Title: A General Framework for Learning under Corruption: Label Noise, Attribute Noise, and Beyond
Venue: NeurIPS 2023
Accepted: no
Keywords: corruption framework

Markov transitions

learning theory
Abstract: Corruption is frequently observed in collected data and has been extensively studied in machine learning under different corruption models.
Despite this, there remains a limited understanding of how these models relate such that a unified view of corruptions and their consequences on learning is still lacking. In this work, we formally analyze corruption models at the distribution level through a general, exhaustive framework based on Markov kernels. We highlight the existence of intricate joint and dependent corruptions on both labels and attributes, which are rarely touched by existing research. Further, we show how these corruptions affect standard supervised learning by analyzing the resulting changes in Bayes Risk. Our findings offer qualitative insights into the consequences of ``more complex'' corruptions on the learning problem, and provide a foundation for future quantitative comparisons. Applications of the framework include corruption-corrected learning, a subcase of which we study in this paper by theoretically analyzing loss correction with respect to different corruption instances.

Abstract

Corruption is frequently observed in collected data and has been extensively studied in machine learning under different corruption models. Despite this, there remains a limited understanding of how these models relate such that a unified view of corruptions and their consequences on learning is still lacking. In this work, we formally analyze corruption models at the distribution level through a general, exhaustive framework based on Markov kernels. We highlight the existence of intricate joint and dependent corruptions on both labels and attributes, which are rarely touched by existing research. Further, we show how these corruptions affect standard supervised learning by analyzing the resulting changes in Bayes Risk. Our findings offer qualitative insights into the consequences of 'more complex' corruptions on the learning problem, and provide a foundation for future quantitative comparisons. Applications of the framework include corruption-corrected learning, a subcase of which we study in this paper by theoretically analyzing loss correction with respect to different corruption instances.

1 Introduction

Machine learning starts with data. The most widespread conception of data defines them as atomic facts, perfectly describing some reality of interest [1]. In learning theories, this is reflected by the often-used assumption that training and test data are drawn independently from the same distribution. The goal of learning is to identify and synthesize patterns based on the knowledge, or information, embedded in these data. In practice, however, corruption regularly occurs in data collection. This creates a mismatch between training and test distributions, forcing us to learn from imperfect facts.

We should thus doubt the view of data as static facts, and consider them as a dynamic element of a learning task [2]. Besides the predictor and the loss function, one may focus on the data dynamics, studying corruptions and intervening in the learning process. Toward this goal, there has been a surge of research in the machine learning community proposing various corruption models, examining and correcting their effects on learning formally or empirically [3, 4, 5, 6, 7, 8]. Nevertheless, it is still unclear how these models relate and whether they characterize all types of corruption. Even though the necessity of investigating this topic is recognized both at a practical [9, 10] and a theoretical [11, 12] level, no standardized way to model and analyze corruption has been so far created [13].

Our primary objective here is to systematically study the problem of learning under corruption, providing a general framework for analysis. Whilst there have been some existing attempts, certain limitations persist in terms of homogeneity and exhaustiveness. A famous early endeavor is Quinonero-Candela et al. [14], grouping together works about the multi-faceted topic of dataset shift, yet not in a unifying or comprehensive manner. Later on, several studies aim to provide a more homogeneous view of corruption, often referred to as noise or distribution shift. However, their frameworks

typically rely on some corruption-invariant assumptions on the marginal or conditional probabilities, 36 and the extent of exhaustiveness is merely conjectured or not considered [15, 16, 17, 18]. 37

In this paper, we take a different point of view from the previous work: we categorize corruption 38 based on its dependence on the feature and label space, rather than relying on the notion of invariance. 39 Our resulting framework is generic, encompassing all possible pairwise stochastic corruptions. 1 The 40 underpinning mathematical tool that enables such exhaustiveness is the Markov kernel. While Markov 41 kernels have been utilized in formalizing corruption [7, 19], their primary focus has been solely on 42 label corruption, attribute corruption, or simple joint corruption. To our knowledge, the proposed 43 framework is novel in the sense of demonstrated exhaustiveness in this domain. Our contributions are 44 summarized as follows: 45

46

47

48

C1 We propose a new taxonomy of corruption in the supervised learning setting (§ 3), hierarchically organized through the notion of dependence (Fig. 1), and connect existing corruption models to this taxonomy (Tab. 1).

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

C2 We analyze the implications of our family of corruptions on learning (§ 4), linking the Bayes risk of the clean and corrupted supervised learning problems through equality results (Theorem 3, Theorem 4, Theorem 5).

C3 We derive corruption-corrected loss functions for different corruption instances within our framework (§ 5). A subcase of these corrections (Theorem 8) generalizes prior results on corruptioncorrected learning in simple label corruption.

Though abstract in general, our results expand upon existing ones on specific corruption models and shed light on the relatively under-explored joint and dependent corruptions.

2 Background

Before introducing our analysis, we review the background framework and notations.

Supervised learning In statistical decision theory [20, 21], a general decision problem can be viewed as a two-player game between nature and decision-maker . Nature chooses its state , then experiment leads to some observations given the state, and the decision-maker picks a suitable action from a fixed set of decision rules . In the specific setting of supervised learning , observations are in the feature space X ⊂ R d , d ≥ 1 , states are in the label space Y , then the experiment E leads to a probability associated with the observation X , given the state Y . Here we focus on the classification task, that is, assuming the label space to be finite . All the stated results can be easily extended to regression cases by considering a continuous label space; we leave it for future application.

To formalize the processes described above, we introduce the Markov kernel.

Definition 1 (Klenke [22]) . A Markov kernel κ from a measurable space ( X 1 , X 1 ) to a measurable space ( X 2 , X 2 ) is a function x 1 ↦→ κ ( x 1 , · ) from X 1 to P ( X 2 ) , the set of probability measures on X 2 , such that κ ( x 1 , B ) is measurable in x 1 for each set B ∈ X 2 . We denote it by κ : X 1 ⇝ X 2 , or more compactly by κ X 1 X 2 . The set of Markov kernels from X 1 to X 2 is referred to as M ( X 1 , X 2 ) .

