Title: Reject option models comprising out-of-distribution detection
Venue: NeurIPS 2023
Accepted: no
Keywords: reject option classification

out-of-distribution detection

selective classification
Abstract: The optimal prediction strategy for out-of-distribution (OOD) setups is a fundamental question in machine learning. In this paper, we address this question and present several contributions. We propose three reject option models for OOD setups: the Cost-based model, the Bounded TPR-FPR model, and the Bounded Precision-Recall model. These models extend the standard reject option models used in non-OOD setups and define the notion of an optimal OOD selective classifier. We establish that all the proposed models, despite their different formulations, share a common class of optimal strategies. Motivated by the optimal strategy, we introduce double-score OOD methods that leverage uncertainty scores from two chosen OOD detectors: one focused on OOD/ID discrimination and the other on misclassification detection. The experimental results consistently demonstrate the superior performance of this simple strategy compared to state-of-the-art methods. Additionally, we propose novel evaluation metrics derived from the definition of the optimal strategy under the proposed OOD rejection models. These new metrics provide a comprehensive and reliable assessment of OOD methods without the deficiencies observed in existing evaluation approaches. 

Abstract

The optimal prediction strategy for out-of-distribution (OOD) setups is a fundamental question in machine learning. In this paper, we address this question and present several contributions. We propose three reject option models for OOD setups: the Cost-based model, the Bounded TPR-FPR model, and the Bounded Precision-Recall model. These models extend the standard reject option models used in non-OOD setups and define the notion of an optimal OOD selective classifier. We establish that all the proposed models, despite their different formulations, share a common class of optimal strategies. Motivated by the optimal strategy, we introduce double-score OOD methods that leverage uncertainty scores from two chosen OOD detectors: one focused on OOD/ID discrimination and the other on misclassification detection. The experimental results consistently demonstrate the superior performance of this simple strategy compared to state-of-the-art methods. Additionally, we propose novel evaluation metrics derived from the definition of the optimal strategy under the proposed OOD rejection models. These new metrics provide a comprehensive and reliable assessment of OOD methods without the deficiencies observed in existing evaluation approaches.

1 Introduction

Most methods for learning predictors from data are based on the closed-world assumption, i.e., the training and the test samples are generated i.i.d. from the same distribution, so-called in-distribution (ID). However, in real-world applications, ID test samples can be contaminated by samples from another distribution, the so-called Out-of-Distribution (OOD), which is not represented in training examples. A trustworthy prediction model should detect OOD samples and reject to predict them, while simultaneously minimizing the prediction error on accepted ID samples.

In recent years, the development of deep learning models for handling OOD data has emerged as a 24 critical challenge in the field of machine learning, leading to an explosion of research papers dedicated 25 to developing effective OOD detection methods (OODD) [10, 11, 4, 3, 12, 8, 1, 17, 16, 19, 20]. 26 Existing methods use various principles to learn a classifier of ID samples and a selective function 27 that accepts the input for prediction or rejects it to predict. We further denote the pair of ID classifier 28 and the selective function as OOD selective classifier, borrowing terminology from the non-OOD 29 setup [7]. There is an agreement that a good OOD selective classifier should reject OOD samples 30 and simultaneously achieve high classification accuracy on ID samples that are accepted [22]. To 31 our knowledge, there is surprisingly no formal definition of an optimal OOD selective classifier. 32 Consequently, there is also no consensus on how to evaluate the OODD methods. The commonly used 33 metrics [21] evaluate only one aspect of the OOD selective classifier, either the accuracy of the ID 34 classifier or the performance of the selective function as an OOD/ID discriminator. Such evaluation 35 is inconclusive and usually inconsistent; e.g., the two most commonly used metrics, AUROC and 36 OSCR, often lead to a completely reversed ranking of evaluated methods (see Sec. 3.4). 37

In this paper, we ask the following question: What would be the optimal prediction strategy for 38 the OOD setup in the ideal case when ID and OOD distributions were known? To this end, we 39 offer the contributions: (i) We propose three reject option models for the OOD setup: Cost-based 40 model, bounded TPR-FPR model, and Bounded Precision-Recall model. These models extend the 41 standard rejection models used in the non-OOD setup [2, 15] and define the notion of an optimal OOD 42 classifier. (ii) We establish that all the proposed models, despite their different formulations, share 43 a common class of optimal strategies. The optimal OOD selective classifier combines a Bayes ID 44 classifier with a selective function based on a linear combination of the conditional risk and likelihood 45 ratio of the OOD and ID samples. This selective function enables a trade-off between distinguishing 46 ID from OOD samples and detecting misclassifications. (iii) Motivated by the optimal strategy, 47 we introduce double-score OOD methods that leverage uncertainty scores from two chosen OOD 48 detectors: one focused on OOD/ID discrimination and the other on misclassification detection. We 49 show experimentally that this simple strategy consistently outperforms the state-of-the-art. (iv) We 50 review existing metrics for evaluation of OODD methods and show that they provide incomplete 51 view, if used separately, or inconsistent view of the evaluated methods, if used together. We propose 52 novel evaluation metrics derived from the definition of optimal strategy under the proposed OOD 53 rejection models. These new metrics provide a comprehensive and reliable assessment of OODD 54 methods without the deficiencies observed in existing approaches. 55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

2 Reject option models for OOD setup

The terminology of ID and OOD samples comes from the setups when the training set contains only ID samples, while the test set contains a mixture of ID and OOD samples. In this paper, we analyze which prediction strategies are optimal on the test samples, but we do not address the problem of learning such strategy. We follow the OOD setup from [5]. Let X be a set of observable inputs (or features), and Y a finite set of labels that can be assigned to in-distribution (ID) inputs. ID samples ( x, y ) ∈ X × Y are generated from a joint distribution p I : X ×Y → R + . Out-of-distribution (OOD) samples x are generated from a distribution p O : X → R + . ID and OOD samples share the same input space X . Let ∅ be a special label to mark the OOD sample. Let ¯ Y = Y ∪ {∅} be an extended set of labels. In the testing stage the samples ( x, ¯ y ) ∈ X × ¯ Y are generated from the joint distribution p : X × ¯ Y → R + defined as a mixture of ID and OOD:

where π ∈ [0 , 1) is the probability of observing the OOD sample. Our OOD setup subsumes the standard non-OOD setup as a special case when π = 0 , and the reject option models that will be introduced below will become for π = 0 the known reject option models for the non-OOD setup.

Our goal is to design OOD selective classifier q : X → D , where D = Y ∪ { reject } , which either predicts a label, q ( x ) ∈ Y , or it rejects the prediction, q ( x ) = reject , when (i) input x ∈ X prevents accurate prediction of y ∈ Y because it is noisy, or (ii) comes from OOD. We represent the selective classifier by the ID classifier h : X → Y , and a stochastic selective function c : X → [0 , 1] that outputs a probability that the input is accepted [7], i.e.,

In the following sections, we propose three reject option models that define the notion of the optimal OOD selective classifier of the form (2) applied to samples generated by (1).

