = u for some unit vector u and b 1 = − u T θ , and we get
[Equation]
p
[Equation]
n
[Equation]
, where
[Equation]
with Z ∼ N (0 , 1) .
Then, by similar calculations in the proof of Theorem 3.
[Equation]
we have
[Equation]
If we can show
[Equation]
the proof will be complete.
Since σ ( · ) − 12 is an odd function, we have
[Equation]
Now consider the function h ( δ )
= (cid:82) σ ( bz + δ ) φ ( z ) dz ,
where b = (cid:112) u T (cid:98) Σ u ≤ √ M
=
O
The fact that | h ( δ ) − 1 / 2 | is sufﬁciently small implies that | δ | =
O (1) .
Then, by the local linearity of h ( δ ) , we have | h ( δ ) − 1 / 2 | =
| h ( δ ) − h (0) | (cid:38) | δ | , which implies (43).
Thus, the proof is complete.