Understanding and Mitigating the Tradeoff Between Robustness and Accuracy
Abstract
Adversarial training augments the training set with perturbations to improve the robust error (over worstcase perturbations), but it often leads to an increase in the standard error (on unperturbed test inputs). Previous explanations for this tradeoff rely on the assumption that no predictor in the hypothesis class has low standard and robust error. In this work, we precisely characterize the effect of augmentation on the standard error in linear regression when the optimal linear predictor has zero standard and robust error. In particular, we show that the standard error could increase even when the augmented perturbations have noiseless observations from the optimal linear predictor. We then prove that the recently proposed robust selftraining (RST) estimator improves robust error without sacrificing standard error for noiseless linear regression. Empirically, for neural networks, we find that RST with different adversarial training methods improves both standard and robust error for random and adversarial rotations and adversarial perturbations in CIFAR10.
1 Introduction
Adversarial training methods (Goodfellow et al., 2015; Madry et al., 2017) attempt to improve the robustness of neural networks against adversarial examples (Szegedy et al., 2014) by augmenting the training set (onthefly) with perturbed examples that preserve the label but that fool the current model. While such methods decrease the robust error, the error on worstcase perturbed inputs, they have been observed to cause an undesirable increase in the standard error, the error on unperturbed inputs (Madry et al., 2018; Zhang et al., 2019; Tsipras et al., 2019).
Previous works attempt to explain the tradeoff between standard error and robust error in two settings: when no accurate classifier is consistent with the perturbed data (Tsipras et al., 2019; Zhang et al., 2019; Fawzi et al., 2018), and when the hypothesis class is not expressive enough to contain the true classifier (Nakkiran, 2019). In both cases, the tradeoff persists even with infinite data. However, adversarial perturbations in practice are typically defined to be imperceptible to humans (e.g. small perturbations in vision). Hence by definition, there exists a classifier (the human) that is both robust and accurate with no tradeoff in the infinite data limit. Furthermore, since deep neural networks are expressive enough to fit not only adversarial but also randomly labeled data perfectly (Zhang et al., 2017), the explanation of a restricted hypothesis class does not perfectly capture empirical observations either. Empirically on Cifar10, we find that the gap between the standard error of adversarial training and standard training decreases as we increase the labeled data size, thereby also suggesting the tradeoff could disappear with infinite data (See Figure 1).
In this work, we provide a different explanation for the tradeoff between standard and robust error that takes generalization from finite data into account. We first consider a linear model where the true linear function has zero standard and robust error. Adversarial training augments the original training set with extra data, consisting of samples where the perturbations are consistent, meaning that the conditional distribution stays constant . We show that even in this simple setting, the augmented estimator, i.e. the minimum norm interpolant of the augmented data (standard + extra data), could have a larger standard error than that of the standard estimator, which is the minimum norm interpolant of the standard data alone. We found this surprising given that adding consistent perturbations enforces the predictor to satisfy invariances that the true model exhibits. One might think adding this information would only restrict the hypothesis class and thus enable better generalization, not worse.
We show that this tradeoff stems from overparameterization. If the restricted hypothesis class (by enforcing invariances) is still overparameterized, the inductive bias of the estimation procedure (e.g., the norm being minimized) plays a key role in determining the generalization of a model.
Figure 2 shows an illustrative example of this phenomenon with cubic smoothing splines. The predictor obtained via standard training (dashed blue) is a line that captures the global structure and obtains low error. Training on augmented data with locally consistent perturbations of the training data (crosses) restricts the hypothesis class by encouraging the predictor to fit the local structure of the high density points. Within this set, the cubic splines predictor (solid orange) minimizes the second derivative on the augmented data, compromising the global structure and performing badly on the tails (Figure 2(b)). More generally, as we characterize in Section 3, the tradeoff stems from the inductive bias of the minimum norm interpolant, which minimizes a fixed norm independent of the data, while the standard error depends on the geometry of the covariates.