2.1 Cost-based rejection model for OOD setup

A classical approach to define an optimal classifier is to formulate it as a loss minimization problem. 78 This requires defining a loss ¯ ℓ : ¯ Y × D → R + for each combination of the label ¯ y ∈ ¯ Y = Y ∪ {∅} 79 and the output of the classifier q ( x ) ∈ D = Y ∪ { reject } . Let ℓ : Y × Y → R + be some application80 specific loss on ID samples, e.g., 0/1-loss or MAE. Furthermore, we need to define the loss for the 81 case where the input is OOD sample ¯ y = ∅ or the classifier rejects q ( x ) = reject . Let ε 1 ∈ R + be 82 the loss for rejecting the ID sample, ε 2 ∈ R + loss for prediction on the OOD sample, and ε 3 ∈ R + 83 loss for correctly rejecting the OOD sample. ℓ , ε 1 , ε 2 and ε 3 can be arbitrary, but we assume that 84

ε 2 > ε 3 . The loss ¯ ℓ is then: 85

Having the loss ¯ ℓ , we can define the optimal OOD selective classifier as a minimizer of the expected 86 risk R ( h, c ) = E x,y ∼ p ( x, ¯ y ) ¯ ℓ (¯ y, ( h, c )( x )) . 87

88

89

Definition 1 (Cost-based OOD model) An optimal OOD selective classifier ( h C , c C ) is a solution to the minimization problem min h,c R ( h, c ) where we assume that both minimizers exist.

An optimal solution of the cost-based OOD model requires three components: The Bayes ID classifier 90

its conditional risk r B ( x ) = ∑ y ∈Y p I ( y | x ) ℓ ( y, h B ( x )) , and the likelihood ratio of the OOD and ID inputs, g ( x ) = p O ( x ) p I ( x ) , which we defined to be g ( x ) = ∞ for p I ( x ) = 0 .

Theorem 1 An optimal selective classifier ( h C , c C ) under the cost-based OOD model is composed of the Bayes classifier (4), h C = h B , and the selective function

where τ is an arbitrary number in [0 , 1] , and ε 1 , ε 2 , ε 3 are losses defining the extended loss (3).

Note that τ can be arbitrary and therefore a deterministic selective function c C ( x ) = [ [ s C ( x ) ≤ ε 1 ] ] is also optimal. An optimal selective function accepts inputs based on the score s C ( x ) , which is a linear combination of two functions, conditional risk r B ( x ) and the likelihood ratio g ( x ) = p O ( x ) /p I ( x ) .

Relation to cost-based model for Non-OOD setup For π = 0 , the cost-based OOD model reduces to the standard cost-based model of the reject option classifier in a non-OOD setup [2]. In the non-OOD setup, we do not need to specify the losses ε 2 and ε 3 and the risk R ( h, c ) simplifies to R ′ ( h, c ) = E x,y ∼ p I ( x,y ) [ ℓ ( y, h ( x )) c ( x ) + ε 1 (1 -c ( x )) ] . The well-known optimal solution is composed of the Bayes classifier h B ( x ) as in the OOD case; however, the selection function c ′ C ( x ) = [ [ r ( x ) ≤ A ] ] accepts the input solely based on the conditional risk r ( x ) .

2.2 Bounded TPR-FPR rejection model

The cost-based OOD model requires the classification loss ℓ for ID samples and defining the costs ε 1 , ε 2 , ε 3 which is difficult in practice because the physical units of ℓ and ε 1 , ε 2 , ε 3 are often different. In this section, we propose an alternative approach which requires only the classification loss ℓ while costs ε 1 , ε 2 , ε 3 are replaced by constraints on the performance of the selective function.

The selective function c : X → [0 , 1] can be seen as a discriminator of OOD/ID samples. Let us consider ID and OOD samples as positive and negative classes, respectively. We introduce three metrics to measure the performance of the OOD selective classifier ( h, c ) . We measure the performance of selective function by the True Positive Rate (TPR) and the False Positive Rate (FPR). The TPR is defined as the probability that ID sample is accepted by the selective function c , i.e.,

̸

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

The FPR is defined as the probability that OOD sample is accepted by the selective function c , i.e., 115

The second identity in (6) and (7) is obtained after substituting the definition of p ( x, ¯ y ) from (1). 116 Lastly, we characterize the performance of the ID classifier h : X → Y by the selective risk 117

defined for non-zero ϕ ( c ) , i.e., the expected loss of the classifier h calculated on the ID samples accepted by the selective function c .

Definition 2 (Bounded TPR-FPR model) Let ϕ min ∈ [0 , 1] be the minimal acceptable TPR and ρ max ∈ [0 , 1] maximal acceptable FPR. An optimal OOD selective classifier ( h T , c T ) under the bounded TPR-FPR model is a solution of the problem

where we assume that both minimizers exist.

Theorem 2 Let ( h, c ) be an optimal solution to (8). Then ( h B , c ) , where h B is the Bayes ID classifier (4), is also optimal to (8).

According to Theorem 2, the Bayes ID classifier h B is an optimal solution to (8) that defines the bounded TPR-FPR model. This is not surprising, but it is a practically useful result, because it allows one to solve (8) in two consecutive steps: First, set h T to the Bayes ID classifier h B . Second, when h T is fixed, the optimal selection function c T is obtained by solving (8) only w.r.t. c which boils down to:

Problem 1 (Bounded TPR-FPR model for known h ( x ) ) Given ID classifier h : X → Y , the optimal selective function c ∗ : X → [0 , 1] is a solution to

Problem 1 is meaningful even if h is not the Bayes ID classifier h B . We can search for an optimal selective function c ∗ ( x ) for any fixed h , which in practice is usually our best approximation of h B learned from the data.

Theorem 3 Let h : X → Y be ID classifier and r : X → R its conditional risk r ( x ) = ∑ y ∈Y p I ( y | x ) ℓ ( y, h ( x )) . Let g ( x ) = p I ( x ) /p I ( x ) be the likelihood ratio of ID and OOD samples. Then, the set of optimal solutions of Problem 1 contains the selective classifier

where decision threshold λ ∈ R , and multiplier µ ∈ R are constants and τ : X → [0 , 1] is a function implicitly defined by the problem parameters.