The Markov kernel generalizes the concept of conditional probability. Looking at the function κ ( · , B ) , it associates different probabilities to the set B given different values of the parameter x 1 . It can transform a distribution µ ∈ P ( X 1 ) into another distribution µκ ∈ P ( X 2 ) , as well as transform a function f : X 2 → R into another function κf : X 1 → R with the following two operators:

provided the integral exists. Next, we define different operations to combine Markov kernels:

P1 Given κ : X 1 ⇝ X 2 and λ : X 1 × X 2 ⇝ X 3 , their chain composition κ ◦ λ : X 1 ⇝ X 3 is defined by ( κ ◦ λ ) f ( x 1 ) := ∫ X 2 κ ( x 1 , dx 2 ) ∫ X 3 λ (( x 1 , x 2 ) , dx 3 ) f ( x 3 ) = κ ( λ f )( x 3 ) where f : X 3 → R is a positive X 3 -measurable function;

P2 For κ : X 1 ⇝ X 2 and λ : X 1 × X 2 ⇝ X 3 , their product composition κ × λ : X 1 ⇝ X 2 × X 3 is ( κ × λ ) f ( x 1 ) := ∫ X 2 κ ( x 1 , dx 2 ) ∫ X 3 λ (( x 1 , x 2 ) , dx 3 ) f ( x 2 , x 3 ) for every f positive X 2 ×X 3 -measurable.

Notice that a probability distribution is a specific instance of a Markov kernel, constant in its parameter 79 space. Therefore, P1 and P2 are well defined for κ ≡ µ ∈ P ( X 2 ) . We can unify the notation of × 80 for distributions thanks to the flexibility of kernels, and consider the µκ as a subcase of µ ◦ κ . 81

Bayes risk Having defined all these objects, a supervised learning problem can be represented by the diagram Y X Y, E h where h is a decision rule chosen in M ( X,Y ) . Its task can be formalized as a risk minimization problem, i.e., finding the optimal action h ∈ H by considering the Bayes Risk (BR) measure

where the notation κ X stands for the kernel κ evaluated on the parameter X , e.g., h X , E Y (this subscript notation will be used throughout), and π is a prior distribution on Y . The function ℓ is asked to be bounded and a proper loss [23, 24], i.e., a loss function ℓ : P ( Y ) × Y → R + whose minimization set contains the ground truth class probability. More formally, we ask for

Since in real-world applications, one deploys a model with only limited representation capacity, we consider the constrained version of BR

We call H the model class . If we fix the joint space to Z = X × Y and the joint probability 82 distribution to P = π Y × E ∈ P ( Z ) , we can refer to a supervised learning problem as the triple 83 ( ℓ, H , P ) . Notice that we can also use an equivalent decomposition of the joint distribution through a 84 posterior kernel F : X ⇝ Y , so that P = π X × F for some prior on the feature space. Hence, each 85 supervised learning problem can have two associated kernels, the experiment E and the posterior 86 one F . We then obtain two views of the learning problem, a generative and a discriminative one, as 87 previously noted by Reid et al. [25]. By means of these, we can define two Conditional BR (CBR): 88

both equal to their corresponding constrained BR. Notice that for Eq. (1) to be well defined, we need 89 at least one minimum of the unconstrained BR to be included in the model class. For our convenience, 90 we ask it to be the h matching the F . 91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

3 A general framework for corruption

In this section, we present a general framework of pairwise corruptions based on the notion of dependence and discuss how existing corruption models fit into this framework as subcategories. First, let us formally define corruption and two additional kernel operations, which will be useful in the buildup of our corruption taxonomy.

Definition 2. A corruption is a Markov kernel κ that sends a probability space ( X × Y, X × Y , P ) into another, ( X × Y, X × Y , ˜ P ) . We write it as κ Z ˜ Z , 2 and call the variables z = ( x, y ) ∈ Z parameters and the differentials d ˜ z = d ˜ xd ˜ y corrupted variables .

The following operations are not considered in the classical probability literature but have been studied in other areas, e.g., through the lens of category theory [26, 27, 28]. Here we rework them to fit our framework.

P3 Given κ : X 1 ⇝ X 2 and λ : X 3 ⇝ X 4 , their superposition (see § S2.1) is equal to κλ : X 1 × X 3 ⇝ X 2 × X 4 as ( κλ ) f ( x 1 , x 3 ) := ∫ X 2 κ ( x 1 , dx 2 ) ∫ X 4 λ ( x 3 , dx 4 ) f ( x 2 , x 4 ) , where f : X 2 × X 4 → R is positive X 2 ×X 4 -measurable;

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

κ

Z

˜

Z

2J

κ

XY

˜

Y

2D-

˜

Y

κ

X

˜

Y

1D-

˜

Y

κ

Y

˜

Y

S-

˜

Y

κ

Y

˜

X

˜

Y

1J-

Y

κ

X

˜

X

˜

Y

1J-

X

κ

XY

˜

X

2D-

˜

X

κ

Y

˜

X

1D-

˜

X

κ

X

˜

X

S-

˜

X

2D-

˜

X

, 2D-

˜

Y

1D-

˜

X

, 2D-

˜

Y

2D-

˜

X

, 1D-

˜

Y

S-

˜

X

, 2D-

˜

Y

2D-

˜

X

, S-

˜

Y

,

S-

˜

X

, S-

˜

Y

1D-

˜

X

, 1D-

˜

Y

Figure 1: Partial orderings on the corruption and combination sets, based on the amount of dependence on the spaces. In the left panel, we underline with dotted nodes the corruptions that cannot be used in any feasible combination. Trivial cases of independence from all parameters or identical kernels are excluded from this analysis.

P4 The pseudo-inverse of a kernel κ : X 1 ⇝ X 2 is defined as κ † : X 2 ⇝ X 1 such that ( κ † ◦ κ ) µ 1 = µ 1 and ( κ ◦ κ † ) µ 2 = µ 2 with µ 1 , µ 2 being the probabilities associated to X 1 , X 2 . In general, the pseudo-inverse is not unique, since it corresponds to a class of equivalence induced by the probability measure on X 1 (see details in § S2.2).

Again, P3 is well defined for κ ≡ µ ∈ P ( X 2 ) . This operation allows for more flexible combinations of kernels, in a 'parallel' fashion. No restriction is imposed on the parameter spaces to be equal, e.g., X 1 = X 2 , or Cartesian products with some space in common, e.g., X 1 = Y 1 × Y 2 , X 2 = Y 1 × Y 3 . When this happens, the action of the two kernels 'superpose' on the same space. In addition, having more than one measure in the integral acting on the same space would make the integral ill-defined, so this case is excluded. Because of these properties, we say that P3 is the operation with the weakest feasibility conditions , i.e., the set of rules to fulfill a well-defined operation.

