Neyman-Pearson Classification under Both Null and Alternative Distributions Shift
Authors: Mohammadreza Mousavi Kalan
Yuyang Deng
Eitan J. Neugut
Samory Kpotufe
Venue: ICLR 2026 Poster (2026)
Keywords: Imbalanced classification
Transfer Learning
Neyman-Pearson Classification.
Primary area: learning theory

Abstract: We consider the problem of transfer learning in Neyman–Pearson classification, where the objective is to minimize the error w.r.t. a distribution $\mu_1$, subject to the constraint that the error w.r.t. a distribution $\mu_0$ remains below a prescribed threshold. While transfer learning has been extensively studied in traditional classification, transfer learning in imbalanced classification such as Neyman–Pearson classification has received much less attention. This setting poses unique challenges, as both types of errors must be simultaneously controlled. Existing works address only the case of distribution shift in $\mu_1$, whereas in many practical scenarios shifts may occur in both $\mu_0$ and $\mu_1$. We derive an adaptive procedure that not only guarantees improved Type-I and Type-II errors when the source is informative, but also automatically adapt to situations where the source is uninformative, thereby avoiding negative transfer. In addition to such statistical guarantees, the procedures is efficient, as shown via complementary computational guarantees.

Published as a conference paper at ICLR 2026
NEYMAN-PEARSON CLASSIFICATION UNDER BOTH NULL AND ALTERNATIVE DISTRIBUTIONS SHIFT
Mohammadreza M. Kalan
Yuyang Deng
Univ Rennes, Ensai, CNRS, CREST-UMR 9194, F-35000 Rennes, France mohammadreza. ka www.e/aaaaaaaaaanaaaaaaaam angensai.
Columbia University, Department of Statistics yd2 82 4 cool umbia a edu
Eitan J. Neugut
Samory Kpotufe
Columbia University, Department of Statistics e eitan. ne umbi edu
Columbia University, Department of Statistics samo ry@col umbi edu
ABSTRACT
We consider the problem of transfer learning in Peearsn classification, where the objective is minimize the error w.r.t. a distribution fl, subject the constraint that the error w.r.t. distribution No remains below prescribed threshold. While transfer learning has been extensively studied in traditional clas- sification, transfer learning in imbalanced classification such as Neyman-Pearson classification has received much less attention. This setting poses unique chal- lenges, as both types of еπrогs must be simultaneously controlled. Existing works address only the case of distribution shift in μn, whereas in many practical scenar- ios shifts may occur in both μo and //1. We derive adaptive procedure that not only guarantees improved Type-I and Type-Il errors when the source is informa- tive, but also automatically adapt to situations where the source is uninformative. thereby avoiding negative transfer. In addition to such statistical guarantees, the procedures is efficient, as shown via complementary computational guarantees.
INTRODUCTION
In many applications, the objective is to learn a decision rule from data that separates two classes with distributions N4 and μ4,, whose sample sizes are often imbalanced. These applications include disease diagnosis (Myszczynska et al. 2020) Bourzac 2014), malware detection in cybersecurity (Alamro et al. K0mar Lim 2019,, and climate science, such as heavy rain detection (Folino) et al.) [023] Frame et al.) 2022). In such settings, the learner must control the error w.r.t. both classes. traditional classification, the goal is to minimize the total loss, and under class imbalance this often leads classifier with large
the rare class. A natural framework to control errors with respect to both classes is Neyman-Pearson (NP) classification (Cannon et al. 2002) Scott Nowak 2005). where the objective
minimize the error with respect μl, subject to the constraint that the eтror with respect to AAg does not exceed a pre-specified threshold a, usually small value (for example, 5% Tong et al. (2018)). ) In NP classification, even when one class is rare relative to the other, the number of available samples from both classes may still be limited. To mitigate data scarcity in the primary classification task (referred to as the targer), where samples from either class may be limited, one can make use of additional datasets (referred to as the source). However, while transfer learning has been extensively studied in traditional classification, how to effectively leverage source data in the NP classification has received far less attention.
In this work, we propose a provable procedure that effectively leverages the source under potential distribution shifts in both μo and μn to improve both Type-I error, i.e., the error w.r.t. μlo, and Type-II error, i.e., the error w.r.t. A11 Prior theoretical works (Kalan et al. 2025 Kalan & Kpotufe, 2024b) address only the special case where the source shares the same Mu with the target and the shift occurs solely in ir However, in many applications, shifts may occur in both distributions, raising the challenge of how exploit the source improve both types of error simultaneously.

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A key difficulty in the presence of a shift in Mo is that a classifier satisfying the source constraint, ie., achieving source Type-I error below a, does not necessarily satisfy the target constraint. Es- pecially when only a few target samples are available from Mo. the set of classifiers that meet the empirical a constraint on these samples may still substantially exceed the true population risk w.r.. (40- Moreover, one cannot straightforwardly use the source rule out classifiers that result in large population error w.r.t o4 in target, since the correspondence between the source constraint and the target constraint is unknown. Thus, the central challenge is how to leverage source samples con- trol the target Type-I error while simultaneously improving the target Type-Il II error- an issue that does not arise in the setting considered by prior works (Kalan et al. 2025 Kalan Kpotufe 2024b).
The procedure proposed in this work adaptively leverages the source without requiring any prior knowledge about its relatedness to the target. When the source is informative, it improves both types of errors on the target; when the source is uninformative, it matches the performance of using only target data, thereby avoiding negative transfer. Moreover, the generalization error bound derived in this work recovers the bounds established in prior works (Kalan et al. 2025 Kalan & Kpotufe
2024b) in the special case where the source and target share the same distribution (Ag. The proposed procedure consists of two stages. In the first stage, it utilizes source class-O samples to determine an effective threshold &g that aligns the source Type-I constraint with the target constraint. This step rules out classifiers that would exceed the target Type-I constraint a, while retaining those capable of achieving low Type-ll error on the target. In the second stage, it leverages source class-1 samples to further reduce the Type-II error on the target while ensuring that the Type-I error remains below the prescribed threshold.
In addition statistical guarantee for the proposed procedure, we establish a computational guar- antee by reformulating the learning procedure as a constrained optimization problem. Specifically, we design a computationally feasible procedure that reduces the learning procedure to a sequence of convex programs and leverages a stochastic convex optimization solver as an algorithmic com- ponent. We show that this optimization procedure results in a model that achieves the statistical guarantee in polynomial time.
2 RELATED WORK
Transfer learning has been extensively studied in traditional classification and regression (Li al.
2022
Cai & Wei. 2021
Kpotufe & Martinet, 2021 Tripuraneni et al. 2020, Mousavi Kalan et al.,
2020
Kalan et al., 2022 Ben-David et al. 2010 2006; Mansour et al. 2009). A popular approach
et al. (024). which assigns weights and 1 1- a to the target and source empirical losses. Another widely used approach is fine-tuning
in vanilla classification is a-ERM (Bu et al.