Recent works (Carmon et al., 2019; Najafi et al., 2019; Uesato et al., 2019) introduced robust selftraining (RST), a robust variant of selftraining that overcomes the sample complexity barrier of learning a model with low robust error by leveraging extra unlabeled data. In this paper, our theoretical understanding of the tradeoff between standard and robust error in linear regression motivates RST as a method to improve robust error without sacrificing standard error. In Section 4.2, we prove that RST eliminates the tradeoff for linear regression—RST does not increase standard error compared to the standard estimator while simultaneously achieving the best possible robust error, matching the standard error (see Figure 2(c) for the effect of RST on the spline problem). Intuitively, RST regularizes the predictions of the robust estimator towards that of the standard estimator on the unlabeled data thereby eliminating the tradeoff.
As previous works only focus on the empirical evaluation of the gains in robustness via RST, we systematically evaluate the effect of RST on both the standard and robust error on Cifar10 when using unlabeled data from Tiny Images as sourced in Carmon et al. (2019). We expand upon empirical results in two ways. First, we study the effect of the labeled training set sizes and and find that the RST improves both robust and standard error over vanilla adversarial training across all sample sizes. RST offers maximum gains at smaller sample sizes where vanilla adversarial training increases the standard error the most. Second, we consider an additional family of perturbations over random and adversarial rotation/translations and find that RST offers gains in both robust and standard error.
2 Setup
We consider the problem of learning a mapping from an input to a target . For our theoretical analysis, we focus on regression where while our empirical studies consider general . Let be the underlying distribution, the marginal on the inputs and the conditional distribution of the targets given inputs. Given training pairs , we use to denote the measurement matrix and to denote the target vector . Our goal is to learn a predictor that (i) has low standard error on inputs and (ii) low robust error with respect to a set of perturbations . Formally, the error metrics for a predictor and a loss function are the standard error
(1) 
and the robust error
(2) 
for consistent perturbations that satisfy
(3) 
Such transformations may consist of small rotations, horizontal flips, brightness or contrast changes (Krizhevsky et al., 2012; Yaeger et al., 1996), or small perturbations in vision (Szegedy et al., 2014; Goodfellow et al., 2015) or word synonym replacements in NLP (Jia & Liang, 2017; Alzantot et al., 2018).
Minimum norm estimators. In this work, we focus on interpolating estimators in highly overparameterized models, motivated by modern machine learning models that achieve near zero training loss (on both standard and extra data). Interpolating estimators for linear regression have been studied in many recent works such as (Ma et al., 2018; Belkin et al., 2018; Hastie et al., 2019; Liang & Rakhlin, 2018; Bartlett et al., 2019). We present our results for interpolating estimators with minimum Euclidean norm, but our analysis directly applies to more general Mahalanobis norms via suitable reparameterization (see Appendix A).
We consider robust training approaches that augment the
standard training data with some extra training data where the rows of consist of vectors in the
set .
(5) 
Notation. For any vector , we use to denote the coordinate of .
3 Analysis in the linear regression setting
In this section, we compare the standard errors of the standard estimator and the augmented estimator in noiseless linear regression. We begin with a simple toy example that describes the intuition behind our results (Section 3.1) and provide a more complete characterization in Section 3.2. This section focuses only on the standard error of both estimators; we revisit the robust error together with the standard error in Section 4.
3.1 Simple illustrative problem
We consider a simple example in 3D where is the true parameter. Let denote the standard basis vectors in . Suppose we have one point in the standard training data . By definition (5), satisfies and hence . However, is unconstrained on the subspace spanned by (the nullspace ). The minnorm objective chooses the solution with . Figure 3 visualizes the projection of various quantities on . For simplicity of presentation, we omit the projection operator in the figure. The projection of onto is the blue dot at the origin, and the parameter error is the projection of onto .
Effect of augmentation on parameter error. Suppose we augment with an extra data point which lies in (black dashed line in Figure 3). The augmented estimator still fits the standard data and thus . Due to fitting the extra data , (orange vector in Figure 3) must also satisfy an additional constraint . The crucial observation is that additional constraints along one direction ( in this case) could actually increase parameter error along other directions. For example, let’s consider the direction in Figure 3. Note that fitting makes have a large component along . Now if is small (precisely, ), has a larger parameter error along than , which was simply zero (Figure 3 (a)). Conversely, if the true component is large enough (precisely, ), the parameter error of along is smaller than that of .