Building a taxonomy of corruptions Corruptions can be naturally classified in different ways, depending on their behavior with respect parameters and corrupted variables. In Fig. 1a, we show all possible non-trivial corruption types, i.e., those that are not identical and not constantly equal to a probability. We classify them based on the number of parameters they depend on , and the type of corrupted variables they result in . Specifically, we employ the following abbreviations: J is short for Joint (both variables are corrupted), S is short for Simple (the parameter and the corrupted variable are the same), and D is short for Dependent (others). We then obtain the classification: 2-parameter joint corruption (2J), 1-parameter joint corruption (1J), 2-parameter dependent corruption (2D), 1-parameter dependent corruption (1D), simple corruption (S), along with an indication of parameter or corrupted space. The general naming rule is {#parameters} + {abbreviation} + {-} + {parameter or corrupted space, depending on where the ambiguity lies}.

We now want to generate all possible corruptions of the type κ Z ˜ Z : X × Y ⇝ ˜ X × ˜ Y . We combine the nodes in Fig. 1a using the superposition operation ( P3 ), obtaining all the feasible combinations included in Fig. 1b. The missing couples are excluded because of P3 's feasibility conditions described above, which, even if weak, still do not allow some corruption pairings. Needing each corrupted variable to appear exactly once , we cannot include the 1-parameter joint corruptions in any factorization of the κ XY ˜ X ˜ Y . It is easy to check that no corruption from (a subset of) { X,Y } to (a subset of) { ˜ X, ˜ Y } can be combined with them. Compatibility problems arise also when trying to combine a simple corruption (S) with a 1-parameter dependent one (1D); we cannot fulfill the feasibility conditions for P3 and obtain a complete joint corruption, since we will be always missing a parameter. We then exclude this combination from our taxonomy. 3

Markov kernels and exhaustiveness Our motivation for formalizing corruptions through Markov kernels is their representation power in terms of couplings. A coupling is formally defined for two probability spaces Σ 1 := ( Z 1 , Z 1 , P 1 ) , Σ 2 := ( Z 2 , Z 2 , P 2 ) as a probability space Σ := ( Z 1 × Z 2 , Z 1 × Z 2 , P ) , such that the marginal probabilities associated to P w.r.t. Z i , i ∈ { 1 , 2 } are

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

the respective P i . By construction, Markov kernels are in bijection with all the possible couplings existent on Z × Z with two fixed probability measures, for us, P, ˜ P . Hence, they represent all possible pairwise dependencies between probability spaces that are stochastic , and for non-stochastic mappings, we are sure to have an alternative Markov kernel representation. 4

In most machine learning research considering corruption, the corruption process typically involves two environments, that is, the training one and the test one. Our definition of corruption (Def. 2) covers all such pairwise cases. Furthermore, one may also apply this framework to settings with more than two spaces, e.g., online learning or learning from multiple different domains [29]. For these cases, we can employ a composed model, where different corruptions are acting together in a 'chained' ( P1 , P2 ) or 'parallel' ( P3 ) fashion and creating more complex patterns. We discuss further possibilities for applying this framework to n > 2 corrupted spaces in § S2.3.

Relations to existing paradigms Next, we examine how existing corruption models fit into our taxonomy. To do so, we reformulate them as specific instantiations of Markov corruptions. This reveals their relationships within the corruption hierarchy presented in Fig. 1a. Our goal here is not to merely demonstrate that a child problem can be solved by a parent one, but rather to gain a deeper understanding of the problem settings. The exhaustiveness of the framework allows us to identify what has been previously overlooked in characterizing all types of corruption. Notably, we highlight the existence of joint and dependent corruptions, which receive far less attention than simple

corruptions, while far greater problems arise in such complicated cases (see § 4). Moreover, we notice 160 that existing categorizations rely mostly on the notion of invariance, i.e., corruptions are defined 161 based on which element of the distributions are preserved. These invariance-based taxonomies have 162 been introduced mainly for robustness and causal analyses. However, they do not have a one-to-one 163 correspondence with ours, and do not allow for a hierarchical nor compositional view of corruption. 164 A summary of representative corruption models in the literature is given in Tab. 1, while all the 165 technical details about correspondences and relations between taxonomies are given in § S1. 166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

4 Consequences of corruption in supervised learning

Traditionally, experiments have been compared through Bayes Risk using what is known as the Data Processing Inequality, or Blackwell-Sherman-Stein Theorem [20, 60]. 5 Recently, in Williamson and Cranko [19], Data Processing Equality results have also been studied within the supervised learning framework. Here we adopt the equality approach to compare the clean and corrupted experiments through Bayes Risk. The equalities formally characterize how the optimization problem is affected by the different kinds of joint corruption in our taxonomy. This gives us a quantitative result in terms of conserved 'information' [19] between corrupted and clean learning problems, and a bridge between the problems themselves.

We rewrite the minimization set of the BR in a more compact way, such as ℓ ◦ H := { ( x, y ) ↦→ ℓ ( h x , y ) | h ∈ H ⊆ M ( X,Y ) } . We define the action of a corruption κ on this set as the set of all the corrupted functions κf, f ∈ ℓ ◦ H . Lastly, we ask f ∗ = ℓ ◦ h ∗ ∈ arg min f E ˜ P [ f ( ˜ X, ˜ Y )] to belong to the constraining space ℓ ◦ H , for reasons already discussed for Eq. (1).

The first two theorems cover the (S, 2D) cases and their subcase (S-˜ X , S-˜ Y ), as proved in [19].

Theorem 3 (BR under (S-˜ X , S-˜ Y ), (2D-˜ X , S-˜ Y ) joint corruption) . Let ( ℓ, H , P ) be a learning problem, E : Y ⇝ X an experiment and κ ˜ X ∈ { κ X ˜ X , κ Y X ˜ X } a corruption. Let κ Y ˜ Y be a simple corruption on Y . Then we can form the corrupted experiment as per the transition diagram 6

Here in Theorem 3 we have shown the BR equality for the experiment E , in line with the Comparison of Experiments and Information Equalities literature mentioned at the beginning of the section. However, for some corruptions the equalities results cannot be stated with E and the Generative CBR, unless ignoring the joint corruption factorization formula (see § S5 for a detailed explanation). We hence use the posterior kernel F defined with the Discriminative CBR (Eq. (1)), and gain more insights about the minimization set while paying a price in elegance of the result.

Theorem 4 (BR under (S-˜ X , 2D-˜ Y ) joint corruption) . Let ( ℓ, H , P ) be a learning problem, F : X ⇝ Y a posterior and κ XY ˜ Y } a Y corruption. Let κ X ˜ X be a simple corruption on X . Then we can form the corrupted experiment as per the transition diagram ˜ X X Y ˜ Y κ X ˜ X F κ ˜ Y κ XY ˜ Y

and obtain

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

We can notice, thanks to Theorems 3, 4, that when corruption involves dependent structures in the factorization, the loss function or the whole minimization set are modified in a parameterized, dependent way. For instance,

with κ XY ˜ Y now viewed as a parameterized label corruption, i.e. ( κ Y ˜ Y ) x . An additional consequence is also that the result can only be given in terms of CBR, Discriminative or Generative. We also see that corruptions on Y only affect the loss function and does not touch the model class, even in the dependent case.

