Original: K NOWLEDGE D ISTILLATION AS S EMIPARAMETRIC I NFERENCE
Revised: K NOWLEDGE D ISTILLATION AS S EMIPARAMETRIC I NFERENCE

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: A popular approach to model compression is to train an inexpensive student model to mimic the class probabilities of a highly accurate but cumbersome teacher model.
Revised: A popular approach to model compression is to train an inexpensive student model to mimic the class probabilities of a highly accurate but cumbersome teacher model.

Original: Surprisingly, this two-step knowledge distillation process often leads to higher accuracy than training the student directly on labeled data.
Revised: Surprisingly, this two-step knowledge distillation process often leads to higher accuracy than training the student directly on labeled data.

Original: To explain and enhance this phenomenon, we cast knowledge distillation as a semiparametric inference problem with the optimal student model as the target, the unknown Bayes class probabilities as nuisance, and the teacher probabilities as a plug-in nuisance estimate.
Revised: To explain and enhance this phenomenon, we cast knowledge distillation as a semiparametric inference problem with the optimal student model as the target, the unknown Bayes class probabilities as nuisance, and the teacher probabilities as a plug-in nuisance estimate.

Original: By adapting modern semiparametric tools, we deriveseveral new guarantees for the prediction error of standard distillation and develop several enhancements with improved guarantees.
Revised: By adapting modern semiparametric tools, we derive new guarantees for the prediction error of standard distillation and develop two enhancements—cross-ﬁtting and loss correction—to mitigate the impact of teacher overﬁtting and underﬁtting on student performance.

Original: We validate our ﬁndings empirically on both tabulardata and image data and observe consistent improvements from our knowledge distillation enhancements.
Revised: We validate our ﬁndings empirically on both tabular and image data and observe consistent improvements from our knowledge distillation enhancements.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Knowledge distillation (KD) (Craven & Shavlik, 1996; Breiman & Shang, 1996; Bucila et al., 2006;
Revised: Knowledge distillation (KD) (Craven & Shavlik, 1996; Breiman & Shang, 1996; Bucila et al., 2006;

Original: Li et al., 2014; Ba & Caruana, 2014; Hinton et al., 2015) is a widely used model compression technique that enables  deployment of ML models on devices such as phones, watches, and virtual assistants (Stock et al., 2020).
Revised: Li et al., 2014; Ba & Caruana, 2014; Hinton et al., 2015) is a widely used model compression technique that enables the deployment of highly accurate predictive models on devices such as phones, watches, and virtual assistants (Stock et al., 2020).

Original: KD works by training a large teacher model and then a smaller student modelthat tries to match the teacher’s predictions.
Revised: KD operates by training a compressed student model to mimic the predicted class probabilities of an expensive, high-quality teacher model.

Original: It is easily applied to any classiﬁcation procedure andany teacher and works remarkably well across a wide variety of domainsincluding computer vision (Hinton et al., 2015), natural language processing (Sanh et al., 2019), and tabular data settings (Liu et al., 2018; Tan et al., 2018; Fakoor et al., 2020).
Revised: Remarkably and across a wide variety of domains (Hinton et al., 2015; Sanh et al., 2019; Jiao et al., 2019; Liu et al., 2018; Tan et al., 2018; Fakoor et al., 2020), this two-step process often leads to higher accuracy than training the student directly on the raw labeled dataset.

Original: Despite KD’s ubiquity and importance, thereasons for its success remains elusive. The practice of KD is often guided by heuristics.
Revised: While the practice of KD is now well developed, a general theoretical understanding of its successes and failures is still lacking.

Original: In practice, KD is often performed in diverse ways, for different tasks and different domains. It is often tailored to particular model types (e.g., CNN, RNNs, Transformer). This precludes a general theoretical understanding of KD.
Revised: As we detail below, a number of authors have argued that the success of KD lies in the more precise “soft labels” provided by the teacher’s predicted class probabilities.

Original: Hinton et al. (2015) remarks that KD works by transferring the “dark knowledge” from the teacher’s predicted class probabilities. As we detail below, several works have tried to explain how this dark knowledge can improve the student’s performance beyond training from scratch.
Revised: Recently,

Original: Our work is motivated by the observation of Menon et al.
Revised: Menon et al.

Original: (2020)  that the teacher probabilities  serve as a proxy for the Bayes probabilities (i.e., the true class probabilities) and that the closer the teacher and Bayes probabilities, the better the student’s performance should be.
Revised: (2020) observed that these teacher probabilities can serve as a proxy for the Bayes probabilities (i.e., the true class probabilities) and that the closer the teacher and Bayes probabilities, the better the student’s performance should be.

Original: Building on this observation, we cast KD as a plug-in approach to semiparametric inference : distillation aims to ﬁt a student model ˆ f in the presence of nuisance (the Bayes probabilities p 0 ) with the teacher’s probabilities ˆ p as a plug-in estimate of p 0 .
Revised: Building on this observation, we cast KD as a plug-in approach to semiparametric inference (Kosorok, 2007): that is, we view KD as ﬁtting a student model ˆ f in the presence of nuisance (the Bayes probabilities p 0 ) with the teacher’s probabilities ˆ p as a plug-in estimate of p 0 .

Original: This insight allows us to adapt modern tools from semiparametric inference to analyze the error of a distilled student in Sec. 3.
Revised: This insight allows us to adapt modern tools from semiparametric inference to analyze the error of a distilled student in Sec. 3.

Original: Our analysis further suggests several enhancements for KD.
Revised: Our analysis also reveals two distinct failure modes of KD: one due to teacher overﬁtting and data reuse and the other due to teacher underﬁtting from model misspeciﬁcation or insufﬁcient training.

Original: For example, in Sec. 4 we show that avoiding sample reuse between the teacher and student through cross-ﬁtting  (see, e.g., Chernozhukov et al., 2018) can improve KD performance. Moreover, a common strategy in semiparametric inference is to improve plug-in estimation by reducing  plug-in bias.
Revised: In Sec. 4, we introduce and analyze two complementary KD enhancements that correct for these failures: cross-ﬁtting —a popular technique from semiparametric inference (see, e.g., Chernozhukov et al., 2018)—mitigates teacher overﬁtting through data partitioning while loss correction mitigates teacher underﬁtting by reducing the bias of the plug-in estimate ˆ p .

Original: The method of orthogonal machine learning (Chernozhukov et al., 2018; Foster & Syrgkanis, 2019)  suggests correcting the teacher’s probabilities by stepping in the y − ˆ p direction where y is an observed label vector.
Revised: The latter enhancement was inspired by the orthogonal machine learning (Chernozhukov et al., 2018; Foster & Syrgkanis, 2019) approach to semiparametric inference which suggests a particular adjustment for the teacher’s log probabilities.

Original: We argue in Sec. 4 thatwhile this  minimizes the teacher bias, the variance could explode.
Revised: We argue in Sec. 4 that this orthogonal correction minimizes the teacher bias but often at the cost of unacceptably large variance.

Original: We instead propose a correction that balances the bias and variance, leading to a better overall generalization bound.
Revised: Our proposed correction avoids this variance explosion by balancing the bias and variance terms in our generalization bounds.

Original: We empirically validate our theoretical analysis in Sec. 5. In a synthetic setting where we have access to the Bayes classiﬁer, our modiﬁed loss yields estimators that are closer to the Bayes classiﬁer.
Revised: In Sec. 5, we complement our theoretical analysis with a pair of experiments demonstrating the value of our enhancements on six real classiﬁcation problems.

Original: Moreover, our enhanced distillation improves dramatically over vanilla distillation in failure regimes of the teacher, e.g., when the teacher is heavily biased or overﬁtted.

Original: On ﬁve real tabular datasets, cross-ﬁtting and corrected distillation improve the student’s performance by up to 4% AUC point compared to vanilla KD.
Revised: On ﬁve real tabular datasets, cross-ﬁtting and loss correction improve student performance by up to 4% AUC over vanilla KD.

Original: Furthermore, on a common image classiﬁcation dataset (CIFAR-10, Krizhevsky & Hinton, 2009), when the teacher model overﬁts, cross-ﬁtting and our corrected loss improve the student accuracy by up to 1.5%. This demonstrates the wide applicability of our enhancements.
Revised: Furthermore, on CIFAR-10 (Krizhevsky & Hinton, 2009), a benchmark image classiﬁcation dataset, our enhancements improve vanilla KD accuracy by up to 1.5% when the teacher model overﬁts.

Original: Related work.
Revised: Related work.

Original: Since we do not aim review the vast literature on KD in its entirety here, we point the interested reader to Gou et al.
Revised: Since we cannot review the vast literature on KD in its entirety, we point the interested reader to Gou et al.

Original: for a good overview of the results.
Revised: for a recent overview of the ﬁeld.

Original: In the following we review theory research studying why distillation works. We further go over some empirical studies and applications ofknowledge distillation in the extended literature review in App. A.
Revised: We devote this section to reviewing theoretical advances in the understanding of KD and summarize complementary empirical studies and applications of in the extended literature review in App. A.

Original: A number of papers have argued that the availability of soft class probabilities from the teacher rather than hard labels enables us to improve training of the student model.
Revised: A number of papers have argued that the availability of soft class probabilities from the teacher rather than hard labels enables us to improve training of the student model.

Original: This was hypothesized in Hinton et al.
Revised: This was hypothesized in Hinton et al.

Original: (2015) with empirical justiﬁcation.
Revised: (2015) with empirical justiﬁcation.

Original: Phuong & Lampert (2019) consider the case where the teacher is a ﬁxed linear classiﬁer and the student is  a linear classiﬁer or a deep linear network.
Revised: Phuong & Lampert (2019) consider the case in which the teacher is a ﬁxed linear classiﬁer and the student is either a linear model or a deep linear network.

Original: They show that the student can learn the teacher perfectly if it has access to training examples more than the ambient dimensions.
Revised: They show that the student can learn the teacher perfectly if the number of training examples exceeds the ambient dimension.

Original: Vapnik & Izmailov (2015) discusses the setting of learning with privileged information where one has additional information at training time which is not available at test time.
Revised: Vapnik & Izmailov (2015) discuss the setting of learning with privileged information where one has additional information at training time which is not available at test time.

Original: Lopez-Paz et al. (2015) draws a connection between this and distillation, arguing that distillation is effective because the teacher learns a better representation allowing the student to learn at a faster rate.
Revised: Lopez-Paz et al. (2015) draw a connection between this and KD, arguing that KD is effective because the teacher learns a better representation allowing the student to learn at a faster rate.

Original: They hypothesize that a teacher’s soft predictions indicating if an example is hard or easy enables this.
Revised: They hypothesize that a teacher’s class probabilities enable student improvement by indicating how difﬁcult each point is to classify.

Original: Tang et al.
Revised: Tang et al.

Original: (2020) argues using empirical evidence that label smoothing and reweighting of training examples using the teacher’s predictions are key to beneﬁts offered by knowledge distillation.
Revised: argue using empirical evidence that label smoothing and reweighting of training examples using the teacher’s predictions are key to the success of KD.

Original: Mobahi et al.
Revised: Mobahi et al.

Original: (2020) analyzed the case of self-distillation where the student and teacher function classes are the same.
Revised: (2020) analyzed the case of self-distillation in which the student and teacher function classes are identical.

Original: They showed that this is equivalent to increasing regularisation strength for kernelized models.
Revised: Focusing on kernel ridge regression models, they proved that self-distillation can act as increased regularization strength.

Original: Bu et al.
Revised: Bu et al.

Original: (2020) considers  model compression in a rate-distortion framework, where the rate is the size of the student model and distortion is the difference in excess risk between the teacher and the student.
Revised: (2020) considers more generic model compression in a rate-distortion framework, where the rate is the size of the student model and distortion is the difference in excess risk between the teacher and the student.

Original: Menon et al.
Revised: Menon et al.

Original: (2020) consider the case of losses such that the population risk is linear in the Bayes class probabilities.
Revised: (2020) consider the case of losses such that the population risk is linear in the Bayes class probabilities.

Original: They consider distilled empirical risk and Bayes distilled empirical risk which are the risk computed using the teacher class probabilities and Bayes class probabilities respectively rather than the observed label.
Revised: They consider distilled empirical risk and Bayes distilled empirical risk which are the risk computed using the teacher class probabilities and Bayes class probabilities respectively rather than the observed label.

Original: They show that the variance of the Bayes distilled empirical risk is lower than the empirical risk.
Revised: They show that the variance of the Bayes distilled empirical risk is lower than the empirical risk.

Original: Then using analysis from Maurer & Pontil (2009); Bennett (1962), they derive the excess risk of the distilled empirical risk as a function of the (cid:96) 2 distance between the teacher’s class probabilities and the Bayes class probabilities.
Revised: Then using analysis from Maurer & Pontil (2009); Bennett (1962), they derive the excess risk of the distilled empirical risk as a function of the (cid:96) 2 distance between the teacher’s class probabilities and the Bayes class probabilities.

Original: We signiﬁcantly depart from Menon et al.
Revised: We signiﬁcantly depart from Menon et al.

Original: (2020) in multiple ways: i) our Thm. 1 allows for the common practice of data re-use, ii) our results cover the standard distillation losses (1) and (2) which are non-linear in p 0 , iii) we use localized Rademacher analysis to achieve tight fast rates for strongly-convex losses,which are commonly used for distillation, and iv) we use techniques from semiparametric inference to improve upon vanilla distillation.
Revised: (2020) in multiple ways: i) our Thm. 1 allows for the common practice of data re-use, ii) our results cover the standard KD losses SEL and ACE which are non-linear in p 0 , iii) we use localized Rademacher analysis to achieve tight fast rates for standard KD losses, and iv) we use techniques from semiparametric inference to improve upon vanilla KD.

Original:  K NOWLEDGE D ISTILLATION B ACKGROUND
Revised:  K NOWLEDGE D ISTILLATION B ACKGROUND

Original: We consider a multiclass classiﬁcation problem with k classes and n training datapoints z i =( x
Revised: We consider a multiclass classiﬁcation problem with k classes and n training datapoints z i =( x

Original: i ,y i ) sampled independently from some distribution P .
Revised: i ,y i ) sampled independently from some distribution P .