Effect of parameter error on standard error. The contribution of different components of the parameter error to the standard error is scaled by the population covariance (see Equation 4). For simplicity, let . In our example, the parameter error along is zero since both estimators interpolate the standard training point . Then, the ratio between and determines which component of the parameter error contributes more to the standard error.
When is ? Putting the two effects together, we see that when is small as in Fig 3(a), has larger parameter error than in the direction . If , error in is weighted much more heavily in the standard error and consequently would have a larger standard error. Precisely, we have
We present a formal characterization of this tradeoff in general in the next section.
3.2 General characterizations
In this section, we precisely characterize when the augmented estimator that fits extra training data points in addition to the standard points has higher standard error than the standard estimator that only fits . In particular, this enables us to understand when there is a “tradeoff” where the augmented estimator has lower robust error than by virtue of fitting perturbations, but has higher standard error. In Section 3.1, we illustrated how the parameter error of could be larger than in some directions, and if these directions are weighted heavily in the population covariance , the standard error of would be larger.
Formally, let us define the parameter errors and . Recall that the standard errors are
(6) 
where is the population covariance of the underlying inputs drawn from .
To characterize the effect of the inductive bias of minimum norm interpolation on the standard errors, we define the following projection operators: , the projection matrix onto and , the projection matrix onto (see formal definition in Appendix B). Since and are minimum norm interpolants, and . Further, in noiseless linear regression, and have no error in the span of and respectively. Hence,
(7) 
Our main result relies on the key observation that for any vector , can be decomposed into a sum of two orthogonal components and such that with and . This is because and thus . Now setting and using the error expressions in Equation 6 and Equation 7 gives a precise characterization of the difference in the standard errors of and .
Theorem 1.
The difference in the standard errors of the standard estimator and augmented estimator can be written as follows.
(8) 
where and .
The proof of Theorem 1 is in Appendix B.3. The increase in standard error of the augmented estimator can be understood in terms of the vectors and defined in Theorem 1. The first term is always positive, and corresponds to the decrease in the standard error of the augmented estimator by virtue of fitting extra training points in some directions. However, the second term can be negative and intuitively measures the cost of a possible increase in the parameter error along other directions (similar to the increase along in the simple setting of Figure 3(a)). When the cost outweighs the benefit, the standard error of is larger. Note that both the cost and benefit is determined by which governs how the parameter error affects the standard error.
We can use the above expression (Theorem 1) for the difference in standard errors of and to characterize different “safe” conditions under which augmentation with extra data does not increase the standard error. See Appendix B.7 for a proof.
Corollary 1.
The following conditions are sufficient for , i.e. the standard error does not increase when fitting augmented data.

The population covariance is identity.

The augmented data spans the entire space, or equivalently .

The extra data is a single point such that is an eigenvector of .
Matching inductive bias. We would like to draw special attention to the first condition. When , notice that the norm that governs the standard error (Equation 6) matches the norm that is minimized by the interpolants (Equation 5). Intuitively, the estimators have the “right” inductive bias; under this condition, the augmented estimator does not have higher standard error. In other words, the observed increase in the standard error of can be attributed to the “wrong” inductive bias. In Section 4, we will use this understanding to propose a method of robust training which does not increase standard error over standard training.
Safe extra points. We use Theorem 1 to plot the safe extra points that do not lead to an increase in standard error for any in the simple 3D setting described in Section 3.1 for two different (Figure 3 (c), (d)). The safe points lie in cones which contain the eigenvectors of (as expected from Corollary 1). The width and alignment of the cones depends on the alignment between and the eigenvectors of . As the eigenvalues of become less skewed, the space of safe points expands, eventually covering the entire space when (see Corollary 1).