The optimal c ∗ ( x ) is based on the score composed of a linear combination of r ( x ) and g ( x ) as in the case of the cost-based model (5). Unlike the cost-based model, the acceptance probability τ ( x ) for boundary inputs X s ( x )= λ = { x ∈ X | s ( x ) = λ } cannot be arbitrary, in general. However, if X is continuous, the set X s ( x )= λ has probability measure zero, up to some pathological cases, and τ ( x ) can be arbitrary, i.e., the deterministic c ∗ ( x ) = [ [ s ( x ) ≤ λ ] ] is optimal. If X is finite, the value of τ ( x ) can be found by linear programming. The linear program and more details on the form of τ ( x ) are in the Appendix.

Relation to Bounded-Abstention model for the non-OOD setup For π = 0 , the bounded TPRFPR model reduces to the bounded-abstention option model for non-OOD setup [15]. Namely, ρ ( c ) ≤ ρ max can be removed because there are no OOD samples, and (8) becomes the boundedabstention model: min h,c R S ( h, c ) , s.t. ϕ ( c ) ≥ ϕ min , which seeks the selective classifier with guaranteed TPR and minimal selective risk. In the non-OOD setup, TPR is called coverage . An optimal solution of the bounded abstention model [6], is composed of the Bayes ID classifier h B , and the same optimal selective function as the TPR-FPR model (9), however, with µ = 0 and τ ( x ) = τ , ∀ x ∈ X , i.e., the score depends only on r ( x ) and an identical randomization is applied in all edge cases [6]. Therefore, r ( x ) is the optimal score to detect misclassified ID samples in non-OOD setup as it allows to achieve the minimal selective risk R S for any fixed coverage (TPR, ϕ ).

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

2.3 Bounded Precision-Recall rejection model

The optimal selective classifier under the bounded TPR-FPR model does not depend on the prior of the OOD samples π , which is useful, e.g., when π is unknown in the testing stage. In the case π is known, it might be more suitable to constrain the precision rather than the FPR, while the constraint on TPR remains the same. In the context of precision, we denote ϕ ( c ) as recall instead of TPR. The precision κ ( c ) is defined as the portion of samples accepted by c ( x ) that are actual ID samples, i.e.,

̸

Definition 3 (Bounded Precision-Recall model) Let κ min ∈ [0 , 1] be a minimal acceptable precision and ϕ min ∈ [0 , 1] minimal acceptable recall (a.k.a. TPR). An optimal selective classifier ( h P , c P ) under the bounded Precision-Recall model is a solution of the problem

where we assume that both minimizers exist.

Theorem 4 Let ( h, c ) be an optimal solution to (10). Then ( h B , c ) , where h B is the Bayes ID classifier (4), is also optimal to (10).

Theorem 4 ensures that the Bayes ID classifier is an optimal solution to (10). After fixing h P = h B , the search for an optimal selective function c leads to:

Problem 2 (Bounded Prec-Recall model for known h ( x ) ) Given ID classifier h : X → Y , the optimal selective function c ∗ : X → [0 , 1] is a solution to

Theorem 5 Let h : X → Y be ID classifier and r : X → R its conditional risk r ( x ) = ∑ y ∈Y p I ( y | x ) ℓ ( y, h ( x )) . Let g ( x ) = p O ( x ) /p I ( x ) be the likelihood ratio of OOD and ID samples. Then, the set of optimal solutions of Problem 2 contains the selective function

where detection threhold λ ∈ R , and multiplier µ ∈ R are constants and τ : X → [0 , 1] is a function implicitly defined by the problem parameters.

2.4 Summary

We proposed three rejection models for OOD setup which define the notion of optimal OOD selective classifier: Cost-based model, Bounded TRP-FPR model, and Bounded Precision-Recall model. We established that all three models, despite different formulation, share the class of optimal prediction strategies. Namely, the optimal OOD selective classifier ( h ∗ , c ∗ ) is composed of the Bayes ID classifier (4), h ∗ = h B , and the selective function

where λ , µ , and τ ( x ) are specific for the used rejection model. However, in all cases, the optimal 185 uncertainty score s ( x ) for accepting the inputs is based on a linear combination of the conditional 186 risk r ( x ) of the ID classifier h ∗ and the OOD/ID likelihood ratio g ( x ) = p O ( x ) /p I ( x ) . On the other 187 hand, from the optimal solution of the well-known Neyman-Person problem [14], it follows that the 188 likelihood ratio g ( x ) is the optimal score of OOD/ID discrimination. Our results thus show that the 189 optimal OOD selective function needs to trade-off the ability to detect the misclassification of ID 190 samples and the ability to distinguish ID from OOD samples. 191

192

193

194

195

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

232

233

234

235

236

237

Single-score vs. double-score OODD methods The existing OODD methods, which we further call single-score methods , produce a classifier h : X → Y and an uncertainty score s : X → R . The score s ( x ) is used to construct a selective function c ( x ) = [ [ s ( x ) ≤ λ ] ] where λ ∈ R is a decision threshold chosen in post-hoc evaluation. Hence, the existing methods effectively produce a set of selective classifiers Q = { ( h, c ) | c ( x ) = [ [ s ( x ) ≤ λ ] ] , λ ∈ R } . In contrast to existing methods, we established that the optimal selective function is always based on a linear combination of two scores: conditional risk r ( x ) and likelihood ratio g ( x ) . Therefore, we propose the double-score method , which in addition to a classifier h ( x ) , produces two scores, s r : X → R and s g : X → R , and uses their combination s ( x ) = s r ( x ) + µs g ( x ) to accept inputs. Formally, the double-score method produces a set of selective classifiers Q = { ( h, c ) | c ( x ) = [ [ s r ( x ) + µs g ( x ) ≤ λ ] ] , µ ∈ R , λ ∈ R } . The double-score strategy can be used to leverage uncertainty scores from two chosen OODD methods: one focused on OOD/ID discrimination and the other on misclassification detection.

3 Post-hoc tuning and evaluation metrics

Let T = (( x i , ¯ y i ) ∈ X × ¯ Y | i = 1 , . . . , n ) be a set of validation examples i.i.d. drawn from a distribution p ( x, ¯ y ) . Given a set of selective classifiers Q , trained by the single-score or double-score OODD method, the goal of the post-hoc tuning is to use T to select the best selective classifier ( h n , c n ) ∈ Q and estimate its performance on unseen samples generated from the same p ( x, ¯ y ) . This task requires a notion of an optimal selective classifier which we defined by the proposed rejection models. In Sec 3.2 and Sec 3.3, we propose the post-hoc tuning and evaluation metrics for the Bounded TPR-FPR and Bounded Precision-Recall models, respectively. In Sec 3.4 we review the existing evaluation metrics for OODD methods and point out their deficiencies. We will exemplify the proposed metrics on synthetic data and OODD methods described in Sec 3.1.