The next theorems cover the factorizations involving 1D corruptions. In the first case, we are again forced to use either E or F , depending on the involved factors. We group the two results in one theorem for brevity.

Theorem 5 (BR under (1D, 2D) joint corruption) . Let ( ℓ, H , P ) be a learning problem, E : Y ⇝ X and F : X ⇝ Y be an experiment and a posterior on it.

Let κ Y ˜ X be a corruption on X and κ XY ˜ Y be a corruption on Y , then we can form the jointly corrupted experiment as per the transition diagram ˜ X Y X ˜ Y κ Y ˜ X E κ XY ˜ Y κ XY ˜ Y and obtain

Let κ X ˜ Y be a corruption on Y and κ XY ˜ X be a corruption on X , then we can form the jointly corrupted posterior as per the transition diagram ˜ Y X Y ˜ X κ X ˜ Y F κ XY ˜ X κ XY ˜ X and obtain

Being the (1D, 1D) a subcase of both previous corruptions, we can prove the result as a simple corollary. Notice that this implies both E and F formulations to hold.

Corollary 6 (BR under (1D, 1D) joint corruption) . Let ( ℓ, H , P ) be a learning problem, E : Y ⇝ X and F : X ⇝ Y be an experiment and a posterior on it. Let κ Y ˜ X be a corruption on X and κ X ˜ Y be a corruption on Y , then we can form the jointly corrupted experiment as per the transition diagram

or equivalently

In all the Theorems involving a 1D corruption, the minimization set is heavily modified. In Eq. (3), the loss function is corrected such that it will be dependent on the parameter x ( ℓ x ), while the whole composition will be evaluated on y instead of x . We the obtain functions of the form ˜ ℓ x ◦ h ( y ) . In

Eq. (4), we instead end up having a minimization space of the form ˜ ( ℓ ◦ h ) y ( x ) . Lastly, both results of Corollary 6 lead to a comparison of performance on the X space instead of Y , with a new loss function that takes in imput y and a probability on X parameterized by y . We can consider the these cases as an expansion of the loss space; more detail will be added in the next section.

The only factorization missing from Fig. 1b is the (2D, 2D) one. Because of its high dependence on the parameters, we could not recover a meaningful decomposition of the effect on ℓ ◦ H . This suggests it to be equivalent to a 2J corruption when looked at through the lens of Bayes Risk. For detailed analysis, see Supplementary material § S3.

5 Corruption-corrected learning

We now leverage our corruption framework for answering the question ' what can we do to ensure unbiased learning from biased data? '. This question has different answers depending on what we mean by unbiased learning. As for the biased data, we assume that biased here refers to non-identical joint corruption acting on a probability, giving us a corrupted training distribution.

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

Past work from Van Rooyen and Williamson [7] and Patrini et al. [34] considered unbiased learning as what is known as generalization , i.e. learn on the corrupted space ˜ P a hypothesis h ∗ such that it is also optimal on the clean distribution P at test time. They choose the approach of corrected learning, which is, correcting the loss function or the model class in order to learn a h ∗ capable to generalize. They both used frameworks related to ours, although only in the presence of simple Y corruption.

We prove similar results to these works for the loss correction task and analyze what we can achieve in other corruption cases described by our taxonomy. In general, we cannot prove generalization but we exhibit a corrected loss allowing the model learned on ˜ P to have the same biases (i.e. loss scores) as the one found for the clean learning problem. To do so, we make use of the pseudo-inverse of a Markov kernel (P4) , as it is more convenient and powerful than the kernel reconstruction introduced in [7]. The results we show here also serve as a first step towards understanding the effect of corruption of the minimization set ℓ ◦H , in the cases where the BR equalities are not giving us much information (i.e. all the cases that are not simple label noise [19]).

Again, in this analysis, we ignore the influence of the data sample and the optimization technique. We use all the assumptions introduced when defining the learning problem in § 2 and the BR results § 4.

The BR equalities for cleaning kernels The theorems proved in § 4 can then be restated, in terms of learning problems and pseudo-inverse κ † : ˜ Z ⇝ Z , as

We will refer here to the pseudo-inverse of our corruption as the cleaning kernel . Notice that the set κ † ( ℓ ◦H ) is not trivially decomposable as ˜ ℓ ◦ ˜ H for some loss and model class. In this case, κ † ( ℓ ◦H ) is said to have no ◦ -factorized structure .

The BR equalities are ensuring the existence of a function f ∗ ∈ ℓ ◦ H ∩ κ † ( ℓ ◦ H ) that minimizes the Bayes Risk, i.e.

Sadly, this is not enough for us to find an optimal hypothesis working for both probability spaces.

Formal results on this optimal h ∈ H for both clean and corrupted spaces only exist for label noise [7, 8]. However, by introducing a few further assumptions, we can get results on which alternative loss to use on train distribution so that the learned h on ˜ P will have the same performance scores as the optimal on ( ℓ, H , P ) . Let us consider the composed representation of the function f ∗ in the test (clean) minimization set, which is f ∗ = ℓ ◦ h ∗ . We want to construct a suitable composed representation for f ∗ also in the space κ † ( ℓ ◦ H ) , namely f ∗ = ˜ ℓ ◦ ˜ h ∗ . We start by fixing a ˜ h ∗ ∈ H of our choice, that if asked to be invertible ( A1 ) identifies the loss function as ˜ ℓ = ℓ ◦ h ∗ ◦ ( ˜ h ∗ ) -1 : P ( Y ) × Y → R + . 7 There can be weaker conditions on ( ˜ ℓ, ˜ h ∗ ) enabling all the following results, but do not investigate the here.

̸

Since in general κ † ( f ∗ ) = f ∗ , we have that: ∃ h ′ ∈ H s.t. κ † ( ℓ ◦ h ′ ) = ˜ ℓ ◦ ˜ h ∗ , where we ask h ′ = h ∗ , otherwise we would be imposing the trivial condition ℓ ◦ h ∗ = ˜ ℓ ◦ ˜ h ∗ = κ † ( ℓ ◦ h ∗ ) , i.e. the corruption is harmless w.r.t. the Bayes Risk value. In order to study the possible loss correction, we choose the corrupted optimum as ˜ h ∗ = h ′ ( A2 ).

