Original: Active Labeling: Streaming Stochastic Gradients
Revised: Active Labeling: Streaming Stochastic Gradients

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AﬃliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: The workhorse of machine learning is stochastic gradient descent.
Revised: The workhorse of machine learning is stochastic gradient descent.

Original: To accessstochastic gradients, it is common to consider iteratively input/output pairs of atraining dataset.
Revised: To accessstochastic gradients, it is common to consider iteratively input/output pairs of atraining dataset.

Original: Interestingly, it appears that one does not need full supervisionto access stochastic gradients, which is the main motivation of this paper.
Revised: Interestingly, it appears that one does not need full supervisionto access stochastic gradients, which is the main motivation of this paper.

Original: Afterformalizing the “active labeling” problem, which generalizes active learning basedon partial supervision, we provide a streaming technique that provably minimizesthe ratio of generalization error over  number of samples.
Revised: Afterformalizing the “active labeling” problem, which focuses on active learning withpartial supervision, we provide a streaming technique that provably minimizesthe ratio of generalization error over the number of samples.

Original: We illustrate our techniquein depth for robust regression.
Revised: We illustrate ourtechnique in depth for robust regression.

Original:  Introduction
Revised:  Introduction

Original: A large amount of the current hype around artiﬁcial intelligence was fueled by the recent successesof supervised learning.
Revised: A large amount of the current hype around artificial intelligence was fueled by the recent successesof supervised learning.

Original: Supervised learning consists in designing an algorithm that maps inputs tooutputs by learning from a set of input/output examples.
Revised: Supervised learning consists in designing an algorithm that maps inputs tooutputs by learning from a set of input/output examples.

Original: When accessing many samples, and givenenough computation power, this framework is able to tackle complex tasks.
Revised: When accessing many samples, and givenenough computation power, this framework is able to tackle complex tasks.

Original: Interestingly, many ofthe diﬃculties arising in practice do not emerge from choosing the right statistical model to solvethe supervised learning problem, but from the problem of collecting and cleaning enough data (see
Revised: Interestingly, many ofthe difficulties arising in practice do not emerge from choosing the right statistical model to solvethe supervised learning problem, but from the problem of collecting and cleaning enough data (see

Original: Chapters 1 and 2 of Géron, 2017, for example).
Revised: Chapters 1 and 2 of Géron, 2017, for example).

Original: Those diﬃculties are not disjoint from the currenttrends towards data privacy regulations (Council of European Union, 2016).
Revised: Those difficulties are not disjoint from the currenttrends toward data privacy regulations (Council of European Union, 2016).

Original: This fact motivates thiswork, where we focus on how to eﬃciently collect information to carry out the learning process.
Revised: This fact motivates thiswork, where we focus on how to efficiently collect information to carry out the learning process.

Original: In this paper, we formalize the “active labeling” problem for weak supervision, where the goal isto learn a target function by acquiring the most informative dataset given a restricted budget forannotation.
Revised: In this paper, we formalize the “active labeling” problem for weak supervision, where the goal isto learn a target function by acquiring the most informative dataset given a restricted budget forannotation.

Original: We focus explicitly on weak supervision that comes as a set of label candidates foreach input, aiming to partially supervise input data in the most eﬃcient way to guide a learningalgorithm.
Revised: We focus explicitly on weak supervision that comes as a set of label candidates foreach input, aiming to partially supervise input data in the most efficient way to guide a learningalgorithm.

Original: We also restrict our study to the streaming variant where, for each input, only a singlepartial information can be collected about its corresponding output.
Revised: We also restrict our study to the streaming variant where, for each input, only a singlepartial information can be collected about its corresponding output.

Original: The crux of this work is toleverage the fact that full supervision is not needed to acquire unbiased stochastic gradients, andperform stochastic gradient descent.
Revised: The crux of this work is toleverage the fact that full supervision is not needed to acquire unbiased stochastic gradients, andperform stochastic gradient descent.

Original: The following summarizes our contributions.
Revised: The following summarizes our contributions.

Original: First, we introduce the “active labeling” problem, which is a relevant theoretical framework that encompasses many useful problems encountered by practitioners trying to annotate their data in the most eﬃcient fashion, as well as its streaming variation, in order to deal withprivacy preserving issues.
Revised: First, we introduce the “active labeling” problem, which is a relevant theoretical framework that encompasses many useful problems encountered by practitioners trying to annotate their data in the most efficient fashion, as well as its streaming variation, in order to deal with privacy preserving issues.

Original: This is the focus of Section 2.
Revised: This is the focus of Section 2.

Original: Then, in Section 3, we give a high level framework to access unbiased stochastic gradientswith weak information only.
Revised: Then, in Section 3, we give a high-level framework to access unbiased stochastic gradientswith weak information only.

Original: This provides a simple solution to the streaming “active labeling”problem.
Revised: This provides a simple solution to the streaming “active labeling”problem.

Original: Finally, we detail this framework for a robust regression task in Section 4, and provide analgorithm whose optimality is proved in Section 5.
Revised: Finally, we detail this framework for a robust regression task in Section 4, and provide analgorithm whose optimality is proved in Section 5.

Original: As a proof of concept, we provide numerical simulations in Section 6.
Revised: As a proof of concept, we provide numerical simulations in Section 6.

Original: We conclude with a high-leveldiscussion around our methods in Section 7.
Revised: We conclude with a high-leveldiscussion around our methods in Section 7.

Original: Related work.
Revised: Related work.

Original: Active query of information is relevant to many settings.
Revised: Active query of information is relevant to many settings.

Original: The most straightforwardapplications are searching games, such as Bar Kokhba or twenty questions (Walsorth, 1882).
Revised: The most straightforwardapplications are searching games, such as Bar Kokhba or twenty questions (Walsorth, 1882).

Original: Werefer to Pelc (2002) for a in-depth survey of such games, especially when liars introduce uncertainty,and their relations with coding on noisy channels.
Revised: Werefer to Pelc (2002) for an in-depth survey of such games, especially when liars introduce uncertainty,and their relations with coding on noisy channels.

Original: But applications are much more diverse, e.g. fornumerical simulation (Chevalier et al., 2014), database search (Qarabaqi and Riedewald, 2014), orshape recognition (Geman and Jedynak, 1993), to name a few.
Revised: But applications are much more diverse, e.g. fornumerical simulation (Chevalier et al., 2014), database search (Qarabaqi and Riedewald, 2014), orshape recognition (Geman and Jedynak, 1993), to name a few.

Original: In terms of motivations, many streams of research can be related to this problem, such as experimentaldesign (Chernoﬀ, 1959), statistical queries (Kearns, 1998; Fotakis et al., 2021), crowd sourcing (Doanet al., 2011), or aggregation methods in weak supervision (Ratner et al., 2020).
Revised: In terms of motivations, many streams of research can be related to this problem, such as experimentaldesign (Chernoff, 1959), statistical queries (Kearns, 1998; Fotakis et al., 2021), crowdsourcing (Doanet al., 2011), or aggregation methods in weak supervision (Ratner et al., 2020).

Original: More precisely,“active labeling” 1 consists in having several inputs and querying partial information on the labels.
Revised: More precisely,“active labeling” 1 consists in having several inputs and querying partial information on the labels.

Original: Itis close to active learning (Settles, 2010; Dasgupta, 2011; Hanneke, 2014), where there are severalinputs but exact outputs are queried; and to active ranking (Valiant, 1975; Ailon, 2011; Bravermanet al., 2019), where partial information is queried but there is only one input.
Revised: Itis close to active learning (Settles, 2010; Dasgupta, 2011; Hanneke, 2014), where there are severalinputs, but exact outputs are queried; and to active ranking (Valiant, 1975; Ailon, 2011; Bravermanet al., 2019), where partial information is queried, but there is only one input.

Original: The streaming variantintroduces privacy preserving constraints, a problem that is usually tackled through the notion ofdiﬀerential privacy (Dwork et al., 2006).
Revised: The streaming variantintroduces privacy preserving constraints, a problem that is usually tackled through the notion ofdifferential privacy (Dwork et al., 2006).

Original: In terms of formalization, we build on the partial supervision formalization of Cabannes et al.(2020), which casts weak supervision as sets of label candidates and generalizes semi-supervisedlearning (Chapelle et al., 2006).
Revised: In terms of formalization, we build on the partial supervision formalization of Cabannes et al.(2020), which casts weak supervision as sets of label candidates and generalizes semi-supervisedlearning (Chapelle et al., 2006).

Original: Finally, our sequential setting with a unique ﬁnal reward is similar tocombinatorial bandits in a pure-exploration setting (Garivier and Kaufmann, 2016; Fiez et al., 2019).
Revised: Finally, our sequential setting with a unique final reward is similar tocombinatorial bandits in a pure-exploration setting (Garivier and Kaufmann, 2016; Fiez et al., 2019).

Original:  The “active labeling” problem
Revised:  The “active labeling” problem

Original: Supervised learning is traditionally modeled in the following manner.
Revised: Supervised learning is traditionally modeled in the following manner.

Original: Consider X an input space, Yan output space, ℓ : Y × Y → R a loss function, and 𝜌 ∈ Δ X × Y a joint probability distribution.
Revised: Consider X an input space, Yan output space, ℓ : Y × Y → R a loss function, and 𝜌 ∈ Δ X × Y a joint probability distribution.

Original: Thegoal is to recover the function
Revised: Thegoal is to recover the function

Original: [Equation]
Revised: [Equation]

Original: yet, without accessing 𝜌 , but a dataset of independent samples distributed according to 𝜌 , D 𝑛 =( 𝑋 𝑖 ,𝑌 𝑖 ) 𝑖 ≤ 𝑛 ∼ 𝜌 ⊗ 𝑛 .
Revised: yet, without accessing 𝜌 , but a dataset of independent samples distributed according to 𝜌 , D 𝑛 =( 𝑋 𝑖 ,𝑌 𝑖 ) 𝑖 ≤ 𝑛 ∼ 𝜌 ⊗ 𝑛 .

Original: In practice, accessing data comes at a cost, and it is valuable to understand thecheapest way to collect a dataset allowing to discriminate 𝑓 ∗ .
Revised: In practice, accessing data comes at a cost, and it is valuable to understand thecheapest way to collect a dataset allowing to discriminate 𝑓 ∗ .