3.1 Synthetic data and exemplar single-score and double-score OODD methods

̸

Let us consider a simple 1-D setup. The input space is X = R and there are three ID labels Y = { 1 , 2 , 3 } . ID samples are generated from p I ( x, 1) = 0 . 3 N ( x ; -1 , 1) , p I ( x, 2) = 0 . 3 N ( x ; 1 , 1) , p I ( x, 3) = 0 . 4 N ( x ; 3 , 1) , where N ( x ; µ, σ ) is normal distribution with mean µ and variance σ . OOD is the normal distribution p O ( x ) = N ( x ; 3 , 0 . 2) , and the OOD prior π = 0 . 25 . We use 0 / 1 -loss ℓ ( y, y ′ ) = [ [ y = y ′ ] ] , i.e., R S is the classification error on accepted inputs. The known ID and OOD alows us to evaluate the Bayes ID classifier h B ( x ) by (4), its conditional risk r B ( x ) = min y ′ ∈Y ∑ y ∈Y p I ( y | x ) ℓ ( y, y ′ ) and the OOD/ID likelihood ratio g ( x ) = p O ( x ) /p I ( x ) .

We consider 3 exemplar single-score OODD methods A, B, C. The methods produce the same optimal classifier h ∗ ( x ) and the selective functions c ( x ) = [ [ r B ( x ) + µg ( x ) ≤ λ ] ] with a different setting of µ . I.e., the method k ∈ { A,B,C } produces the set of selective classifiers Q k = { ( h ∗ ( x ) , c ( x )) | c ( x ) = [ [ r B ( x ) + µ k g ( x ) ≤ λ ] ] , λ ∈ R } , where the constant µ k is defined as follows:

Method A( ∞ ): µ = ∞ , s ( x ) = g ( x ) . This corresponds to the optimal OOD/ID discriminator.

Method B( 0 . 2 ): µ = 0 . 2 , s ( x ) = r B ( x ) + 0 . 2 g ( x ) . Combination of method A and C.

Method C( 0 ): µ = 0 , s ( x ) = r B ( x ) . This corresponds to the optimal misclassification detector.

We also consider a double-score method, Method D( R ), which outputs the same optimal classifier h ∗ ( x ) , and scores s r ( x ) = r ( x ) and s g ( x ) = g ( x ) . I.e., Method D( R ) produces the set of selective classifiers Q D = { ( h ∗ ( x ) , c ( x )) | c ( x ) = [ [ r ( x ) + µg ( x ) ≤ λ ] ] , µ ∈ R , λ ∈ R } . Note that we have shown that Q D contains an optimal selective classifier regardless of the reject option model used.

3.2 Bounded TPR-FPR rejection model

The bounded TPR-FPR model is defined using the selective risk R S ( h, c ) , TPR ϕ ( c ) and FPR ρ ( c ) the value of which can be estimated from the validation set T as follows:

̸

where I I = { i ∈ { 1 , . . . , n } | ¯ y i = ∅} and I O = { i ∈ { 1 , . . . , n } | ¯ y i = ∅} are indices of ID and OOD samples in T , respectively.

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

Table 1: Evalution of the examplar single-score methods A, B, C and the proposed double-score method D on synthetic data using the proposed metrics and the existing ones. The selective risk correponds to the classification error on accepted ID samples.

Given the target TPR ϕ min ∈ (0 , 1] and FPR ρ max ∈ (0 , 1] , the best selective classifier ( h n , c n ) out of Q is found by solving:

Proposed evaluation metric If problem (13) is feasible, R S n ( h n , c n ) is reported as the performance estimator of OODD method producing Q . Otherwise, the method is marked as unable to achieve the target TPR and FPR. Tab. 1 shows the selective risk for the methods A-D at the target TPR ϕ min = 0 . 7 and FPR ρ max = 0 . 2 . The minimal R S n is achieved by method D( R ), followed by B(0.2) and A( ∞ ), while C(0) is unable to achieve the target TPR and FPR. One can visualize R S n in a range of operating points while bounding only ρ max or ϕ min . E.g., by fixing ρ max we can plot R S n as a function of attainable values of ϕ n by which we obtain the Risk-Coverage curve, known from non-OOD setup, at ρ max . Recall that TPR is coverage. See Appendix for Risk-Coverage curve at ρ max for methods A-D.

ROC curve The problem (13) can be infeasible. To choose a feasible target on ϕ min and ρ max , it is advantageous to plot the ROC curve, i.e., values of TPR and FPR attainable by the classifiers in Q . For single-score methods, the ROC curve is a set of points obtained by varying the decision threshold: ROC( Q ) = { ( ϕ n ( h, c ) , ρ n ( h, c )) | c ( x ) = [ [ s ( x ) ≤ λ ] ] , λ ∈ R } . In case of double-score methods, we vary ρ max ∈ [0 , 1] and for each ρ max we choose the maximal feasible ϕ n . I.e., ROC curve is ROC( Q ) = { ( ϕ, ρ max ) | ϕ = max ( h,c ) ∈Q ϕ n ( h, c ) s.t. ρ n ( h, c ) ≤ ρ max , ρ max ∈ [0 , 1] } . See Appendix for ROC curve of the methods A-D. In Tab. 1 we report the Area Under ROC curve (AUROC) which is a commonly used summary of the entire ROC curve. The highest AUROC achieved Methods A( ∞ ) and E( R ). Recall that Method A( ∞ ) uses the optimal ID/OOD discriminator and the proposed Method E( R ) subsumes A( ∞ ).

3.3 Bounded Precision-Recall rejection model

Let κ n ( c ) = (1 -π ) ϕ n ( c ) / ((1 -π ) ϕ n ( c ) + πρ n ( c )) be the sample precision of the selective function c . Given the target recall ϕ min ∈ (0 , 1] and precision κ min ∈ (0 , 1] , the best selective classifier ( h n , c n ) out of Q is found by solving

Proposed evaluation metric If problem (14) is feasible, R S n ( h n , c n ) is reported as the performance estimator of OODD method which produced Q . Otherwise, the method is marked as unable to achieve the target Precison/Recall. Tab. 1 shows the selective risk for the methods A-D at the Precision κ min = 0 . 9 and recall ϕ max = 0 . 7 . The minimal R S n is achieved by the proposed method D( R ), followed by B(0.2) and A( ∞ ), while method C(0) is unable to achieve the target Precision/Recall. Note that single-score methods A-C achieve the same R S n under both TPR-FPR and Prec-Recall models while the results for double-score method D( R ) differ. The reason is that both models share the same constraint ϕ n ≥ 0 . 7 (TPR is Recall) which is active, while the other two constraints are not active because R S n is a monotonic function w.r.t. the value of the decision threshold.