Original: Each feature vector x belongs to a set X , each label vector y ∈{ e 1 ,...,e j }⊂{ 0 , 1 } k is a one-hot encoding of the class label, and the conditional probability of observing each label is the Bayes class probability function p 0 ( x )
Revised: Each feature vector x belongs to a set X , each label vector y ∈{ e 1 ,...,e j }⊂{ 0 , 1 } k is a one-hot encoding of the class label, and the conditional probability of observing each label is the Bayes class probability function p 0 ( x )

Original: =
Revised: =

Original: E [ Y | X = x ] .
Revised: E [ Y | X = x ] .

Original: Our aim is to identify a scoring rule f : X → R k that minimizes a prediction loss on average under the distribution P .
Revised: Our aim is to identify a scoring rule f : X → R k that minimizes a prediction loss on average under the distribution P .

Original: Knowledgedistillation.
Revised: Knowledgedistillation.

Original: Knowledgedistillation(KD)isatwo-steptrainingprocesswhereoneﬁrstuses alabeleddatasettotrainateachermodelandthentrainsastudentmodeltopredicttheteacher’spredictedclassprobabilities.
Revised: Knowledgedistillation(KD)isatwo-steptrainingprocesswhereoneﬁrstuses alabeleddatasettotrainateachermodelandthentrainsastudentmodeltopredicttheteacher’spredicted class probabilities.

Original: Typicallytheteachermodelislargerandmorecumbersome,whilethestudentis smallerandmoreefﬁcient.
Revised: Typically the teacher model is larger and more cumbersome, while the student is smaller and more efﬁcient.

Original: Knowledgedistillationwasﬁrstmotivatedbymodelcompression(Bucila etal. ,2006),toﬁndcompactyethigh-performingmodelstobedeployed(suchasonmobiledevices).
Revised: Knowledge distillation was ﬁrst motivated by model compression (Bucila et al., 2006), to ﬁnd compact yet high-performing models to be deployed (such as on mobile devices).

Original: In training the student to match the teacher’s prediction probability, there are several types of loss functions that are commonly used.
Revised: In training the student to match the teacher’s prediction probability, there are several types of loss functions that are commonly used.

Original: Let ˆ p ( x ) ∈
Revised: Let ˆ p ( x ) ∈

Original: R k be the teacher’s vector of predicted class probabilities, f ( x ) ∈
Revised: R k be the teacher’s vector of predicted class probabilities, f ( x ) ∈

Original: R k be the student model’s output, and [ k ] (cid:44) { 1 , 2 ,...,k } .
Revised: R k be the student model’s output, and [ k ] (cid:44) { 1 , 2 ,...,k } .

Original: The most popular distillation loss functions 1 (cid:96) ( z ; f ( x ) , ˆ p ( x )) include the squared error logit  loss (Ba & Caruana, 2014) 
Revised: The most popular distillation loss functions 1 (cid:96) ( z ; f ( x ) , ˆ p ( x )) include the squared error logit (SEL) loss (Ba & Caruana, 2014) (cid:96) [Equation] ( x

Original: [Equation]
Revised: [Equation]

Revised: (SEL) and the annealed cross-entropy (ACE) loss (Hinton et al., 2015)

Original: [Equation]
Revised: [Equation]

Revised: =

Revised: [Equation]

Revised: l

Revised: [Equation]

Original:  for an inverse temperature β > 0 .
Revised: [Equation] x [Equation] (cid:17) (ACE) for an inverse temperature β > 0 .

Original: These loss functions measure the divergence between the probabilities predicted by the teacher and the student.
Revised: These loss functions measure the divergence between the probabilities predicted by the teacher and the student.

Original: A student model trained with knowledge distillation often performs better than the same model trained from scratch (Bucila et al., 2006; Hinton et al., 2015).
Revised: A student model trained with knowledge distillation often performs better than the same model trained from scratch (Bucila et al., 2006; Hinton et al., 2015).

Original: In Secs.
Revised: In Secs.

Original: 3 and 4, we will adapt modern tools from semiparametric inference to understand and enhance this phenomenon.
Revised: 3 and 4, we will adapt modern tools from semiparametric inference to understand and enhance this phenomenon.

Original:  D ISTILLATION AS S EMIPARAMETRIC I NFERENCE
Revised:  D ISTILLATION AS S EMIPARAMETRIC I NFERENCE

Original: In semiparametric inference (Kosorok, 2007), one aims to estimate a target parameter or function f 0 , but that estimation depends on an auxiliary nuisance function p 0 that is unknown and not of primary interest.
Revised: In semiparametric inference (Kosorok, 2007), one aims to estimate a target parameter or function f 0 , but that estimation depends on an auxiliary nuisance function p 0 that is unknown and not of primary interest.

Original: We cast the knowledge distillation process as a semiparametric inference problem, by treating the unknown Bayes class probabilities p 0 as nuisance and the teacher’s predicted probabilities as a plug-in estimate of that nuisance.
Revised: We cast the knowledge distillation process as a semiparametric inference problem, by treating the unknown Bayes class probabilities p 0 as nuisance and the teacher’s predicted probabilities as a plug-in estimate of that nuisance.

Original: This perspective allows us bound the generalization of the student in terms of the mean squared error (MSE) between the teacher and the Bayes probabilities.
Revised: This perspective allows us bound the generalization of the student in terms of the mean squared error (MSE) between the teacher and the Bayes probabilities.

Original: In the next section (Sec. 4) we use techniques from semiparametric inference to enhance the performance of the student.
Revised: In the next section (Sec. 4) we use techniques from semiparametric inference to enhance the performance of the student.

Original: The interested reader could consult Tsiatis (2007) for more details on semiparametric inference.
Revised: The interested reader could consult Tsiatis (2007) for more details on semiparametric inference.

Original: Our analysis starts from taking the following perspective on distillation.
Revised: Our analysis starts from taking the following perspective on distillation.

Original: For a given pointwise loss function (cid:96) ( z ; f ( x ) ,p 0 ( x )) , we view the goal of the student as minimizing an oracle population loss over a function class F ,[Equation] ( Z
Revised: For a given pointwise loss function (cid:96) ( z ; f ( x ) ,p 0 ( x )) , we view the goal of the student as minimizing an oracle population loss over a function class F ,

Original: [Equation]
Revised: [Equation]

Original: f ( X ) ,p 0 ( X ))] with f 0 (cid:44) argmin f ∈F L D ( f,p 0 ) .

Original: The main hurdle is that this is objective depends on the unknown Bayes probabilities p 0 .
Revised: The main hurdle is that this is objective depends on the unknown Bayes probabilities p 0 .

Original: We view the teacher’s model ˆ p as an approximate version of p 0 and bound the distillation error of the student as a function of the teacher’s estimation error.
Revised: We view the teacher’s model ˆ p as an approximate version of p 0 and bound the distillation error of the student as a function of the teacher’s estimation error.

Original: Typical semiparametric inference considers cases where f 0 is a ﬁnite dimensional parameter; however recent work of Foster & Syrgkanis (2019) extends this framework to inﬁnite dimensional models f 0 and to develop statistical learning theory with a nuisance component framework.
Revised: Typical semiparametric inference considers cases where f 0 is a ﬁnite dimensional parameter; however recent work of Foster & Syrgkanis (2019) extends this framework to inﬁnite dimensional models f 0 and to develop statistical learning theory with a nuisance component framework.

Original: The distillation problem ﬁts exactly into this setup.
Revised: The distillation problem ﬁts exactly into this setup.

Original: Bounds on vanilla distillation with data re-use As a ﬁrst step we derive a vanilla bound on the error of the distilled student model without any further modiﬁcations of the distillation process, i.e., we assume that the student is trained on the same data as the teacher and is trained by running empirical risk minimization (ERM) on the plug-in loss, plugging in the teacher’s model instead of p 0 , i.e., 
Revised: Bounds on vanilla KD As a ﬁrst step we derive a vanilla bound on the error of the distilled student model without any further modiﬁcations of the distillation process, i.e., we assume that the student is trained on the same data as the teacher and is trained by running empirical risk minimization (ERM) on the plug-in loss, plugging in the teacher’s model instead of p 0 , i.e., ˆ f =argmin f ∈F L n ( f, ˆ p ) for L n ( f, ˆ p ) (cid:44) E n [ (cid:96) ( Z

Original: [Equation]
Revised: [Equation]

Original:  where E n [ X ]= 1 n (cid:80) ni
Revised: f ( X ) , ˆ p ( X ))] (Vanilla KD) where E n [ X ]= 1 n (cid:80) n i =1 X

Original: =1 X i denotes the empirical expectation of a random variable.
Revised: i denotes the empirical expectation of a random variable.

Original: Technical deﬁnitions Before presenting our main theorem we introduce some technical notation.
Revised: Technical deﬁnitions Before presenting our main theorem we introduce some technical notation.

Original: For a vector valued function f that takes as input a random variable X , we use the shorthand notation (cid:107) f (cid:107) p,q (cid:44) (cid:107)(cid:107) f
Revised: For a vector valued function f that takes as input a random variable X , we use the shorthand notation (cid:107) f (cid:107) p,q (cid:44) (cid:107)(cid:107) f

Original: ( X ) (cid:107) p (cid:107) L q = E (cid:2) (cid:107) f
Revised: ( X ) (cid:107) p (cid:107) L q = E (cid:2) (cid:107) f

Original: ( X ) (cid:107) qp (cid:3) 1 /q .
Revised: ( X ) (cid:107) qp (cid:3) 1 /q .

Original: Let ∇ φ and ∇ π denote the partial derivatives of (cid:96) ( z ; φ,π ) , with respect to its second and third input correspondingly and ∇ φπ the Jacobian of cross partial derivatives,
Revised: Let ∇ φ and ∇ π denote the partial derivatives of (cid:96) ( z ; φ,π ) , with respect to its second and third input correspondingly and ∇ φπ the Jacobian of cross partial derivatives,

Original: i.e., [ ∇ φπ (cid:96) ( z ; φ,π )] i,j = ∂ 

Revised: i.e., [ ∇ φπ (cid:96) ( z ; φ,π )] i,j = ∂

Original: ∂φ j ∂π i (cid:96) ( z ; φ,π ) .
Revised: ∂φ j ∂π i (cid:96) ( z ; φ,π ) .

Original: Finally, let q f,p ( x )=
Revised: Finally, let q f,p ( x )=

Original: E [ ∇ φπ (cid:96) ( Z ; f ( X ) ,p ( X )) | X = x ] and γ f,p ( x )=
Revised: E [ ∇ φπ (cid:96) ( Z ; f ( X ) ,p ( X )) | X = x ] and γ f,p ( x )=

Original: E U ∼ Unif ([0 , 1])
Revised: E U ∼ Unif ([0 , 1])

Original: [ q f,Up +(1 − U ) p 0 ( x )] .
Revised: [ q f,Up +(1 − U ) p 0 ( x )] .

Original: Critical radius Finally, we need to deﬁne the notion of the critical radius (see e.g., Wainwright (2019)) of a function class, which typically provides tight learning rates for statistical learning theory tasks.
Revised: Critical radius Finally, we need to deﬁne the notion of the critical radius (see, e.g., Wainwright (2019, 14.1.1)) of a function class, which typically provides tight learning rates for statistical learning theory tasks.

Original: For any function class F we deﬁne the localized Rademacher complexity as:
Revised: For any function class F we deﬁne the localized Rademacher complexity as:

Original: [Equation]
Revised: [Equation]

Original: where (cid:15) i are i.i.d.
Revised: where (cid:15) i are i.i.d.

Original: random variables taking values equiprobably in {− 1 , 1 } .
Revised: random variables taking values equiprobably in {− 1 , 1 } .

Original: The critical radius of a class F , taking values in [ − H,H ] , is the smallest solution δ n to the inequality R ( δ ; F ) ≤ δ 2 H .
Revised: The critical radius of a class F , taking values in [ − H,H ] , is the smallest solution δ n to the inequality R ( δ ; F ) ≤ δ 2 H .

Original: Theorem 1 (Vanilla distillation analysis) .
Revised: Theorem 1 (Vanilla KD analysis) .

Original: Assume that f 0 ∈ F and F is a convex set  and that the teacher estimatesa model ˆ p ∈P on the same sample as the student.
Revised: Suppose f 0 belongs to a convex set F satisfying the (cid:96) 2 /(cid:96) 4 ratio condition sup f ∈F (cid:107) f − f 0 (cid:107) 2 , 4 / (cid:107) f − f 0 (cid:107) 2 , 2 ≤ C and that the teacher estimates ˆ p ∈ P from the same dataset used to train the student.

Original: Let δ n,ζ = δ n + c 0 (cid:113) log( c 1 /ζ )
Revised: Let δ n,ζ = δ n + c 0 (cid:113) log( c 1 /ζ )

Original: n forsome universal constants c 0 ,c 1 and δ nis an upper bound on the critical radius  the function class:
Revised: n for universal constants c 0 ,c 1 and δ n an upper bound on the critical radius of the function class

Original: [Equation]
Revised: [Equation]

Original: Assume that, for all z ∈Z and π , the loss (cid:96) ( z ; φ,π ) is σ -strongly convex with respect to φ , uniformly bounded in [ − H,H ] for all z,φ,πand let: µ ( z ) = sup φ (cid:107)∇ φ (cid:96) ( z ; φ, ˆ p ( x )) (cid:107) 2 .
Revised: Let µ ( z ) = sup φ (cid:107)∇ φ (cid:96) ( z ; φ, ˆ p ( x )) (cid:107) 2 , and assume that, for all z ∈ Z and π , the loss (cid:96) ( z ; φ,π ) is σ -strongly convex in φ and uniformly bounded in [ − H,H ] for all z,φ,π .