(Vrbančić & Podgorelec 2020), where model trained on the source domain is further refined using target data. However, NP classification requires controlling the Type-I error, leading to a constrained problem that minimizes the Type-II error under a Type-I constraint. These unconstrained methods do not address this requirement.
A closely related work is Hanneke & Kpotufe (2019), which studies transferability across domains under distribution shift and develops an adaptive transfer learning approach. They introduce a notion of transfer distance that quantifies how performance on the source domain transfers to the target, establish minimax rates under this measure, and show that their procedure adapts to the unknown transfer distance in balanced classification. However, their setting does not address imbalanced problems, where controlling both Type-I and Type-Il errors simultancously is the main challenge.
The problem of NP classification when the underlying distributions are unknown except through samples was first formulated by Cannon et al. (2002); Scoot& & Nowak (2005). who considered em- pirical risk minimization with respect to one class while constraining the empirical error on the other below pre-specified threshold. RRigollet& & Tong (2011) studied the same problem under a surrogate convex loss. Tong (2013) studied a nonparametric NP classification framework and established rates of convergence for plug-in approach based estimating the class distributions. More recently, Kalan& & Kpotufe (2024a) derived distribution-fia minimax rates for NP classification and showed that, unlike traditional classification, the problem exhibits a dichotomy between fast and slow rates.
Related to this work, in the context of transfer learning in NP classification, Kalan & Kpotuic (224b) derived minimax rates for 0-1 loss. Building this, Kalan et al. (025) introduced an
2

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implementable transfer learning procedure for NP classification using surrogate loss and derived upper bound guarantees. Both Kalan & Kpotufe (2024b); Kalan et al. (025)) focused on the re- stricted setting where the source and target share the class-0 distribution Ao and the shift occurs only in 141- Moreover, they established only statistical guarantees without addressing computational aspects. However, in this work, we consider the general setting where distribution shifts may oceur in both and μ11 We propose an adaptive procedure with statistical guarantees that also recov- ers the special case in Kalan & Kpotufe (2024b); Kalan et al. (2025). On the computational side, we reformulate the learning problem within two-stage convex programming framework and de- velop a concrete stochastic optimization procedure that outputs model with the desired excess-risk guarantee with a bound on the gradient complexity of the procedure.
3 SETUP
Let (X, Σ) be a measurable space, and let H denote a hypothesis class of measurable functions h X' IR. In the binary classification setting, each h H induces a decision rule of the form (1((x) 2 0}. which assigns a sample to either class 0 or class 1. In this paper, we adopt the NP classification framework. Classes and are generated according to distributions po and μl, respectively, and the goal is to learn a function h H. from samples drawn from these distributions. The classifier h is required to keep the error w.r.t. class (i.c., the Type-I error) below a pre-specified threshold a, while minimizing the error w.r.t class 1 (i.e., the Type-II error). We first introduce loss function w.r.t. which the Type-I and Type-II errors are defined.
Definition 1 (Surrogate Loss). We call function & TR. Rt an L-Lipschitz surrogate loss whenever the following hold: yp is monotone nondecreasing and normalized by :p00) = 1; it satisfies the Lipschitz bound (p(x)-<((p)
< Ll for all æ, y E IR; and there exists some constant C such that, for every h E H. and x E X. we have max(<p(h<z)), ((-h(x))) C.
standard assumption in learning theory, ensur- ing that the empirical loss uniformly concentrates around the population counterpart (Bartlett & Mendelson 2002). This assumption not restrictive in practice. Many commonly used surrogate losses (such as logistic or hinge losses) are clipped in implementations for numerical stability. which directly enforces boundedness. Moreover, boundedness naturally holds when the hypothesis class norm-bounded, as is typical for linear models or neural networks with bounded parameters.
The boundedness requirement in Definition
Next, we define Type-I and Type-Il errors w.r.t. a surrogate loss.
Definition 2. For surrogate loss sp. the T-Type- and sy-Tye-lll errors of h are defined as Po (h) = Ep. [x(())) and R. (h) = Ep, [xx))())) respectively.
In particular, for the indicator loss p2() 1 1[2 2 0), the emrors Rap,es and Rap. reduce to the standard Type-I and Type-ll errors.
The NP classification problem w.r.t. surrogate loss yp and pre-specified threshold is defined
min Rap (h) s.t. R.p (h) S a
(1)
NEH
and we denote by h (not necessarily unique) solution to (1. Furthermore, note that when H
contains all measurable functions from X' to R and to is the indicator loss, the NP Lemma (Lehmann)
& Lehhaann 1986 characterizes the solution to (1 as h; 21
pa()) }} 1, for suitable X,
pa(=)
under mild regularity conditions, where Po and pl denote the class densities of classes.
In a practical setting, the learner does not have access to the distributions He and μl except through observing i.i.d. samples from each class: (x(")) )m
y ~ 1o and (x(") y 141- Then, the corre-
(
sponding empirical T-Type-II and T--Tpe-III errors are
1 no
Rep (h)
(h(x!"))
Rep (h)
- M(x(")).
T20
71
showed that by the following constrained empirical risk Cannon et al. 2002 Scott Nowak
2005
minimization
= arg min Ri (h) s.t. R (h) S a + co
(2)
WCH
3

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where eo
for an appropriate constant Č one obtain that Re (h) Ropm (lad) 2 toe
and Ro (h)
V
3.1 TRANSFER LEARNING SETUP
We consider a transfer learning setting with source and target domains, where for each D E {S,T} the class and class distributions are denoted by Ho.D and M1.D. The NP classification problem w.r.t surrogate loss yp and pre-specified threshold in each domain D (S,T) is defined as
min R (...).)()))) s.I. Ripsoo (h) S a NEW
and we denote by Di a (not necessarily unique) solution to 3). Moreover, the learner has access
to i.i.d samples from each class in both domains: (x(0. no.
~ NA,DD and (x(1. D1.D ~ P1.D
di=l
Ji=l
for D E (). Then, the corresponding empirical yo T-Tppe-l and y T-TTypelIl errors for D (S,T) are
No. D Σ
R1,D
Rep.a.D (h)
)h(x!".D1))
R
(h)
Σ d ((((((.D))
TAO
TOL
The goal of the learner in transfer learning setting is learn function h H using no,s, n1.s, NO,T, R1, samples from the source and target domains, such that it performs well on the target domain. Specifically, the learner's goal to minimize the target g-Type- excess error
(h) = [Re.e (h) - Rup. ()
[u]+ == max u),
subject to the pp TyTee- error constraint Ro. (h) S + E0,7, where 60,7 is of order Ro,
1/22
In this paper. we aim to develop an adaptive procedure that, without requiring any prior knowledge about the relatedness between source and target, effectively leverages both types of samples with two guarantees. First, regardless of whether the source is related to the target, it is as good as using only target samples and ignoring the source. Hence, it avoids negative transfer. Second, whenever the source related to the target, it improves upon the performance of using only target samples.