Original: We shall suppose that the input data ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 are easy to collect, yet that labeling those inputs to getoutputs ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 demands a high amount of work.
Revised: We shall suppose that the input data ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 are easy to collect, yet that labeling those inputs to getoutputs ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 demands a high amount of work.

Original: For example, it is relatively easy to scrap the webor medical databases to access radiographies, but labeling them by asking radiologists to recognizetumors on zillions of radiographies will be both time-consuming and expensive.
Revised: For example, it is relatively easy to scrap the web ormedical databases to access radiography images, but labeling them by asking radiologists to recognizetumors on zillions of radiographs will be both time-consuming and expensive.

Original: As a consequence,we assume the ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 given but the ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 unknown .
Revised: As a consequence, weassume the ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 given but the ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 unknown .

Original: As getting information on the labels comes at acost ( e.g. , paying a pool of label workers, or spending your own time), given a budget constraint, whatinformation should we query on the labels?
Revised: As getting information on the labels comes at acost ( e.g. , paying a pool of label workers, or spending your own time), given a budget constraint, whatinformation should we query on the labels?

Original: To quantify this problem, we will assume that we can sequentially and adaptively query 𝑇 informationof the type 1 𝑌 𝑖𝑡 ∈ 𝑆 𝑡 , for any index 𝑖 𝑡 ∈ { 1 , · · · , 𝑛 } and any set of labels 𝑆 𝑡 ⊂ Y (belonging to a speciﬁedset of subsets of Y ).
Revised: To quantify this problem, we will assume that we can sequentially and adaptively query 𝑇 informationof the type 1 𝑌 𝑖𝑡 ∈ 𝑆 𝑡 , for any index 𝑖 𝑡 ∈ { 1 , · · · , 𝑛 } and any set of labels 𝑆 𝑡 ⊂ Y (belonging to a specifiedset of subsets of Y ).

Original: Here, 𝑡 ∈ { 1 , · · · ,𝑇 } indexes the query sequence, and 𝑇 ∈ N is a ﬁxed budget.
Revised: Here, 𝑡 ∈ { 1 , · · · ,𝑇 } indexes the query sequence, and 𝑇 ∈ N is a fixed budget.

Original: The goal is to optimize the design of the sequence ( 𝑖 𝑡 , 𝑆 𝑡 ) in order to get the best estimate of 𝑓 ∗ interms of risk minimization (1).
Revised: The goal is to optimize the design of the sequence ( 𝑖 𝑡 , 𝑆 𝑡 ) in order to get the best estimate of 𝑓 ∗ interms of risk minimization (1).

Original: In the following, we give some examples to make this setting moreconcrete.
Revised: In the following, we give some examples to make this setting moreconcrete

Original: Example 1 (Classiﬁcation with attributes) .
Revised: Example 1 (Classification with attributes) .

Original: Suppose that a labeler is asked to provide ﬁne-grainedclasses on images (Krause et al., 2016; Zheng et al., 2019), such as the label “caracal” in Figure 6.
Revised: Suppose that a labeler is asked to provide fine-grainedclasses on images (Krause et al., 2016; Zheng et al., 2019), such as the label “caracal” in Figure 6.

Original: This would be diﬃcult for many people.
Revised: This would be difficult for many people.

Original: Yet, it is relatively easy to recognize that the imagedepicts a “feline” with “tufted-ears” and “sandy color”.
Revised: Yet, it is relatively easy to recognize that the imagedepicts a “feline” with “tufted-ears” and “sandy color”.

Original: As such, a labeler can give the weakinformation that 𝑌 belongs to the set “feline”, 𝑆 1 = { “cat”, “lion”, “tiger” , . . . } , and the set “tuftedears”, 𝑆 2 = { “Great horned owl”, “Aruacana chicken” , . . . } .
Revised: As such, a labeler can give the weakinformation that 𝑌 belongs to the set “feline”, 𝑆 1 = { “cat”, “lion”, “tiger” , . . . } , and the set “tuftedears”, 𝑆 2 = { “Great horned owl”, “Aruacana chicken” , . . . } .

Original: This is enough to recognize that𝑌 ∈ 𝑆 1 ∩ 𝑆 2 = { “caracal” } .
Revised: This is enough to recognize that𝑌 ∈ 𝑆 1 ∩ 𝑆 2 = { “caracal” } .

Original: The question 1 𝑌 ∈ 𝑆 1 , corresponds to asking if the image depicts a feline.
Revised: The question 1 𝑌 ∈ 𝑆 1 , corresponds to asking if the image depicts a feline.

Original: Literature on hierarchical classiﬁcation and autonomic taxonomy construction provides interestingideas for this problem ( e.g. , Cesa-Bianchi et al., 2006; Gangaputra and Geman, 2006).
Revised: Literature on hierarchical classification and autonomic taxonomy construction provides interestingideas for this problem ( e.g. , Cesa-Bianchi et al., 2006; Gangaputra and Geman, 2006).

Original: Example 2 (Ranking with partial ordering) .
Revised: Example 2 (Ranking with partial ordering) .

Original: Consider a problem where for a given input 𝑥 , characterizing a user, we are asked to deduce their preferences over 𝑚 items.
Revised: Consider a problem where for a given input 𝑥 , characterizing a user, we are asked to deduce their preferences over 𝑚 items.

Original: Collecting such a label requiresto know the exact ordering of the 𝑚 items induced by a user.
Revised: Collecting such a label requiresknowing the exact ordering of the 𝑚 items induced by a user.

Original: This might be hard to ask for.
Revised: This might be hard to ask for.

Original: Instead,one can easily ask the user which items they prefer in a collection of  few items.
Revised: Instead,one can easily ask the user which items they prefer in a collection of a few items.

Original: The user’s answerwill give weak information about the labels, which can be modelled as knowing 1 𝑌 𝑖 ∈ 𝑆 = 1 , for 𝑆 theset of total orderings that satisfy this partial ordering.
Revised: The user’s answerwill give weak information about the labels, which can be modeled as knowing 1 𝑌 𝑖 ∈ 𝑆 = 1 , for 𝑆 the setof total orderings that satisfy this partial ordering.

Original: We refer the curious reader to active rankingand dueling bandits for additional contents (Jamieson and Nowak, 2011; Bengs et al., 2021).
Revised: We refer the curious reader to active ranking anddueling bandits for additional contents (Jamieson and Nowak, 2011; Bengs et al., 2021).

Original: Example 3 (Pricing a product) .
Revised: Example 3 (Pricing a product) .

Original: Suppose that we want to sell a product at a consumer characterizedby some features 𝑥 , this consumer is ready to pay a price 𝑦 ∈ R for this product.
Revised: Suppose that we want to sell a product to a consumer characterizedby some features 𝑥 , this consumer is ready to pay a price 𝑦 ∈ R for this product.

Original: We price it 𝑓 ( 𝑥 ) ∈ R ,and we observe 1 𝑓 ( 𝑥 ) <
Revised: We price it 𝑓 ( 𝑥 ) ∈ R ,and we observe 1 𝑓 ( 𝑥 ) <

Original: 𝑦 , that is if the consumer is willing to buy this product at this price tag or not(Cesa-Bianchi et al., 2019; Liu et al., 2021).
Revised: 𝑦 , that is if the consumer is willing to buy this product at this price tag or not(Cesa-Bianchi et al., 2019; Liu et al., 2021).

Original: Although, in this setting, the goal is often to minimizethe regret, which contrasts with our pure exploration setting.
Revised: Although, in this setting, the goal is often to minimizethe regret, which contrasts with our pure exploration setting.

Original: As a counter-example, our assumptions are not set to deal with missing data, i.e. if some coordinatesof some input feature vectors 𝑋 𝑖 are missing (Rubin, 1976).
Revised: As a counter-example, our assumptions are not set to deal with missing data, i.e. if some coordinatesof some input feature vectors 𝑋 𝑖 are missing (Rubin, 1976).

Original: Typically, this happens when inputdata comes from diﬀerent sources ( e.g. , when trying to predict economical growth from countryinformation that are self-reported).
Revised: Typically, this happens when input datacomes from different sources ( e.g. , when trying to predict economic growth from country informationthat is self-reported).

Original: Streaming variation.
Revised: Streaming variation.

Original: The special case of the active labeling problem we shall consider consists inits variant without resampling.
Revised: The special case of the active labeling problem we shall consider consists inits variant without resampling.

Original: This corresponds to the online setting where one can only ask onequestion by sample, formally 𝑖 𝑡 = 𝑡 .
Revised: This corresponds to the online setting where one can only ask onequestion by sample, formally 𝑖 𝑡 = 𝑡 .

Original: This setting is particularly appealing for privacy concerns, insettings where the labels ( 𝑌 𝑖 ) contain sensitive information that should not be revealed totally.
Revised: This setting is particularly appealing for privacy concerns, insettings where the labels ( 𝑌 𝑖 ) contain sensitive information that should not be revealed totally.

Original: Forexample, some people might be more comfortable giving a range over a salary rather than the exactvalue; or in the context of polling, one might not call back a previous respondent characterized bysome features 𝑋 𝑖 to ask them again about their preferences captured by 𝑌 𝑖 .
Revised: Forexample, some people might be more comfortable giving a range over a salary rather than the exactvalue; or in the context of polling, one might not call back a previous respondent characterized bysome features 𝑋 𝑖 to ask them again about their preferences captured by 𝑌 𝑖 .

Original: Similarly, the streamingsetting is relevant for web marketing, where inputs model new users visiting a website, queries modelsets of advertisements chosen by an advertising company, and one observes potential clicks.
Revised: Similarly, the streamingsetting is relevant for web marketing, where inputs model new users visiting a website, queries modelsets of advertisements chosen by an advertising company, and one observes potential clicks.

Original:  Weak information as stochastic gradients
Revised:  Weak information as stochastic gradients

Original: In this section, we discuss how stochastic gradients can be accessed through weak information.
Revised: In this section, we discuss how stochastic gradients can be accessed through weak information.

Original: Suppose that we model 𝑓 = 𝑓 𝜃 for some Hilbert space Θ (cid:51) 𝜃 .
Revised: Suppose that we model 𝑓 = 𝑓 𝜃 for some Hilbert space Θ ∋ 𝜃 .

Original: With some abuse of notations, let usdenote ℓ ( 𝑥, 𝑦, 𝜃 ) : =
Revised: With some abuse of notations, let usdenote ℓ ( 𝑥, 𝑦, 𝜃 ) : =

Original: ℓ ( 𝑓 𝜃 ( 𝑥 ) , 𝑦 ) .
Revised: ℓ ( 𝑓 𝜃 ( 𝑥 ) , 𝑦 ) .