Original: Moreover, suppose that the function class f satisﬁes the (cid:96) 2 /(cid:96) 4 ratio condition: sup f ∈F (cid:107) f − f 0 (cid:107) 2 , 4 (cid:107) f − f 0 (cid:107) 2 , 2 ≤ C. Then  ˆ f(3) satisﬁes[-(cid:107) ˆ f − f 0 (cid:107) 22 , 2 ≤ 1 σ 2 O ( δ 2 n,ζ C 2 H 2 (cid:107) µ (cid:107) 24 + (cid:107) γ (cid:62) f 
-]
Revised: Then the Vanilla KD ˆ f satisﬁes[-(cid:107) ˆ f − f 0 (cid:107) 22 , 2 ≤ 1 σ 2 O ( δ 2 n,ζ C 2 H 2 (cid:107) µ (cid:107) 24 + (cid:107) γ (cid:62) f 
-]

Original: ˆ p (ˆ p − p 0 )
Revised: [Equation] [Equation]

Original: (cid:107) 22 , 2 ) with probability  1 − ζ.
Revised: 2 ) with probability at least 1 − ζ.

Original: Remark The proof is in App.
Revised: Thm. 1, proved in App.

Original: D.
Revised: C, shows that vanilla distillation yields an accurate student whenever the teacher generalizes well (i.e., (cid:107) ˆ p − p 0 (cid:107) 2 , 2 is small) and the student and teacher model classes F and P are not too complex.

Original: We note that the (cid:96) 2 /(cid:96) 4 ratio requirement can be removed, albeit the ﬁnal bound will depend on a uniform bound (cid:107) µ (cid:107) ∞
Revised: The (cid:96) 2 /(cid:96) 4 ratio requirement can be removed at the expense of replacing (cid:107) µ (cid:107) 4 by (cid:107) µ (cid:107) ∞

Original: =
Revised: =

Original: sup z | µ ( z ) | , instead of (cid:107) µ (cid:107) 4 .
Revised: sup z | µ ( z ) | in the ﬁnal bound.

Original: Moreover, we note thatthestrongconvexityrequirementfor (cid:96) issatisﬁedbyallstandarddistillationobjectives, asitisstrong convexity with respect to the output of f and not the parameters of f .
Revised: Moreover, we highlight that the strong convexity requirement for (cid:96) is satisﬁed by all standard distillation objectives including SEL and ACE, as it is strong convexity with respect to the output of f and not the parameters of f .

Original: Even this requirement could be removed, but this would yield slow rate bounds of the form: (cid:107) ˆ f − f 0 (cid:107) 22 , 

Revised: Even this requirement could be removed, but this would yield slow rate bounds of the form: (cid:107) ˆ f − f 0 (cid:107) 22 ,

Original: =
Revised: =

Original: O ( δ n,ζ + (cid:107) γ (cid:62) f 0 , ˆ p (ˆ p − p 0 )
Revised: O ( δ n,ζ + (cid:107) γ (cid:62) f 0 , ˆ p (ˆ p − p 0 )

Original: (cid:107) 22 , 2 ) .
Revised: (cid:107) 22 , 2 ) .

Original: Thm. 1 shows that the standard approach to distillation can provably work to provide an accurate student and potentially improve upon training from scratch. For instance, if the teacher is very accurate, i.e., ˆ p ≈ p 0 , then the term that remains contains the fourth moment of the loss gradient (cid:107) µ (cid:107) 4 . If we were to replace the teacher ˆ p ( x ) with the raw labels y , then we would have gotten a dependence on the fourth moment of the loss gradient ˜ µ ( z ) (cid:44) sup f ∈F (cid:107)∇ φ (cid:96) ( z ; f ( x ) ,y ) (cid:107) 2 , which now contains extra variation due to the variance of the labels y . Thus in practice it can be that (cid:107) µ (cid:107) 4 (cid:28)(cid:107) ˜
Revised: Failure modes of vanilla KD Thm.

Original: µ (cid:107) 4 . Drawbacks of vanilla distillation There are two major drawbacks of Thm. 1, corresponding to two distinct sources of KD student error: error due to teacheroverﬁtting and error due toteacher underﬁtting. First, the function class G contains the variation of the student’s model over the whole P , the function class of the teacher.
Revised: 1 also hints at two distinct ways in which vanilla distillation could fail. First, since the student only learns from the teacher and does not have access to the original labels, we would expect the student to be erroneous when the teacher probabilities are inaccurate due to model misspeciﬁcation, an overly restrictive teacher function class, or insufﬁcient training.

Original: If P is a large model space, with a large critical radius, then G will inherit this complexity and the ﬁnal bound will depend on the sample complexity of the teacher. Second, if the teacher is not very accurate, i.e., (cid:107) ˆ p − p 0 (cid:107) 2 , 2 is large, then the student’s model will be signiﬁcantly impacted. In the next section, we use techniques from semiparametric inference to tackle both of these problems. We show two examples to illustrate how the problems of teacher overﬁtting/underﬁtting could manifest themselves in knowledge distillation.
Revised: Prop. 2, proved in App.

Original: These examples serve to lower bound student performance, in the worst case, by the teacher’s critical radius and the teacher’s class probability MSE, matching the upper bounds given in Thm. 1. However, we note that in other better-case scenarios vanilla distillation can perform better than the upper-bounding Thm. 1 would imply.
Revised: D, conﬁrms that, in the worst case, student error suffers from inaccuracy due to this teacher underﬁtting even when both the student and teacher belong to low complexity model classes.

Original: Example 1 (Impact of teacher overﬁtting on vanilla distillation) .
Revised: Proposition 2 (Impact of teacher underﬁtting on vanilla KD) .

Original: Suppose that the Bayes probabilityfunction p 0 has Lipschitz gradient and, for known (cid:15)> 0 , belongs to the set [Equation] Suppose moreover that X ∈ R d has Lebesgue density bounded away from 0 and ∞ and that (cid:15) < 14 E [ p 0 ,j ( X )(1 − p 0 ,j ( X )) 2 ] E [(1 − p 0 ,j ( X )) /p 0 ,j ( X )] for each j .
Revised: There exists a classiﬁcation problem in which the following properties all hold simultaneously with high probability for f 0 =log( p 0 ) :

Original: Consider the teacher estimates ˆ pj ( x  ) = max( (cid:15), ˜ p
Revised: • The teacher learns ˆ p ( x )= 1 n (1+ λ ) (cid:80) ni

Original: j ( x )){+=
 y i+} for ˜ p the Nadaraya-Watson kernel smoothing estimator (Nadaraya, 1964; Watson, 1964) [Equation] if
Revised: {+=
 y i+} for all x ∈X via ridge regression with λ =

Original: [Equation]
Revised: Θ(1 /n 1 / 4 ) .

Original: ) otherwise with [-kernel K ( x ) = (cid:107) x (cid:107) − a 
 I ] (cid:107)[-x
Revised: • Vanilla KD with [-kernel K ( x ) = (cid:107) x (cid:107) − a 
 I

Original:  (cid:107) 2 ≤ 1] , a ∈ (0 , d/ 2) , and h = n − 1 / (4+ d ) . Notably, ˜ p exactly interpolates the observed labels (i.e., ˜ p ( x i )  = y i ) 
Revised: ˆ f − f 0 (cid:107) 22 , 2 ≥ (cid:107) γ (cid:62) f 0 ,p 0 (ˆ p − p 0 ) (cid:107) 22 , 2 = Ω( 1 √ n ) , matching the dependence of the Thm.

Original: but, by Belkin et al. (2019, Thm. 1), also satisﬁes
Revised: 1 upper bound up to a constant factor.

Original: E [ (cid:107) p 0 − ˆ  p  (cid:107) 22 , 2] = O (  n− 4 / (4+ d ) ) .
Revised: • Enhanced KD with SEL loss, ˆ γ ( t ) = diag( 1ˆ p ( t ) ) , and constant ˆ f satisﬁes (cid:107) ˆ f − f 0 (cid:107) 22 , 2 = O ( 1 n ) .

Original: Consider now a student that learns a constant prediction rule via distilled ERM (3) with the squared error logit loss (1) and [Equation]
Revised: Second, the critical radius in Thm. 1 depends on the complexity of the teacher model class P .

Original: [log( (cid:15) ) , 0] k for all x,x (cid:48) ∈X} . Since the student only has access to  the teacher’s training set probabilities, its estimate ˆ f ( x )= 1 n (cid:80) ni =1 log(max( y i ,(cid:15) )) is inconsistent for the optimal constant rule f 0 ( x )= E [log( p 0 ( X ))] as f 0 ,j ( x ) − E ˆ f j ( x )= E [log
Revised: If P has a large critical radius, then the student error bound suffers due to potential teacher overﬁtting even if the teacher generalizes well .

Original: [Equation] p 0 ,j ( X )) − log
Revised: Prop. 3, proved in App.

Original: [Equation] max [Equation] Y j ,
Revised: E, shows that, in the worst case, this teacher overﬁtting penalty is unavoidable and does in fact lead to increased student error.

Original: [Equation] ))] ≥ E [ p [Equation] j
Revised: This occurs as the student only has access to the teacher’s training set probabilities which, due to overﬁtting, need not reﬂect its test set probabilities.

Original: [Equation] X [Equation] ] by Taylor’s theorem with Lagrange remainder.
Revised: Proposition 3 (Impact of teacher overﬁtting on vanilla KD) .

Original: This non-vanishing student error reﬂects the nonvanishing critical radius δ n of the composite student-teacher function class Gdeﬁned in Thm. 1; since the student function class F has low complexity, the complexity ofG is driven by the highly ﬂexible interpolating teacher. We will demonstrate how to avoid this dependence on teacher complexity in Sec. 4.2.
Revised: There exists a classiﬁcation problem in which the following properties all hold simultaneously with high probability for f 0 =log( p 0 ) : • The critical radius δ n of the teacher-student function class G in Thm. 1 is a non-vanishing constant, due to the complexity of the teacher’s function class.

Original: Example 2 (Impact of teacher underﬁtting on vanilla distillation) . Suppose that p 0 belongs to the set (4) and does not vary with x . Consider the  constant teacher estimate ˆ p ( x )=
Revised: • The Vanilla KD error (cid:107) ˆ f − f 0 (cid:107) 22 , 2 for constant ˆ f with SEL loss is lower bounded by a non-vanishing constant, matching the δ n dependence of the Thm. 1 upper bound up to a constant factor.

Original: ¯ y 1+ λ obtained via ridge regression with regularization strength λ and [-¯ y (cid:44) 1 n (cid:80) ni =
-]
Revised: • Enhanced KD with SEL loss, ˆ γ ( t ) =0 , and [-¯ y (cid:44) 1 n (cid:80) ni =
-]constant ˆ f satisﬁes (cid:107) ˆ f CF − f 0 (cid:107) 22 , 2 =

Original: y i . A constant student prediction rule trained via distilled ERM (3) with the squared error logit loss (1) yields ˆ f ( x )=log( ¯ y 1+ λ ) for all x ∈X and satisﬁes [Equation] by the concavity of the logarithm.
Revised: O ( n − 4 / (4+ d ) ) .

Original: [-Since P (¯ y j − p 0 ,j ( x ) ≥ θp 0 ,j ( x )) ≤ ζk for θ ≥ (cid:113) 2(1 − p 0 ( x )) np 
 ( x ) log( kζ ) +1 np 0 ( x ) log( kζ )-] byBernstein’s inequality (Bernstein, 1946), we have P ( (cid:107) f 0 − ˆ f (cid:107) 22 , 2 ≥ (cid:107) γ (cid:62) f 0 ,p 0 ( p 0 − ˆ p ) (cid:107) 22 , 2 ) ≥ 1 − ζ matching the teacher MSE dependence of the Thm.upper bound whenever λ ≥ (cid:113) 2 n(cid:15) log( kζ )+ 43 1 n(cid:15) log( kζ ) .
Revised: [-Since P (¯ y j − p 0 ,j ( x ) ≥ θp 0 ,j ( x )) ≤ ζk for θ ≥ (cid:113) 2(1 − p 0 ( x )) np 
 ( x ) log( kζ ) +1 np 0 ( x ) log( kζ )-]These examples serve to lower bound student performance in the worst case by the teacher’s critical radius and class probability MSE, matching the upper bounds given in Thm. 1.

Original: Moreover, since limsup n [Equation]
Revised: However, we note that in other better-case scenarios vanilla distillation can perform better than the upper-bounding Thm.would imply.

Original: with probability 1 by the law of the iterated logarithm, (cid:107) f [Equation] (min(1 ,λ 2 )) with probability 1 whenever λ ≥ (cid:113) [Equation]
Revised: In the next section, we adapt and generalize techniques from semiparametric inference to mitigate the effects of teacher overﬁtting and underﬁtting in all cases.

Original:  E NHANCING K NOWLEDGE D ISTILLATION
Revised:  E NHANCING K NOWLEDGE D ISTILLATION

Original: To address the two distinct inefﬁciencies of vanilla distillation revealed in Sec. 3, we will adapt and generalize two distinct techniques from semiparametric inference: orthogonal correction and cross-ﬁtting.
Revised: To address the two distinct inefﬁciencies of vanilla distillation revealed in Sec. 3, we will adapt and generalize two distinct techniques from semiparametric inference: orthogonal correction and cross-ﬁtting.

Original: 4. C OMBATING TEACHER UNDERFITTING WITH γ - CORRECTION
Revised: 4. C OMBATING TEACHER UNDERFITTING WITH LOSS CORRECTION

Original: We can view the plug-in distillation loss (cid:96) ( z ; f ( x ) , ˆ p ( x )) as a zeroth order Taylor approximation to the ideal loss (cid:96) ( z ; f ( x ) ,p 0 ( x )) around ˆ p .
Revised: We can view the plug-in distillation loss (cid:96) ( z ; f ( x ) , ˆ p ( x )) as a zeroth order Taylor approximation to the ideal loss (cid:96) ( z ; f ( x ) ,p 0 ( x )) around ˆ p .