Kalan et al. 2025); Kalan & Kpotufe (2024b) develop adaptive procedures for the special case where Mo,7 Ho.s. They stablish the bounds R. (h) Z nn
where no denotes the number of
samples from ni = Po.s = Mo,r, and
( S min
Remi r (lis,a) Rig. (IT,,)
1/93
11,.5
where p is the transfer exponent (Hanneke & Kpotufe 2019), which quantifies the relatedness be- tween source and target.
Challenges of transfer learning in general setting: When Ho.s * Mo.T the feasible set in (3) under the target constraint need not coincide with its source counterpart. Therefore, a classifier that satisfies the source T-Type-I at level a need not satisfy the target constraint. More precisely, let HD(a) = { E H R.. (h) < a} for D E (S. If po.s + pa.r, then h E Hs(a) does not imply E (a)) Furthermore, since the learner only observes samples rather than having direct access to the dis- tributions, the empirical set Hr(a) == {n h E H: Rip.po (h) S + co.r } may yield a (p- -y-TyeeII error substantially larger than a, especially when ng,r is small and consequently co, is large. In addition, one does not know in advance an appropriate value a' E [0, 1] in the source domain such that exploiting Hss(a') = {1 H Pep.on.s (h) < a' t where 60. /
é_ would lead
to improved y I performance target. Choosing too small an a' source may reduce the T error in target, but at the cost of a substantially larger Ty-pee-lI error in target.
These challenges motivate the need for an adaptive transfer learning procedure, which we develop in the next section.

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4 ADAPTIVE TRANSFER LEARNING PROCEDURE
We start with defining the smallest a' E [0, 1] such that the constrained set in the source, i.. Hs(a') contains the optimal function in the target as:
= inf HHs(a)) (a) #
(4)
where (a) C H. denote the set of solutions of Target problem (3). Next, we define the set of functions whose empirical yo- g-Type-I error the target satisfies the a constraint, and whose empir- ical co-Type-Il error in the target domain does not exceed that of the corresponding empirical risk minimizer. Specifically,
HaG = {n Ha((a) : R (M) S Re (hr a - + 6
(5)
where 61,7 = /aL.T and 1 = arg min Ra (h) The population y p-Type-I error (e
of functions in R can substantially exceed when the number of target samples from μO,T, i.e., NO,T. is small. To address this, we exploit the source domain by restricting it to a suitable subset, thereby retaining only functions with lower T-Type-I error. Specifically, we first define
ãs = inf {a' e [a. 1] Hs(a') n HiGT + 03.
(6)
and then introduce the restricted set Fi! Hss(ás) n Rir,. We will show formally later that the functions in HL' satisfy the required bound on the T-Type-I error. At this stage, we need further restrict H'' to achieve small T--Type-Il error. For this purpose, we introduce the following sets for D E (S:
Mi, == {h H'' R.p. (h) S Ri (Hl' + 2
C and R Ro (Hl' min Rep Now, equipped with these definitions, hER
where 61
v
we propose the following transfer learning procedure, which outputs a function h E H.
If Hi Pl. # 0, then choose h E Híis Hh.r.
Otherwise, choose = arg min Re (h).
(8)
heW
MAIN RESULTS
5.1
GENERALIZATION ERROR BOUNDS
We begin with the definition of Rademacher complexity, capacity measure used to ensure uniform convergence guarantees between the empirical and population risks.
Definition 3 (Rademacher Complexity (Bartlett & Mendelson 2002)): ). Let XI,..., be i.i.d. random variables drawn from distribution μ X. Let 01..... on be independent Rademacher variables, i.e. random signs with Pr(a, = +1) 1) = Pr(a; = -1)
The empirical Rademacher
complexity of H is defined as RH, = E2 sup E ox)(X.) khe
The Rademacher complexity
H is then R, (HL = Exp (70)
Assumption (Class Complexity). We assume that the complexity of the hypothesis class satisfies R, (H) < Big for some constant By capturing the complexity of H.
v
Note that when the input features are bounded, many hypothesis classes used in practice-such as linear models or neural networks with bounded 2018). Furthermore, we make the following convexity assumption on the hypothesis class, which, together with the convexity of the loss, ensures that small deviations in the T-Type-I error do not lead to large deviations in the i- IyIpee-III error. The bounds can also be derived without this assumption, though it yields tighter bounds.
Assumption (Golowich al.)
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Assumption 2 (Class Convexity). The class H is convex: given any two hypotheses h h2 E H and any (0, 1), we have 0h1 + (1 - h) E H.
Note that polynomial regression functions and majority votes over a set of basis functions are ex- amples that satisfy Assumption 2 In contrast, neural network class with fixed architecture is generally not closed under convex combinations. Nevertheless, because the Rademacher complex- ity of a class matches that its convex hull, the convex hull of a neural network class can be considered instead, which is convex (Kalan et al. 2025 Rigollet & Tong 2011)
Next, we need a notion of distance between the source and target that translates the performance of a function h E H on the source to its performance the target. We first introduce general notion of distance and then derive bounds on the generalization error in terms of this notion. This formulation is broad enough to also yield bounds expressed through the commonly used notion of transfer exponent (Hanneke & Kpotufe 2019: Kalan & Kpotufe 2024b).
Let us define HiSTTa hEHs(a)nRRrlo
arg min Repai. (h), which denotes the set of solutions minimizing
the source 40- T--Tye-III risk within the intersection of a feasible sets in the source and target. Among these solutions, let T arg maXkeRH Rep (h) denote the pivoting function serving as the reference for comparing errors across domains in the definition of transfer distance. We then define the excess error of any hypothesis h H w.r.t. this pivoting function, for D E (S,T), , as
Eh (s,TTa)) == Rup.u, (h) - Re (hi,TT.)) Using this, define the transfer modulus, which translates the performance of a function h H from source to target:
Definition 4 (Transfer Modulus). For & 2 0, the transfer moduli for s--Type- and s-Type-ll errors are defined as follows
(e) = suu (EIT(hh hāsrra)): h H. Ei h hasr,a)) se).
S-T TT = sup ( (h) : h E H. R.p (M) (b)5 5e).
Now, equipped with these definitions, we state the following theorem, which provides upper bounds on the generalization error of the proposed transfer learning procedure in Section
P0.S1
MO,T
Mo.S.