272

273

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

318

319

320

321

322

Precision-Recall (PR) curve To choose feasible bounds on κ min and ϕ min before solving (14), one can plot the PR curve, i.e., the values of precision and recall attainable by the classifiers in Q . For single-score methods, the PR curve is a set of points obtained by varying the decision threshold: PR( Q ) = { ( κ n ( h, c ) , ϕ n ( h, c )) | c ( x ) = [[ s ( x ) ≤ λ ] ] , λ ∈ R } . In case of doublescore methods, we vary ϕ min ∈ [0 , 1] and for each ϕ min we choose the maximal feasible κ n , i.e., PR( Q ) = { ( κ, ϕ min ) | κ = max ( h,c ) ∈Q κ n ( h, c ) s.t. ϕ n ( h, c ) ≥ ϕ min , ϕ min ∈ [0 , 1] } . See Appendix for PR curve of the methods A-D. We compute the Area Under the PR curve and report it for Methods A-D in Tab. 1. Rankings of the methods w.r.t AUPR and AUROC are the same.

3.4 Shortcomings of existing evaluation metrics

The most commonly used metrics to evaluate OODD methods are the AUROC and AUPR [10, 13, 3, 12, 1, 16]. Both metrics measure the ability of the selective function c ( x ) to distinguish ID from OOD samples. AUROC and AUPR are often the only metrics reported although they completely ignore the performance of the ID classifier. Our synthetic example shows that high AUROC/AUPR is not a precursor of a good OOD selective classifier. E.g., Method A( ∞ ), using optimal OOD/ID discriminator, attains the highest (best) AUROC and AUPR (see Tab. 1), however, at the same time Method A( ∞ ) achieves the highest (worst) R S n under both rejection models, and it is also the worst misclassification detector according to the OSCR score defined below.

The performance of the ID classifier h ( x ) is usually evaluated by the ID classification accuracy (a.k.a. closed set accuracy) [13, 3] and by the OSCR score [4, 8, 1]. The ID accuracy measures the performance of h ( x ) assuming all inputs are accepted, i.e., c ( x ) = 1 , ∀ x ∈ X , hence it says nothing about the performance on the actually accepted samples like R S n . E.g., Methods A-D in our synthetic example use the same classifier h ( x ) and hence have the same ID accuracy, however, they perform quite differently in terms of the other more relevant metrics, like R S n or OSCR. The OSCR score is defined as the area under CCR versus FPR curve [21], where the CCR stands for the correct classification rate on the accepted ID samples; in case of 0/1-loss CCR = 1 -R S n . The CCR-FPR curve evaluates the performance of the ID classifier on the accepted samples, but it ignores the ability of c ( x ) to discriminate OOD and ID samples as it does not depend on TPR. E.g., Method D(0), using the optimal misclassification detector, achieves the highest (best) OSCR score; however, at the same time, it has the lowest (worst) AUROC and AUPR.

Other, less frequently used metrics involve: F1-score, FPR @ TPRx, TNR @ TPRx, CCR @ FPRx [10, 8, 1, 21, 16]. All these metrics are derived from either ROC, PR or CCR-FPR curve, and hence they suffer with the same conceptual problems as AUROC, AUPR and OSCR, respectively.

We argue that the existing metrics evaluate only one aspect of the OOD selective classifier, namely, either the ability to disciminate ID from OOD samples, or the performance of ID classifier on the accepted (or on possibly all) ID samples. We show that in principle there can be methods that are best OOD/ID discriminators but the worst misclassification detectors and vice versa. Therefore, using individual metrics can (and often does) provide inconsistent ranking of the evaluated methods.

3.5 Summary

We propose novel evaluation metrics derived from the definition of the optimal strategy under the proposed OOD rejection models. The proposed metrics simultaneously evaluate the classification performance on the accepted ID samples and they guarantee the perfomance of the OOD/ID discriminator, either via constraints in TPR-FPR or Precision-Recall pair. Advantages of the proposed metrics come at a price. Namely, we need to specify feasible target TPR and FPR, or Precision and Recall, depending on the model used. However, feasible values of TPR-FPR and Prec-Recall pairs can be easily read out of the ROC and PR curve, respectively. We argue that setting these extra parameters is better than using the existing metrics that provide incomplete, if used separately, or inconsistent, if used in combination, view of the evaluated methods.

Another issue is solving the problems (13) and (14) to compute the proposed evaluation metrics and figures. Fortunately, both problems lead to optimization w.r.t one or two varibales in case of the single-score and double-score methods, respectively. A simple and efficient algorithm to solve the problems in O ( n log n ) time is provided in Appendix.

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

4 Experiments

In this section, we evaluate single-score OODD methods and the proposed double-score strategy, using the existing and the proposed evaluation metrics. We use MSP [10], MLS [9], ODIN [11] as baselines and REACT [17], KNN [19], VIM [20] as repesentatives of recent single-score approaches. We evaluate two instances of the double-score strategy. First, we combine the scores of MSP [10] and KNN [18] and, second, scores of MSP and VIM [20]. MSP score is asymptotically the best misclassification detector, while KNN and VIM are two best OOD/ID discriminators according to their AUROC. We always use the ID classifier of the MSP method. The evaluation data and implementations of OODD methods are taken from OpenOOD benchmark [21]. Because the datasets have unrealistically high portion of OOD samples, e.g., π > 0 . 5 , we use metrics that do not depend on π . Namely, AUROC and OSCR as the most frequently used metrics, and the proposed selective risk at TPR and FPR. We use 0/1-loss, hence the reported selective risk is the classification error on accepted ID samples with guranteed TPR and FPR. In all experiments we fix the target TPR to 0.8 while FPR is set for each database to the highest FPR attained by all compared methods.

Results are presented in Tab. 2. It is seen that the single-score methods with the highest AUROC and OSCR are always different, which prevents us to create a single conclusive ranking of the evaluated approaches. MSP is almost consistently the best misclassification detector according to OSCR. The best OOD/ID discriminator is, according to AUROC, one of the recent methods: REACT, KNN, or VIM. The proposed double-score strategy, KNN+MSP and VIM+MSP, consistently outperforms the other approaches in all metrics.

5 Conclusions

This paper introduces novel reject option models which define the notion of the optimal prediction 344 strategy for OOD setups. We prove that all models, despite their different formulations, share the 345 same class of optimal prediction strategies. The main insight is that the optimal prediction strategy 346 must trade-off the ability to detect misclassified examples and to distinguish ID from OOD samples. 347 This is in contrast to existing OOD methods that output a single uncertainty score. We propose a 348 simple and effective double-score strategy that allows us to boost performance of two existing OOD 349 methods by combining their uncertainty scores. Finally, we suggest improved evaluation metrics 350 for assessing OOD methods that simultaneously evaluate all aspects of the OOD methods and are 351 directly related to the optimal OOD strategy under the proposed reject option models. 352

353

354

355

356

357

358

359

360

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

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

References

Guangyao Chen, Peixi Peng, Xiangqian Wang, and Yonghong Tian. Adversarial reciprocal points learning for open set recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence , 44(11):80658081, 2022.