Local versus global structure. We now tie our analysis back to the cubic splines interpolation problem from Figure 2. The inputs can be appropriately rotated and scaled such that the cubic spline interpolant is the minimum Euclidean norm interpolant (as in Equation 5). Under this transformation, the different eigenvectors of the nullspace of the training data represent the “local” high frequency components with small eigenvalues or “global” low frequency components with large eigenvalues (see Figure 4). An augmentation that encourages the fitting local components in could potentially increase the error along other global components (like the increase in error along in Figure 3(a)). Such an increase, coupled with the fact that global components have larger eigenvalue in , results in the standard error of being larger than that of . See Figure 8 and Appendix C.3.1 for more details. This is similar to the recent observation that adversarial training with perturbations encourages neural networks to fit the high frequency components of the signal while compromising on the lowfrequency components (Yin et al., 2019).
Model complexity. Finally, we relate the magnitude of increase in standard error of the augmented estimator to the complexity of the true model.
Proposition 1.
For a given ,
for some scalar that depends on .
In other words, for a large increase in standard error upon augmentation, the true parameter needs to be sufficiently more complex (in the norm) than the standard estimator . For example, the construction of the cubic splines interpolation problem relies on the underlying function (staircase) being more complex with additional local structure than the standard estimator—a linear function that fits most points and can be learned with few samples. Proposition 1 states that this requirement holds more generally. The proof of Proposition 1 appears in Appendix B.5. A similar intuition can be used to construct an example where augmentation can increase standard error for minimum norm interpolants when is dense (Appendix G).
4 Robust selftraining
We now use insights from Section 3 to construct estimators with low robust error without increasing the standard error. While Section 3 characterized the effect of adding extra data in general, in this section we consider robust training which augments the dataset with extra data that are consistent perturbations of the standard training data .
Since the standard estimator has small standard error, a natural strategy to mitigate the tradeoff is to regularize the augmented estimator to be closer to the standard estimator. The choice of distance between the estimators we regularize is very important. Recall from Section 3.1 that the population covariance determines how the parameter error affects the standard error. This suggests using a regularizer that incorporates information about .
We first revisit the recently proposed robust selftraining (RST) (Carmon et al., 2019; Najafi et al., 2019; Uesato et al., 2019) that incorporates additional unlabeled data via pseudolabels from a standard estimator. Previous work only focused on the effectiveness of RST in improving the robust error. In Section 4.2, we prove that in linear regression, RST eliminates the tradeoff between standard and robust error (Theorem 4.2). The proof hinges on the connection between RST and the idea of regularizing towards the standard estimator discussed above. In particular, we show that the RST objective can be rewritten as minimizing a suitable induced distance to the standard estimator.
In Section 4.3, we expand upon previous empirical RST results for Cifar10 across various training set sizes and perturbations (rotations/translations in addition to ). We observe that across all settings, RST substantially improves the standard error while also improving the robust error over the vanilla supervised robust training counterparts.
4.1 General formulation of RST
We first describe the general twostep robust selftraining (RST) procedure (Carmon et al., 2019; Uesato et al., 2019) for a parameteric model :

Perform standard training on labeled data to obtain .

Perform robust training on both the labeled data and unlabeled inputs with pseudolabels generated from the standard estimator .
The second stage typically involves a combination of the standard loss and a robust loss . The robust loss encourages invariance of the model over perturbations , and is generally defined as
(9) 
It is convenient to summarize the robust selftraining estimator as the minimizer of a weighted combination of four separate losses as follows. We define the losses on the labeled dataset as
The losses on the unlabeled samples which are psuedolabeled by the standard estimator are
Putting it all together, we have
(10)  
for fixed scalars .
4.2 Robust selftraining for linear regression
We now return to the noiseless linear regression as described in Section 2 and specialize the general RST estimator described in Equation (10) to this setting. We prove that RST eliminates the decrease in standard error in this setting while achieving low robust error by showing that RST appropriately regularizes the augmented estimator towards the standard estimator.
Our theoretical results hold for RST procedures where the pseudolabels can be generated from any interpolating estimator satisfying . This includes but is not restricted to the mininumnorm standard estimator defined in (5). We use the squared loss as the loss function . For consistent perturbations , we analyze the following RST estimator for linear regression
(11) 
Figure 5 shows the four losses of RST in this special case of linear regression.