#L,
F1.5
h5
hi
H",
Figure 1: In this figure, consider one target and two sources that all share the distribution fi, while μio differs across them. All distributions Gaussian with the same variance but different means. The optimal NP classifiers are denoted by hi,,. hs and h52.a- Moreover, we assume there are sufficiently many target samples such that H..r does not intersect with h or hig... which implies that ass > ass =
Theorem Suppose that the hypothesis class H. satisfies Assumptions 7 and 2 Moreover, let 8 O and G,D va,, for E {0, 1} and E {S,T), where Č &B + 2
č
Assume further that no, 2 n1,T, that Hr(a/8 is nonempty, and that no.r is large enough so that C0. S Ta Furthermore, let & be a convex surrogate loss function, and let h denote the hypothesis
16-
returned by the procedure in Section 4 Then, with probability at least 1 - 46, we have:
(h) S min {EL.T Rip,irr (K'sT,a)) Rup. (hT,a) + + o5 (Ae1,))
min(a= + 2E0.T, o0
S-T (a 260,s)} ifa 2 as,
Rap (h) 5
min(a + 2 de S-T (ās + 2c0.s)), if < ag-
where e is a universal constant and âg the empirical threshold defined in (6) In particular, if o.S, then âg S ag.

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Note that the assumption RO,T 2 n1rr typically holds in practice, since in the NP setting class usually corresponds to the rare class, in contrast to class O. Furthermore, without this assumption, only the bound for the -p-Type-Il error would change, where the term E1,7 would be replaced by e- (fo.T + 61,r).
Remark 1. In the special case where M po,r, Theorem 7 recovers the bounds in Kalan et al. y ; Kalan & Kpotufe (2024b). In this setting, have as S by ( and og -- reduces to the identity function. Consequently, the bound for the y- Type-l error simplifies to a + eg. where co 00,5 co.r. Regarding the 4-1ype-ll excess ervor bound, in this case have h'g hig.ar Furthermore, in the case where source and target distributions are identical, i.e., 14.9 = pa,r for E {0,1}, 13, as B1, + oo, the G-Type-ll excess error converges while the upper bound on the ---Type-l error converges to a.
R1,55
Remark 2. Theorem Saanremm@uuaee guarantees the adaptivity of the proposed transfer learning procedure with- out requiring any prior knowledge of the relatedness between source and target. It recovers the performance of using only target data, thereby avoiding negative transfer when the source is un- informative. Moreover, whenever the source is related and informative, the procedure leverages it to achieve better performance compared using only the target, in terms of both up --Tpee- and Sp sp-Type-ll errors.
Remark 3. In Theorem the source effect captured by the functions Di- and &5 6--T. In contrast. Kalan et al.) 20 Kalan & Kpotufe use the notion of transfer exponent (aannkee) Kpotufe (019)) to characterize the source effect. In Appendix we show that Theorem 7 can also be expressed in terms of the transfer exponent.
Remark Figure 7 illlusraates the effect of as on reducing the target Type-I error. In this example, H = {1 2 {}: L E R} and we consider one targel and two sources that all share the distribution (A,, while pa differs across them. All distributions are Gaussian with the same variance but different means. For source-l, as, = a, which implies that it does not shrink Harr since H.s (āts) 5 Haarr and thus source-1 cannot reduce the target Type-I error. In contrast, for source-2, ass, > a, and Ht' Hs((as)) n Hia yields an improvement in the larget Type-I error.
5.2 COMPUTATIONAL GUARANTEE FOR THE TRANSFER LEARNING PROCEDURE
In this subsection, we provide a computational guarantee for the transfer learning procedure intro- duced in Section 4 by reformulating it as an optimization procedure. The optimization procedure consists of two stages: an auxiliary algorithm for computing ås in ( and a main algorithm for solving the final model i in (8). For this optimization procedure, we consider a parameterized hypothesis class H (hg : 0 E RR. |10|| < BJ.
Notations. ||-I| denotes the Euclidean (&2) norm. In this subsection, R (0) is a short hand for (ho). The notation 2() denotes the evaluation of the function f at 2, parameterized by 6.
We make the following assumption on the loss function yp and the hypothesis class.
Assumption (Convexity, Bounded and Lipschitz Gradient). We assume that ((((y 1)ha(z)) (x)) is convex in A for any x X and y E {0, 1}. In addition, the gradient is bounded, i.e. (2y 1)ha(z))1| < G, for all x E X and 0 with [10]] < B, and the gradient is Lipschitz continuous, i.e.
[|Vo<p((2)
1))+(x)) Vop((2g - 1)h (x))I < H|10 - 0'1l,
for all x E X, y E {0, 1}, and 0, 0' with ||]]], ||0'1 < B.
The above assumption is standard and holds for common loss functions and hypothesis classes, such as the logistic loss, linear classifiers, and majority votes over a set of basis functions.
First Stage: Computing ås. Define (0) = Rep.o.r (0) a C0.T, (0.s(0, a') =
Rep.ro.s (0) -a' - -p,s. and (0) = Rep.i.r Rup. (0= TT.a-co.7 6 where 0r aa arg mine Rep, (6) s.t. (0) Rep.oo.r a - C0, S 0. By definition, âg is the small- est a' 2 a such that the intersection of the sets {0: ( < 0), (0 (f, a') < 0} and {0: (0) < 0) is non-empty. This can be formulated as the following optimization problem:
min d' S.l. a' 2 a, a0) < 0.
(10)
a'

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where a'') = max {90. (0), (0, a²), 91. (0)). Problem 10 is equivalent to the follow-
ing minimax problem, in the sense that the primal solution of (1I coincides with the solution of I0 (Boyd & Vandenberghe, 2004, Section 5.2.3):
min a' Ag (0,
(11)
a' 2'20,0 X20
Algorithm 1: cP-solver(f.g.f.c)
Input: objective f(0) = 1
|2 Ecez, [(0; (); constraint g(0) [2,1 Ecez, g(0: slackness
5. emror tolerance e, failure probability 8.
Initialize: (8u. Ao) = (0, 0), 17 VN(c,,)" G supe.cris max(
(,)11,,
G p² = B² + dh
r()) Cg = supaic Ig(0; {) and
GC / log 1/2 + G
x² +
G r(a)
Cg Vog/log log
o( G Cop log log 8)
222( +
6²
for t t= 0. O..... NCc, 5) - do
Sample {s. Sg.t uniformly from 2g and 2g. respectively
0,+1 = 0, - 77( V (0,; ()))) ))+ + Xe (g(6,; )g
Oc+1 0/+1/ max { |00+ 111 /B}
Ac+1 [[1 A + 7g (g( Oz Sa.+]]
Output: i projection of N 1= Os onto the set (0 ((0) -
To solve 11 we need an approximation of a. This can be obtained by solving:
ming Re (0) s.t. 90, (0) S O, which is equivalent to ming max R (0) + (0). Fol- lowing (Mahdavi et al., 2012), we solve this problem using Stochastic Gradient Descent-Ascent (SGDA) method with projection step, as described in Algorithm (aathhmm[pPPSollll CDCCCClllve lver. This algorithm solves convex programs which takes objective and constraint (possibly inexact) and returns an e ac- curate solution. Since only have access inexact constraint g. the final projection step is onto {0) (() S } to allow some slackness.