C. Chow. On optimum recognition error and reject tradeoff. IEEE Transactions on Information Theory , 16(1):41-46, 1970.

Terrance DeVries and Graham W Taylor. Learning confidence for out-of-distribution detection in neural networks. arXiv preprint arXiv:1802.04865 , 2018.

Akshay Raj Dhamija, Manuel Günther, and Terrance Boult. Reducing network agnostophobia. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 31. Curran Associates, Inc., 2018.

Zhen Fang, Yixuan Li, Jie Lu, Jiahua Dong, Bo Han, and Feng Liu. Is out-of-distribution detection learnable? In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems , volume 35, pages 37199-37213. Curran Associates, Inc., 2022.

Vojtech Franc, Daniel Prusa, and Vaclav Voracek. Optimal strategies for reject option classifiers. Journal of Machine Learning Research , 24(11):1-49, 2023.

Y. Geifman and R. El-Yaniv. Selective classification for deep neural networks. In Advances in Neural Information Processing Systems 30 , pages 4878-4887, 2017.

Federica Granese, Marco Romanelli, Daniele Gorla, Catuscia Palamidessi, and Pablo Piantanida. Doctor: A simple method for detecting misclassification errors. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems , volume 34, pages 5669-5681. Curran Associates, Inc., 2021.

Dan Hendrycks, Steven Basart, Mantas Mazeika, Andy Zou, Joseph Kwon, Mohammadreza Mostajabi, Jacob Steinhardt, and Dawn Song. Scaling out-of-distribution detection for real-world settings. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research , pages 8759-8773. PMLR, 17-23 Jul 2022.

Dan Hendrycks and Kevin Gimpel. A baseline for detecting misclassified and out-of-distribution examples in neural networks. In Proceedings of International Conference on Learning Representations , 2017.

Shiyu Liang, Yixuan Li, and R. Srikant. Enhancing the reliability of out-of-distribution image detection in neural networks. In International Conference on Learning Representations , 2018.

Andrey Malinin and Mark Gales. Predictive uncertainty estimation via prior networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 31. Curran Associates, Inc., 2018.

Lawrence Neal, Matthew Olson, Xiaoli Fern, Weng-Keen Wong, and Fuxin Li. Open set learning with counterfactual images. In Vittorio Ferrari, Martial Hebert, Cristian Sminchisescu, and Yair Weiss, editors, Computer Vision - ECCV 2018 , pages 620-635, Cham, 2018. Springer International Publishing.

Jerzy Neyman and Egon Person. On the use and interpretation of certain test criteria for purpose of statistical inference. Biometrica , pages 175-240, 1928.

T. Pietraszek. Optimizing abstaining classifiers using ROC analysis. In Proceedings of the 22nd International Conference on Machine Learning , page 665-672, 2005.

Yue Song, Nicu Sebe, and Wei Wang. Rankfeat: Rank-1 feature removal for out-of-distribution detection. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems , volume 35, pages 17885-17898. Curran Associates, Inc., 2022.

Yiyou Sun, Chuan Guo, and Yixuan Li. React: Out-of-distribution detection with rectified activations. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems , 2021.

Yiyou Sun, Chuan Guo, and Yixuan Li. React: Out-of-distribution detection with rectified activations. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems , volume 34, pages 144-157. Curran Associates, Inc., 2021.

403

404

405

406

407

408

409

410

411

412

413

414

415

Yiyou Sun, Yifei Ming, Xiaojin Zhu, and Yixuan Li. Out-of-distribution detection with deep nearest neighbors. In Kamalika Chaudhuri, Stefanie Jegelka, Le Song, Csaba Szepesvari, Gang Niu, and Sivan Sabato, editors, Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research , pages 20827-20840. PMLR, 17-23 Jul 2022.

Haoqi Wang, Zhizhong Li, Litong Feng, and Wayne Zhang. Vim: Out-of-distribution with virtual-logit matching. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) , pages 4911-4920, 2022.

Jingkang Yang, Pengyun Wang, Dejian Zou, Zitang Zhou, Kunyuan Ding, Wenxuan Peng, Haoqi Wang, Guangyao Chen, Bo Li, Yiyou Sun, Xuefeng Du, Kaiyang Zhou, Wayne Zhang, Dan Hendrycks, Yixuan Li, and Ziwei Liu. Openood: Benchmarking generalized out-of-distribution detection. In Conference on Neural Information Processing Systems (NeurIPS 2022) Track on Datasets and Benchmar , 2022.