Original: We aimat to minimizing R ( 𝜃 ) =
Revised: We aim to minimize R ( 𝜃 ) =

Original: E ( 𝑋,𝑌 ) [ ℓ ( 𝑋,𝑌, 𝜃 )] .
Revised: E ( 𝑋,𝑌 ) [ ℓ ( 𝑋,𝑌, 𝜃 )] .

Original: Assume that R is diﬀerentiable (or subdiﬀerentiable) and denote its gradients by ∇ 𝜃 R .
Revised: Assume that R isdifferentiable (or sub-differentiable) and denote its gradients by ∇ 𝜃 R .

Original: Deﬁnition 1 (Stochastic gradient) .
Revised: Definition 1 (Stochastic gradient) .

Original: A stochastic gradient of R is any random function 𝐺 : Θ → Θ suchthat E [ 𝐺 ( 𝜃 )]
Revised: A stochastic gradient of R is any random function 𝐺 : Θ → Θ suchthat E [ 𝐺 ( 𝜃 )]

Original: = ∇ 𝜃 R ( 𝜃 ) .
Revised: = ∇ 𝜃 R ( 𝜃 ) .

Original: Given some stepsize function 𝛾 : N → R ∗ , a stochastic gradient descent(SGD) is a procedure, ( 𝜃 𝑡 )
Revised: Given some step size function 𝛾 : N → R ∗ , a stochastic gradient descent(SGD) is a procedure, ( 𝜃 𝑡 )

Original: ∈
Revised: ∈

Original: Θ N , initialized with some 𝜃 0 and updated as 𝜃 𝑡 + 1 =
Revised: Θ N , initialized with some 𝜃 0 and updated as 𝜃 𝑡 + 1 =

Original: 𝜃 𝑡 − 𝛾 ( 𝑡 ) 𝐺 ( 𝜃 𝑡 ) ,where the realization of 𝐺 ( 𝜃 𝑡 ) given 𝜃 𝑡 is independent of the previous realizations of 𝐺 ( 𝜃 𝑠 ) given 𝜃 𝑠 .
Revised: 𝜃 𝑡 − 𝛾 ( 𝑡 ) 𝐺 ( 𝜃 𝑡 ) ,where the realization of 𝐺 ( 𝜃 𝑡 ) given 𝜃 𝑡 is independent of the previous realizations of 𝐺 ( 𝜃 𝑠 ) given 𝜃 𝑠 .

Original: In supervised learning, SGD is usually performed with the stochastic gradients ∇ 𝜃 ℓ ( 𝑋,𝑌, 𝜃 ) .
Revised: In supervised learning, SGD is usually performed with the stochastic gradients ∇ 𝜃 ℓ ( 𝑋,𝑌, 𝜃 ) .

Original: Moregenerally, stochastic gradients are given by
Revised: Moregenerally, stochastic gradients are given by

Original: [Equation]
Revised: [Equation]

Original: for 𝜏 : T → Θ with T ⊂ 2 Θ a set of subsets of Θ , and 𝑇 a random variable on T , such that
Revised: for 𝜏 : T → Θ with T ⊂ 2 Θ a set of subsets of Θ , and 𝑇 a random variable on T , such that

Original: [Equation]
Revised: [Equation]

Revised: Stated otherwise, if you have a way to image a vector 𝜃 from partial measurement 1 𝜃 ∈ 𝑇 such thatyou can reconstruct this vector in a linear fashion (3), then it provides you a generic strategy to get aunbiased stochastic estimate of this vector from a partial measurement (2).

Original: For 𝜓 : Y → Θ a function from Y to Θ ( e.g. , 𝜓 = ∇ 𝜃 ( 𝑋, · , 𝜃 ) ), a question 1 𝜓 ( 𝑌 )∈ 𝑇 translates into aquestion 1 𝑌 ∈ 𝑆 for some set 𝑆 = 𝜓 − 1 ( 𝑇 ) ⊂
Revised: For 𝜓 : Y → Θ a function from Y to Θ ( e.g. , 𝜓 = ∇ 𝜃 ( 𝑋, · , 𝜃 ) ), a question 1 𝜓 ( 𝑌 )∈ 𝑇 translates into aquestion 1 𝑌 ∈ 𝑆 for some set 𝑆 = 𝜓 − 1 ( 𝑇 ) ⊂

Original: Y , meaning that the stochastic gradient (2) can be evaluatedfrom a single query.
Revised: Y , meaning that the stochastic gradient (2) can be evaluatedfrom a single query.

Original: As a proof of concept, we derive a generic implementation for 𝑇 and 𝜏 in
Revised: As a proof of concept, we derive a generic implementation for 𝑇 and 𝜏 in

Original: Appendix B. This provides a generic SGD scheme to learn functions from weak queries.
Revised: Appendix B. This provides a generic SGD scheme to learn functions from weak queries when there isno constraints on the sets to query.

Original: Remark 2 (Cutting plane methods) .
Revised: Remark 2 (Cutting plane methods) .

Original: While we provide here a descent method, one could also developcutting-plane/ellipsoid methods to localize 𝜃 ∗ according to weak information, which corresponds tothe techniques developed for pricing by Cohen et al.
Revised: While we provide here a descent method, one could also developcutting-plane/ellipsoid methods to localize 𝜃 ∗ according to weak information, which corresponds tothe techniques developed for pricing by Cohen et al.

Original: (2020) and related literature.
Revised: (2020) and related literature.

Original:  Median regression
Revised:  Median regression

Original: In this section, we focus on eﬃciently acquiring weak information providing stochastic gradients forregression problems.
Revised: In this section, we focus on efficiently acquiring weak information providing stochastic gradients forregression problems.

Original: In particular, we motivate and detail our methods for the absolute deviation loss.
Revised: In particular, we motivate and detail our methods for the absolute deviation loss.

Original: Motivated by seminal works on censored data (Tobin, 1958), we shall suppose that we query halfspaces .
Revised: Motivated by seminal works on censored data (Tobin, 1958), we shall suppose that we query halfspaces .

Original: For an output 𝑦 ∈ Y
Revised: For an output 𝑦 ∈ Y

Original: = R 𝑚 , and any hyper-plane 𝑧 +
Revised: = R 𝑚 , and any hyper-plane 𝑧 +

Original: 𝑢 ⊥ ⊂ R 𝑚 for 𝑧 ∈ R 𝑚 , 𝑢 ∈ S 𝑚 − 1 , we canask a labeler to tell us which half-space 𝑦 belongs to.
Revised: 𝑢 ⊥ ⊂ R 𝑚 for 𝑧 ∈ R 𝑚 , 𝑢 ∈ S 𝑚 − 1 , we canask a labeler to tell us which half-space 𝑦 belongs to.

Original: Formally, we access the quantity sign ((cid:104) 𝑦 − 𝑧, 𝑢 (cid:105))for a given unit cost .
Revised: Formally, we access the quantity sign (⟨ 𝑦 − 𝑧, 𝑢 ⟩)for a given unit cost .

Original: Least-squares.
Revised: Least-squares.

Original: For regression problems, it is common to look at the mean square lossℓ
Revised: For regression problems, it is common to look at the mean square loss

Original: [Equation]
Revised: [Equation]

Original: where 𝐷 𝑓 𝜃 ( 𝑥 ) ∈ Y ⊗ Θ denotes the Jacobian of 𝜃 → 𝑓 𝜃 ( 𝑥 ) .
Revised: where 𝐷 𝑓 𝜃 ( 𝑥 ) ∈ Y ⊗ Θ denotes the Jacobian of 𝜃 → 𝑓 𝜃 ( 𝑥 ) .

Original: In rich parametric models, it is preferableto ask questions on 𝑌 ∈ Y rather than on gradients in Θ which is a potentially much bigger space.
Revised: In rich parametric models, it is preferableto ask questions on 𝑌 ∈ Y rather than on gradients in Θ which is a potentially much bigger space.

Original: If we assume that 𝑌 and 𝑓 𝜃 ( 𝑋 ) are bounded in ℓ 2 -norm by 𝑀 ∈ R + , we can adaptEqs. (2) and (3)through the fact that for any 𝑧 ∈ Y , such that (cid:107) 𝑧 (cid:107) ≤ 2 𝑀 , as proven in Appendix B, 
Revised: Ifwe assume that 𝑌 and 𝑓 𝜃 ( 𝑋 ) are bounded in ℓ 2 -norm by 𝑀 ∈ R + , we can adapt (2) and (3) throughthe fact that for any 𝑧 ∈ Y , such that ∥ 𝑧 ∥ ≤ 2 𝑀 , as proven in Appendix B, [Equation]

Original: E 𝑈,𝑉 (cid:2) 1 (cid:104) 𝑧,𝑈 (cid:105)≥ 𝑉 · 𝑈 (cid:3) = 𝑐 1 · 𝑧, where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: for 𝑈 uniform on the sphere S 𝑚 − 1 and 𝑉 uniform on [ 0 , 2 𝑀 ] .
Revised: for 𝑈 uniform on the sphere S 𝑚 − 1 and 𝑉 uniform on [ 0 , 2 𝑀 ] .

Original: Applied to 𝑧 = 𝑓 𝜃 ( 𝑋 ) − 𝑌 , it designs a
Revised: Applied to 𝑧 = 𝑓 𝜃 ( 𝑋 ) − 𝑌 , it designs an

Original: SGD procedure by querying information of the type 1 (cid:104) 𝑌,𝑈 [Table] < 
Revised: SGD procedure by querying information of the type 1 ⟨ 𝑌,𝑈 ⟩ < ⟨ 𝑓 𝜃 ( 𝑋 ) ,𝑈 ⟩− 𝑉 .

Original: [Table]

Original: 𝑓

Original: [Table]

Original: [Equation]

Original: [Table]

Original: [Equation]

Original: A case for median regression.
Revised: A case for median regression.

Original: Motivated by robustness purposes, we will rather expand on medianregression.
Revised: Motivated by robustness purposes, we will rather expand on medianregression.

Original: In general, we would like to learn a function that, given an input, replicates the outputof I/O samples generated by the joint probability 𝜌 .
Revised: In general, we would like to learn a function that, given an input, replicates the outputof I/O samples generated by the joint probability 𝜌 .