Once an approximation Ôr,a- is obtained, we return to replacing (1.r(0) with (0,r(0) = m. (0) Re ( (0T 6,,,T, and again call Algorithm I to compute an approximation of âg. The following key quantity will appear in the analysis of Algorithm [
Definition 5. We define r(g(0)) + inf [[||]| E 0g(0), (0) = 0, 0 E dom(g))
The above notion, introduced by Mahdavi et al. (2012), measures the magnitude of the constraint gradient along the boundary of the constraint It is used in the convergence analysis to control the distance between the returned solution and the solution prior to the final projection step.
Second Stage: Solving the Final Model h. We reduce the final step of the procedure in Sec- tion which checks whether Hi,s and Hir intersect, as defined in (8) to a constrained op- timization problem. We define Gäg (0) = max (9u,r(0), go.s (8, åg) ( The set {0 : Iāg (0) < 0} coincides with the hypothesis class Ft' defined in Section We also define
(0) = R. (0) - R. 1am..(07.ce -21,D
(12)
9as (0) 0. g (0) < 0} coincides with the hypothesis class Hi' as in (8) Then, the second stage of the algorithm is to solve the following problem:
where 03.45 = arg ming R, p (0) s.t. gas (0) S 0 for D E (S,T). The set {0
<
min R (0) s.t. (0) <
(13)
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Algorithm 2: NP-Trrnnsfe--Leaming
Initialize: Gy + 2C +
er + (0) rág + r(g'(00 r(ggs + r(1z(0)), 13, + r(gs.r
Define the slackness function: E(c,r) min 28 (+ G2 /18
Tolerance setup:
es.r + 6,S, e min(f(es,, rs,)), 61Th) 6s 6 é1.
+ min(((((fee rs,r). (eee (l) (()r,
T,a-5,,r min((((s,,, rs,r), Eles, rr)
on
0 + CP-Solver R Do O, eTa-co
(0) + (0) (0) Re 100,a-00 61,. (0, 6.
+ max(ouu(()), su.s a), 01.2 (8)
Compute à: LL CP-Solver (a', g' a0(,,''),
eas (
Define the joint constraint: â (0) max (go. (0), s0,s (0, @), (n)r
Solve and S subproblems:
8- + CP-Sel ver R da. s(eT, rT), ex
8/ + CP-Solver RRpmsss ga, Elésir's, (s
Define the final constraint:
(0) ( R (0,) 2 : 188,7 (0) (a)) (7)(
Final solve h:
i CP-Solver (R IS,T, Eles,T, es.T.
Output: If Rp..i, (0) Rep.u (8's) > 2e1,s: 0 - OG else: 0 +
where (0) max {yan 9z) To approximate 03 in 12) for D E (S,T), again call Algorithm to solve ming.ga, (0)50 Rippar (f) to a certain accuracy prescribed in Algorithm 2 At last, if the approximate solution of (13) also has small source empirical Type-ll error, we return it; otherwise, we simply return target model 67. The whole algorithm (first stage + second stage) is presented in Algorithm
Theorem 2. Let Assumption hold.
With properly chosen on in Algorithm 7 then the Algorithm outputs á such that, with probability at least 1 56, the hypothesis hg satisfies the statistical guar- antee in Theorem and the number of stochastic gradient evaluations is bounded by
3 (
(CGA/L00g
B2
G²
c. .. log
G.
1 min
min
G
log
2
min
Cn
Tmin
for some C > O, where <min - min fei,s, El,,s, (8,T), Ele's, r's r's) Elerr (ag)) and Tiin = min (FT, Tägs ryy "'s, where & is defined in Algorithm
Remark (Computational Complexity). The proposed algorithm outputs model that satisfies the learning guarantee as in Theorem]] [ and the total number of (stochastic) grudient evaluations scales as min e1.s }. Roughly speaking, we call CP-So ver procedures multiple times to solve different convex programs to different accuracies, and the final gradient complexity is dominated by the worst one of them.
EXPERIMENTS
In this section, implement the proposed algorithm on two climate datasets: Yu et al. (2023), ) with 124 features, NASA POWER (2024), with features, for heavy rain detection, where source and target correspond to different locations. We multilayer perceptron with hidden layers and

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Published as a conference paper at ICLR 2026
ReLU activation. In these experiments, the number of target samples is fixed no,7 11,7 40, while the number of source samples nu.s = 11,s varies from 50 to 950, with a = 0.1. Each case is run for 10 trials, and the trained model is evaluated
target test set of size n0,7 11, 1700.
In Figure 2 the proposed Transfer Learning Algorithm (TLA) keeps the Type-I error close to the threshold, whereas the Only Target method exceeds it more and also performs poorly the Type- error. this figure, the source is informative, and TLA leverages it to outperform Only Target method without requiring prior knowledge of the relatedness between source and target.
In Figure 3, the source is not informative, and the Only Source method suffers from a high Type-II error. TLA, however, adaptively avoids negative transfer and achieves performance comparable to the Only Target method, corroborating the result in Theorem [ tive, our algorithm adaptively exploits it, and when not, it avoids negative transfer, even in the presence of potential shifts in both class-O and class-1 distributions.
Hence, when the source is informa-
Moreover, we conduct additional experiments on synthetic Gaussian data, provided in Appendix
Only Target
Only Source
TLA -x Only Target
Only Source
Target Type- II aroo
0.25
Tärget Type-1 error
0.17
0.2
0.15
0.14
0.12
5 5-10-= 1002
0.10
2- 10
0.08
250
450
650
850
250
450
650
850
R0,8
R1,8
N0,5 711,5
Figure The performance our algorithm (TLA), along with two baselines using only source data only target data-on Climate data Yuu et al. 2023). is evaluated under a Type-I error threshold of a = 0.1. In this experiment, we fix No.7 n1.T 40 and vary the number of source samples nius #1.5.
uniin
Only Targel
Only
- Only Target
Only
Target Type-Il II error
0.2
0.15
0.1
Target
10
0.2
2-10
450
650
850
850
no.55
12,,8
710,3 7k1,3
Figure 3: The performance our algorithm (TLA), along with two usine only source data only target data-on on Climate data (NASA POWER 2024), is evaluated under Type-T error threshold of a = 0.1. In this experiment, we fix Ao,7
R1,r 40 and vary the number of source samples Ro.s = 11,8.
7 CONCLUSION AND FUTURE DIRECTIONS
This work introduces an adaptive transfer learning procedure for NP classification under distribu- tion shifts both class-O and class-1 distributions. The method provides statistical guarantees that recover existing results when only the class-1 distribution shifts, and it additionally offers compu- tational guarantees through a reduction to convex programs. The procedure adapts to the unknown relatedness between source and target, improving performance when the source is informative while matching the target-only method when
not. natural future direction is to establish match- ing lower bounds for the general setting in which both class-0 and class-1 distributions may shift. Such lower bounds complement the upper bounds this work and clarify the fundamental limits of transfer in NP classification.