Original: An ideal ﬁrst-order approximation would take the form
Revised: An ideal ﬁrst-order approximation would take the form

Original: [Equation]
Revised: [Equation]

Revised: z

Revised: [Equation]

Revised: f

Revised: [Equation]

Original: However, its computation also requires knowledge of p 0 .
Revised: However, its computation also requires knowledge of p 0 .

Original: Nevertheless, since p 0 ( x )=
Revised: Nevertheless, since p 0 ( x )=

Original: E [ Y | X = x ] , we can always construct an unbiased estimate of the ideal ﬁrst order term by replacing p 0 ( x ) with y : 
Revised: E [ Y | X = x ] , we can always construct an unbiased estimate of the ideal ﬁrst order term by replacing p 0 ( x ) with y : [Equation] ortho ( [Equation]

Revised: x

Original: [Equation]
Revised: [Equation]

Revised: , ˆ p

Revised: [Equation]

Revised: x

Revised: [Equation]

Original:  For standard distillation base losses like (1) and (2), the orthogonal loss (5) has an especially simple form, as ∇ π (cid:96) ( z ; f ( x ) , ˆ p ( x )) is linear in f .
Revised: = [Equation] [Equation] For standard distillation base losses like SEL and ACE, the orthogonal loss (1) has an especially simple form, as ∇ π (cid:96) ( z ; f ( x ) , ˆ p ( x )) is linear in f .

Original: Indeed, this is true more generally for the following class of Bregman divergence losses.
Revised: Indeed, this is true more generally for the following class of Bregman divergence losses.

Original: Deﬁnition 1 (Bregman divergence losses) .
Revised: Deﬁnition 1 (Bregman divergence losses) .

Original: Any Bregman divergence loss function of the form (cid:96) ( z ; f ( x ) ,p ( x ))
Revised: Any Bregman divergence loss function of the form (cid:96) ( z ; f ( x ) ,p ( x ))

Original: (cid:44) Ψ( f ( x ))
Revised: (cid:44) Ψ( f ( x ))

Original: − Ψ( g ( p ( x )))
Revised: − Ψ( g ( p ( x )))

Original: −(cid:104)∇ g Ψ( g ( p ( x ))) ,f ( x ) − g ( p ( x ))
Revised: −(cid:104)∇ g Ψ( g ( p ( x ))) ,f ( x ) − g ( p ( x ))

Original: (cid:105) has (cid:96) ortho ( z ; f ( x ) ,p( x ))
Revised: (cid:105) has (cid:96) ortho ( z ; f ( x ) ,p

Original: = (cid:96) ( z ; f ( x ) ,p ( x )) +( y − p ( x )) (cid:62) ∇ p g ( p ( x )) (cid:62) ∇ 2 gg Ψ( g ( p ( x )))
Revised: [Equation]

Original: f ( x )+const (6) with the second term bilinear in f ( x ) and y − p ( x ) .
Revised: +const [Equation] ) with the second term bilinear in f ( x ) and y − p ( x ) .

Original: For the squared error logit loss (1) , Ψ( s )=
Revised: For the SEL loss, Ψ( s ) =

Original: 12 (cid:107) s (cid:107) 22 , g ( p ) =
Revised: 12 (cid:107) s (cid:107) 22 , g ( p ) =

Original: log( p ) , and the correction matrix ∇ p g ( p ( x )) (cid:62) ∇ 2 gg Ψ( g ( p ( x )))
Revised: log( p ) , and the correction matrix ∇ p g ( p ( x )) (cid:62) ∇ 2 gg Ψ( g ( p ( x )))

Original: =
Revised: =

Original: diag( 1 p ( x ) ) .
Revised: diag( 1 p ( x ) ) .

Original: Similarly, the annealed cross-entropy loss(2) falls into the class of Bregman divergence losses.
Revised: Similarly, the ACE loss falls into the class of Bregman divergence losses.

Original: We will show that orthogonal correction (5) can signiﬁcantly improve student bias due to teacher underﬁtting; however, for our standard distillation losses (1) and (2), the same orthogonal correction term often introduces unreasonably large variance due to division by small probabilities appearing in the correction matrix (see Deﬁnition 1).
Revised: We will show that orthogonal correction (1) can signiﬁcantly improve student bias due to teacher underﬁtting; however, for our standard distillation losses (SEL and ACE), the same orthogonal correction term often introduces unreasonably large variance due to division by small probabilities appearing in the correction matrix (see Deﬁnition 1).

Original: To grant ourselves more ﬂexibility in balancing bias and variance, we propose and analyze a family of γ -corrected losses , parameterized by a matrix valued functionγ : X → R k × R k :
Revised: To grant ourselves more ﬂexibility in balancing bias and variance, we propose and analyze a family of γ -corrected losses , parameterized by a matrix valued function

Original: [Equation]
Revised: [Equation]

Original: (cid:96) ( [Equation] f
Revised: →

Original: [Equation]
Revised: [Equation]

Original: x
Revised: ×

Original: [Equation]
Revised: [Equation]

Original: ,p

Original: [Equation]

Original: +

Original: [Equation]
Revised: [Equation]

Original: p

Original: [Equation] ) (cid:62) γ ( x [Equation] f ( x ) to mimic the bilinear structure of Bregman orthogonal losses (6).
Revised: to mimic the bilinear structure of Bregman orthogonal losses (2).

Original: Note that we can always recover the vanilla distillation loss by taking γ ≡ 0 .
Revised: Note that we can always recover the vanilla distillation loss by taking γ ≡ 0 .

Original: We denote the associated population and empirical risks by
Revised: We denote the associated population and empirical risks by

Original: L D ( f,p,γ ) (cid:44) E [ (cid:96) γ ( Z ; f ( X ) ,p ( X ))] and L n ( f,p,γ ) (cid:44) E n [ (cid:96) γ ( Z ; f ( X ) ,p ( X ))] .
Revised: [Equation]

Original: Observe that at p 0 the correction term is mean-zero and hence L D ( f,p 0 ,γ ) is independent of γ
Revised: Observe that at p 0 the correction term is mean-zero and hence L D ( f,p 0 ,γ ) is independent of γ

Original: L D ( f,p 0 ) (cid:44) E [ (cid:96) ( Z ; f ( X ) ,p 0 ( X )) ]= L D ( f,p 0 ,γ ) for all γ.
Revised: [Equation]

Original: The γ -corrected loss has strong connections to the literature on Neyman orthogonality (Chernozhukov et al., 2018;
Revised: The γ -corrected loss has strong connections to the literature on Neyman orthogonality (Chernozhukov et al., 2018;

Original: Chernozhukov et al., 2016; Nekipelov et al., 2018; Chernozhukov et al., 2018; Foster & Syrgkanis, 2019).
Revised: Chernozhukov et al., 2016; Nekipelov et al., 2018; Chernozhukov et al., 2018; Foster & Syrgkanis, 2019).

Original: In particular, if the function γ is set appropriately, then one can show that the γ -corrected loss function satisﬁes the condition of a Neyman orthogonal loss deﬁned by Foster &
Revised: In particular, if the function γ is set appropriately, then one can show that the γ -corrected loss function satisﬁes the condition of a Neyman orthogonal loss deﬁned by Foster &

Original: Syrgkanis (2019).
Revised: Syrgkanis (2019).

Original: We begin our analysis by showing a general lemma for any estimator ˆ f , which adapts the main theorem of Foster & Syrgkanis (2019) to account for approximate orthogonality; the proof can be found in App.
Revised: We begin our analysis by showing a general lemma for any estimator ˆ f , which adapts the main theorem of Foster & Syrgkanis (2019) to account for approximate orthogonality; the proof can be found in App.

Original: C.
Revised: F.

Original: Lemma 2 (Algorithm-agnostic analysis) .
Revised: Lemma 4 (Algorithm-agnostic analysis) .

Original: Consider any estimation algorithm that produces an estimate ˆ f with small plug-in excess risk, i.e.,
Revised: Consider any estimation algorithm that produces an estimate ˆ f with small plug-in excess risk, i.e.,

Original: [Equation]
Revised: [Equation]

Original: and that the loss L D is σ -strongly convex with respect to f and F is a convex set. Then:
Revised: If the loss L D is σ -strongly convex with respect to f and F is a convex set, then

Original: [Equation]
Revised: [Equation]

Original: If, further, sup z,φ,π,i ∈
Revised: If, in addition, sup z,φ,π,i ∈

Original: [ d ] (cid:107)∇ φ i ππ (cid:96) ( z ; φ,π ) (cid:107) op ≤ M , then:
Revised: [ d ] [Equation] φ i ππ (cid:96) ( z ; φ,π ) (cid:107) op ≤ M , then

Original: [Equation]
Revised: [Equation]

Revised: p

Original:  Connection to Neyman orthogonality Note that if we set γ = q f 0 , ˆ p , thenwe obtain that the MSE error of the student  depends  on the  MSEerror of the teacher only in a second-order manner! Setting γ = q f 0 , ˆ p renders the loss function Neyman orthogonal (Foster & Syrgkanis, 2019).
Revised: [Equation] [Table] Connection to Neyman orthogonality Remarkably, if we set γ = q f 0 , ˆ p , then the γ -corrected loss is Neyman orthogonal (Foster & Syrgkanis, 2019), and the student MSE bound depends only on the squared MSE of the teacher.

Original: Moreover, q f 0 , ˆ p is an observable quantity for any Bregman divergence loss (Deﬁnition 1) as q f 0 , ˆ p is independent of f 0 .
Revised: Moreover, q f 0 , ˆ p is an observable quantity for any Bregman divergence loss (Deﬁnition 1) as q f 0 , ˆ p is independent of f 0 .

Original: However, we note that this setting of the γ can lead to larger variance, i.e., the achievable excess risk can be much larger than the excess risk without the γ -correction.
Revised: However, we note that this setting of the γ can lead to larger variance, i.e., the achievable excess risk can be much larger than the excess risk without the correction.

Original: For instance, in the case of the squared error logit loss(1) q f 0 , ˆ p ( x ) = 1ˆ p ( x ) , which can be excessively large when ˆ p is close to 0 , leading to a large increase in the variance of our loss.
Revised: For instance, in the case of the SEL loss q f 0 , ˆ p ( x ) = 1ˆ p ( x ) , which can be excessively large when ˆ p is close to 0 , leading to a large increase in the variance of our loss.

Original: Thus, in a departure from the standard approach in semiparametric inference, we will be choosing γ in practice to balance bias and variance.
Revised: Thus, in a departure from the standard approach in semiparametric inference, we will be choosing γ in practice to balance bias and variance.

Original: Example instantiation of student’s estimation algorithm
Revised: Example instantiation of student’s estimation algorithm

Original: If we use plug-in empirical risk minimization , i.e., ˆ f = argmin f ∈F L n ( f, ˆ p,γ ) , to estimate f 0 with ˆ p estimated on an independent sample, then the results of Maurer & Pontil (2009) directly imply that as long as the loss function (cid:96) ( z ; φ,π ) is uniformly bounded in [ − H,H ] , then w.p. 1 − δ :
Revised: If we use plug-in empirical risk minimization , i.e., ˆ f = argmin f ∈F L n ( f, ˆ p,γ ) , to estimate f 0 with ˆ p estimated on an independent sample, then the results of Maurer & Pontil (2009) directly imply that as long as the loss function (cid:96) ( z ; φ,π ) is uniformly bounded in [ − H,H ] , then, with probability at least 1 − δ ,

Original: [Equation]
Revised: [Equation]

Original: where τ ( n )= N ∞ (1 /n, F , 2 n ) and N ∞ ( (cid:15), F ,m ) is the (cid:96) ∞ empirical covering number of function class
Revised: where τ ( n )= N ∞ (1 /n, F , 2 n ) and N ∞ ( (cid:15), F ,m ) is the (cid:96) ∞ empirical covering number of function class

Original: F in the worst-case over all realizations of m samples and at approximation level (cid:15) .
Revised: F in the worst-case over all realizations of m data points and at approximation level (cid:15) .

Original: This result has two drawbacks: it is a slow rate result that scales as 1 / √ n for parametric of bounded VC-dimension classes and it requires the student to be ﬁt on a completely separate sample than the teacher.
Revised: This result has two drawbacks: it is a slow rate result that scales as 1 / √ n for parametric of bounded VC-dimension classes, and it requires the student to be ﬁt on a completely separate dataset from the teacher’s.

Original: In the next theorem, we address both of these drawbacks: i) we invoke localized Rademacher complexity analysis to provide a fast rate result which would be of the order of 1 /n for VC or parametric function classes, ii) we use a more sophisticated sample-splitting technique called cross-ﬁtting, which allows the student to be trained on all the available data.
Revised: In the next theorem, we address both of these drawbacks: i) we invoke localized Rademacher complexity analysis to provide a fast rate result which would be of the order of 1 /n for VC or parametric function classes, ii) we use a more sophisticated sample-splitting technique called cross-ﬁtting, which allows the student to be trained on all the available data.

Original: 4. C OMBATING TEACHER OVERFITTING WITH CROSS - FITTING
Revised: 4. C OMBATING TEACHER OVERFITTING WITH CROSS - FITTING

Original: We now provide a more sophisticated version of sample splitting to make use of all datapoints in our ﬁnal estimation, while at the same time not taking a hit by the sample complexity of the teacher’s function space.
Revised: We now provide a more sophisticated version of sample splitting to make use of all data points in our ﬁnal estimation, while at the same time not taking a hit by the sample complexity of the teacher’s function space.

Original: This approach is referred to as cross-ﬁtting  in the semiparametric inference literature (see e.g., Chernozhukov et al.
Revised: This approach is referred to as cross-ﬁtting (CF) in the semiparametric inference literature (see, e.g., Chernozhukov et al.

Original: Partition the dataset into B equally sized partitions P 1 ,...,P B 2.
Revised: Partition the dataset into B equally sized partitions P 1 ,...,P B 2.

Original: For each fold t ∈ [ B ] estimate ˆ p ( t ) and ˆ γ ( t ) using all the out-of-fold samples 3.
Revised: For each fold t ∈ [ B ] estimate ˆ p ( t ) and ˆ γ ( t ) using all the out-of-fold samples 3.