̸

Loss corrections We now try to formalize how to define a suitable loss for the corrupted learning problem, such that the optimal hypothesis is learned in the clean learning space. The problem setting gives us access to ℓ, κ † given by the problem, and ˜ h ∗ chosen by us. We want to find a way to retrieve a suitable h ∗ for the clean distribution. That means, the loss correction task here is finding a formulation of ˜ ℓ that depends on ℓ, κ † . An essential preliminary result, for which the proof is given in § S4.1, is

Lemma 7. The feasible factorization of a Markov kernel κ is also a valid factorization for its pseudo-inverse κ † , both for the full kernel or considering their parameterized versions.

We then give the correction results (proof in § S4.2), and discuss them.

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

Theorem 8. Let ( ℓ, H , P ) be a clean learning problem and ( κ † ( ℓ ◦ H ) , κP ) its associated corrupted one, not necessarily with a ◦ -factorized structure. Let κ † be the joint cleaning kernel reversing κ , such that assumptions A1 and A2 hold for the said problems. The factorization of κ † is assumed to be feasible and to have an equality result of the form Eq. (5) . We write κ † ( dz, ˜ z ) = κ X ( dx, · ) κ Y ( dy, · ) , with ( · ) some feasible parameters. Hence, we can prove the following points:

When κ is either ( id , SY ) or ( id , 2DY ), we can write the corrected loss as

† X X ˜ ℓ ( h (˜ x ) , ˜ y ) = ( κ Y ℓ ) ( h (˜ x ) , ˜ y ) ∀ (˜ x, ˜ y ) ∈ ˜ X × ˜ Y ,

with κ Y ℓ = κ Y ˜ x ℓ for the second case.

When κ † is (SX , SY ), (2DX , SY ) or (SX , 2DY ), we have ˜ ℓ (˜ x, ˜ y, h ) = E u ∼ κ X h (˜ x ) [ κ Y ℓ ( u , ˜ y )] ∀ (˜ x, ˜ y ) ∈ ˜ X × ˜ Y ,

with κ X ˜ x h (˜ x )( A ) := κ X ( h -1 ( A ) , ˜ x ) , A ⊂ P ( Y ) being the push-forward probability measure of κ X ( · , ˜ x ) through h , h seen as a function. For the cases that involve a 2D corruption, we have κ Y ℓ = κ Y ˜ x ℓ for the former κ † factorization, κ X h (˜ x ) = κ X ˜ y h (˜ x ) for the latter.

When κ is a (1DX , 1DY ) corruption, we can write the corrected loss as

† with κ X ˜ x h (˜ y )( B ) := κ X ( h -1 ( B ) , ˜ y ) , B ⊂ P ( X ) .

When

κ † is a (2D, 1D) corruption, we can write the corrected loss as ˜ ℓ (˜ x, ˜ y, h ) = E u ∼ κ X h (˜ y ) [ κ ˜ x Y ℓ ( u , ˜ y )] , ˜ ℓ (˜ x, ˜ y, h ) = E u ∼ κ X ˜ y h (˜ x ) [ κ Y ℓ ( u , ˜ x )] ∀ (˜ x, ˜ y ) ∈ ˜ X × ˜ Y . for the (1DX , 2DY ), (2DX , 1DY ) respectively.

When minimized, the corrected losses will by construction give back the hypothesis ˜ h ∗ . Since ℓ ◦ h ∗ = ˜ ℓ ◦ ˜ h ∗ , the learned ˜ h ∗ has on the clean distribution the optimum performance we wanted to achieve with the original loss function ℓ . Hence, we achieve unbiased learning in the sense of matching scores and in the distributional sense.

The corrections found by the theorem are more complex than the ones defined in previous work [7, 34], i.e., the first part of point 1. In the second part, we characterize the effect of a more 'dependent' Y cleaning kernel, i.e. closer to the root in Fig. 1a. When also κ X is non-trivial in the factorization, we have an action on h . Then, the corrected functions lie in a larger function space than the usual one, the one of positive, bounded functions ℓ : X × Y ×H→ R + .

The result additionally underlines how the cleaning kernel affects the a hypothesis on X : it induces a set of 'reachable predictions' from h through κ † , depending on the outcome of the stochastic process κ † : ˜ X ⇝ X . The push-forward probability measures are probabilities on a set of probabilities . For instance, in point 2 we have κ X ˜ y h (˜ x ) ∈ P ( P ( Y )) , while for point 3 we have κ X ˜ x h (˜ x ) ∈ P ( P ( X )) .

6 Conclusions

We proposed a comprehensive and unified framework for corruption using Markov kernels, systematically studying corruption in three key aspects: classification, consequence, and correction. We established a new taxonomy of corruption, enabling qualitative comparisons between corruption models in terms of the corruption hierarchy. To gain a deeper quantitative understanding of corruption, we analyzed the consequences of different corruptions from an information-theoretic standpoint by proving Data Processing Equalities for Bayes Risk. As a consequence of them, we obtained loss correction formulas that gives us more insights into the effect of corruption on losses.

Throughout the work, we consider data as probability distributions, implicitly assuming that each dataset has an associated probabilistic generative process. We treat corruption as Markov kernels, assuming full access to their actions, and analyze the consequences of corruption through Bayes risks without accounting for sampling or optimization. Bridging the gap between the distributional-level and the sample-level results would be the next step for this study, which requires tailored ad-hoc analyses. Other directions for making this framework more practically usable include developing quantitative methods to compare corruption severity and investigating the effects of optimization algorithms on the analysis.

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

References

Mary Poovey et al. A history of the modern fact: Problems of knowledge in the sciences of wealth and society . University of Chicago Press, 1998.

Robert Williamson. Process and Purpose, Not Thing and Technique: How to Pose Data Science Research Challenges. Harvard Data Science Review , 2(3), sep 30 2020. https://hdsr.mitpress.mit.edu/pub/f2cllynw.

Hidetoshi Shimodaira. Improving predictive inference under covariate shift by weighting the log-likelihood function. Journal of statistical planning and inference , 90(2):227-244, 2000.

Xingquan Zhu and Xindong Wu. Class noise vs. attribute noise: A quantitative study. The Artificial Intelligence Review , 22(3):177, 2004.

Nagarajan Natarajan, Inderjit S Dhillon, Pradeep K Ravikumar, and Ambuj Tewari. Learning with noisy labels. Advances in neural information processing systems , 26, 2013.

Aditya Menon, Brendan Van Rooyen, Cheng Soon Ong, and Bob Williamson. Learning from corrupted binary labels via class-probability estimation. In International conference on machine learning , pages 125-134. PMLR, 2015.

Brendan Van Rooyen and Robert C Williamson. A theory of learning with corrupted labels. J. Mach. Learn. Res. , 18(1):8501-8550, 2017.