Jingkang Yang, Kaiyang Zhou, Yixuan Li, and Ziwei Liu. Generalized out-of-distribution detection: A survey, 2022.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider 3 rejection models in the OOD detection setting and establish theorems about the optimal rules for these models. Due to the commonly used metrics that may partially only focus on either OOD detection or misclassification, they proposed a double-score method to consider both aspects.\n\nSTRENGTHS:\n* The authors extend the rejection in the non-OOD setting to the OOD detection setting.\n* The 3 different rejection models share similar optimal strategies, and those strategies outperform other baselines.\n* Since previous works only focus on either OOD detection or misclassification, the authors proposed a novel double-score method to consider both aspects.\n\nWEAKNESSES:\n* Now that we know all the distributions, including OOD and ID, why put effort into eq 3? Can’t we just treat augmented data of OOD + ID as the new collection with $K + 1$ \"inlier\" classes, then follow the regular rejection analysis under the closed-world assumption?\n\n* The optimal rules heavily rely on the clear information (the distribution in the theorem or estimated from the sample) on OOD data. For example, the prior $\\pi$ in the bounded precision-recall rejection model, the conditional probability (of OOD) in the bounded TPR-FPR rejection model, or even both in the cost-based model. However, it is challenging to access OOD data and estimate these probabilities in real-world applications within an open-world setting. This raises concerns about the practical guidance provided by these optimal rules.\n\n* I have a reservation about the constraints in equation 8 or the setup of problem 1. In practice, it is more likely the threshold or the selective function is determined by only one constraint (even in your proof of theorem 2, there is nowhere about the constraint $\\rho(c)$). For example, once the $\\lambda$ is determined by one constraint, it does not necessarily satisfy another one, which is similar to the trade-off between the type I and type II errors or Neyman-Pearson classification. A similar argument applies to Problem 2.\n\n* Optimal rules: the authors claim the optimal rules related to the Bayes classifier (lines 126, 170), but from theorems, it holds under the condition that the given $(h, c)$ is optimal. If we cannot find the optimal $h$, can we still say the Bayes classifier is optimal?\n\nQUESTIONS:\n* Line 137: typos in \"Let $g(x)$ …. samples\" $\\rightarrow$ $p_O(x)/p_I(x)$... ratio of OOD and ID samples.\n\n* Theorem 3: Do you mean any $h$ is optimal even if it is not the Bayes classifier? The optimality looks like is conditional on the given $h$, how could you guarantee the picked $h$ associated with the corresponding $c$ simultaneously achieve to the minimizer?\n\n* Line 157: What does this notation mean: (TPR, $\\phi$)?\n\n* Eq 12: Since $h^*=h_B$, do you mean $r_B(x)$ instead of $r(x)$?\n\n* Line 301 (FPR@TPRx, TNR@TPRx, CCR@FPRx): These metrics also control one metric @ percentage x and then compared to another one, depending on their definitions of positive/negative for the ID and OOD, which is similar to the idea of your proposed problem 1/2, like controlling two metrics (actually can only achieve one constraint in general) and compare the selective risk.\n\n* Line 441: Could you explain why $\\min_{q\\in \\mathcal{Y}}R_x(q)$ is in this form (are you saying the smallest risk is always achieved by making predictions even for OOD data?), while later you use $r(x)$ instead of $r_B(x)$ in $\\min_{q\\in \\mathcal{Y}}R_x(q)$ (1st equality below line 443). \n\n* Line 449: You miss the subscript $I$ under $p$.\n\nLIMITATIONS:\nSee my concerns and questions above.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides a unified viewpoint of reject model evaluation metrics under OOD-ness. Despite different performance metrics being used in practice, they can all be factorized into both the misclassification prediction on ID datapoints as well as the discrimination performance between ID and OOD. The experimental section provides some evidence that this proposed two score decomposition can provide competitive OOD detection and input rejection performance.\n\nSTRENGTHS:\n- OOD detection and classifying with a reject option are both timely and important issues in trustworthy machine learning. ML algorithms need to be aware of their limitations and signal if they are deployed on previously unseen data.\n- The unified framework / viewpoint makes sense and yields an interesting decomposition result showing that the optimal uncertainty score is a function of both misclassification prediction on ID datapoints as well as distinction performance between ID and OOD. Although this appears like an intuitive desiderata for a prediction uncertainty score, explicitly tying it back to the different rejection models seems new.\n\nWEAKNESSES:\n- The paper assumes that the OOD distribution is known. This is a strong assumption and severely limits the applicability of the approach in practical settings where knowing or explicitly estimating this distribution is often prohibitive.\n- Even though the paper does provide a nice unification of three performance measures for reject option models in an OOD setup (cost-based rejection, TPR-FPR rejection, precision-recall rejection), these metrics are not new and routinely used in existing applications. To me it seems like this kind of unified viewpoint would be better suited for a survey-style paper.\n- Section 3 could have been motivated better. To me, it appears a bit sudden without much justification or transition from the previous section.\n- Section 3.1 could have significantly profited from a figure showcasing the 1D Gaussian example for added intuition.\n- While the optimal uncertainty score is unified across error models, the experimental section showing the efficacy of the double score method is very lackluster. Details about models, training methods, and hyper-parameters area all missing (and also not documented in the appendix). Results are also based on a single run which makes it impossible to judge the statistical significance of the presented results. Moreover, the considered datasets do not all share the same sample shapes (e.g. MNIST vs CIFAR-10) which makes me wonder how OOD scores were obtained here in the first place. Real life shifts, like from the WILDS dataset collection would have been a better place to assess OOD-ness of samples. I am not convinced by this evaluation.\n\n### Post-rebuttal\n\nI have read the author's rebuttal and found that although many of my concerns were adequately addressed, the experimental documentation issue remained unaddressed. I increased my score only slightly as a result.\n\nQUESTIONS:\n- Can you add additional motivation for Section 3?\n- Can you provide more experimental details?\n\nLIMITATIONS:\nWeaknesses of the approach are not adequately discussed in the paper. See weaknesses and questions above.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes three reject option models and introduces double-score OOD methods that consistently outperform state-of-the-art methods. The authors also propose novel evaluation metrics for comprehensive and reliable assessment of OOD methods. The proposed metrics simultaneously evaluate the classification performance on the accepted ID samples and guarantee the performance of the OOD/ID discriminator, either via constraints in TPR-FPR or Precision-Recall pair. The authors argue that setting these extra parameters is better than using the existing metrics that provide incomplete if used separately, or inconsistent, if used in combination, view of the evaluated methods.  Overall, the paper's contributions provide a significant improvement in OOD detection and evaluation.\n\nSTRENGTHS:\nThe paper proposes three reject option models for OOD setups, which extend the standard reject option models. These models define the notion of an optimal OOD selective classifier and establish that all the proposed models, despite their different formulations, share a common class of optimal strategies. This is an original and creative approach to the problem of OOD detection, and the proposed models are well-motivated and clearly explained. The paper introduces double-score OOD methods that leverage uncertainty scores from two chosen OOD detectors: one focused on OOD/ID discrimination and the other on misclassification detection. The paper proposes novel evaluation metrics derived from the definition of the optimal strategy under the proposed OOD rejection models. These metrics provide a comprehensive and reliable assessment of OOD methods. \n\nOverall, the paper is well-written and easy to follow, with clear explanations of the proposed models, methods, and evaluation metrics.\n\nWEAKNESSES:\nOne weakness of the paper in the experimental results is that the dataset used in the experiments is relatively small, and the proposed methods do not show significant advantages over the baseline. This raises questions about the generalizability of the proposed methods to larger and more diverse datasets. Additionally, the paper could benefit from a more detailed analysis of the limitations of the proposed methods and how they can be improved in future work.\n\nQUESTIONS:\n1. The dataset used in the experiments of the paper is relatively small, and the proposed methods do not show significant advantages over the baseline in terms of experimental results. It would be helpful for the quality of the paper if the authors could provide more compelling experimental results or demonstrate the effectiveness of the proposed methods on larger datasets.\n\nLIMITATIONS:\nThe authors did not explicitly discuss the limitation of the proposed method.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

comments: SUMMARY:
This paper addresses out-of-distribution detection (ID/OOD discrimination) and misclassification detection (selection classification). The authors introduce double-score OOD methods that leverage uncertainty scores from OOD detector and misclassification detector. For evaluation metric, this paper proposes to use PR curve.

STRENGTHS:
(1) Existing works mainly focus on OOD detection, which this paper simultaneously considers OOD and selection classification. This is more practical and useful for high risk applications.

(2) Theoretical analysis of OOD detection and selective classification is valuable.

WEAKNESSES:
(1) The analysis of Bayes-optimal OOD selective classifier as well as the Bayes-optimal misclassification selective classifier has also been investigated in [1]. The authors are suggested to demonstrate the difference.