Obtaining this specialized estimator from the general RST estimator in Equation (10) involves the following steps. First, for convenience of analysis, we assume access to the population covariance via infinite unlabeled data and thus replace the finite sample losses on the unlabeled data by their population losses . Second, the general RST objective minimizes some weighted combination of four losses. When specializing to the case of noiseless linear regression, since , rather than minimizing , we set the coefficients on the losses such that the estimator satisfies a hard constraint . This constraint which enforces interpolation on the labeled dataset allows us to rewrite the robust loss (Equation 9) on the labeled examples equivalently as a selfconsistency loss defined independent of labels.
Since is invariant on perturbations by definition, we have and thus we introduce a constraint in the estimator.
For the losses on the unlabeled data, since the pseudolabels are not perfect, we minimize in the objective instead of enforcing a hard constraint on . However, similarly to the robust loss on labeled data, we can reformulate the robust loss on unlabeled samples as a selfconsistency loss that does not use pseudolabels. By definition, and thus we enforce in the specialized estimator.
We now study the standard and robust error of the linear regression RST estimator defined above in Equation (4.2). {restatable}[]theoremlinearx Assume the noiseless linear model . Let be an arbitrary interpolant of the standard data, i.e. . Then
Simultaneously, . See Appendix D for a full proof.
The crux of the proof is that the optimization objective of RST is an inductive bias that regularizes the estimator to be close to the standard estimator, weighing directions by their contribution to the standard error via . To see this, we rewrite
By incorporating an appropriate induced regularizer while satisfying constraints on the robust losses, RST ensures that the standard error of the estimator never exceeds the standard error of . The robust error of any estimator is lower bounded by its standard error, and this gap can be arbitrarily large for the standard estimator. However, the robust error of the RST estimator matches the lower bound of its standard error which in turn is bounded by the standard error of the standard estimator and hence is small. To provide some graphical intuition for the result, see Figure 2 that visualizes the RST estimator on the cubic splines interpolation problem that exemplifies the increase in standard error upon augmentation. RST captures the global structure and obtains low standard error by matching (straight line) on unlabeled inputs. Simultaneously, RST enforces invariance on local transformations on both labeled and unlabeled inputs, and obtains low robust error by capturing the local structure across the domain.
Implementation of linear RST. The constraint on the standard loss on labeled data simply corresponds to interpolation on the standard labeled data. The constraints on the robust selfconsistency losses involve a maximization over a set of transformations. In the case of linear regression, such constraints can be equivalently represented by a set of at most linear constraints, where is the dimension of the covariates. Further, with this finite set of constraints, we only require access to the covariance in order to constrain the population robust loss. Appendix D gives a practical iterative algorithm that computes the RST estimator for linear regression reminiscent of adversarial training in the semisupervised setting.
Method  Robust Test Acc.  Standard Test Acc. 
Standard Training  0.8%  95.2% \rdelim}33mm[] 
PGAT (Madry et al., 2018)  45.8%  87.3% 
TRADES (Zhang et al., 2019)  55.4%  84.0% 
Standard SelfTraining  0.3%  96.4% \rdelim}53mm[] 
Robust Consistency Training (Carmon et al., 2019)  56.5%  83.2% 
RST + PGAT (this paper)  58.5%  91.8% 
RST + TRADES (this paper) (Carmon et al., 2019)  63.1%  89.7% 
Interpolated AT (Lamb et al., 2019) 
45.1%  93.6% \rdelim}43mm[] 
Neural Arch. Search (Cubuk et al., 2017)  50.1%  93.2% 
Method  Robust Test Acc.  Standard Test Acc. 
Standard Training  0.2%  94.6% \rdelim}33mm[] 
Worstof10  73.9%  95.0% 
Random  67.7%  95.1% 
RST + Worstof10 (this paper)  75.1%  95.8% \rdelim}23mm[] 
RST + Random (this paper)  70.9%  95.8% 
Worstof10 (Engstrom et al., 2019) 
69.2%  91.3% \rdelim}33mm[] 
Random (Yang et al., 2019) 
58.3%  91.8% 
4.3 Empirical evaluation of RST
Carmon et al. (2019) empirically evaluate RST with a focus on studying gains in the robust error. In this work, we focus on both the standard and robust error and expand upon results from previous work. Carmon et al. (2019) used TRADES (Zhang et al., 2019) as the robust loss in the general RST formulation (10); we additionally evaluate RST with Projected Gradient Adversarial Training (AT) (Madry et al., 2018) as the robust loss. Carmon et al. (2019) considered and perturbations. We study rotations and translations in addition to perturbations, and also study the effect of labeled training set size on standard and robust error. Table 1 presents the main results. More experiment details appear in Appendix D.3.