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ACKNOWLEDGMENT
Samory Kpotufe and Yuyang Deng were funded by the NSF through the Learning the Earth with Ar- tificial Intelligence and Physies (LEAP) Science and Technology Center (STC) (Award #2019625).
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Seyed Mohammadreza Mousavi Kalan, Mahdi Soltanolkotabi, and A Salman Avestimehr. Statis- tical minimax lower bounds for transfer learning in linear binary classification. 2022 IEEE International Symposium Information Theory (ISIT), pp. 282-287. IEEE, 2022.
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Mohammadreza Mousavi Kalan, Zalan Fabian, Salman Avestimehr, and Mahdi Soltanolkotabi. Min- imax lower bounds for transfer learning with linear and one-hidden layer neural networks. Ad- vances Neural Information Processing Systems, 33:1959-1969, 2020.
Monika A Myszczynska, Poojitha N Ojamics, Alix MB Lacoste, Daniel Neil, Amir Saffari, Richard Mead, Guillaume M Hauthergue, Joanna D> Holbrook, and Laura Ferraiuolo. Applications of machine learning to diagnosis and treatment of neurodegenerative diseases. Nature reviews neu- rology, 16(8):440 456, 2020.
NASA POWER Nasa power: Prediction of worldwide energy resource api, 2024. URL https: 7/power. larc. nasa gww. wttpaaa.ooo/ppp/temttrraaaaaaaaa hatpo//aaaaaalllp/pooo/a
Philippe Rigollet and Xin Tong. Neyman-pearson classification, convexity and stochastic con- straints. Journal of machine learning reseurch, 12(10), 2011.
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Sungduk Yu, Walter Hannah, Liran Peng, Jerry Lin, Mohamed Aziz Bhouri, Ritwik Gupta, Bjöm Lütjens, Justus C Will, Gunnar Behrens, Julius Busecke, al. Climsim: A large multi-scale dataset for hybrid physics-ml climate emulation. Advances in neural information processing 2023.
systems,
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A EXPRESSING THEOREM 1 IN TERMS OF TRANSFER EXPONENT
Theorem uses the functions ØS- and 00 to capture the source effect in the generaliztion bound. This formulation is broad enough yield bounds expressed through the notion of transfer exponent (Hanneke & Kpotufe, 2019 Kalan & Kpotufe 2024b).
Regarding the so-Type-II excess error, assume that there exist constants C₁ 0 and pt such that for all E H. ( his.r.))) 2 C1 [EI.((h (,sTT,a)+
Then, by Definition for any 2 O we have (e) S C2/P1 st/Ps, and consequently the term (e-+Tc- (,- in Theorem can be replaced with c.S the T-Tyye- emor, first we define the minimal source level that guarantees level target
1/p1 for some numerical constant c. Regarding
d'sT == an E [0, o (a') S a}.
Assume that there exist constants Co > 0 and pa > such that for all H
[Re.Po.s s (h) ( 2 Co Co (h) a1 00
Then the Ty-pe error bound takes the form
1/P0
a + min 60, c. cos a
a 2 as,
Rop (h) s
1/000
a + min C,,,; e. (las ( a3-r(a))+
os
a < as.
In particular, if Ho.s Ho,r then 0S = a, and the bound matches that of Kalan et al. (2025).
Kalan & Kpotufe (2024b).
B PROOF OF THEOREM
We begin with two lemmas. The first establishes uniform concentration bound for the difference between empirical and population errors. The second shows that the p--Type-II eπтоr can be con- trolled by the deviation in the T-Tpee error, provided that both the hypothesis class H and the loss are convex.
Lemma 1. Let 5 and let H be hypothesis class satisfying Assumption Denote by Reg. the empirical error computed from n i.idd. samples drawn from distribution μt, where i may be either Hi or ft. Then, with probability at least 1 5,
4ByL + /2llog(2/))
sup (h)
(
NEW
Vn
where the constant C' is as defined in Definition
Proof. See the proof of Proposition I in Kalan et al. 2025
which relies on symmetrization and Chapter 26).
McDiarmid's inequality (Shalev-Shwartz Ben- Ben-David 2014
Lemma 2. Suppose that the hypothesis class H. satisfies Assumption 2 and that the surrogate loss in Definition is convex. Further assume that (r(a/8) is nonempty and that E S Io. Then,
8
Rig.tr (hira--aax)
Rep.nr (hr,a) $ c. Go,
(14)
for some numerical constant c.
Proof. See the proof of Theorem 2 in Kalan et al. 2025), which shows that
((a) inf Rep (h)
heHr(a)
non-increasing convex function [0, 1].
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Consider the event that Lemma I holds simultaneously for the distributions μ4,D with i E (0, 1) and D E {S,T]; this event oceurs with probability at least 46. We divide the proof into two cases: (i) ag, and (ii) as.
Case I: a 2 ag..
First, we show that dg From 6 know that âg Hence, it suffices to establish that Hy(a) n Harr + 0. Under the considered event, we have
Hs(a) 2 Hs(a) 2 Hs(as),
so by 4 there exists h (a) such that hū E Hs(a)) On the other hand, h4 E Hagrs since
Repporr (1T S Ry (ITTa) + corr S a + co.r
(15)
and
o (hT,a) S Repir (TT + C1T Re 11T
+ 261,T,
(16)
which implies that
n and Hs(a) Har + 0.
(17)
Regarding the Ty-pe-- error bound in Theorem 1 have h Hir S H.' Hslaa) Hagp which implies that
Rap (h) S 250.s and Repe (h) S aa- 2
Together with Definition this yields the bound stated in Theorem
Regarding the -y-Typ-lIl excess error bound in Theorem first consider the case hTS.T $ H'. Since h E Hs(a) S Als(a) and h5,T,a E Hr(a)= S Hr(a), follows from 5 that B > Rgm ( + 6C1.T-
(18)
Thus,
G,7 + Rep (H'.T,)) 2 Re (hS,T
> Rep (hr + 661T
Remr -oo)) + 5EIT
2 Re (1i,,a) + 561,
(19)
where the last inequality the fact that
Rew.o. (ar S Rep.or (hT, + C0, S a
which implies a E Hr(a), together with the definition hūr Therefore,
arg min Re (h). hEHr(a)
Re (h'sT, - Rep.n (hr,a) > 4EI,T
(20)
On the other hand, for every Hy' we have
(1) Rep.u1 (h) Re (h) + E1,T B (Hl) + 3eT R.p (hi + 3ETT
$ R (hr,a) 4E
where inequality (1) holds because hia He by (17) and m (Hl) min Rown (h). Hence, heW
wh E MiT. (h)r(h) s 461.T
(21)
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Combining this with 220, conclude the g-Type-II excess error bound in Theorem for the case where H H.'