Original: Estimate ˆ f by minimizing the empirical loss:
Revised: Estimate ˆ f by minimizing the empirical loss:

Original: [Equation]
Revised: [Equation]

Original: In other words, the nuisance estimates ˆ γ ( t ) , ˆ p ( t ) that are evaluated on the samples in fold t , are estimated based only on samples outside of P t .
Revised: In other words, the nuisance estimates ˆ γ ( t ) , ˆ p ( t ) that are evaluated on the data points in fold t , are estimated based only on samples outside of P t .

Original: Theorem 3 (Cross-ﬁtted ERM analysis) .
Revised: Theorem 5 (Enhanced KD analysis) .

Original: Assume that f 0 ∈F and F is a convex set.
Revised: Assume that f 0 ∈ F and F is a convex set.

Original: Let δ n,ζ = δ n + c 0 (cid:113) log
Revised: Let δ n,ζ = δ n + c 0 (cid:113) log

Original: c 1 /ζ )
Revised: c 1 /ζ )

Original: n for universal constants c 0 ,c 1 and δ n an upper bound on the critical radii of the classes:
Revised: n for universal constants c 0 ,c 1 and δ n an upper bound on the critical radii of the classes:

Original: [Equation]
Revised: [Equation]

Original: 1] } for each t ∈ [ B ] .
Revised: ] } for each t ∈ [ B ] .

Original: Assume that, for all z ∈Z , φ , and π , the loss (cid:96) ( z ; φ,π ) is uniformly bounded in [ − H,H ] and σ -strongly convex with respect to φ , and let µ ( z )
Revised: Assume that, for all z ∈Z , φ , and π , the loss (cid:96) ( z ; φ,π ) is uniformly bounded in [ − H,H ] and σ -strongly convex with respect to φ , and let µ ( z )

Original: =
Revised: =

Original: sup f ∈F ,t ∈
Revised: sup f ∈F ,t ∈

Original: [ B ] (cid:13)(cid:13) ∇ φ (cid:96) ( z ; f ( X ) , ˆ p ( t ) ( x ))
Revised: [ B ] (cid:13)(cid:13) ∇ φ (cid:96) ( z [Equation] f ( X ) , [Equation]

Revised: p ( t ) ( x ))

Original: (cid:13)(cid:13) 2 .
Revised: (cid:13)(cid:13) 2 .

Original: Moreover, suppose that the function class f satisﬁes the (cid:96) 2 /(cid:96)4 ratio condition: sup f ∈F (cid:107)
Revised: Moreover, suppose that the function class f satisﬁes the (cid:96) 2 /(cid:96)

Original: f − f 0 (cid:107) 2 , 4 (cid:107) f − f 0 (cid:107) 2 , 2 ≤ C.
Revised: [Equation]

Original: If ˆ f is the output of the cross-ﬁtted empirical risk minimization algorithm on a fresh set of n samples, then, w.p. 1 − ζ :
Revised: If ˆ f is the output of Enhanced KD, then, with probability at least 1 − ζ ,

Original: [Equation]
Revised: [Equation]

Original: The proof is found in App.
Revised: The proof is found in App.

Original: E. Observe that unlike Thm. 1, the function classes G (ˆ p ( t ) , ˆ γ ( t ) ) in the latter theorem do not vary the teacher’s model over P but rather only vary f ∈F and evaluate p at the out-of-fold estimate  p ( t ) .
Revised: G. Observe that unlike Thm. 1, the function classes G (ˆ p ( t ) , ˆ γ ( t ) ) in the latter theorem do not vary the teacher’s model over P but rather only vary f ∈F and evaluate p at the out-of-fold estimate ˆ p ( t ) .

Original: Since in practice the teacher’s model can be quite complex, getting rid of this dependence on the sample complexity of the teacher’s function space can bring immense improvement.
Revised: Since in practice the teacher’s model can be quite complex, getting rid of this dependence on the sample complexity of the teacher’s function space can bring immense improvement.

Original: Thus the critical radius of G (ˆ p ( t ) , ˆ γ ( t ) ) , can be signiﬁcantly smaller than G .
Revised: Thus the critical radius of G (ˆ p ( t ) , ˆ γ ( t ) ) , can be signiﬁcantly smaller than G .

Original: For instance, if P and  F are VC-subgraph classes with VC dimensions d P (cid:29) d F  , correspondingly, then the critical radius of 
Revised: For instance, suppose that the loss function (cid:96) ˆ γ ( t ) ( z ; f, ˆ p ( t ) ) is L -Lipschitz with respect to f and that F is a VC-subgraph class with VC dimension d F . Then the critical radius of the function class G (ˆ p ( t ) , ˆ γ ( t ) ) is of order (cid:112) d F log( n ) /n for any choice of ( However, under the same conditions, the critical radius of the teacher-student function class G in

Revised: Thm. 1 will still depend on the teacher’s function space.

Original: G (ˆ p ( t ) , ˆ γ ( t ) ) , will be of the order (cid:112) d Flog( n ) /n , while that of G  of the  orderof (cid:112) d P log( n ) /n .
Revised: If P is also a VC-subgraph class with VC dimensions d P (cid:29) d F , then the critical radius of G will be of the much larger order (cid:112) d P log( n ) /n .

Original: Moreover, we can see in the bound of Thm.
Revised: In addition, we can see in the bound of Thm.

Original: 3 the interplay between bias and variance introduced by γ .
Revised: 5 the interplay between bias and variance introduced by γ .

Original: In particular, the part of the bound that depends on ˆ γ ( t ) can be further simpliﬁed as:
Revised: In particular, the part of the bound that depends on ˆ γ ( t ) can be further simpliﬁed as

Original: [Equation]
Revised: [Equation]

Original: where the ﬁrst part corresponds to thebias increase  due to γ , while the second part the bias decrease.
Revised: where the terms respectively encode the increase in variance and decrease in bias from employing loss correction. Notably, Thm. 5 implies that CF without γ -correction (i.e., ˆ γ ( t ) ( x )=0 ) is sufﬁcient to reduce student error due to teacher overﬁtting but may still be susceptible to excessive student error due to teacher underﬁtting. These qualitative predictions accord with our experimental observations in Sec. 5 and Fig.

Original: 4. B IASED STOCHASTIC GRADIENT DESCENT ANALYSIS
Revised: 4. B IASED STOCHASTIC GRADIENT DESCENT ANALYSIS

Original: When the set of candidate prediction rules f θ is parameterized by a vector θ ∈ R d , we may alternatively ﬁt θ via stochastic gradient descent (SGD) (Robbins & Monro, 1951; Bottou & Bousquet, 2008) on the γ -corrected objective L D ( f θ , ˆ p, ˆ γ ) .
Revised: When the set of candidate prediction rules f θ is parameterized by a vector θ ∈ R d , we may alternatively ﬁt θ via stochastic gradient descent (SGD) (Robbins & Monro, 1951; Bottou & Bousquet, 2008) on the γ -corrected objective L D ( f θ , ˆ p, ˆ γ ) .

Original: With a minibatch size of 1 and a starting point θ 0 , the parameter updates take the form
Revised: With a minibatch size of 1 and a starting point θ 0 , the parameter updates take the form

Original: [Equation]
Revised: [Equation]

Original: Ideally, these updates would converge to a minimizer of the ideal risk L ( θ ; p 0 ) =
Revised: Ideally, these updates would converge to a minimizer of the ideal risk L ( θ ; p 0 ) =

Original: L D ( f θ ,p 0 ) .
Revised: L D ( f θ ,p 0 ) .

Original: Our next result shows that, if the teacher ˆ p is independent of ( W t ) t ∈ [ n ] , then the SGD updates (8) have excess ideal risk governed by a bias term ζ (ˆ γ ) and a variance term σ (ˆ γ ) 2 /n .
Revised: Our next result shows that, if the teacher ˆ p is independent of ( W t ) t ∈ [ n ] , then the SGD updates (4) have excess ideal risk governed by a bias term ζ (ˆ γ ) and a variance term σ (ˆ γ ) 2 /n .

Original: Here, σ 20 ( θ ) represents the baseline stochastic gradient variance that would be incurred if SGD were run directly on the ideal risk L ( θ ; p 0 ) rather than our surrogate risk.
Revised: Here, σ 20 ( θ ) represents the baseline stochastic gradient variance that would be incurred if SGD were run directly on the ideal risk L ( θ ; p 0 ) rather than our surrogate risk.

Original: Our proof in App.
Revised: Our proof in App.

Original: F builds upon the biased SGD bounds of Ajalloeian & Stich (2020).
Revised: H builds upon the biased SGD bounds of Ajalloeian & Stich (2020).

Original: Theorem 4 (Biased SGD analysis) .
Revised: Theorem 6 (Biased SGD analysis) .

Original: Suppose that the loss L ( θ ; p 0 ) is λ -strongly smooth in θ .
Revised: Suppose that the loss L ( θ ; p 0 ) is λ -strongly smooth in θ .

Original: Deﬁne the bias and root-variance parameters
Revised: Deﬁne the bias and root-variance parameters

Original: [Equation]
Revised: [Equation]

Original: for σ 20 ( θ ) (cid:44) (cid:80)
Revised: for σ 20 ( θ ) [Equation] d [Equation]

Original: i ∈
Revised: Var[

Original: [ d ] Var[ ∇ θ i (cid:96) ( W ; f θ ( X ) , p 0 ( X ))] the unbiased SGD variance.
Revised: [Equation] W ; f θ ( X ) , p 0 ( X ))] the unbiased SGD variance.

Original: If
Revised: If

Original: F {+
+}
Revised: F {+
+}

Original: [Equation]

Original: =
Revised: =

Original: [Equation] − [Equation] ∈ R d L ( θ ; p 0 ) , then the iterates { θ t } n t =1 of the γ -corrected SGD algorithm satisfy
Revised: L ( θ 0 ; p 0 ) − min θ ∈ R d L ( θ ; p 0 ) , then the iterates { θ t } nt =1 of the γ -corrected SGD algorithm satisfy

Original: [Equation]
Revised: [Equation]

Original: If, in addition, L ( θ ; p 0 ) is µ -strongly convex in θ , then the iterates satisfy
Revised: If, in addition, L ( θ ; p 0 ) is µ -strongly convex in θ , then the iterates satisfy

Original: [Equation]
Revised: [Equation]

Original: Similar to Thm.
Revised: Similar to Thm.

Original: 3, the bound in Thm. 4, portrays the interplay of bias and variance as ˆ γ ranges from 0 to q f θ , ˆ p (which we remind that for any Bregman loss is independent of f θ ).
Revised: 5, the bound in Thm. 6, portrays the interplay of bias and variance as ˆ γ ranges from 0 to q f θ , ˆ p (which we remind that for any Bregman loss is independent of f θ ).

Original: In particular, the part of the bound for strongly convex losses that depends on ˆ γ can be further simpliﬁed to:
Revised: In particular, the part of the bound for strongly convex losses that depends on ˆ γ can be further simpliﬁed to:

Original: [Equation]
Revised: [Equation]

Original: This has a very intuitive form: the ﬁrst term is the impact of ˆ γ ( X ) on the variance, which is also related to the square of the noise of y , divided by the standard error scaling.
Revised: This has a very intuitive form: the ﬁrst term is the impact of ˆ γ ( X ) on the variance, which is also related to the square of the noise of y , divided by the standard error scaling.

Original: The second controls how ˆ γ improves the bias introduced by the error in the teacher’s ˆ p .
Revised: The second controls how ˆ γ improves the bias introduced by the error in the teacher’s ˆ p .

Original: Both the variance and the bias are re-weighted based on the gradient of θ , which expresses that some regions of X are more important in identifying the parameters and therefore, when we minimize the average bias-variane trade-off, we should be re-weighting the samples based on how informative they are about the parameters θ .

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We empirically validate that the two components of our solution, cross-ﬁtting and γ -correction, can address the two failure modes of KD (teacher overﬁtting/underﬁtting), in a variety of settings: (i) a synthetic setting that validates the theory in Secs. 3 and 4, (ii) tabular data showing that cross-ﬁtting and γ -correction yield improvement, and (iii) image data to show the wide applicability of our methods.
Revised: We complement our theoretical analysis with a pair of experiments demonstrating the practical beneﬁts of cross-ﬁtting and loss correction on six real-world classiﬁcation tasks.

Original: Throughout, we use the squared error logit loss (1).
Revised: Throughout, we use the SEL loss.

Original: Additional details and results are in App.
Revised: Code to replicate all experiments can be found at github.com/microsoft/semiparametric distillation, and supplementary experimental details and results can be found in App.

Original: G.
Revised: I.

Original: Selecting the  correction matrix ˆ γ Motivated by the analyses in Sec. 4, for each training point ( x,y ) , we will select our correction matrix ˆ γ ( x ) to balance bias and variance by minimizing a pointwise upper bound on the γ -correction error (9) (ideally with a closed-form solution to avoid excessive computational overhead).
Revised: Selecting the loss correction matrix ˆ γ Motivated by the analyses in Sec. 4, for each training point ( x,y ) , we will select our correction matrix ˆ γ ( x ) to balance bias and variance by minimizing a pointwise upper bound on the loss correction error (5) (ideally with a closed-form solution to avoid excessive computational overhead).

Original: 2 To eliminate dependence on the unobserved p 0 , we observe that the bias term (cid:107) ( γ f θ , ˆ p ( x ) − ˆ γ ( x )) (cid:62) (ˆ p ( x ) − p 0 ( x ))
Revised: 3 To eliminate dependence on the unobserved p 0 , we observe that the bias term (cid:107) ( γ f θ , ˆ p ( x ) − ˆ γ ( x )) (cid:62) (ˆ p ( x ) − p 0 ( x ))

Original: (cid:107) 

Revised: (cid:107)

Original: = O ( (cid:107) q f θ , ˆ p ( x ) − ˆ γ ( x ) (cid:107) 2op ) up to additive terms independent of ˆ γ .
Revised: = O ( (cid:107) q f θ , ˆ p ( x ) − ˆ γ ( x ) (cid:107) 2op ) up to additive terms independent of ˆ γ .