Both RST+AT and RST+TRADES have lower robust and standard error than their supervised counterparts AT and TRADES across all perturbation types. This mirrors the theoretical analysis of RST in linear regression (Theorem 4.2) where the RST estimator has small robust error while provably not sacrificing standard error, and never obtaining larger standard error than the standard estimator.
Effect of labeled sample size. Recall that our work motivates studying the tradeoff between robust and standard error while taking generalization from finite data into account. We showed that the gap in the standard error of a standard estimator and that of a robust estimator is large for small training set sizes and decreases as the labeled dataset is larger (Figure 1). We now study the effect of RST as we vary the training set size in Figure 6. We find that RST+AT has lower standard error than standard training across all sample sizes for small , while simultaneously achieving lower robust error than AT (see Appendix E.2.1). In the small data regime where vanilla adversarial training hurts the standard error the most, we find that RST+AT gives about 3x more absolute improvement than in the large data regime. We note that this set of experiments are complementary to the experiments in (Schmidt et al., 2018) which study the effect of the training set size only on robust error.
Effect on transformations that do not hurt standard error. We also test the effect of RST on perturbations where robust training slightly improves standard error rather than hurting it. Since RST regularizes towards the standard estimator, one might suspect that the improvements from robust training disappear with RST. In particular, we consider spatial transformations that consist of simultaneous rotations and translations. We use two common forms of robust training for spatial perturbations, where we approximately maximize over with either adversarial (worstof10) or random augmentations (Yang et al., 2019; Engstrom et al., 2019). Table 1 (right) presents the results. In the regime where vanilla robust training does not hurt standard error, RST in fact further improves the standard error by almost 1% and the robust error by 23% over the standard and robust estimators for both forms of robust training. Thus in settings where vanilla robust training improves standard error, RST seems to further amplify the gains while in settings where vanilla robust training hurts standard error, RST mitigates the harmful effect.
Comparison to other semisupervised approaches. The RST estimator minimizes both a robust loss and a standard loss on the unlabeled data with pseudolabels (bottom row, Figure 5). Both of these losses are necessary to simultaneously improve both the standard and robust error over the vanilla supervised robust training. Standard selftraining, which only uses standard loss on unlabeled data, has very high robust error (). Similarly, Robust Consistency Training, an extension of Virtual Adversarial Training (Miyato et al., 2018) that only minimizes a robust selfconsistency loss on unlabeled data, marginally improves the robust error but actually hurts standard error. See Table 1.
Complementary methods for robustness and accuracy. In Table 1, we also report the standard and robust errors of other methods that improve the tradeoff between standard and robust error. Interpolated Adversarial Training (IAT) (Lamb et al., 2019) considers a different training algorithm based on Mixup, and Neural Architecture Search (NAS) (Cubuk et al., 2017) uses RL to search for more robust architectures. RST, IAT and NAS are incomparable as they find different tradeoffs between standard and robust error. However, we believe that since RST provides a complementary statistical perspective on the tradeoff, it can be combined with methods like IAT or NAS to see further gains. We leave this to future work.
5 Conclusion
We studied the commonly observed increase in standard error upon adversarial training taking generalization from finite data into account. We showed that augmenting training data with perturbations, like in adversarial training can surprisingly increase the standard error even in a simple setting of noiseless linear regression where the true linear function has zero standard and robust error. Our analysis reveals that the interplay between the inductive bias of models and the underlying geometry of the inputs causes the standard error to increase even when the augmented data is perfectly labeled. This insight provides a method that provably eliminates the increase in standard error upon augmentation in linear regression by incorporating an appropriate regularizer based on the geometry of the inputs. While not immediately apparent, we show that this is a special case of the recently proposed robust selftraining (RST) procedure that uses additional unlabeled data. Previous works view RST as a method to improve the robust error by effectively using more samples. Our work provides some theoretical justification for why RST improves both the standard and robust error thereby mitigating the tradeoff between accuracy and robustness in practice. How to best utilize unlabeled data, and whether sufficient unlabeled data would completely eliminate the tradeoff remain open questions.