Original: In many instances, 𝑋 does not characterize allthe sources of variations of 𝑌 , i.e. input features are not rich enough to characterize a unique output,leading to randomness in the conditional distributions ( 𝑌 | 𝑋 ) .
Revised: In many instances, 𝑋 does not characterize allthe sources of variations of 𝑌 , i.e. input features are not rich enough to characterize a unique output,leading to randomness in the conditional distributions ( 𝑌 | 𝑋 ) .

Original: When many targets can be linked to avector 𝑥 ∈ X , how to deﬁne a consensual 𝑓 ( 𝑥 ) ?
Revised: When many targets can be linked to avector 𝑥 ∈ X , how to define a consensual 𝑓 ( 𝑥 ) ?

Original: For analytical reasons, statisticians tend to use theleast-squares error which corresponds to asking for 𝑓 ( 𝑥 ) to be the mean of the distribution ( 𝑌 | 𝑋 = 𝑥 ) .
Revised: For analytical reasons, statisticians tend to use theleast-squares error which corresponds to asking for 𝑓 ( 𝑥 ) to be the mean of the distribution ( 𝑌 | 𝑋 = 𝑥 ) .

Original: Yet, means are known to be too sensitive to rare but large outputs (see e.g., Huber, 1981), and cannotbe deﬁned as good and robust consensus in a world of heavy-tailed distributions.
Revised: Yet, means are known to be too sensitive to rare but large outputs (see e.g. , Huber, 1981), and cannotbe defined as good and robust consensus in a world of heavy-tailed distributions.

Original: This contrasts withthe median, which, as a consequence, is often much more valuable to summarize a range of value.
Revised: This contrasts withthe median, which, as a consequence, is often much more valuable to summarize a range of values.

Original: For instance, median income is preferred over mean income as a population indicator (see e.g. , US
Revised: For instance, median income is preferred over mean income as a population indicator (see e.g. , US

Original: Census Bureau, 2021).
Revised: Census Bureau, 2021).

Original: Median regression.
Revised: Median regression.

Original: The geometric median is variationally deﬁned through the absolute deviationloss, leading to
Revised: The geometric median is variationally defined through the absolute deviationloss, leading to

Original: [Equation]
Revised: [Equation]

Original: Similarly to the least-squares case, we can access weakly supervised stochastic gradients through thefact that for 𝑧 ∈ S 𝑚 − 1 , as shown in Appendix B,
Revised: Similarly to the least-squares case, we can access weakly supervised stochastic gradients through thefact that for 𝑧 ∈ S 𝑚 − 1 , as shown in Appendix B,

Original: [Equation] where 
Revised: E 𝑈 [ sign (⟨ 𝑧,𝑈 ⟩) · 𝑈 ] = 𝑐 2 · 𝑧, where 𝑐 2 = E 𝑈

Original: [Equation] where 𝑈 is uniformly drawn on the sphere S 𝑚 − 1 , and Γ is the gamma function.
Revised: [ sign (⟨ 𝑒 1 ,𝑈 ⟩) · ⟨ 𝑒 1 ,𝑈 ⟩] = √ 𝜋 Γ ( 𝑚 − 12 ) 𝑚 Γ ( 𝑚 2 ) , (5) where 𝑈 is uniformly drawn on the sphere S 𝑚 − 1 , and Γ is the gamma function.

Original: This suggests
Revised: This suggests

Original: Algorithm 1.
Revised: Algorithm 1.

Original: Algorithm Data: A model 𝑓 𝜃 for 𝜃 ∈ Θ , some data ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 , a labeling budget 𝑇 , a stepsize rule 𝛾 : N → R +
Revised: Data: A model 𝑓 𝜃 for 𝜃 ∈ Θ , some data ( 𝑋 𝑖 ) 𝑖 ≤ 𝑛 , a labeling budget 𝑇 , a step size rule 𝛾 : N → R +

Original: Result: A learnt parameter ˆ 𝜃 and the predictive function ˆ 𝑓 = 𝑓 ˆ 𝜃 .
Revised: Result: A learned parameter ˆ 𝜃 and the predictive function ˆ 𝑓 = 𝑓 ˆ 𝜃 .

Original: Initialize 𝜃 0 .
Revised: Initialize 𝜃 0 .

Original: for 𝑡 ← 1 to 𝑇 do
Revised: for 𝑡 ← 1 to 𝑇 do

Original: Sample 𝑈 𝑡 uniformly on S 𝑚 − 1 .
Revised: Sample 𝑈 𝑡 uniformly on S 𝑚 − 1 .

Original: Query 𝜀 = sign ((cid:104) 𝑌 𝑡 − 𝑧,𝑈 𝑡 (cid:105)) for 𝑧 = 𝑓 𝜃 𝑡 − 1 ( 𝑋 𝑡 ) .
Revised: Query 𝜀 = sign (⟨ 𝑌 𝑡 − 𝑧,𝑈 𝑡 ⟩) for 𝑧 = 𝑓 𝜃 𝑡 − 1 ( 𝑋 𝑡 ) .

Original: Update the parameter 𝜃 𝑡 = 𝜃 𝑡 − 1 + 𝛾 ( 𝑡 ) 𝜀 · 𝑈 (cid:62) 𝑡 ( 𝐷 𝑓 𝜃 𝑡 − 1 ( 𝑋 𝑡 )) .
Revised: Update the parameter 𝜃 𝑡 = 𝜃 𝑡 − 1 + 𝛾 ( 𝑡 ) 𝜀 · 𝑈 ⊤ 𝑡 ( 𝐷 𝑓 𝜃 𝑡 − 1 ( 𝑋 𝑡 )) .

Original: Output ˆ 𝜃 = 𝜃 𝑇 , or some average, e.g. , ˆ 𝜃 = 𝑇 − 1 (cid:205)
Revised: Output ˆ 𝜃 = 𝜃 𝑇 , or some average, e.g. , ˆ 𝜃 = 𝑇 − 1 (cid:205)

Original: 𝑇𝑡 = 1 𝜃 𝑡 .
Revised: 𝑇 𝑡 = 1 𝜃 𝑡 .

Original:  Statistical analysis
Revised:  Statistical analysis

Original: In this section, we quantify the performances of Algorithm 1 by proving optimal rates of convergencewhen the median regression problem is approached with (reproducing) kernel.
Revised: In this section, we quantify the performance of Algorithm 1 by proving optimal rates of convergencewhen the median regression problem is approached with (reproducing) kernels.

Original: For simplicity, we willassume that 𝑓 ∗ can be parameterized by a linear model (potentially of inﬁnite dimension).
Revised: For simplicity, wewill assume that 𝑓 ∗ can be parametrized by a linear model (potentially of infinite dimension).

Original: Assumption 1.
Revised: Assumption 1.

Original: Assume that the solution 𝑓 ∗ : X → R 𝑚 of the median regression problemEqs. (1) and(4) can be parameterized by some separable Hilbert space H , and a bounded feature map 𝜑 : X → H ,such that, for any 𝑖 ∈ [ 𝑚 ] , there exists some 𝜃 ∗ 𝑖 ∈ H such that (cid:104) 𝑓 ∗ (·) , 𝑒 𝑖 (cid:105) Y =
Revised: Assume that the solution 𝑓 ∗ : X → R 𝑚 of the median regression problem (1) and (4)can be parametrized by some separable Hilbert space H , and a bounded feature map 𝜑 : X → H ,such that, for any 𝑖 ∈ [ 𝑚 ] , there exists some 𝜃 ∗ 𝑖 ∈ H such that ⟨ 𝑓 ∗ (·) , 𝑒 𝑖 ⟩ Y =

Original: (cid:10) 𝜃 ∗ 𝑖 , 𝜑 (·) (cid:11) H , where( 𝑒 𝑖 ) is the canonical basis of R 𝑚 .
Revised: (cid:10) 𝜃 ∗ 𝑖 , 𝜑 (·) (cid:11) H , where( 𝑒 𝑖 ) is the canonical basis of R 𝑚 .

Original: Written into matrix form, there exists 𝜃 ∗ ∈ Y ⊗ H , such that𝑓 ∗ (·)
Revised: Written into matrix form, there exists 𝜃 ∗ ∈ Y ⊗ H , such that𝑓 ∗ (·)

Original: =
Revised: =

Original: 𝜃 ∗ 𝜑 (·) .
Revised: 𝜃 ∗ 𝜑 (·) .

Original: The curious reader can easily relax this assumption in the realm of reproducing kernel Hilbert spacesfollowing the work of Pillaud-Vivien et al.
Revised: The curious reader can easily relax this assumption in the realm of reproducing kernel Hilbert spacesfollowing the work of Pillaud-Vivien et al.

Original: Under the linear model of Assumption 1, Algorithm 1is speciﬁed with 𝑢 (cid:62) 𝐷 𝑓 𝜃 ( 𝑥 ) =
Revised: Under the linear model of Assumption 1, Algorithm 1is specified with 𝑢 ⊤ 𝐷 𝑓 𝜃 ( 𝑥 ) =

Original: 𝑢 ⊗ 𝜑 ( 𝑥 ) .
Revised: 𝑢 ⊗ 𝜑 ( 𝑥 ) .

Original: Note that rather than working with Θ = Y ⊗ H which ispotentially inﬁnite-dimensional, empirical estimates can be represented in the ﬁnite-dimensionalspace Y ⊗ Span { 𝜑 ( 𝑋 𝑖 )} 𝑖 ≤ 𝑛 , and well approximated by small-dimensional spaces to ensure eﬃcientcomputations (Williams and Seeger, 2000; Meanti et al., 2020).
Revised: Note that rather than working with Θ = Y ⊗ H which ispotentially infinite-dimensional, empirical estimates can be represented in the finite-dimensionalspace Y ⊗ Span { 𝜑 ( 𝑋 𝑖 )} 𝑖 ≤ 𝑛 , and well approximated by small-dimensional spaces to ensure efficientcomputations (Williams and Seeger, 2000; Meanti et al., 2020).

Original: One of the key points of SGD is that gradient descent is so gradual that one can use noisy or stochasticgradients without loosing statistical guarantees while speeding up computations.
Revised: One of the key points of SGD is that gradient descent is so gradual that one can use noisy or stochasticgradients without loosing statistical guarantees while speeding up computations.