Original: We introduce a tunable hyperparameter α> 0 to trade off between this bias bound and the variance term in (9) and select ˆ γ ( x )=
Revised: We introduce a tunable hyperparameter α> 0 to trade off between this bias bound and the variance term in (5) and select ˆ γ ( x )=

Original: diag( v ( x )) to minimize (cid:107) ˆ γ ( x ) (cid:107) 2op

Original: E [ (cid:107) y − ˆ p ( x ) (cid:107) 22 | x ]+ α (cid:107) q f θ

Original: [Equation]

Original: ˆ

Original: [Equation] ( x ) − [Equation] x ) (cid:107) 2op
Revised: diag( v ( x )) to minimize:

Original: [Equation]
Revised: [Equation]

Original: (cid:107)

Original: [Equation]

Original: ( x ) (cid:107) 22 E [ (cid:107)

Original: [Equation]

Original: −

Original: [Equation]

Original: [-( x ) (cid:107) 
-]
Revised: [-( x ) (cid:107) 
-]

Original: | x ]+ α (cid:107) 1ˆ p ( x

Original: [Equation]

Original: − [Equation] ( x ) (cid:107) 22 . Since the conditional expectation involves the unknown quantity p 0 , we approximate E [ (cid:107) y − ˆ p ( x ) (cid:107) 22 | x ] by its sample (cid:107) y − ˆ p ( x ) (cid:107) 2 2 . 

Revised: Since the conditional expectation involves the unknown quantity p 0 , we approximate E [ (cid:107) y − ˆ p ( x ) (cid:107) 22 | x ] by its sample (cid:107) y − ˆ p ( x ) (cid:107) 22 .

Original: This objective is quadratic in v ( x ) and thus has a closed-form solution.
Revised: This objective is quadratic in v ( x ) and thus has a closed-form solution.

Original: Moreover, once we solve for ˆ γ ( x ) , the student’s γ -corrected objective can be shown to be equivalent to a square loss with labels log( p ( x ))
Revised: Given ˆ γ ( x ) , the student’s loss-corrected objective is equivalent to a square loss with labels log( p ( x ))

Original: +ˆ γ ( x ) (cid:62) ( y − p ( x )) .
Revised: +ˆ γ ( x ) (cid:62) ( y − p ( x )) .

Original: Validation of theoretical analysis.
Revised: Tabular data.

Original: When the Bayes probability p 0 ( x ) is known, we validate that the corrected loss function yields a student model whose accuracy is closer to the performance of the Bayes classiﬁer, compared to a student model trained from scratch. The synthetic data is generated from a mixture of two Gaussians with equal weights, where the Bayes classiﬁer is known. The student model is a linear logistic regression model, which can capture the
Revised: We ﬁrst validate our KD enhancements on ﬁve real-world tabular datasets—FICO (FIC),

Original: Bayes classiﬁer. To simulate the teacher probabilities, we take the true probability p 0 ( x ) and move it closer to either 0 or 1, whichever is closer to p 0 ( x ) . We compare 4 training objectives for the student model: (i) trained from scratch (ii) trained with vanilla KD (iii) trained with γ -corrected KD loss (iv) trained to match the Bayes probability p 0 ( x ) . Figure 1 shows that the corrected KD loss brings the student’s accuracy close to the case where it is trained on the true probabilities.
Revised: StumbleUpon (Eve; Liu et al., 2017), and Adult, Higgs, and MAGIC from Dheeru & Karra Taniskidou (2017)—with random forest (Breiman, 2001) students and teachers.

Original: [Figure]
Revised: In Fig.

Original: Tabular data. We validate cross-ﬁtting and corrected KD loss on ﬁve real-world tabular datasets datasets: Adult (Dheeru & Karra Taniskidou, 2017), FICO (FIC), Higgs (Dheeru & Karra Taniskidou, 2017), MAGIC (Dheeru & Karra Taniskidou, 2017), and StumbleUpon (Eve; Liu et al., 2017).
Revised: 1a, we examine the impact of varying student model capacity for a ﬁxed high-capacity teacher with 500 trees on FICO.

Original: Cross-ﬁtting can improve the student performance by up to 4% AUC points when the teacher model likely overﬁts, while γ -corrected KD loss is most helpful when the teacher has high bias.
Revised: This setting lends itself to teacher overﬁtting, and we ﬁnd that cross-ﬁtting consistently improves upon vanilla KD by up to 4 AUC percentage points.

Original: We validate that cross-ﬁtting can improve KD when the teacher overﬁts, and γ -correction is helpful when the teacher has high bias. Both teacher and student models are random forest.
Revised: In Fig.

Original: In the ﬁrst case (Fig. 2a), we use a large teacher (500 trees) and vary the student’s capacity, while in the second case (Fig. 2b) we vary the teacher’s bias by limiting the teacher’s max tree depth. Figure 2 shows the result of the ﬁrst case on the FICO dataset, and the results of the second case on the Adult dataset.
Revised: 1b we explore the impact of teacher underﬁtting by limiting the teacher’s maximum tree depth on Adult. Here we observe consistent gains from loss correction with student performance exceeding even that of the teacher for smaller maximum tree depths.

Original: The results on the rest of the  datasets are similar (App.
Revised: Analogous results for the remaining datasets can be found in App.

Original: G.2). When the teacher overﬁts, cross-ﬁtting shows signiﬁcant improvement over using the same teacher on the training points. The γ correction successfully reduces the teacher’s bias and thus improves the student’s performance. The effect is most pronounced when the teacher has large bias (low tree depth).
Revised: I.

Original: [Figure]
Revised: [Figure]

Original: Image data.
Revised: Image data.

Original: On the image classiﬁcation dataset CIFAR-10 (Krizhevsky & Hinton, 2009), we show that cross-ﬁtting can improve the student’s performance in the case where the teacher overﬁts.
Revised: We next validate our KD enhancements on the image classiﬁcation dataset CIFAR- (Krizhevsky & Hinton, 2009).

Original: We use ResNet-8 as the student and ResNet-14/20/32/44/56 (He et al., 2016) as the teacher.
Revised: We pair a residual network (ResNet-8) student with teacher networks of varying depths (ResNet-14/20/32/44/56) (He et al., 2016).

Original: It has been observed that larger or deeper teachers might not yield better students, as the teacher might overﬁt to the training set (Cho & Hariharan, 2019; M¨uller et al., 2019).
Revised: It has been observed that larger and deeper teachers need not yield better students, as the teacher might overﬁt to the training set (Cho & Hariharan, 2019; M ¨ uller et al., 2019).

Original: To simulate overﬁtting, we turn off data augmentation (random horizontal ﬂipping and cropping).
Revised: To induce this overﬁtting, we turn off data augmentation (random horizontal ﬂipping and cropping).

Original: We compare students trained with vanilla KD objective with and without cross-ﬁtting and γ -correction in Fig.
Revised: We compare students trained with vanilla KD and enhanced KD with and without loss correction in Fig.

Original: Cross-ﬁtting can reduce the effect ofteacher’s overﬁtting, especially for deep teacher  models.
Revised: We ﬁnd that cross-ﬁtting consistently reduces the effect of teacher overﬁtting with largest impact realized for the deepest models.

Original: This effect is most evident in the test loss, where thestudent that is trained with vanilla KD has much higher loss than the student trained with KD and cross-ﬁtting.
Revised: This effect is most evident in the test loss, where the vanilla KD student incurs signiﬁcantly larger loss than the cross-ﬁtted student.

Original: The γ -correction consistently provides a small performance boost.
Revised: For both accuracy and test loss, employing loss correction on top of cross-ﬁtting provides an additional small performance boost.

Original: [Figure]
Revised: Ablation on the effect of the loss correction hyperparameter α The hyperparameter α controls the tradeoff between bias and variance.

Original: App.
Revised: When α is very small, the objective is close to the vanilla KD objective. When α is large, the objective is closer to the Neyman-orthogonal loss.

Original: G.3 shows the effect of varyingthe hyperparameter α .
Revised: In Figure 3, we show the effect of varying α , with ResNet-8 as the student and ResNet-20 as the teacher, on the CIFARdataset.

Original: Large  α (closer to orthogonal loss) leads to high variance and  lower test accuracy.
Revised: Large values of α lead to high variance and thus lower test accuracy.

Original: Intermediate  α improves on the vanilla KD  objective ( α =0 ). 
Revised: Intermediate values of α improves on the vanilla KD objective, which corresponds to α =0 and on the orthogonal objective ( α = ∞ ). The test accuracy drops sharply beyond some threshold of α as the variance becomes too high (due to the terms q ˆ p ( x )=diag , causing training to become unstable [Figure]

Revised: [Figure]

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: We provide a new analysis of knowledge distillation under the lens of semiparametric inference.
Revised: We developed a new analysis of knowledge distillation under the lens of semiparametric inference.

Original: By viewing the KD process as learninga student model withthe true class probabilities as a nuisance function and the teacher’s probabilities as a plug-in estimator, we obtain generalization bounds on the student model.
Revised: By framing the KD process as learning with plug-in estimation in the presence of nuisance, we obtained new generalization bounds for distillation and new lower bounds highlighting the susceptibility of KD to teacher overﬁtting and underﬁtting.

Original: This perspective also suggests two techniques to improve KD, namely cross-ﬁtting and γ -correction, which we show leads to better KD performance, both theoretically and empirically.
Revised: To address these failure modes, we introduced two complementary KD enhancements—cross-ﬁtting and loss correction—which improve student performance both in theory and in practice.

Original: This also opens the door to combining KD with other techniques and corrections in the semiparametric literature, such asthose based on targeted maximum likelihoodframework (Van Der Laan & Rubin, 2006).
Revised: A tantalizing open question is whether other techniques from semiparametric inference, such as targeted maximum likelihood (Van Der Laan & Rubin, 2006), can be used to improve

Original: We are excited about future work on unraveling many surprising phenomena in KD, such as the  success of self-distillation (Furlanello et al., 2018) and noisy student training (Xie et al., 2020).
Revised: KD performance. In addition, a natural direction for future work is exploring whether a semiparametric perspective can explain the surprising success of self-distillation (Furlanello et al., 2018) and noisy student training (Xie et al., 2020) through which students routinely outperform their teachers.

Original: A A DDITIONAL LITERATURE REVIEW
Revised: A E XTENDED LITERATURE REVIEW

Original: There has been a lot of work on knowledge distillation in the last half of a decade. Gou et al. (2020) gives a  survey of the various directions that have been studied in this regard.
Revised: We point the interested reader to Gou et al. (2020) for a sweeping survey of the many developments in knowledge distillation over the past half decade.

Original: Inthis appendix, we sketch some directions. In addition to  references discussing theoretic aspects discussed in Sec. 1,here we mention some work empirically studying why distillation works.
Revised: In addition to the references discussing theoretical aspects of knowledge distillation provided in Sec. 1, we highlight here a number of empirical investigations of why distillation works.

Original: Cho & Hariharan (2019) show that larger teacher models do not necessarily improve the performance of student models as parsimonious student models are not able to mimic the teacher model.
Revised: Cho & Hariharan (2019) show that larger teacher models do not necessarily improve the performance of student models as parsimonious student models are not able to mimic the teacher model.

Original: They suggest early stopping in training large teacher neural networks as means of regularizing.
Revised: They suggest early stopping in training large teacher neural networks as means of regularizing.

Original: Cheng et al.
Revised: Cheng et al.

Original: (2020) show that when applied to image data, distillation allows the student neural net to learn multiple visual concepts simultaneously, while in knowledge distillation neural networks learn concepts sequentially.
Revised: (2020) show that when applied to image data, distillation allows the student neural net to learn multiple visual concepts simultaneously, while in knowledge distillation neural networks learn concepts sequentially.

Original: Knowledge distillation has been used in various other settings such as adversarial attacks (Papernot et al., 2016b; Ross &
Revised: Knowledge distillation has been used in various other settings such as adversarial attacks (Papernot et al., 2016b; Ross &

Original: Doshi-Velez, 2017; Gil et al., 2019; Goldblum et al., 2020), data security (Papernot et al., 2016a; Lopes et al., 2017; Wang et al., 2019), image processing (Li & Hoiem, 2017; Wang et al., 2017; Chen et al., 2018; Li et al., 2017), natural language processing (Nakashole & Flauger, 2017; Mou et al., 2016;
Revised: Doshi-Velez, 2017; Gil et al., 2019; Goldblum et al., 2020), data security (Papernot et al., 2016a; Lopes et al., 2017; Wang et al., 2019), image processing (Li & Hoiem, 2017; Wang et al., 2017; Chen et al., 2018; Li et al., 2017), natural language processing (Nakashole & Flauger, 2017; Mou et al., 2016;

Original: Hu et al., 2018; Freitag et al., 2017), speech processing (Chebotar & Waters, 2016;
Revised: Hu et al., 2018; Freitag et al., 2017), speech processing (Chebotar & Waters, 2016;

Original: Lu et al., 2017; Watanabe et al., 2017; Oord et al., 2018;
Revised: Lu et al., 2017; Watanabe et al., 2017; Oord et al., 2018;

Original: Shen et al., 2018).
Revised: Shen et al., 2018).