Next, consider the case harraa E H'. We first analyze the case where Hi,s n Hí.r * 0, in which the learner selects à Hiss n Mr By 21). we know that Ei,r(h) S 461,T. Moreover, since h Hí,ss have
Re (h) s Remis + E1S S R pm (H') + 331,5
2 R s (bis,T + 351,5
S Ru s (li,.T,a) + 461.8
(22)
where inequality (1) holds because HISTTO Hl' and R (Fl') min (pmm.s(h) Hence, hER
E1 (ishi)s,TTT)) = RAp., (h) R. (KG,T, S 4c1,s, which, together with Definition 4 implies
(23)
(h) S R
Re (1T,,a) +
(24)
This establishes the Ty-Tye- excess error bound in Theorem In the case H''ss n Hi,r = 0, the learner selects h = arg min Rem (h). Let h's denote function belonging to the set Hiss Since
NEW
H s H'' 0, we obtain
R (K's) R. + 261
(25)
We claim that Rep (N's) > Re (h). Otherwise, we would have
Re (h's) < Rup (I's) E1, S Re + E1, S R (h) 221.5
which contradicts (25 Therefore,
(h)) Re (h) Rup
Rup.ur (1's) - Re (1T;aa)
< R. (h$s.Taa)
(1T,,,)) + Φs ( (4c1,s).
(26)
where the last inequality follows from 24). which holds for every E Hy Hence, together with (h) S 46,,,, we conclude the T-Tyye-III excess error bound in Theorem Case II: a < as.
First, we show that âs S as. Since as a, by (6) suffices to show that (ss(ss n H. 0. By there exists hi,a E (a) such that hi,, E Hss(ss) S Hs(as). Moreover, using (15) and
16 we have hr.a E Ha,T, which completes the argument.
Regarding the o-Tyee-I error bound in Theorem I we have h Hir S H. Hs(ās) Ha.T. which implies that
Ro. (h) ás + 2 and P (h) S a 2E0.7.
Together with Definition this yields the bound stated in Theorem [
Regarding the x Ty excess enror bound in Theorem first consider the case h's & H.. Since TT,.T. E Hs(a) C Hs((ā)) Hs(ās) and h E Hr(a) S Hr((a), it follows, by the same reasoning in 18 and (9 that (20 holds. On the other hand, for all h E H we have Rep < R (h) + +EIT S Remir (hira- 7EI.
s Repmr (hi, 2c0,7 + 7EI.T
< Re (1T..a-20.77 + 8E1.T
s Rep (1i + C 61.T
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for some numerical constant c, where in the last inequality we used Lemma 2 together with the assumptions <0,r Ta and no,T 2 n1,r- Therefore, for all h Hirr H' HA,T obtain
(1,r(h) S 61,T
(27)
which, together with 20 yields the bound stated in Theorem for the case where Ti, & H'.
Next, consider the case h E H' We first analyze the case where Hiss n Hi, # 0, in which the learner selects h Hiss H'r By (27). we have (i,r(h) S C1,T Furthermore, since h Mis and h H'' which in turn establishes the p-Type-II excess error bound in Theorem
apply the same reasoning in 22 and 23 to obtain 24
In the case H1.s 7 Hi = 0, the learner selects = arg min Rep (4). The same reasoning as in
NEK
the case 2 as yields (26). and together with 27). this implies the yo- -yTTye-III excess error bound in Theorem 1
C PROOF OF THEOREM 2
In this section we will present the proof of Theorem 2 the convergence rate of our optimization procedure. We first introduce the following Theorem that establishes the convergence of our convex program solver (Algorithm D. which is an important component of our main algorithm.
G² Theorem 3. Let 0* arg ming ((0) s.I. g(0) and p² = B + rx. Assume f annd g are convex,
C, C,V3log7
G' Lipschitz. И smooth and supa.c (g c) S Cg. Suppose Eo (x+v22))VN' For Algorithm 1
assuming g(0) = 5()) - for some e E [0, col. if we choose n = VN'
co
where
VG 32vNeo -- 2H
On S min 2/3C,' 12v6(1 /2p X G 3log z 12V6C,
1og log
then with probability at least 1 we have
GC log + +
p + - C log 1
f(0) f(0))
G+ Vocy (C+Co/188)
VN - -2HEO)
(r 2HE
VN
That is, we choose & en and eo min
(G+ Co log
4H J. know f(0)
I+He
f(0)) I c and the number of total gradient evaluations is bounded by
log I
[CG log o²
G
Cg
Vogloog
log
N > G G G
Go
e
To prove Theorem 3 we need the following lemma.
Lemma 3. For Algorithm I least 1 8:
the following statement holds true for any X 2 0 with probability at
(f(0x) f(0))) + (g(0T)
2n//V X²
2
/² + 1)C + 1
C / 30g
V2p + V2p + x*)
/ /3lg ?
(X* /2p)= +
nN
VN
C 3log 2
+ =oo + v2p) +
+ X.
N
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Proof. Define L(), = Ao(0) 1メ²
- and let X arg maxx2o min f(0) + Ag(O). We first
show that |10, -01-0 + || - -X-11" < 2p", for any t E [T'. We prove this by induction. Assume
this holding for t, and for + we have
||0+11 11 = |10 19x- 1
= 110g 1 2(n9.. -0) 11 11
where 9: ( (5.e) + de A
Then for t + 1, we have:
10,+1 1- 12 - 0. 1-11 22m (V)(v + (v 6, - 2 1/ + G2)
I ||0 8¹11 2n (L(O,, Aa) L(0", x)) 2 + 1/
where Pt (0:) Xe (++) gt |. and last step we use the convexity of L(, Aa). We know that
Elg.] = f((,)) + X, V 0.). Elexp (p + +,,G eex((1)
Similarly, have:
D 2 Mi 1²2 + (0( kn,z) ydc, A, >, -²) + 1//g(00; Co.e)) >ee2 S m)) 212 -17 (L(0 X²) L(fe.
+ 2v 20 + 2v 21)/1 .+ + C2?
where
a. = - = (g(()) 5(0)) = and at last step we use the 1-strong--oonaavity of L(O,,)) ). It is easy to see that Blep (u /C,] S exp(1).
Putting pieces together we have
|10++1 1- + | - xa12 < (1x- 1²12 110 110--001)) ( (L(0,, L(0", A.))
+ 2v 2mppp + 22 + + 2n + +1 + G2)
$ (In- 212+ -10 -10--0 2n (L(O,, x²) L(0", X1))
2ー
+ 2 + 12 x + 2v 2mppppp + -: + ED)
where the last step is due to
I(O", Ae) ((0.) X²)(0.) (1(0')+++)))))))) X,)(())) 7(1²)² 2Xp
Performing telescoping sum yields:
|00+1 1-1 + 1 2112 < (106 -0-1 + 12o 121) +272 + + + V2p x2
+ 2v 2np Σ Po + 2v2mp Σ 9: + 2v 2mpteo + Nom({*)
Due to Lemma 4 of Lan 2012), we know with probability 1- 5/2.