Original: This is especiallytrue when minimizing convex functions that are nor strongly-convex, i.e. , bounded below by aquadratic, nor smooth, i.e. , with Lipschitz-continuous gradient (see, e.g. , Bubeck, 2015).
Revised: This is especiallytrue when minimizing convex functions that are nor strongly-convex, i.e. , bounded below by aquadratic, nor smooth, i.e. , with Lipschitz-continuous gradient (see, e.g. , Bubeck, 2015).

Original: In particular,the following theorem, proven in Appendix A.1, states that Algorithm 1 minimizes the populationrisk at a speed at least proportional to 𝑂 ( 𝑇 − 1 / 2 ) .
Revised: In particular,the following theorem, proven in Appendix A.1, states that Algorithm 1 minimizes the populationrisk at a speed at least proportional to 𝑂 ( 𝑇 − 1 / 2 ) .

Original: Theorem 1 (Convergence rates) .
Revised: Theorem 1 (Convergence rates) .

Original: Under Assumption 1, and under the knowledge of 𝜅 and 𝑀 tworeal values such that E [(cid:107) 𝜑 ( 𝑋 )(cid:107) 2 ] ≤ 𝜅 2 and (cid:107) 𝜃 ∗ (cid:107) ≤ 𝑀 , with a budget 𝑇 ∈ N , a constant stepsize𝛾 = 𝑀𝜅 √ 𝑇 and the average estimate ˆ 𝜃 = 1 𝑇 (cid:205)
Revised: Under Assumption 1, and under the knowledge of 𝜅 and 𝑀 tworeal values such that E [∥ 𝜑 ( 𝑋 )∥ 2 ] ≤ 𝜅 2 and ∥ 𝜃 ∗ ∥ ≤ 𝑀 , with a budget 𝑇 ∈ N , a constant step size𝛾 = 𝑀𝜅 √ 𝑇 and the average estimate ˆ 𝜃 = 1 𝑇 (cid:205)

Original: 𝑇 − 1 𝑡 = 0 𝜃 𝑡 , Algorithm 1 leads to an estimate 𝑓 that suﬀers froman excess of risk
Revised: 𝑇 − 1 𝑡 = 0 𝜃 𝑡 , Algorithm 1 leads to an estimate 𝑓 that suffers froman excess of risk

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  where the expectation is taken with respect to the randomness of ˆ 𝜃 that depends on the dataset ( 𝑋 𝑖 ,𝑌 𝑖 )as well as the questions ( 𝑖 𝑡 , 𝑆 𝑡 ) 𝑡 ≤ 𝑇 .
Revised: (6) where the expectation is taken with respect to the randomness of ˆ 𝜃 that depends on the dataset ( 𝑋 𝑖 ,𝑌 𝑖 )as well as the questions ( 𝑖 𝑡 , 𝑆 𝑡 ) 𝑡 ≤ 𝑇 .

Original: While we give here a result for a ﬁxed stepsize, one could retake the extensive literature on SGD toprove similar results for decaying stepsizes that do not require to know the labeling budget in advance( e.g. setting 𝛾 ( 𝑡 ) ∝ 𝑡 − 1 / 2 at the expense of an extra term in log ( 𝑇 ) in front of the rates), as well asdiﬀerent averaging strategies (see e.g. , Bach, 2023).
Revised: While we give here a result for a fixed step size, one could retake the extensive literature on SGD toprove similar results for decaying step sizes that do not require to know the labeling budget in advance ( e.g. setting 𝛾 ( 𝑡 ) ∝ 𝑡 − 1 / 2 at the expense of an extra term in log ( 𝑇 ) in front of the rates), as well asdifferent averaging strategies (see e.g. , Bach, 2023).

Original: In pratice, one might not know a priori theparameter 𝑀 but could nontheless ﬁnd the right scaling for 𝛾 based on cross-validation.
Revised: In practice, one might not know a priori theparameter 𝑀 but could nonetheless find the right scaling for 𝛾 based on cross-validation.

Original: The rate in 𝑂 ( 𝑇 − 1 / 2 ) applies more broadly to all the strategies described in Section 3 as long as theloss ℓ and the parametric model
Revised: The rate in 𝑂 ( 𝑇 − 1 / 2 ) applies more broadly to all the strategies described in Section 3 as long as theloss ℓ and the parametric model

Original: 𝑓 𝜃 ensure that R ( 𝜃 ) is convex and Lipschitz-continuous.
Revised: 𝑓 𝜃 ensure that R ( 𝜃 ) is convex and Lipschitz-continuous.

Original: Although the constants appearing in front of rates depend on the complexity to reconstruct the full gradient∇ 𝜃 ℓ ( 𝑓 𝜃 ( 𝑋 𝑖 ,𝑌 𝑖 )) from the reconstruction scheme (3).
Revised: Althoughthe constants appearing in front of rates depend on the complexity to reconstruct the full gradient∇ 𝜃 ℓ ( 𝑓 𝜃 ( 𝑋 𝑖 ,𝑌 𝑖 )) from the reconstruction scheme (3).

Original: Those constants correspond to the secondmoment of the stochastic gradient.
Revised: Those constants correspond to the secondmoment of the stochastic gradient.

Original: For example, for the least-squares technique described earlier onewould have to replace 𝑐 2 by 𝑐 1 in Eq.
Revised: For example, for the least-squares technique described earlier onewould have to replace 𝑐 2 by 𝑐 1 in (6).

Original: Theorem 2, proven in Appendix A.3, states that any algorithm that accesses a fully supervisedlearning dataset of size 𝑇 cannot beat the rates in 𝑂 ( 𝑇 − 1 / 2 ) , hence any algorithm that collects weakerinformation on ( 𝑌 𝑖 ) 𝑖 ≤ 𝑇 cannot display better rates than the ones veriﬁed by Algorithm 1.
Revised: Theorem 2, proven in Appendix A.3, states that any algorithm that accesses a fully supervisedlearning dataset of size 𝑇 cannot beat the rates in 𝑂 ( 𝑇 − 1 / 2 ) , hence any algorithm that collects weakerinformation on ( 𝑌 𝑖 ) 𝑖 ≤ 𝑇 cannot display better rates than the ones verified by Algorithm 1.

Original: This provesminimax optimality of our algorithm up to constants.
Revised: This provesminimax optimality of our algorithm up to constants.

Original: Theorem 2 (Minimax optimality) .
Revised: Theorem 2 (Minimax optimality) .

Original: Under Assumption 1 and the knowledge of an upper bound on(cid:107) 𝜃 ∗ (cid:107) ≤ 𝑀 , assuming that 𝜑 is bounded by 𝜅 , there exists a universal constant 𝑐 3 such that for anyalgorithm A that takes as input D 𝑇 =
Revised: Under Assumption 1 and the knowledge of an upper bound on∥ 𝜃 ∗ ∥ ≤ 𝑀 , assuming that 𝜑 is bounded by 𝜅 , there exists a universal constant 𝑐 3 such that for anyalgorithm A that takes as input D 𝑇 =

Original: ( 𝑋 𝑖 ,𝑌 𝑖 ) [Equation] 𝑇 ∼ 𝜌 ⊗ 𝑇 for any 𝑇 ∈ N and output a parameter 𝜃 ,
Revised: ( 𝑋 𝑖 ,𝑌 𝑖 ) 𝑖 ≤ 𝑇 ∼ 𝜌 ⊗ 𝑇 for any 𝑇 ∈ N and output a parameter 𝜃 ,

Original: [Equation]
Revised: [Equation]

Original: The supremum over 𝜌 ∈ M 𝑀 has to be understood as the supremum over all distributions 𝜌
Revised: The supremum over 𝜌 ∈ M 𝑀 has to be understood as the supremum over all distributions 𝜌

Original: ∈
Revised: ∈

Original: Δ X
Revised: Δ X

Original: × Ysuch that the problem deﬁned through the risk R ( 𝑓 ; 𝜌 ) :
Revised: × Ysuch that the problem defined through the risk R ( 𝑓 ; 𝜌 ) :

Original: = E 𝜌 [ ℓ ( 𝑓 ( 𝑋 ) ,𝑌 )] is minimized for 𝑓 𝜌 thatveriﬁes Assumption 1 with (cid:107) 𝜃 ∗ (cid:107) bounded by a constant 𝑀 .
Revised: = E 𝜌 [ ℓ ( 𝑓 ( 𝑋 ) ,𝑌 )] is minimized for 𝑓 𝜌 thatverifies Assumption 1 with ∥ 𝜃 ∗ ∥ bounded by a constant 𝑀 .

Original: The same theorem applies for least-squares with a diﬀerent universal constant.
Revised: The same theorem applies for least-squares with a different universal constant.

Original: It should be notedthat minimax lower bounds are in essence quantifying worst cases of a given class of problems.
Revised: It should be notedthat minimax lower bounds are in essence quantifying worst cases of a given class of problems.

Original: Inparticular, to prove Theorem 2, we consider distributions that lead to hard problems; more speciﬁcally,we assumed the variance of the conditional distribution ( 𝑌 | 𝑋 ) to be high.
Revised: Inparticular, to prove Theorem 2, we consider distributions that lead to hard problems; more specifically,we assumed the variance of the conditional distribution ( 𝑌 | 𝑋 ) to be high.

Original: The practitioner shouldkeep in mind that it is possible to add additional structure on the solution, leverage active learningor semi-supervised strategy such as uncertainty sampling (Nguyen et al., 2021), or Laplacianregularization (Zhu et al., 2003; Cabannes et al., 2021a), and reduce the optimal rates of convergence.
Revised: The practitioner shouldkeep in mind that it is possible to add additional structure on the solution, leverage active learningor semi-supervised strategy such as uncertainty sampling (Nguyen et al., 2021), or Laplacianregularization (Zhu et al., 2003; Cabannes et al., 2021a), and reduce the optimal rates of convergence.

Original: To conclude this section, let us remark that most of our derivations could easily be reﬁned forpractitioners facing a slightly diﬀerent cost model for annotation.
Revised: To conclude this section, let us remark that most of our derivations could easily be refined forpractitioners facing a slightly different cost model for annotation.

Original: In particular, they might preferto preform batches of annotations before updating 𝜃 rather than modifying the question strategyafter each input annotation.
Revised: In particular, they might preferto perform batches of annotations before updating 𝜃 rather than modifying the question strategyafter each input annotation.

Original: This would be similar to mini-batch in gradient descent.
Revised: This would be similar to mini-batching in gradient descent.