Original: B G LOSSARY
Revised: B G LOSSARY

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] [Table]
Revised: model f

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] [Equation]
Revised: model

Original: [Table] [Equation] [Table]
Revised: ˆ f

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: ∇ φ Partial derivative of (cid:96) ( z,φ,π ) with respect to the second input ∇ π

Original:  derivative 
Revised: Partial derivative of (cid:96) ( z,φ,π ) with respect to the third input ∇

Original: [Table]
Revised: [Equation]

Revised: Localized Rademacher complexity of function class F δ n

Original: [Table]
Revised: [Table]

Revised: radius γ -corrected loss (cid:96) γ ( z

Revised: [Equation]

Revised: f ( x

Revised: [Equation]

Revised: p ( x ))

Revised: [Equation]

Revised: Population γ -risk L D ( f,p,γ )

Revised: [Equation]

Revised: X

Revised: [Equation]

Revised: p

Revised: [Equation]

Revised: X )

Revised: [Equation]

Revised: Empirical

Revised: [Table]

Revised: [Table]

Revised: ,γ )

Revised: [Equation]

Revised: C P ROOF OF T HM . 1: V ANILLA DISTILLATION ANALYSIS

Revised: First by localized Rademacher analysis (see Lemma 11 of Foster & Syrgkanis (2019)) if δ n upper bounds the critical radius of the function class G and if we denote with (cid:96) ˆ p,f ( w )= (cid:96) ( w ; f ( x ) , ˆ p ( x )) , then w.p. 1 − ζ :

Original: to the second
Revised: [Equation] Moreover, if we let µ ( w )= sup φ (cid:107)∇ φ (cid:96) ( w ; φ, ˆ p ( x )) (cid:107) 2 , then we have by Cauchy-Scwarz inequality: [Equation]

Revised: If we further assume that the function class F satisﬁes an (cid:96) 2 /(cid:96) 4 condition that:

Revised: [Equation]

Revised: then:

Revised: [Equation]

Revised: Plugging in the bound above to Lemma 4 (which holds irrespective of whether data re-use, sample splitting, or cross-ﬁtting is employed) and applying again the arithmetic-geometric mean inequality yields σ

Original: [Table]
Revised: [Equation]

Revised: D P ROOF OF P ROP . 2: I MPACT OF TEACHER UNDERFITTING ON VANILLA DISTILLATION

Revised: Suppose that, for known (cid:15)> 0 , p 0 belongs to the set

Revised: [Equation]

Revised: } and does not vary with x .

Revised: As all quantities in this proof are independent of x , we will omit the dependence on x whenever appropriate.

Revised: Consider the constant teacher estimate ˆ p = ¯ y 1+ λ obtained via ridge regression with regularization strength λ and ¯ y (cid:44) 1 n (cid:80) ni

Original: derivative{+=
 y i . A constant student prediction rule in [Equation]+}
Revised: {+=
 y i . A constant student prediction rule in [Equation]+}

Revised: trained via Vanilla KD with SEL loss yields ˆ f ( x )=log( ¯ y 1+ λ ) and satisﬁes

Original: [Table]
Revised: [Equation]

Revised: by the concavity of the logarithm.

Revised: Since

Revised: [Equation]

Revised: by Bernstein’s inequality (Bernstein, 1946), we have

Revised: P

Original: (cid:96) (
Revised: [Equation] (cid:107) [Equation] [Equation]

Revised: ˆ

Revised: [Equation]

Revised: 0 , ∀

Original: [Table]
Revised: [Equation] [ k ]) [Equation] λp

Revised: [Equation]

Revised: ≥ ¯ y j − p 0 ,j , ∀ j ∈ [ k ]) ≥ 1 − ζ whenever λ ≥ (cid:113) 2 n(cid:15) log( kζ )+ 43 1 n(cid:15) log( kζ ) .

Revised: Moreover, since limsup n →∞

Revised: [Equation]

Revised: √

Revised: [Equation]

Revised: ) =1 with probability 1 by the law of the iterated logarithm, (cid:107) γ (cid:62)

Original: [Table]
Revised: [Equation] [Equation] p [Equation]

Original: to
Revised: ( p 0 − ˆ p ) (cid:107) 22 , 2 =Ω(min(1 ,λ 2 )) with probability 1 whenever λ ≥ (cid:113) 2loglog( n ) n(cid:15) . The choice λ [Equation] max

Revised: [Equation]

Revised: now yields the ﬁrst two advertised claims.

Revised: The ﬁnal claim follows directly from Thm.

Revised: 5 with B = O (1) as ˆ γ ( t ) =

Revised: γ f 0 , ˆ p and the critical radius of G (ˆ p ( t ) , ˆ γ ( t ) ) satisﬁes δ n/B

Revised: = O ( (cid:112)

Revised: Bk/n ) by Wainwright (2019, Ex. 13.8).

Revised: E P ROOF OF P ROP . 3: I MPACT OF TEACHER OVERFITTING ON VANILLA DISTILLATION

Revised: Suppose that the Bayes probability function p 0 has Lipschitz gradient and, for known (cid:15)> 0 , belongs to the set

Revised: P = { p : p j ( x ) ∈ [ (cid:15), 1] , ∀ x

Original: [Table]
Revised: [Equation] j ∈ [ k ] } . Suppose moreover that X ∈ R d has Lebesgue density bounded away from 0 and ∞ and that (cid:15)< 14 E [ p 0 ,j ( X )(1 − p 0 ,j ( X )) 2 ] E [(1 − p 0 ,j ( X )) /p 0 ,j ( X )] for each j . Consider the teacher estimates ˆ p j ( x )=max( (cid:15), ˜ p

Revised: j ( x )) for ˜ p the Nadaraya-Watson kernel smoothing estimator (Nadaraya, 1964;

Revised: Watson, 1964)

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: y i if x = x i [Equation] ni =

Original: [Equation]
Revised: [Equation]

Revised: ) / (cid:80) ni =1 K (( x − x i )

Revised: /h )

Original: {+otherwise with kernel K ( x ) = (cid:107) x (cid:107) − a 
+}
Revised: {+otherwise with kernel K ( x ) = (cid:107) x (cid:107) − a 
+}

Revised: I [ (cid:107) x

Original: [Table]
Revised: (cid:107) 2 ≤ 1] , a ∈ (0 ,d/ 2) , and h = n − 1 / (4+ d ) . By Belkin et al. (2019, Thm. 1), the teacher satisﬁes E [ (cid:107) p 0 − ˆ p (cid:107) 22 , 2 ]= O ( n − 4 / (4+ d ) ) .

Revised: Now instantiate the notation of Thm. 1, and consider a student prediction rule trained to learn a constant prediction rule via Vanilla

Revised: KD with the SEL loss and

Original: [Equation]
Revised: [Equation]

Revised: all x

Original: [Table]
Revised: [Equation]

Revised: Since ˜ p exactly interpolates the observed labels (i.e., ˜ p ( x i ) =

Original: Localized Rademacher complexity of  function class F
Revised: y i ), the critical radius of the teacher-student function class G satisﬁes δ n = Ω(1) . Moreover, since the student only has access to the teacher’s training set probabilities, its estimate ˆ f ( x )= 1 n (cid:80) ni

Revised: =1 log(max( y i ,(cid:15) )) is inconsistent for the optimal constant rule f 0 ( x )=

Revised: E [log( p 0 ( X ))] as f 0 ,

Revised: [Equation]

Revised: ( x )

Revised: [Equation]

Revised: x )

Revised: [Equation]

Original: [Table]
Revised: [log [Equation] p 0 , [Equation]

Revised: X )

Original: [Equation]
Revised: [Equation]

Original: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: ] by Taylor’s theorem with Lagrange remainder. This non-vanishing student error reﬂects the non-vanishing critical radius δ n of the composite student-teacher function class G deﬁned in Thm. 1; since the student function class F has low complexity, the complexity of G is driven by the highly ﬂexible interpolating teacher. Next, instantiate the notation of Thm. 5, and consider a student prediction rule ˆ f trained via Enhanced KD with SEL loss, ˆ γ ( t ) =0 ,

Original: [Equation]
Revised: B = O (1) , and F (6).

Original: [Table]
Revised: The critical radius of G (ˆ p ( t ) , ˆ γ ( t ) ) satisﬁes δ n/B

Original: [Equation]
Revised: = O ( (cid:112)

Original: [Table]
Revised: Bk/n ) by Wainwright (2019, Ex. 13.8). Moreover, each cross-ﬁtted teacher satisﬁes E [ (cid:107) p 0 − ˆ p ( t ) (cid:107) 22 , 2 ]=

Original: [Equation]
Revised: O

Original: [Table]
Revised: n − 4 / (4+ d ) ) by Belkin et al. (2019, Thm. 1), so, by Chebyshev’s and Jensen’s inequalities, with probability at least 1 − ζ/ 2 ,

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: for all t. Therefore, Thm. 5 implies that

Original: [Equation]
Revised: [Equation]

Revised: with probability at least 1 − ζ .

Original: C P ROOF OF L EMMA 2: A LGORITHM - AGNOSTIC ANALYSIS
Revised: F P ROOF OF L EMMA 4: A LGORITHM - AGNOSTIC ANALYSIS

Original: First we deﬁne for any functional L ( f ) the Frechet derivative as:
Revised: First we deﬁne for any functional L ( f ) the Frechet derivative as:

Original: [Equation]
Revised: [Equation]

Original: When L is an operator of the form: E [ g ( f ( X ))], then: D f L ( f )
Revised: When L is an operator of the form: E [ g ( f ( X ))]

Original: [ ν ]= E [ ∇ g ( f ( X )) (cid:62) ν ( X )] .
Revised: [Equation] then: D f L ( f )[ ν ]= E [ ∇ g ( f ( X )) (cid:62) ν ( X )] .

Original: By σ -strong convexity of L D , 4 we have that
Revised: By σ -strong convexity of L D [Equation] 6 we have that

Original: [Equation]
Revised: [Equation]

Revised: Furthermore, our excess risk assumption and the optimality of f 0 give us σ

Original: [Equation]
Revised: [Equation]

Original: By Taylor’s theorem with integral remainder
Revised: By Taylor’s theorem with integral remainder,

Original: [Equation]
Revised: [Equation]

Original: whenever ∇ φπ (cid:96) is well-deﬁned.
Revised: whenever ∇ φπ (cid:96) is well-deﬁned.

Original: We can now invoke the expansion (10) and Cauchy-Schwarz to obtain the bound
Revised: We can now invoke the expansion (7) and Cauchy-Schwarz to obtain the bound

Original: [Equation]
Revised: [Equation]

Original: Thus combining all the above inequalities: σ
Revised: Thus combining all the above inequalities: σ

Original: [Equation]
Revised: [Equation]

Original: By an AM-GM inequality, for all a,b ≥ 0 : a · b ≤ 12 ( 2 σ a 2 + σ 2 b 2 ) .
Revised: By an AM-GM inequality, for all a,b ≥ 0 : a · b ≤ 1 2 ( 2 σ a 2 + σ 2 b 2 ) .

Original: Applying this to the product of norms on the RHS and re-arranging yields
Revised: Applying this to the product of norms on the RHS and re-arranging yields

Original: [Equation]
Revised: [Equation]

Original: To get the ﬁnal ineguality, observe that:
Revised: To get the ﬁnal ineguality, observe that:

Original: [Equation]
Revised: [Equation]

Original: Moreover, by the boundedness of the third derivative, we have:
Revised: Moreover, by the boundedness of the third derivative, we have:

Original: [Equation]
Revised: [Equation]

Original: Combining all the above yields the ﬁnal bound.
Revised: Combining all the above yields the ﬁnal bound.

Original: D P ROOF OF T HEOREM
Revised: G P ROOF OF T HM . 5: C ROSS - FITTED ERM ANALYSIS

Original: Proof First by localized Rademacher analysis (see Lemma 11 of Foster & Syrgkanis (2019)) if δ n upper bounds the critical radius of the function class G and if we denote with (cid:96) ˆ p,f ( w )= (cid:96) ( w ; f ( x ) , ˆ p ( x )) , then w.p. 1 − ζ : [Equation]

Original: Moreover, if we let µ ( w )= sup φ (cid:107)∇ φ (cid:96) ( w ; φ, ˆ p ( x )) (cid:107) 2 , then we have by Cauchy-Scwarz inequality: [Equation] −
Revised: Let L n,t denote the empirical loss over the samples in the t -th fold and ˆ p ( t ) , ˆ γ ( t ) the nuisance functions used on the samples in the k -th fold.

Original: [Equation] If we further assume that  the function classF satisﬁes an (cid:96) 2 /(cid:96) 4 condition that: [Equation]
Revised: For any t ∈ [ K ] and conditional on ˆ p ( t ) , ˆ γ ( t ) , suppose that δ n upper bounds the critical radius of the function class

Original: then: [Equation] Plugging in the bound above to Lemma 2 (which holds irrespective of data re-use or sample splitting or cross-ﬁtting) and applying again the AM-GM inequality yields:
Revised: G (ˆ p ( t ) , ˆ γ ( t ) ) , then by Lemma 11 of Foster & Syrgkanis (2019), 7 if we denote with (cid:96) t,f ( w )= (cid:96) ˆ γ ( t ) ( w ; f ( x ) , ˆ p ( t ) ( x

Original: [Equation]
Revised: [Equation]

Original: E P ROOF OF T HM . 3: C ROSS - FITTED ERM ANALYSIS

Original: Let L n,t denote the empirical loss over the samples in the t -th fold and ˆ p ( t ) , ˆ γ ( t ) the nuisance functions used on the samples in the k -th fold.