(2) The experiments are insufficient, only evaluating on cifar-10 and mnist. Results on CIFAR-100 is valuable.

(3) What is the advantage of PR curve over AURC (risk-coverage) curve [2] for evaluating rejection model?

[1] Narasimhan, H., Menon, A. K., Jitkrittum, W., & Kumar, S. (2023). Learning to reject meets OOD detection: Are all abstentions created equal?. arXiv preprint arXiv:2301.12386.

[2] Geifman, Y., & El-Yaniv, R. (2017). Selective classification for deep neural networks. Advances in neural information processing systems, 30.

QUESTIONS:
A recent work [3] also addresses the selective classification and OOD detection at the same time. it is worth mentioning it.

[3] Zhu, F., Cheng, Z., Zhang, X. Y., & Liu, C. L. (2023). OpenMix: Exploring Outlier Samples for Misclassification Detection. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 12074-12083).

LIMITATIONS:
n/a
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-07

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents a formal analysis of three distinct models for classifiers with a reject option in the presence of out-of-distribution inputs at test time. All three models, viz. 1) cost-based rejection, 2) bounded TPR-FPR, and 3) bounded precision-recall, share the same form of optimal selective classifier (see Section 2.4). \n\nThis selective classifier consists of the Bayes in-distribution classifier $h_B(x)$ (which is optimal for a given in-distribution), and a selection function score that is a linear combination of the conditional risk $r(x)$ and the likelihood-ratio of OOD to ID $g(x)$. Based on this analysis, the authors point out the limitations of current OOD detection methods which only have a single score, and instead propose double-score OOD detection methods which can focus on both mis-classification detection and ID/OOD separation.\n\nUsing a concrete synthetic example, they discuss the limitations of existing metrics such as AUROC and AUPR in evaluating selective classifiers in the OOD setting. They propose a novel metric (one each for the bounded TPR-FPR and bounded precision-recall models) which calculates the selective risk, subject to a given minimum TPR and maximum FPR (or a given minimum recall and minimum precision). The proposed metric is shown to be better at capturing the overall performance of selective classifiers in the OOD setting.\n\nSTRENGTHS:\n1. The proposed reject option models and evaluation metrics are developed systematically and connected well with summary sections. Found the paper to be well written. \n\n2. The analysis builds upon the prior work [6] which deals with reject option models but not in the OOD setting. In this work, they consider rejecting inputs due to both mis-classification and being OOD.\n\n3. Although the proposed work makes a strong assumption of a known OOD distribution, the analysis is useful to draw insights about the need for a double-score OOD detection method, and also to highlight the shortcomings of existing evaluation metrics.\n\n[6] https://www.jmlr.org/papers/volume24/21-0048/21-0048.pdf\n\nWEAKNESSES:\n1. The analysis and proposed new metrics (parts of the new metrics like the FPR and precision) depend upon knowledge of the OOD distribution. This is a strong assumption for practical settings. \n\n2. For the proposed double-score OODD method, it seems to me that we need access to OOD data in order to set the hyper-parameter $\\mu$ in the combined score $s_r(x) + \\mu s_g(x)$. The authors should clarify if this is set based on validation data from a different OOD distribution than the test data.\n\n3. Minor: it is a bit tedious to keep track of all the notations needed for the analysis.\n\n**Update after author-reviewer discussions:** \\\nI have read the authors rebuttal. The paper lacks sufficient discussion on the experiments. Not enough details are provided in the appendix as well. Hence, I decrease my rating to 6.\n\nQUESTIONS:\n1. What is the metric OSCR on line 37? This is defined much later in the paper, please add a reference earlier. \n\n2. Suggestion: “Selection function” seems like a better term than “Selective function”. \n\n3. Line 28: “rejects it to predict” → “rejects it from prediction”.\n\n4. Line 87: in the definition of the expected risk, the expectation should be over $(x, \\bar{y}) \\sim p(x, \\bar{y})$.\n\n5. Some notations are not defined, e.g, the indicator function $c_C(x) = [[ s_C(x) \\leq \\epsilon_1 ]]$ on line 96 (likewise on line 145). The logical AND symbol $\\wedge$ in Eqn (3).\n\n6. In the equation for $R^S(h, c)$ on line 117, the arguments of the loss function seem to be swapped. Should be $\\ell(y, h(x))$.\n\n7. In equation (8), it’s not clear to me what the sets $\\mathcal{Y}^{\\mathcal{X}}$ and $[0, 1]^{\\mathcal{X}}$ mean. Please clarify.\n\n8. Statement of Theorem 3 has a minor typo: should be $g(x) = p_O(x) / p_I(x)$.\n\n9. The validation set $\\mathcal{T}$ at the start of Section 3 is assumed to have OOD samples from the unknown OOD distribution $p_O(x)$. The OOD samples from the test set follow this same distribution. Is this a reasonable assumption to make for post-hoc tuning? Would not be true in most practical scenarios. \n\n10. Typo on line 230: should be $h^*(x)$.\n\n11. Lines 257, 258: the method should be $D(\\mathbb{R})$ and not $E(\\mathbb{R})$.\n\n12. For the proposed double-score OODD method denoted by $D(\\mathbb{R})$, how is the constant $\\mu$ set (e.g. in Table 1)? Is it set by searching for the value that leads to the smallest selective risk (in this case selective error rate) subject to the TPR/FPR constraints? If this is the case, then it could be considered unfair since the OOD distribution in the test dataset is very likely to be different from that of the validation dataset $\\mathcal{T}$. Could you clarify this point?\n\n13. In Table 1, why are the AUROC values the same for the methods $A(\\infty)$ and $D(\\mathbb{R})$? Same question for the AUPR metric.\n\n14. For the experiments in Table 2, can you explain how the methods KNN+MSP and VIM+MSP are combined for the proposed double-score method? As described on lines 202 - 203, one of the OOD methods is used for OOD/ID discrimination and other used for misclassification detection. How do you set the constant $\\mu$ used for combining the scores?\n\n15. In Table 2, why is the selective risk at given TPR and FPR the same (or almost the same) for multiple OOD datasets? For example, for CIFAR-10 as in-distribution, all the methods (including the proposed KNN+MSP and VIM+MSP) have the same selective risk across all three OOD datasets.\n\nLIMITATIONS:\nSome limitations of the proposed work are mentioned in Section 3.5. Another limitation to include is the fact that the proposed analysis and novel metrics depend on the OOD distribution which is usually unknown. One could use a validation set (as in the paper) with a mix of in-distribution and auxiliary OOD data, but the distribution of auxiliary OOD data in the validation and test sets should be different. \n\nNegative societal impacts has not been discussed, but may not be applicable here.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-17"}