Original: Indeed, thedependency of our result on the annotation cost model and on Assumption 1 should not be seen as alimitation but rather as a proof of concept.
Revised: Indeed, thedependency of our result on the annotation cost model and on Assumption 1 should not be seen as alimitation but rather as a proof of concept.

Original:  Numerical analysis
Revised:  Numerical analysis

Original: In this section, we illustrate the diﬀerences between our active method versus a classical passivemethod, for regression and classiﬁcation problems.
Revised: In this section, we illustrate the differences between our active method versus a classical passivemethod, for regression and classification problems.

Original: Extensive details are provided in Appendix E.
Revised: Extensive details are provided in Appendix E.

Original: Our code is available online at anonymized-url .
Revised: Our code is available online at anonymized-url .

Original: Let us begin with the regression problem that consists in estimating the function 𝑓 ∗ that maps𝑥 ∈ [ 0 , 1 ] to sin ( 2 𝜋𝑥 ) ∈ R .
Revised: Let us begin with the regression problem that consists in estimating the function 𝑓 ∗ that maps𝑥 ∈ [ 0 , 1 ] to sin ( 2 𝜋𝑥 ) ∈ R .

Original: Such a regular function, which belongs to any Hölder or Sobolev classesof functions, can be estimated with the Gaussian kernel, which would ensure Assumption 1, andthat corresponds to a feature map 𝜑
Revised: Such a regular function, which belongs to any Hölder or Sobolev classesof functions, can be estimated with the Gaussian kernel, which would ensure Assumption 1, andthat corresponds to a feature map 𝜑

Original: such that 𝑘 ( 𝑥, 𝑥 (cid:48) ) 
Revised: such that 𝑘 ( 𝑥, 𝑥 ′ ) :

Original: : = (cid:104) 𝜑

Original:  ( 𝑥 ) , 𝜑 ( 𝑥 (cid:48) ) (cid:105) = exp (− | 𝑥 − 𝑥 (cid:48) | /( 2 𝜎 2 ))
Revised: = ⟨ 𝜑 ( 𝑥 ) , 𝜑 ( 𝑥 ′ )⟩ = exp (− | 𝑥 − 𝑥 ′ | /( 2 𝜎 2 ))

Original: forany bandwidth parameter 𝜎 > 0.
Revised: forany bandwidth parameter 𝜎 > 0.

Original: On Figure 1, we focus on estimating 𝑓 ∗ given data ( 𝑋 𝑖 )
Revised: On Figure 1, we focus on estimating 𝑓 ∗ given data ( 𝑋 𝑖 )

Original: 𝑖 ∈[ 𝑇 ] thatare uniform on [ 0 , 1 ] in the noiseless setting where 𝑌 𝑖 = 𝑓 ∗ ( 𝑋 𝑖 ) , based on the minimization of theabsolute deviation loss.
Revised: 𝑖 ∈[ 𝑇 ] thatare uniform on [ 0 , 1 ] in the noiseless setting where 𝑌 𝑖 = 𝑓 ∗ ( 𝑋 𝑖 ) , based on the minimization of theabsolute deviation loss.

Original: The passive baseline consists in randomly choosing a threshold 𝑈 𝑖 ∼ N ( 0 , 1 )and acquiring the observations ( 1 𝑌 𝑖 >𝑈 𝑖 )
Revised: The passive baseline consists in randomly choosing a threshold 𝑈 𝑖 ∼ N ( 0 , 1 ) and acquiring the observations ( 1 𝑌 𝑖 >𝑈 𝑖 )

Original: 𝑖 ∈
Revised: 𝑖 ∈

Original: [ 𝑇 ] that can be cast as the observation of the half-space𝑆 𝑖 = (cid:8) 𝑦 ∈ Y (cid:12)(cid:12) 1 𝑦>𝑈 𝑖
Revised: [ 𝑇 ] that can be cast as the observation of the half-space𝑆 𝑖 = (cid:8) 𝑦 ∈ Y (cid:12)(cid:12) 1 𝑦>𝑈 𝑖

Original: = 1 𝑌 𝑖 >𝑈 𝑖 (cid:9) = :
Revised: = 1 𝑌 𝑖 >𝑈 𝑖 (cid:9) = :

Original: 𝑠 ( 𝑌 𝑖 ,𝑈 𝑖 ) .
Revised: 𝑠 ( 𝑌 𝑖 ,𝑈 𝑖 ) .

Original: In this noiseless setting, a good baseline to learn 𝑓 ∗ fromthe data ( 𝑋 𝑖 , 𝑆 𝑖 ) is provided by the inﬁmum loss characterization (see Cabannes et al., 2020) 
Revised: In this noiseless setting, a good baseline to learn 𝑓 ∗ fromthe data ( 𝑋 𝑖 , 𝑆 𝑖 ) is provided by the infimum loss characterization (see Cabannes et al., 2020) [Equation] arg

Original: [Equation]
Revised: [Equation]

Original:  where the distribution over 𝑋 corresponds to the marginal of 𝜌 over X , and the distribution over( 𝑆 | 𝑋 = 𝑥 ) is the pushforward of 𝑈 ∼ N ( 0 , 1 ) under 𝑠 ( 𝑓 ∗ ( 𝑥 ) , ·) .
Revised: 𝑦 )] , where the distribution over 𝑋 corresponds to the marginal of 𝜌 over X , and the distribution over( 𝑆 | 𝑋 = 𝑥 ) is the pushforward of 𝑈 ∼ N ( 0 , 1 ) under 𝑠 ( 𝑓 ∗ ( 𝑥 ) , ·) .

Original: The left plot on Figure 1 correspondsto an instance of SGD on such an objective based on the data ( 𝑋 𝑖 , 𝑆 𝑖 ) , while the right plot correspondsto Algorithm 1.
Revised: The left plot on Figure 1 correspondsto an instance of SGD on such an objective based on the data ( 𝑋 𝑖 , 𝑆 𝑖 ) , while the right plot correspondsto Algorithm 1.

Original: We take the same hyperparameters for both plots, a bandwidth 𝜎 = 0 .
Revised: We take the same hyperparameters for both plots, a bandwidth 𝜎 = 0 .

Original: 2 and a SGDstepsize 𝛾 = 0 .
Revised: 2 and an SGDstep size 𝛾 = 0 .

Original: We refer the curious reader to Figure 7 in Appendix E for plots illustrating thestreaming history, and to Figure 9 for “real-world” experiments.
Revised: We refer the curious reader to Figure 7 in Appendix E for plots illustrating thestreaming history, and to Figure 10 for “real-world” experiments.

Original: [Figure]
Revised: [Figure]

Original: To illustrate the versatility of our method, we approach a classiﬁcation problem through the mediansurrogate technique presented in Proposition 3.
Revised: To illustrate the versatility of our method, we approach a classification problem through the mediansurrogate technique presented in Proposition 3.

Original: To do so, we consider the classiﬁcation problemwith 𝑚 ∈ N classes, X = [ 0 , 1 ] and the conditional distribution ( 𝑌 | 𝑋 ) linearly interpolating between
Revised: To do so, we consider the classification problemwith 𝑚 ∈ N classes, X = [ 0 , 1 ] and the conditional distribution ( 𝑌 | 𝑋 ) linearly interpolating between

Original: Dirac in 𝑦 1 , 𝑦 2 and 𝑦 3 respectively for 𝑥 = 0, 𝑥 = 1 / 2 and 𝑥 = 1 and the uniform distribution for𝑥 = 1 / 4 and 𝑥 = 3 / 4; and 𝑋 uniform on X \ ([ 1 / 4 − 𝜀, 1 / 4 + 𝜀 ] ∪ [ 3 / 4 − 𝜀, 3 / 4 + 𝜀 ]) .[Figure]
Revised: Dirac in 𝑦 1 , 𝑦 2 and 𝑦 3 respectively for 𝑥 = 0, 𝑥 = 1 / 2 and 𝑥 = 1 and the uniform distribution for𝑥 = 1 / 4 and 𝑥 = 3 / 4; and 𝑋 uniform on X \ ([ 1 / 4 − 𝜀, 1 / 4 + 𝜀 ] ∪ [ 3 / 4 − 𝜀, 3 / 4 + 𝜀 ]) .

Original:  Discussion
Revised:  Discussion

Original: 7.1 Discrete output problems
Revised: 7.1 Discrete output problems

Original: Learning problems with discrete output spaces are not as well understood as regression problems.
Revised: Learning problems with discrete output spaces are not as well understood as regression problems.

Original: This is a consequence ot the complexity of dealing with combinatorial structures in contrast withcontinuous metric spaces.
Revised: This is a consequence of the complexity of dealing with combinatorial structures in contrast withcontinuous metric spaces.

Original: In particular, gradients are not deﬁned for discrete output models.
Revised: In particular, gradients are not defined for discrete output models.

Original: Thecurrent state-of-the-art framework to deal with discrete output problems is to introduce a continuoussurrogate problem whose solution can be decoded as a solution on the original problem (Bartlett et al.
Revised: Thecurrent state-of-the-art framework to deal with discrete output problems is to introduce a continuoussurrogate problem whose solution can be decoded as a solution on the original problem (Bartlett et al.

Original: For example, one could solve a classiﬁcation task with a median regression surrogate problem,which is the object of the next proposition, proven in Appendix C.
Revised: For example, one could solve a classification task with a median regression surrogate problem,which is the object of the next proposition, proven in Appendix C.

Original: Proposition 3 (Consistency of median surrogate) .
Revised: Proposition 3 (Consistency of median surrogate) .

Original: The classiﬁcation setting where Y is a ﬁnite space,and ℓ : Y × Y → R is the zero-one loss ℓ ( 𝑦, 𝑧 )
Revised: The classification setting where Y is a finite space,and ℓ : Y × Y → R is the zero-one loss ℓ ( 𝑦, 𝑧 )

Original: = 1 𝑦 ≠ 𝑧 can be solved as a regression task through thesimplex embedding of Y in R Y with the orthonormal basis ( 𝑒 𝑦 ) 𝑦 ∈ Y .
Revised: = 1 𝑦 ≠ 𝑧 can be solved as a regression task through thesimplex embedding of Y in R Y with the orthonormal basis ( 𝑒 𝑦 ) 𝑦 ∈ Y .