Aditya Krishna Menon, Brendan Van Rooyen, and Nagarajan Natarajan. Learning from binary labels with instance-dependent noise. Machine Learning , 107(8):1561-1595, 2018.

Andrey Malinin, Neil Band, German Chesnokov, Yarin Gal, Mark JF Gales, Alexey Noskov, Andrey Ploskonosov, Liudmila Prokhorenkova, Ivan Provilkov, Vatsal Raina, et al. Shifts: A dataset of real distributional shift across multiple large-scale tasks. arXiv preprint arXiv:2107.07455 , 2021.

Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Balsubramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of inthe-wild distribution shifts. In International Conference on Machine Learning , pages 5637-5664. PMLR, 2021.

Xiao-Li Meng. Enhancing (publications on) data quality: Deeper data minding and fuller data confession. Journal of the Royal Statistical Society Series A: Statistics in Society , 184(4):1161-1175, 2021.

Negar Rostamzadeh, Ben Hutchinson, Christina Greer, and Vinodkumar Prabhakaran. Thinking beyond distributions in testing machine learned models. In NeurIPS 2021 Workshop on Distribution Shifts: Connecting Methods and Applications .

Nithya Sambasivan, Shivani Kapania, Hannah Highfill, Diana Akrong, Praveen Paritosh, and Lora M Aroyo. "Everyone wants to do the model work, not the data work": Data Cascades in High-Stakes AI. In proceedings of the 2021 CHI Conference on Human Factors in Computing Systems , pages 1-15, 2021.

Joaquin Quinonero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. Dataset shift in machine learning . Mit Press, 2008.

José A. Sáez. Noise models in classification: Unified nomenclature, extended taxonomy and pragmatic categorization. Mathematics , 10(20), 2022. ISSN 2227-7390. doi: 10.3390/math10203736.

Adarsh Subbaswamy, Bryant Chen, and Suchi Saria. A unifying causal framework for analyzing dataset shift-stable learning algorithms. Journal of Causal Inference , 10(1):64-89, 2022.

Jose G Moreno-Torres, Troy Raeder, Rocío Alaiz-Rodríguez, Nitesh V Chawla, and Francisco Herrera. A 357 unifying view on dataset shift in classification. Pattern recognition , 45(1):521-530, 2012. 358

Meelis Kull and Peter Flach. Patterns of dataset shift. In First International Workshop on Learning over 359 Multiple Contexts (LMCE) at ECML-PKDD , 2014. 360

Robert C Williamson and Zac Cranko. Information processing equalities and the information-risk bridge. 361 arXiv preprint arXiv:2207.11987 , 2022. 362

Erik Torgersen. Comparison of statistical experiments , volume 36. Cambridge University Press, 1991. 363

Albert N Shiryaev and Vladimir G Spokoiny. Statistical Experiments And Decision, Asymptotic Theory , 364 volume 8. World Scientific, 2000. 365

Achim Klenke. Probability Theory: A Comprehensive Course . Springer, 2007. 366

Mark D Reid and Robert C Williamson. Composite binary losses. The Journal of Machine Learning 367 Research , 11:2387-2422, 2010. 368

Robert Williamson, Elodie Vernet, Mark Reid, et al. Composite multiclass losses. 2016. 369

Mark Reid, Robert Williamson, et al. Information, divergence and risk for binary experiments. 2011. 370

Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Ohad Kammar. Bayesian inversion by ω -complete 371 cone duality. In 27th International Conference on Concurrency Theory , 2016. 372

Kenta Cho and Bart Jacobs. Disintegration and bayesian inversion via string diagrams. Mathematical 373 Structures in Computer Science , 29(7):938-971, 2019. 374

375

376

Bart Jacobs and Fabio Zanasi. The logical essentials of bayesian reasoning. Foundations of Probabilistic Programming , pages 295-331, 2020.

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Vaughan. A theory of learning from different domains. Machine Learning , 79:151-175, 2010.

George Shackelford and Dennis Volper. Learning k-dnf with noise in the attributes. In Proceedings of the first annual workshop on Computational learning theory , pages 97-103, 1988.

Sally A. Goldman and Robert H. Sloan. Can pac learning algorithms tolerate random attribute noise? Algorithmica , 14(1):70-84, 1995.

Dana Angluin and Philip Laird. Learning from noisy examples. Machine Learning , 2:343-370, 1988.

Avrim Blum and Tom Mitchell. Combining labeled and unlabeled data with co-training. In Proceedings of the eleventh annual conference on Computational learning theory , pages 92-100, 1998.

Giorgio Patrini, Alessandro Rozza, Aditya Krishna Menon, Richard Nock, and Lizhen Qu. Making deep neural networks robust to label noise: A loss correction approach. In Proceedings of the IEEE conference on computer vision and pattern recognition , pages 1944-1952, 2017.

Leon A Gatys, Alexander S Ecker, and Matthias Bethge. A neural algorithm of artistic style. arXiv preprint arXiv:1508.06576 , 2015.

Justin Johnson, Alexandre Alahi, and Li Fei-Fei. Perceptual losses for real-time style transfer and superresolution. In Computer Vision-ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part II 14 , pages 694-711. Springer, 2016.

Eric Grinstein, Ngoc QK Duong, Alexey Ozerov, and Patrick Pérez. Audio style transfer. In 2018 IEEE international conference on acoustics, speech and signal processing (ICASSP) , pages 586-590. IEEE, 2018.

Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199 , 2013.

Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. arXiv preprint arXiv:1412.6572 , 2014.

Nicolas Papernot, Patrick McDaniel, Somesh Jha, Matt Fredrikson, Z Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. In 2016 IEEE European symposium on security and privacy (EuroS&P) , pages 372-387. IEEE, 2016.

Alexey Kurakin, Ian J Goodfellow, and Samy Bengio. Adversarial examples in the physical world. In Artificial intelligence safety and security , pages 99-112. Chapman and Hall/CRC, 2018.

Dan Hendrycks, Kevin Zhao, Steven Basart, Jacob Steinhardt, and Dawn Song. Natural adversarial examples. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 15262-15271, 2021.

Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with bounded instance and label-dependent label noise. In International Conference on Machine Learning , pages 1789-1799. PMLR, 2020.

Yu Yao, Tongliang Liu, Mingming Gong, Bo Han, Gang Niu, and Kun Zhang. Instance-dependent labelnoise learning under a structural causal model. Advances in Neural Information Processing Systems , 34: 4409-4420, 2021.

Qizhou Wang, Bo Han, Tongliang Liu, Gang Niu, Jian Yang, and Chen Gong. Tackling instance-dependent label noise via a universal probabilistic model. In Proceedings of the AAAI Conference on Artificial Intelligence , volume 35, pages 10183-10191, 2021.