Original: For any t ∈ [ K ] and conditional on ˆ p ( t ) , ˆ γ ( t ) , suppose that δ n upper bounds the critical radius of the function class

Original: G (ˆ p ( t ) , ˆ γ ( t ) ) , then by Lemma 11 of Foster & Syrgkanis (2019), 6 if we denote with (cid:96) t,f ( w )= (cid:96) ˆ γ ( t ) ( w ; f ( x ) , ˆ p ( t ) ( x )) , w.p. 1 − ζ : (cid:12)(cid:12)(cid:12) L n,t ( ˆ f, ˆ p ( t ) , ˆ γ ( t ) )

Original: − L n,t ( f 0 , ˆ p ( t ) , ˆ γ ( t ) ) − ( L D ( ˆ f, ˆ p ( t ) , ˆ γ ( t ) )

Original: − L D ( f 0 , ˆ p ( t ) , ˆ γ ( t ) ))

Original: (cid:12)(cid:12)(cid:12) ≤ O (cid:16) δ n/B,ζ (cid:107) (cid:96) t,f − (cid:96) t,f 0 (cid:107) 2 , 2 + δ 2 n/B,ζ (cid:17)

Original: Moreover, we have that by the deﬁnition of cross-ﬁtted ERM:
Revised: Moreover, we have that by the deﬁnition of cross-ﬁtted ERM:

Original: [Equation]
Revised: [Equation]

Original: Thus we have that w.p. 1 − ζB :B
Revised: Thus we have that w.p. 1 − ζB :

Original: B (cid:88) t =

Original: L D ( ˆ f, ˆ p ( t ) , ˆ γ ( t ) )

Original: − L D

Original: [Equation]
Revised: [Equation]

Original: f 0 , ˆ p ( t ), ˆ γ ( t ) )
Revised: Moreover, if we let µ ( w )=sup φ,t (cid:13)(cid:13) ∇ φ (cid:96) ( w ; φ, ˆ p ( t ) ( x )) (cid:13)(cid:13) 2 , then we have by Cauchy-Schwarz inequality:

Original: [Equation]
Revised: [Equation]

Original: Moreover
Revised: −

Original: [Equation]
Revised: [Equation]

Original: if we let µ ( w )=sup [Equation]
Revised: ≤

Original: [Equation]
Revised: [Equation]

Original: ∇ φ (cid:96) ( w ; φ, ˆ p ( t ) ( x )) (cid:13)(cid:13) 2 , then we have by Cauchy-Schwarz inequality: [Equation]
Revised: µ (cid:107) [Equation]

Original: If we further assume that the function class F satisﬁes an (cid:96) 2 /(cid:96) 4 condition that:
Revised: If we further assume that the function class F satisﬁes an (cid:96) 2 /(cid:96) 4 condition that:

Original: [Equation]
Revised: [Equation]

Original: then w.p. 1 − ζ :
Revised: then w.p [Equation] 1 − ζ :

Original: [Equation]
Revised: [Equation]

Original: Applying Lemma 2 for any ˆ p ( t ) , ˆ γ ( t ) and averaging the ﬁnal inequality we get:
Revised: Applying Lemma 4 for any ˆ p ( t ) , ˆ γ ( t ) and averaging the ﬁnal inequality we get:

Original: [Equation]
Revised: [Equation]

Original: Plugging in the bound above to Lemma 2 and applying the AM-GM inequality and Jensen’s inequality, yields:
Revised: Plugging in the bound above to Lemma 4 and applying the AM-GM inequality and Jensen’s inequality

Original: [Equation]
Revised: [Equation]

Original: t = (cid:114) E (cid:104)(cid:13)(cid:13) ( Y − ˆ p ( t ) ( X )) (cid:62) ˆ γ ( t ) ( X ) (cid:13)(cid:13) 4 2 (cid:105) (cid:33)(cid:33) +σ O (cid:32) B B (cid:88) t
Revised: yields:

Original: = (cid:107) ( γ f 0 , ˆ p ( t ) − ˆ γ ( t ) ) (cid:62) (ˆ p ( t ) − p 0 ) (cid:107) 22 ,(cid:33) .
Revised: [Equation]

Original: F P ROOF OF T HM . 4: B IASED SGD ANALYSIS
Revised: H P ROOF OF T HM . 6: B IASED SGD ANALYSIS

Original: Below, for any integer s , we deﬁne the operator norm of any vector v ∈ R s and any tensor T operating on R s as (cid:107) v (cid:107) op (cid:44) (cid:107) v (cid:107)and (cid:107) T (cid:107) op
Revised: Below, for any integer s , we deﬁne the operator norm of any vector v ∈ R s and any tensor T operating on R s as (cid:107) v (cid:107) op (cid:44) (cid:107) v (cid:107)and (cid:107) T (cid:107) op

Original: (cid:44) sup v : (cid:107) v (cid:107) 2 =1 (cid:107) T
Revised: (cid:44) sup v : (cid:107) v (cid:107) 2 =1 (cid:107) T

Original: [ v ] (cid:107) op .
Revised: [ v ] (cid:107) op .

Original: Recall the deﬁnition
Revised: Recall the deﬁnition

Original: [Equation]
Revised: [Equation]

Original: Observe that since E [ Y | X = x ]= p 0 ( x ) , we can write for any γ :
Revised: Observe that since E [ Y | X = x ]= p 0 ( x ) , we can write for any γ :

Original: [Equation]
Revised: [Equation]

Original: Thus we also have that:
Revised: Thus we also have that:

Original: [Equation]
Revised: [Equation]

Original: Given this observation, we can decompose the gradient that is used in our SGD algorithm into a bias and variance component, when viewed from the perspective of a biased SGD algorithm for the population oracle loss:
Revised: Given this observation, we can decompose the gradient that is used in our SGD algorithm into a bias and variance component, when viewed from the perspective of a biased SGD algorithm for the population oracle loss:

Original: [Equation]
Revised: [Equation]

Original: The following two lemmas bound the gradient bias and noise terms.
Revised: The following two lemmas bound the gradient bias and noise terms.

Original: Lemma 5 (Gradient bias) .
Revised: Lemma 7 (Gradient bias) .

Original: If sup x,φ,π (cid:107) E [ ∇ ππφ (cid:96) ( W ; φ,π ) | X
Revised: If sup x,φ,π (cid:107) E [ ∇ ππφ (cid:96) ( W ; φ,π ) | X

Original: = x ] (cid:107) op ≤ M , then for any parameter vector θ and functions p and γ , we have:
Revised: = x ] (cid:107) op ≤ M , then for any parameter vector θ and functions p and γ , we have:

Original: [Equation]
Revised: [Equation]

Original: Proof
Revised: Proof

Original: By Taylor’s theorem with integral remainder and Lagrange remainder respectively the SGD bias for each parameter i takes the form
Revised: By Taylor’s theorem with integral remainder and Lagrange remainder respectively the SGD bias for each parameter i takes the form

Original: [Equation]
Revised: [Equation]

Original: Furthermore, our operator norm assumption and Cauchy-Schwarz imply
Revised: Furthermore, our operator norm assumption and Cauchy-Schwarz imply

Original: [Equation]
Revised: [Equation]

Original: Thus
Revised: Thus

Original: [Equation]
Revised: [Equation]

Original: by the triangle inequality and Jensen’s inequality we ﬁnd that
Revised: by the triangle inequality and Jensen’s inequality we ﬁnd that

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Revised: ∇ θ f (cid:62) θ ( q f θ ,p − γ ) (cid:62) ( p − p 0 ) (cid:13)(cid:13) 2 , 2 + M 2 (cid:107)∇ θ f θ (cid:107) F, 2 (cid:107) p − p 0 (cid:107) 22 , 4 .

Original: Lemma 6 (Gradient Variance) .
Revised: Lemma 8 (Gradient Variance) .

Original: Deﬁne For any parameter θ and functions p and γ , 
Revised: Deﬁne For any parameter θ and functions p and γ , (cid:113) E

Original: [Equation]{+[ (cid:107) n ( W ; θ,p,γ ) (cid:107) 22 ] ≤ σ 0 ( θ )+ (cid:113) E [ (cid:107)∇ θ f θ ( X ) (cid:62) γ ( X ) (cid:62) ( Y − p 
 ( X )) (cid:107) 22 ]+ (cid:13)(cid:13) ∇ θ f (cid:62)+}
Revised: {+[ (cid:107) n ( W ; θ,p,γ ) (cid:107) 22 ] ≤ σ 0 ( θ )+ (cid:113) E [ (cid:107)∇ θ f θ ( X ) (cid:62) γ ( X ) (cid:62) ( Y − p 
 ( X )) (cid:107) 22 ]+ (cid:13)(cid:13) ∇ θ f (cid:62)+}

Revised: θ ( γ f θ ,p − γ ) (cid:62) ( p − p 0 )

Revised: (cid:13)(cid:13) 2 2 , 2 .

Original: Proof
Revised: Proof

Original: For each i ∈ [ d ] , deﬁne the shorthand
Revised: For each i ∈ [ d ] , deﬁne the shorthand

Original: [Equation]
Revised: [Equation]

Original: X

Original: [Equation] p [Equation] X )) and 
Revised: for some convex combination ¯ p ( X ) of p ( X ) and p 0 ( X ) .

Original: [Equation] [Table] E
Revised: We begin by bounding the target expectation using Cauchy-Schwarz

Original: [Equation]
Revised: [Equation]

Original: We next employ the law of total variance to rewrite the variance terms:
Revised: We next employ the law of total variance to rewrite the variance terms:

Original: [Equation]
Revised: [Equation]

Original: [Table] Var [Table] Finally, we control Var[ Z i ] using Cauchy-Schwarz (cid:115)(cid:88)
Revised: Finally, we control Var[ Z i ] using Cauchy-Schwarz (cid:115)(cid:88)

Original: [Table]
Revised: [Equation]

Original: The two claims of Thm. 4 now follow from Theorems 2 and 3 of Ajalloeian & Stich (2020) respectively, with the parameters σand ζ instantiated with quantities σ 2 ( γ ) and ζ ( γ ) of Lemmas 5 and6.
Revised: The two claims of Thm. 6 now follow from Theorems 2 and 3 of Ajalloeian & Stich (2020) respectively, with the parameters σand ζ instantiated with quantities σ 2 ( γ ) and ζ ( γ ) of Lemmas 7 and

Original: G E XPERIMENT D ETAILS AND A DDITIONAL R ESULTS G. S YNTHETIC EXPERIMENT
Revised: . I E XPERIMENT D ETAILS AND A DDITIONAL R ESULTS I. T ABULAR DATA

Original: The synthetic data is generated from a distribution P that is a mixture of two Gaussians with equal weights. withmeans at ± ( r,r,...,r ) ∈ R 10 , where r ∈ R controls the separation between the two Gaussian distributions. For each x , the label y is index of the Gaussian from which x was drawn.
Revised: We use cross-ﬁtting with 10 folds.

Original: Given this generative process, the Bayes classiﬁer has the form p 0 ( x )= p (y =1 | x )= 1 1+ e θTx for some vector θ ∈ R 10 . We set r =0 . 25 , and train with gradient descent for 40 epochs, with learning rate 0 .
Revised: The student is trained using the SEL loss with clipped teacher class probabilities max(ˆ p ( x ) ,(cid:15) ) for (cid:15) =10 − 3 .

Original: The λ hyperparameter of the γ -correction was chosen by cross-validation.
Revised: The λ hyperparameter of the loss correction was chosen by cross-validation with 5 folds.

Original: We repeat the experiments 5 times to measure the mean and standard deviation.
Revised: We repeat the experiments 5 times to measure the mean and standard deviation.

Original: G. T ABULAR DATA

Original: We use cross-ﬁtting with 10 folds.

Original: The λ hyperparameter of the γ -correction was chosen by cross-validation.

Original: We repeat the experiments 5 times to measure the mean and standard deviation.

Original: For the overﬁtting experiment, we use a random forest with 500 trees as the teacher, and a random forest with 1-40 trees as the student.
Revised: For the overﬁtting experiment, we use a random forest with 500 trees as the teacher and a random forest with 1-40 trees as the student.

Original: The student is trained to match the class log probability log(ˆ p ( x )) of the teacher.

Original: When ˆ p ( x )=0 , we clip its value to 10 − 3 .

Original: We also evaluate the impact of γ correction compared to without correction (i.e., training on just the teacher’s probabilities), especially when the teacher has high bias.

Original: We vary the teacher’s bias by limited the  tree depthin the teacher (from 1 to 20).
Revised: We also evaluate the impact of teacher underﬁtting by limiting the teacher’s maximum tree depth (from 1 to 20).

Original: Lower depth corresponds to more bias.
Revised: Lower depth corresponds to greater underﬁtting.

Original: The teacher has 100 trees and the student has 10 trees.
Revised: The teacher has 100 trees, and the student has 10 trees.

Original: For all of the dataset, γ correction successfully reduces the teacher’s bias and thus improves the student’s performance.
Revised: For all of the datasets, loss correction successfully mitigates the teacher’s underﬁtting and thus improves the student’s performance.

Original: The effect is most pronounced when the teacher has large bias (low tree depth).
Revised: The effect is most pronounced when the teacher underﬁts more heavily (has lower tree depth).

Original: We show the full results for all 5 of the datasets in Figs.
Revised: We show the full results for all 5 of the datasets in Figs.

Original: 4 and5.
Revised: 4 and

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: G. E XPERIMENT ON IMAGE DATA (CIFAR-10)
Revised: I. E XPERIMENT ON IMAGE DATA (CIFAR-10)

Original: We use SGD with initial learning rate 0.1 and momentum 0.9, batch size 128, to train for 200 epochs.
Revised: We use SGD with initial learning rate 0.1 and momentum 0.9, batch size 128, to train for 200 epochs.

Original: We use the standard learning rate decay schedule, where the learning rate is divided by 5 at epoch 60, 120, and 160.
Revised: We use the standard learning rate decay schedule, where the learning rate is divided by 5 at epoch 60, 120, and 160.

Original: Ablation on the effect of hyperparameter α

Original: The hyperparameter α controls the tradeoff between bias and variance.

Original: When α is very small, the objective is close to the vanilla KD objective.

Original: When α is large, the objective is closer to the Neyman-orthogonal loss.

Original: In Figure 6, we show the effect of varying α , with ResNet-8 as the student and ResNet-20 as the teacher, on the CIFARdataset.

Original: Large values of α lead to high variance and thus lower test accuracy.

Original: Intermediate values of α improves on the vanilla

Original: KD objective, which corresponds to α =0 .

Original: [-The test accuracy drops sharply beyond some threshold of α as the variance becomes too high (due to the terms q ˆ p ( x )=diag (cid:16) 
-]
Revised: [-The test accuracy drops sharply beyond some threshold of α as the variance becomes too high (due to the terms q ˆ p ( x )=diag (cid:16) 
-]

Original: ˆ p 1 ( x ) ,..., 1 ˆ p K

Original: ( x ) (cid:17) ), causing training to become unstable.

Original: [Figure]