Appendix A Transformations to handle arbitrary matrix norms
Consider a more general minimum norm estimator of the following form. Given inputs and corresponding targets as training data, we study the interpolation estimator,
(12) 
where is a positive definite (PD) matrix that incorporates prior knowledge about the true model. For simplicity, we present our results in terms of the norm (ridgeless regression) as defined in Equation 12. However, all our results hold for arbitrary –norms via appropriate rotations. Given an arbitrary PD matrix , the rotated covariates and rotated parameters maintain and the norm of parameters simplifies to .
Appendix B Standard error of minimum norm interpolants
b.1 Projection operators
The projection operators and are formally defined as follows.
(13)  
(14) 
b.2 Invariant transformations may have arbitrary nullspace components
We show that the transformations which satisfy the invariance condition where is a transformation of may have arbitrary nullspace components for general transfomation mappings . Let and be the column space and nullspace projections for the original data . The invariance condition is equivalent to
(15) 
which implies that as long as , then for any choice of nullspace component , there is a choice of which satisfies the condition. Thus, we consider augmented points with arbitrary components in the nullspace of .
b.3 Proof of Theorem 1
Inequality (8) follows from
(16) 
by decomposition of where and . Note that the error difference does scale with , although the sign of the difference does not.
b.4 Proof of Corollary 1
Corollary 1 presents three sufficient conditions under which the standard error of the augmented estimator is never larger than the standard error of the standard estimator .
b.5 Proof of Proposition 1
The proof of Proposition 1 is based on the following two lemmas that are also useful for characterization purposes in Corollary B.7.
Lemma 1.
If a PSD matrix has nonequal eigenvalues, one can find two unit vectors for which the following holds
(19) 
Hence, there exists a combination of original and augmentation dataset such that condition (19) holds for two directions and .
Note that neither nor can be eigenvectors of in order for both conditions in equation (19) to hold. Given a population covariance, fixed original and augmentation data for which condition (19) holds, we can now explicitly construct for which augmentation increases standard error.
Lemma 2.
Assume are fixed. Then condition (19) holds for two directions and iff there exists a such that for some . Furthermore, the norm of needs to satisfy the following lower bounds with
(20) 
where are constants that depend on .
Proposition 1 follows directly from the second statement of Lemma 2 by minimizing the bound (20) with respect to which is a free parameter to be chosen during construction of (see proof of Lemma (2). The minimum is attained for . We hence conclude that needs to be sufficiently more complex than a good standard solution, i.e. where is a constant that depends on the .
b.6 Proof of technical lemmas
In this section we prove the technical lemmas that are used to prove Theorem 1.
Proof of Lemma 2
Any vector can be decomposed into orthogonal components . Using the minimumnorm property, we can then always decompose the (rotated) augmented estimator and true parameter by
where we define “ext” as the set of basis vectors which span and respectively “rest” for . Requiring the standard error increase to be some constant can be rewritten using identity (16) as follows
(21) 
The left hand side of equation (21) is always positive, hence it is necessary for this equality to hold with any , that there exists at least one pair such that and one direction of the iff statement is proved.
For the other direction, we show that if there exist and for which condition (19) holds (wlog we assume that the ) we can construct a for which the inequality (8) in Theorem 1 holds as follows:
It is then necessary by our assumption that for at least some . We can then set such that , i.e. that the augmented estimator is not equal to the standard estimator (else obviously there can be no difference in error and equality (21) cannot be satisfied for any desired error increase ).
The choice of minimizing that also satisfies equation (21) is an appropriately scaled vector in the direction of where we define and . Defining for convenience and then setting
(22) 
which is welldefined since , yields a such that augmentation increases standard error. It is thus necessary for that