Nathalie Japkowicz and Shaju Stephen. The class imbalance problem: A systematic study. Intelligent data analysis , 6(5):429-449, 2002.

Haibo He and Edwardo A Garcia. Learning from imbalanced data. IEEE Transactions on knowledge and data engineering , 21(9):1263-1284, 2009.

Mateusz Buda, Atsuto Maki, and Maciej A Mazurowski. A systematic study of the class imbalance 422 problem in convolutional neural networks. Neural networks , 106:249-259, 2018. 423

Zachary Lipton, Yu-Xiang Wang, and Alexander Smola. Detecting and correcting for label shift with black 424 box predictors. In International conference on machine learning , pages 3122-3130. PMLR, 2018. 425

Gilles Blanchard, Marek Flaska, Gregory Handy, Sara Pozzi, and Clayton Scott. Classification with 426 asymmetric label noise: Consistency and maximal denoising. Electronic Journal of Statistics , 10(2): 427 2780-2824, 2016. 428

Julian Katz-Samuels, Gilles Blanchard, and Clayton Scott. Decontamination of mutual contamination 429 models. Journal of machine learning research , 20(41), 2019. 430

431

432

Arthur Gretton, Alex Smola, Jiayuan Huang, Marcel Schmittfull, Karsten Borgwardt, and Bernhard Schölkopf. Covariate shift by kernel mean matching. Dataset shift in machine learning , 3(4):5, 2009.

433

434

435

436

437

438

439

440

441

442

443

444

445

446

447

448

449

450

451

452

453

Masashi Sugiyama and Motoaki Kawanabe. Machine learning in non-stationary environments: Introduction to covariate shift adaptation . MIT press, 2012.

Tianyi Zhang, Ikko Yamane, Nan Lu, and Masashi Sugiyama. A one-step approach to covariate shift adaptation. In Asian Conference on Machine Learning , pages 65-80. PMLR, 2020.

Kun Zhang, Bernhard Schölkopf, Krikamol Muandet, and Zhikun Wang. Domain adaptation under target and conditional shift. In International conference on machine learning , pages 819-827. PMLR, 2013.

Mingming Gong, Kun Zhang, Tongliang Liu, Dacheng Tao, Clark Glymour, and Bernhard Schölkopf. Domain adaptation with conditional transferable components. In International conference on machine learning , pages 2839-2848. PMLR, 2016.

Xiyu Yu, Tongliang Liu, Mingming Gong, Kun Zhang, Kayhan Batmanghelich, and Dacheng Tao. Labelnoise robust domain adaptation. In International conference on machine learning , pages 10913-10924. PMLR, 2020.

Gerhard Widmer and Miroslav Kubat. Learning in the presence of concept drift and hidden contexts. Machine learning , 23:69-101, 1996.

Jie Lu, Anjin Liu, Fan Dong, Feng Gu, Joao Gama, and Guangquan Zhang. Learning under concept drift: A review. IEEE Transactions on Knowledge and Data Engineering , pages 1-1, 2018. doi: 10.1109/tkde. 2018.2876857.

Brendan Van Rooyen et al. Machine learning via transitions . PhD thesis, The Australian National University, 2015.

Bianca Zadrozny. Learning and evaluating classifiers under sample selection bias. In Proceedings of the twenty-first international conference on Machine learning , page 114, 2004.

Masashi Sugiyama, Matthias Krauledat, and Klaus-Robert Müller. Covariate shift adaptation by importance 454 weighted cross validation. Journal of Machine Learning Research , 8(5), 2007. 455

Ievgen Redko, Emilie Morvant, Amaury Habrard, Marc Sebban, and Younès Bennani. A survey on domain 456 adaptation theory: learning bounds and theoretical guarantees. arXiv preprint arXiv:2004.11829 , 2020. 457

Christopher M Bishop and Nasser M Nasrabadi. Pattern recognition and machine learning , volume 4. 458 Springer, 2006. 459

Madelyn Glymour, Judea Pearl, and Nicholas P Jewell. Causal inference in statistics: A primer . John 460 Wiley & Sons, 2016. 461