Original: More precisely, if 𝑔 ∗ : X → R Yis the minimizer of the median surrogate risk R 𝑆 ( 𝑔 ) =
Revised: More precisely, if 𝑔 ∗ : X → R Yis the minimizer of the median surrogate risk R 𝑆 ( 𝑔 ) =

Original: E [(cid:107) 𝑔 ( 𝑋 ) − 𝑒 𝑌 (cid:107)] , then 𝑓 ∗ : X → Y deﬁnedas 𝑓 ∗ ( 𝑥 ) =
Revised: E [∥ 𝑔 ( 𝑋 ) − 𝑒 𝑌 ∥] , then 𝑓 ∗ : X → Y definedas 𝑓 ∗ ( 𝑥 ) =

Original: arg max 𝑦 ∈ Y 𝑔 ∗ 𝑦 ( 𝑥 ) minimizes the original risk R ( 𝑓 ) =
Revised: arg max 𝑦 ∈ Y 𝑔 ∗ 𝑦 ( 𝑥 ) minimizes the original risk R ( 𝑓 ) =

Original: E [ ℓ ( 𝑓 ( 𝑋 ) ,𝑌 )] .
Revised: E [ ℓ ( 𝑓 ( 𝑋 ) ,𝑌 )] .

Original: More generally, any discrete output problem can be solved  reusing the consistent least-squaressurrogate of Ciliberto et al.
Revised: More generally, any discrete output problem can be solved by reusing the consistent least-squaressurrogate of Ciliberto et al.

Original: Algorithm 1 can be adapted to the least-squares problem based onspeciﬁcations at the beginning of Section 4.
Revised: Algorithm 1 can be adapted to the least-squares problem based on [Figure] specifications at the beginning of Section 4.

Original: This allows to use our method in an oﬀ-the-shelve fashionfor all discrete output problems.
Revised: This allows using our method in an off-the-shelve fashionfor all discrete output problems.

Original: In this setting, Theorem 1 can be reﬁned under margin conditionswhere our approach would exhibit exponential convergence rates as illustrated on Figure 2.
Revised: In this setting, Theorem 1 can be refined under margin conditionswhere our approach would exhibit exponential convergence rates as illustrated on Figure 2.

Original: As a sidenote, while we are not aware of any generic theory encompassing the absolute-deviation surrogate of
Revised: As a sidenote, while we are not aware of any generic theory encompassing the absolute-deviation surrogate of

Original: Proposition 3, we showcase its superiority over least-squares on at least two types of problems on
Revised: Proposition 3, we showcase its superiority over least-squares on at least two types of problems on

Original: Figures 3 and 4 in Appendix C.
Revised: Figures 3 and 4 in Appendix C.

Original: 7.2 Supervised learning baseline with resampling
Revised: 7.2 Supervised learning baseline with resampling

Original: When resampling is allowed a simple baseline for the active labeling problem is provided by supervisedlearning.
Revised: When resampling is allowed a simple baseline for the active labeling problem is provided by supervisedlearning.

Original: In regression problems with the query of any half-space, a method that consists in annotatingeach ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 ( 𝑇,𝜀 ) up to precision 𝜀 , before using any supervised learning method to learn 𝑓 from( 𝑋 𝑖 ,𝑌 𝑖 ) 𝑖 ≤ 𝑛 ( 𝑇,𝜀 ) could acquire 𝑛 ( 𝑇, 𝜀 ) (cid:39) 𝑇 / 𝑚 log 2 ( 𝜀 − 1 ) datapoints with a dichotomic search along alldirections, assuming 𝑌 𝑖 bounded or sub-Gaussian.
Revised: In regression problems with the query of any half-space, a method that consists in annotatingeach ( 𝑌 𝑖 ) 𝑖 ≤ 𝑛 ( 𝑇,𝜀 ) up to precision 𝜀 , before using any supervised learning method to learn 𝑓 from( 𝑋 𝑖 ,𝑌 𝑖 ) 𝑖 ≤ 𝑛 ( 𝑇,𝜀 ) could acquire 𝑛 ( 𝑇, 𝜀 ) ≃ 𝑇 / 𝑚 log 2 ( 𝜀 − 1 ) data points with a dichotomic search along alldirections, assuming 𝑌 𝑖 bounded or sub-Gaussian.

Original: In term of minimax rates, such a procedure cannotperform better than in 𝑛 ( 𝑇, 𝜀 ) − 1 / 2 + 𝜀 , the ﬁrst term being due to the statistical limit in Theorem 2,the second due to the incertitude 𝜀 on each 𝑌 𝑖 that transfers to the same level of incertitude on 𝑓 .
Revised: In terms of minimax rates, such a procedure cannotperform better than in 𝑛 ( 𝑇, 𝜀 ) − 1 / 2 + 𝜀 , the first term being due to the statistical limit in Theorem 2,the second due to the incertitude 𝜀 on each 𝑌 𝑖 that transfers to the same level of incertitude on 𝑓 .

Original: Optimizing with respect to 𝜀 yields a bound in 𝑂 ( 𝑇 − 1 / 2 log ( 𝑇 ) 1 / 2 ) .
Revised: Optimizing with respect to 𝜀 yields a bound in 𝑂 ( 𝑇 − 1 / 2 log ( 𝑇 ) 1 / 2 ) .

Original: Therefore, this not-so-naivebaseline is only sub-optimal by a factor log ( 𝑇 ) 1 / 2 .
Revised: Therefore, this not-so-naivebaseline is only suboptimal by a factor log ( 𝑇 ) 1 / 2 .

Original: In the meanwhile, Algorithm 1 can be rewrittenwith resampling, as well as Theorem 1, which we prove in Appendix A.2.
Revised: In the meanwhile, Algorithm 1 can be rewrittenwith resampling, as well as Theorem 1, which we prove in Appendix A.2.

Original: Hence, our technique willstill achieve minimax optimality for the problem “with resampling”.
Revised: Hence, our technique willstill achieve minimax optimality for the problem “with resampling”.

Original: In other terms, by decidingto acquire more imprecise information, our algorithm reduces annotation cost for a given level ofgeneralization error (or equivalently reduces generalization error for a given annotation budget) by afactor log ( 𝑇 ) 1 / 2 when compared to this baseline.
Revised: In other terms, by decidingto acquire more imprecise information, our algorithm reduces annotation cost for a given level ofgeneralization error (or equivalently reduces generalization error for a given annotation budget) by afactor log ( 𝑇 ) 1 / 2 when compared to this baseline.

Original: The picture is slightly diﬀerent for discrete-output problems.
Revised: The picture is slightly different for discrete-output problems.

Original: If one can ask any question 𝑠 ∈ 2 Y thenwith a dichotomic search, one can retrieve any label with log 2 ( 𝑚 ) questions.
Revised: If one can ask any question 𝑠 ∈ 2 Y thenwith a dichotomic search, one can retrieve any label with log 2 ( 𝑚 ) questions.

Original: Hence to theoreticallybeat the fully supervised baseline with the SGD method described in Section 3, one would have toderive a gradient strategy (2) with a small enough second moment ( e.g. , for convex losses that arenon-smooth nor strongly convex, the increase in the second moment compared to the usual stochasticgradients should be no greater than log 2 ( 𝑚 ) 1 / 2 ).
Revised: Hence, to theoreticallybeat the fully supervised baseline with the SGD method described in Section 3, one would have toderive a gradient strategy (2) with a small enough second moment ( e.g. , for convex losses that arenon-smooth nor strongly convex, the increase in the second moment compared to the usual stochasticgradients should be no greater than log 2 ( 𝑚 ) 1 / 2 ).

Original: How to best reﬁne our technique to better take intoaccount the discrete structure of the output space is an open question.
Revised: How to best refine our technique to better take intoaccount the discrete structure of the output space is an open question.

Original: Introducing bias that does notmodify convergence properties while reducing variance eventually thanks to importance sampling isa potential way to approach this problem.
Revised: Introducing bias that does notmodify convergence properties while reducing variance eventually thanks to importance sampling isa potential way to approach this problem.

Original: A simpler idea would be to remember information of thetype 𝑌 𝑖 ∈ 𝑠 to restrict the questions asked in order to locate 𝑓 𝜃 𝑡 ( 𝑋 𝑖 ) − 𝑌 𝑖 when performing stochasticgradient descent with resampling.
Revised: A simpler idea would be to remember information of the type 𝑌 𝑖 ∈ 𝑠 to restrict the questions asked in order to locate 𝑓 𝜃 𝑡 ( 𝑋 𝑖 ) − 𝑌 𝑖 when performing stochasticgradient descent with resampling.

Original: Combinatorial bandits might also provide helpful insights on thematter.
Revised: Combinatorial bandits might also provide helpful insights on thematter.

Original: Ultimately, we would like to build an understanding of the whole distribution ( 𝑌 | 𝑋 ) and notonly of 𝑓 ∗ ( 𝑋 ) as we explore labels in order to reﬁne this exploration.
Revised: Ultimately, we would like to build an understanding of the whole distribution ( 𝑌 | 𝑋 ) and notonly of 𝑓 ∗ ( 𝑋 ) as we explore labels in order to refine this exploration.

Original: 7.3 Min-max approaches
Revised: 7.3 Min-max approaches

Original: Min-max approaches have been popularized for searching games and active learning, where onesearches for the question that minimizes the size of the space where a potential guess could lie underthe worse possible answer to that question.
Revised: Min-max approaches have been popularized for searching games and active learning, where onesearches for the question that minimizes the size of the space where a potential guess could lie underthe worst possible answer to that question.

Original: A particularly well illustrative example is the solution ofthe Mastermind game proposed by Knuth (1977).
Revised: A particularly well illustrative example is the solution ofthe Mastermind game proposed by Knuth (1977).

Original: While our work leverages plain SGD, one couldbuild on the vector ﬁeld point-of-view of gradient descent (see, e.g. , Bubeck, 2015) to tackle min-maxconvex concave problems with similar guarantees.
Revised: While our work leverages plain SGD, one couldbuild on the vector field point-of-view of gradient descent (see, e.g. , Bubeck, 2015) to tackle min-maxconvex concave problems with similar guarantees.