Vladimir Igorevich Bogachev. Measure theory , volume 1. Springer, 2007. 462

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nEven though there have been a lot of studies on learning in noisy settings, they are often disparate, and the authors aim to present a unified view of a diverse class of corruptions and their implications on learning.\n\nSTRENGTHS:\n- The paper offers an interesting direction of looking at existing learning formulations in a noisy setting. It attempts to unify a more extensive class in a single umbrella and then analyzes the different settings through the unified framework.\n- They propose a new taxonomy of corruption in classification sitting and attempt to link the existing works to the taxonomy.\n- The authors also derive a loss correction formula, which would help infer the consequences of different types of corruption on learning.\n\nWEAKNESSES:\nI encourage the authors to discuss the practical implications of the paper's theory, which would strengthen the submission.\n- It would also be helpful to understand how much insight their work provides compared to the corresponding works for each problem. For instance, what advantages does unifying the noises & studying them offer as opposed to studying them in separation?\n- One of the issues is how realistic the assumption is that we know the process, which adds to the corruption. Does the Markov kernel capture all the classes of often encountered corruptions (or what assumptions does it impose on each class of noises)?\n- In practice, the models often pick up spurious correlations in the data. Does that also fit into the taxonomy considered? How does the bias problem (regarding the fairness of predictors) fit into the framework (even if that can be considered a noise-generating process)?\n\nQUESTIONS:\nRefer to weaknesses\n\nLIMITATIONS:\nThe authors do talk about limitations, but it isn't complete. I encourage the authors to refer to the questions & address them in the list of limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a theoretical framework for analyzing corruptions in machine learning using Markov kernels. Within this framework, the authors establish a taxonomy of various types of corruptions and provide data processing equalities for each category. Additionally, the paper derives corrected loss functions for different types of corruptions within their theoretical framework.\n\nSTRENGTHS:\nThe authors categorize corruptions in a rigorous and systematic manner. This classification and the theoretical view from Markov kernels can potentially help future research future research in understanding and studying different types of corruptions. The derived corrected loss can potentially be used to improve performance of machine learning algorithms in the presence of corruptions.\n\nWEAKNESSES:\nOverall, I find it hard to appreciate the contribution of the paper. It lacks clear intuition and practical implications, making it difficult to understand its usefulness. Theorems are presented without concrete insights, and it is unclear how the results can guide real-world practice.\n1. In Section 4, the paper discusses the \"consequences\" of corruption in supervised learning. However, there is a lack of concrete interpretations of these consequences. How do Theorems 3, 4, 5 translate to real-world machine learning scenarios? What can we deduce about the nature of these corruptions and their implications? \n2. The paper introduces a corrected loss function, but it is not well-explained how it operates or why it is effective. What is the underlying intuition behind this correction and how does it improve the performance under label noise?\n3. It would be beneficial if the authors could provide concrete examples to illustrate how the corrected loss function can be implemented. For instance, what would be the formula of the loss in the context of cross-entropy loss under label noise? Right now it is even unclear whether these losses are feasible and whether they require any information that is not observable.\n4. It would also be valuable to include experiments that demonstrate the effectiveness of the derived corrected loss function.\n\nQUESTIONS:\nSee questions raised in Weaknesses.\n\nLIMITATIONS:\nThe authors have adequately discussed some limitations in section 6. For other limitations, see Weaknesses.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis study addresses the prevalent issue of data corruption in machine learning and the lack of a unified understanding of how different corruption models interrelate. The authors analyze corruption models at the distribution level using a comprehensive framework based on Markov kernels, providing a fresh perspective on the issue. They draw attention to joint and dependent corruptions on both labels and attributes, which existing research often overlooks. By examining the effects of these corruptions on supervised learning through Bayes Risk changes, the authors gain insights into the implications of complex corruptions. The proposed framework also has applications in corruption-corrected learning, which the authors explore in this paper. However, the paper does not have any empirical contributions, and has several theoretical discussions that need further clarifications.\n\nSTRENGTHS:\n1. The paper delves into the crucial problem of data corruption in machine learning, offering a comprehensive framework based on Markov kernels to analyze corruption models at the distribution level.\n\n2. It paves a way for future quantitative comparisons by delivering a foundational understanding of more complex corruptions' consequences.\n\n3. The authors' taxonomy of corruption is comprehensive and organized hierarchically through the notion of dependence, helping to relate existing corruption models.\n\nWEAKNESSES:\n1. the paper does not involve any empirical studies, which is very rare for this venue, although there are precedents. Most theoretical papers still choose to validate their finding with some small-scale experiments, the authors does not seem to clarify a particular reasons that they do not need any empirical evidence to support their findings. \n\n    - This seems a particular issue when the authors explicitly suggest that the theoretical framework can help answer \"what can we do to ensure unbiased learning from biased data\" (line 228), and mentions two previous works [7] and [34], where [34] has a decent amount of empirical works. \n\n2. on the theoretical end, there are several questions need further clarifications, potentially paint a significant issue of rigor of the results. (see below questions)\n\nQUESTIONS:\n1. there are several parts of the theoretical discussions need further clarifications that potentially paint a challenge to the rigor of the discussed results. \n\n    - In the manuscript, there are two assumptions discussed A1 at line 259, and A2 at line 265. However, such assumptions are used again in appendix (line 746, A1-A3)\n  \n         - (minor) it seems the appendix are re-using the notations A1, A2 to discuss different assumptions, and certain naming conventions need to be avoided to improve the rigor of the work\n         - (major) the appendix proof suggests that the proved results Theorems are proved under assumptions A1-A3, yet the main theorem body in the main manuscript has no such mentions of the assumptions (within theorem body), which could potentially mislead future readers. \n         - Also, A1 in appendix does not seem to be a conventionally rigorious way of stating an assumption. \n\n2. As the theorem body are usually preferred to be self-contained, it seems not so conventional to use footnote to explain notations. \n\n\n3. line 262-265 seems very confusing, the first part seems to suggest we need $h'\\neq h^\\star$, and the second half assumes $h' =  h^\\star$ as A2. \n     - if I understand correctly, A2 is a very strong assumption, it might be crucial to talk about its practical implications. (also, another reason that some empirical evidence will be needed.)\n\n4. if possible, please try to talk about the practical implications of all the assumptions.\n\nLIMITATIONS:\nno explicit discussions of limitations, but there are discussions about future directions, which might be interpreted as the limitation discussion.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a framework for systematically analyzing and categorizing corruption models in machine learning. The authors propose a new taxonomy of corruption based on its dependence on the feature and label space, rather than relying on invariance assumptions. They utilize the concept of Markov kernels to provide an exhaustive framework that includes all possible pairwise stochastic corruptions. The authors also derive corruption-corrected loss functions within the framework.\n\nSTRENGTHS:\n- A novel unification of existing data corruption models: The proposed theoretical model based on the Markov kernel is capable of modeling various existing data corruption and distribution shifts. This offers a new perspective to understand how different data dynamics connect to each other.\n\n- Insights on how to correct the errors: The authors also derive loss correction formulas for a few instances of the unified data corruption model. These results give an abstract way to study data corruption-aware learning.\n\nWEAKNESSES:\n- Unaccessible to a wider community: One major concern with this paper is whether it is accessible to the NeurIPS community. This paper seems to be a pure theory paper: it is symbol-heavy, theory-oriented, and without any empirical support. The theory community (e.g., COLT) might appreciate it more, and I am a bit worried about the NeurIPS community. See my questions in the next section for more details.\n\n- Unclear concrete applications of the proposed theoretical models: It is unclear how useful the unification is. Do its insights give statistically or computationally more efficient algorithms for corruption-aware learning? Or does it implies one corruption instance subsume another and thus focusing on it is enough? Or does it reveal new corruption instances not studied before but important in practice? A good paper, in my opinion, should give useful advice to its readers.\n\nQUESTIONS:\nA few concrete questions regarding the weakness points.\n\n- How does the Kernel parameter correspond to the original parameter in the existing corruption model? For example, the authors claim that 1D-X, 2D-Y correspond to target shift/label shift. Given the label shift matrix p(y=j|y=i), how does the markov kernel look like?\n\n- How is the derived corruption-aware learning loss different from the existing method? For example, [49] proposes a correction algorithm for label shift with sample complexity guarantee. How to compare [49]'s algorithm with the derived approach? When should which be used? \n\n- Could you give a running example throughout the paper?  For example, give a concrete label shift example (with the shift matrix), show its corresponding kernel, compare the corresponding corrected loss with existing correction algorithms, and justify why the new techniques are desired.\n\n- How does the proposed model connect to modern data distribution shifts? For example, sparse joint shift [r1] was a recently proposed distribution shift model measuring the joint shift of covariate and label. What is the Markov kernel for this model? Where should it be in Table 1 and Figure 1? Does the kernel model reveal new connections between sparse joint shifts and more traditional shifts such as covariate shifts?\n\n[r1] Chen et al, Estimating and Explaining Model Performance When Both Covariates and Labels Shift, 2022.\n\n- More generally, what are some new data corruption instances offered by the new model? Could you clearly state how it is different from the existing ones, and why it might be encountered in practice?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}