Original: In particular, we could design weakly supervisedlosses 𝐿 ( 𝑓 ( 𝑥 ) , 𝑠 ; 1 𝑦 ∈ 𝑠 ) and min-max games where a prediction player aims at minimizing such a losswith respect to the prediction 𝑓 , while the query player aims at maximizing it with respect to thequestion 𝑠 , that is querying information that best elicit mistakes made by the prediction player.
Revised: In particular, we could design weakly supervisedlosses 𝐿 ( 𝑓 ( 𝑥 ) , 𝑠 ; 1 𝑦 ∈ 𝑠 ) and min-max games where a prediction player aims at minimizing such a losswith respect to the prediction 𝑓 , while the query player aims at maximizing it with respect to thequestion 𝑠 , that is querying information that best elicit mistakes made by the prediction player.

Original: Forexample, the dual norm characterization of the norm leads to the following min-max approach to themedian regression
Revised: Forexample, the dual norm characterization of the norm leads to the following min-max approach to themedian regression

Original: [Equation]
Revised: [Equation]

Original: Such min-max formulations would be of interest if they lead to improvement of computational andstatistical eﬃciencies, similarly to the work of Babichev et al.
Revised: Such min-max formulations would be of interest if they lead to improvement of computational andstatistical efficiencies, similarly to the work of Babichev et al.

Original: For classiﬁcation problems, thefollowing proposition introduces such a game and suggests its suitability.
Revised: For classification problems, thefollowing proposition introduces such a game and suggests its suitability.

Original: Its proof can be found in
Revised: Its proof can be found in

Original: Appendix D.
Revised: Appendix D.

Original: Proposition 4.
Revised: Proposition 4.

Original: Consider the classiﬁcation problem of learning 𝑓 ∗ : X → Y where Y is of ﬁnitecardinality, with the 0-1 loss ℓ ( 𝑧, 𝑦 )
Revised: Consider the classification problem of learning 𝑓 ∗ : X → Y where Y is of finitecardinality, with the 0-1 loss ℓ ( 𝑧, 𝑦 )

Original: = 1 𝑧 ≠ 𝑦 , minimizing the risk (1) under a distribution 𝜌 on X × Y .
Revised: = 1 𝑧 ≠ 𝑦 , minimizing the risk (1) under a distribution 𝜌 on X × Y .

Original: Introduce the surrogate score functions 𝑔 : X → Δ Y ; 𝑥
Revised: Introduce the surrogate score functions 𝑔 : X → Δ Y ; 𝑥

Original: → 𝑣 where 𝑣 = ( 𝑣 𝑦 ) 𝑦
Revised: → 𝑣 where 𝑣 = ( 𝑣 𝑦 ) 𝑦

Original: ∈ Y is a family ofnonnegative weights that sum to one, as well as the surrogate loss function 𝐿 : Δ Y × S × {− 1 , 1 } → 319 R ; ( 𝑣, 𝑆, 𝜀 ) =
Revised: ∈ Y is a family ofnon-negative weights that sum to one, as well as the surrogate loss function 𝐿 : Δ Y × S × {− 1 , 1 } → 323 R ; ( 𝑣, 𝑆, 𝜀 ) =

Original: 𝜀 ( 1 − 2 (cid:205) 𝑦 ∈ 𝑆 𝑣 𝑦 ) , and the min-max gamemin 𝑔: X → Δ Y max 𝜇 : X → Δ S E ( 𝑋 [Equation] 𝑌
Revised: 𝜀 ( 1 − 2 (cid:205) 𝑦 ∈ 𝑆 𝑣 𝑦 ) , and the min-max gamemin 𝑔

Original: [Equation]
Revised: [Equation]

Original: E

Original: [Equation]

Original: ∼ 𝜇

Original: [Equation]

Original: [Equation]

Original: 𝐿 ( 𝑓 (

Original: [Equation]

Original: [Equation] 𝑆 [Equation] When S contains the singletons and with the low-noise condition thatalmost everywhere 321 P ( 𝑌 ≠ 𝑓 ∗ ( 𝑥 ) | 𝑋 = 𝑥 ) < 1 / 2 , then 𝑓 ∗ can be learned through the relation 𝑓 ∗ ( 𝑥 ) =
Revised: When S contains the singletons and with the low-noise condition that P ( 𝑌 ≠ 𝑓 ∗ ( 𝑥 ) | 𝑋 = 𝑥 ) < 1 / 2almost everywhere, then 𝑓 ∗ can be learned through the relation 𝑓 ∗ ( 𝑥 ) =

Original: arg min 𝑦 ∈ Y 𝑔 ∗ ( 𝑥 ) 𝑦for the unique minimizer 𝑔 ∗ of (8) .
Revised: arg min 𝑦 ∈ Y 𝑔 ∗ ( 𝑥 ) 𝑦 for theunique minimizer 𝑔 ∗ of (8) .

Original: Moreover, the minimization of the empirical version of thisobjective with the stochastic gradient updates for saddle point problems provides a natural “activelabeling” scheme to ﬁnd this 𝑔 ∗ .
Revised: Moreover, the minimization of the empirical version of this objectivewith the stochastic gradient updates for saddle point problems provides a natural “active labeling”scheme to find this 𝑔 ∗ .

Original:  Conclusion
Revised:  Conclusion

Original: We have introduced the “active labeling” problem, which corresponds to “active partially supervisedlearning”.
Revised: We have introduced the “active labeling” problem, which corresponds to “active partially supervisedlearning”.

Original: We provided a solution to this problem based on stochastic gradient descent.
Revised: We provided a solution to this problem based on stochastic gradient descent.

Original: Although ourmethod can be used for any discrete output problem, we detailed how it works for median regression,where we show that it optimizes the generalization error for a given annotation budget.
Revised: Although ourmethod can be used for any discrete output problem, we detailed how it works for median regression,where we show that it optimizes the generalization error for a given annotation budget.

Original: In a nearfuture, we would like to focus on better exploiting the discrete structure of classiﬁcation problems,eventually with resampling strategies.
Revised: In a nearfuture, we would like to focus on better exploiting the discrete structure of classification problems,eventually with resampling strategies.

Original: Understanding more precisely the key issues in applications concerned with privacy, and studyinghow weak gradients might provide a good trade-oﬀ between learning eﬃciently and revealing toomuch information also provide interesting follow-ups.
Revised: Understanding more precisely the key issues in applications concerned with privacy, and studyinghow weak gradients might provide a good trade-off between learning efficiently and revealing toomuch information also provide interesting follow-ups.

Original: Finally, regarding dataset annotation, exploringdiﬀerent paradigms of weakly supervised learning would lead to diﬀerent active weakly supervisedlearning frameworks.
Revised: Finally, regarding dataset annotation, exploringdifferent paradigms of weakly supervised learning would lead to different active weakly supervisedlearning frameworks.

Original: While this work is based on partial labeling, similar formalizations could bemade based on other weak supervision models, such as aggregation ( e.g. , Ratner et al., 2020), orgroup statistics (Dietterich et al., 1997).
Revised: While this work is based on partial labeling, similar formalization could bemade based on other weak supervision models, such as aggregation ( e.g. , Ratner et al., 2020), orgroup statistics (Dietterich et al., 1997).

Original: In particular, annotating a huge dataset is often done bybagging inputs according to predicted labels and correcting errors that can be spotted on those bagsof inputs (Deng et al., 2009).
Revised: In particular, annotating a huge dataset is often done by bagging inputs according to predicted labels and correcting errors that can be spotted on those bagsof inputs (Deng et al., 2009).

Original: We left for future work the study of variants of the “active labeling”problem that model those settings. 
Revised: We left for future work the study of variants of the “active labeling”problem that model those settings. Checklist

Revised: For all authors...

Revised: (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s

Revised: contributions and scope?

Revised: [Yes]

Revised: (b) Did you describe the limitations of your work?

Revised: [Yes] See discussion section.

Revised: (c) Did you discuss any potential negative societal impacts of your work?

Revised: [N/A]

Revised: This work

Revised: aims at developping advanced techniques to learn without too much supervision.

Revised: Such

Revised: a quest of increasing AI systems capability at a reduced human labor cost is associated

Revised: with broad societal issues.

Revised: Those questions being really generic, we did not mentionthem in the main text.

Revised: (d) Have you read the ethics review guidelines and ensured that your paper conforms to

Revised: them?

Revised: [Yes]

Revised: If you are including theoretical results...

Revised: (a) Did you state the full set of assumptions of all theoretical results?

Revised: [Yes]

Revised: (b) Did you include complete proofs of all theoretical results?

Revised: [Yes]

Revised: If you ran experiments...

Revised: (a) Did you include the code, data, and instructions needed to reproduce the main

Revised: experimental results (either in the supplemental material or as a URL)?

Revised: [Yes](b) Did you specify all the training details (e

Revised: g., data splits, hyperparameters, how theywere chosen)?

Revised: [Yes]

Revised: (c) Did you report error bars (e.g., with respect to the random seed after running experiments

Revised: multiple times)?

Revised: [Yes]

Revised: (d) Did you include the total amount of compute and the type of resources used (e

Revised: g., typeof GPUs, internal cluster, or cloud provider)?

Revised: [N/A]

Revised: The experiments were run on a

Revised: personal laptop and did not require many charges.

Revised: Indeed, the amount of compute for

Revised: experiments were similar to the amount used to write this paper.

Revised: If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...

Revised: (a) If your work uses existing assets, did you cite the creators?

Revised: [Yes] Although we have

Revised: not

Revised: cited

Revised: the creators of some L A TEX libraries we used such as

Revised: Michael

Revised: Sharpe and the

Revised: newtx package which we used for fonts in our text

Revised: (b) Did you mention the license of the assets?

Revised: [N/A] Numpy and LIBSVM are under

Revised: Berkeley Software Distribution licenses (respectively the liberal and revised ones),

Revised: Python and matplotlib are under the Python Software Foundation license.

Revised: (c) Did you include any new assets either in the supplemental material or as a URL?

Revised: [Yes]

Revised: (d) Did you discuss whether and how consent was obtained from people whose data you’re

Revised: using/curating?

Revised: [N/A]

Revised: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Revised: information or offensive content?

Revised: [N/A]

Revised: If you

Revised: used

Revised: crowdsourcing or conducted research with human subjects...

Revised: (a) Did you include the full text of instructions given to participants and screenshots, if

Revised: applicable?

Revised: [N/A]

Revised: (b) Did you describe any potential participant risks, with links to Institutional Review

Revised: Board (IRB) approvals, if applicable?

Revised: [N/A]

Revised: (c) Did you include the estimated hourly wage paid to participants and the total amount

Revised: spent on participant compensation?

Revised: [N/A]