2 N-1 Pt S (1 N /
q: S v /N2, Σ
33og 2
(28)
8'
8'
t=
1=0
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Published as a conference paper at ICLR 2026
Putting pieces together yields:
|10++1 - 8¹112= + 1 - 1912 - AAu - x- + 110 1² + 2nn2 9 + (1 + x2p+2'802
2
+ +22nnp N(1+ +
+ +2 3log
8
2 + +2V2mpNee + TTm() + 2v 212pVVNC,1 3 3log 5
(29)
Since we choose 7 =
Su and y = Go where VN
P
P
G, S min
2/3Cg 12v6(1 + v2p+ x2) /300333033 2 12v60, v log
32v/e Ne
we conclude that [|0++1 0- + Du+1 - X-12 5 2pp.
Now by similar analysis we have that for any 2 0
|10+1 11 + Me+1 2 s 110: 0| + 1A 1/2 - (2 L(0 Ar))
-
C27 + 1/ + 6² + (0 (g.t) g(Os), de 2)
+ +2)(g(0)) - ((.)) d -- A) + 21p 10: 1'
$ 110,- - 0011²++ |4, - []2-2n(L(00) A) L(0", ,(0)
+ C2 + (1 +1 x2) + 2ng (A. + A)
+ +neo (A: + A) 2V2mppp.
Since Aig 1 S 22p we know d < x + V2p. Hence we have
A) - |10++1 - 0²1 + Me+1 - 212 < A/- - 212 + 11: - 0- - ()
xxe)
+ 2n 2g + (1 2,) + -214 (x* + v22p)
+ 21/00 (x- + / + 27901 + 2no + 2V/2ppP.
Performing telescoping sum yields:
N Σ L(e,, A) - L(0", (00 S 2
1
(120 A13 + 100 0113 +
10
1
1
Σ (1 1+ 02
N
2N "
N-1 1
1 N-1
N-1
Σ qt (x- + v2p)
Σ
x + v2p)
Σ ged
+
+ N
c
+ N
N
t=0
t=0
N-1
+ < + 2p N
Σ Pt-
t=
By the definition of Lagrangian, we have
1
(f(O.) + (g(O.) - 7x² - = de Ax
+ -7x3)
N
2
1=0
50
N-1
1
(140 - 212 + 11uu - -8-13) + 1
N 10
+
1
Σ
N
2N
1=0
1 N-1
1 N-1
1 N-1
qo (x- + (x*+
v20) + cu (X- + v2p)
+
Σ
qi+
co + V2p
Σ
Pr-
N
N
N
t=0
1=
t=1
18

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Published as a conference paper at ICLR 2026
Evoking the bound from
28 yields:
Σ
(f(01) + xo((1) -x-- + (t)
2
2
1=0
N-1
$
2n
1 (1u0 -112 + 110, - 00113) + 1
N N02
2N
1
"
Σ
(1+ (1)
t=
1
2
(X* (x+22V2)) + ED (x- + v2p)
+ N V T03/
V 3log
1
2 TC2 1 3log
V2p + v2p + X G
3 3
N V
5 +
N
Plugging in Ao= = 0, 0g = 0 and re-arranging the terms yields:
N-1 1
N-1 1
(f(O)) - 5(0))) +
Σ
xg(O.) -
7 + 2n x²
N
N
2
t=0
0²
22p(1 + V2p + X*G
3 3og
+ 112 +
2N Σ
(n(1 + 21) -x,)) +
T/T
VN
Og C//3log log?
(Coy3log 2
+
(x* + 20)) + =0(A + +2v2p)= +
+ A.
VN
By our choice, 7 = G2 so we have
N
N-1 xg(O,) -
(r
- f(00))(000)) + N
+
N X²
2
0=
< p²
+ +n0 + +n + Cgy log
v2p(1 + v2p + X*)
/
3log 3AA
(A* v2p)
r/T
VN
VN
Cg v 3 3
+ (o + V2p) +
+ E0
Define Or 2/2 NN1 Gr, and then by Jensen's 's inequality we have
(7 +
(f(8r) - -(())) + +g((r))
2²
2
2nNV
< 0² nT + tn² +n0² +
Cop 3 3og 2
V2p(1 + v2p x2) v
3log
(A* v20)
VN
V/V
C c 3log 2
+ + + v2p) +
+ Ev X
N
19

---

Published as a conference paper at ICLR 2026
C.1 PROOF OF THEOREM! 3
Proof. Note that Lemma 3 holds for any À 2 O. Now let's discuss by cases. If Oy is in the constraint
set, then 0 = Or and we simply set À = 0 and get the convergence:
-(0r)) + Ar(Or) - 2 +
2n 2²
1² + '12 n12 + Coy 3 log 2B
V2p(1 v2p + GO 3log aid
(X* + v2p)
nT
VN
VN
Cg 3 log
+ =(x* + 2p) +
+e
A.
N
0r is not in the constraint set, we set À _(0r)
which yields:
(f(0)) (²(Or))² 2(7 + )
I p² + +n0 + +ng +
Co Coy3n 3log 3
v2p(1+ V2p + 22p+
log 641
(X+ +
1/T
VN
VN
C
3 on
g(Or) 7 + 7f
cu Xx + v20) +
+c
(30)
Since Or is not in the constraint set and 0 is the projection of it onto inexact constraint set ((0) S
-5. by KKT condition we know g(f) = -5 and 0r - 0 Vg(θ) for some >> where
g' g'( = g(0) + 6. Defining& A = ô c, and due to our choice of we know 2 0. Then we have
g(Or) = ğ(Or) g'(0)
5(0T) π(0) (g' (0())) g(OT))
((0T) ī(0) (8 cc)
2 (V)(0), 0r 0) - A = v 0T - 0/1 -A
where the inequality is due to convexity of g(.). Let 0g be such that x(0o) = 0, and then we have
min (((0)=0
[V((0)] 2 min
Vv(())| |Vv](0) V
r'(0)
2 r 2L (g(0g) (()) 2Lğ(0)
= r 21(c 5) 5)
2 r 2LED
so g(0r)= 2 (r - 2ee) 0, - øl A. On the other hand, since is the projection of 6y onto
constraint set, and 0 in the constraint set, we know
0 8- 2 S 0r - 18-12 1|2=$ s 2p².
Hence we also know
((0r) σ(0r) g'(0)
= g(0r) - ((0) -
g(0r) 6(0) ((0 9()) 8
< G |0r é +
20

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Published as a conference paper at ICLR 2026
Plugging the upper and lower bound of g(0r) into 30
yields:
- ((0r)- + vN ((r 2HE
10 Ō|| - A)²
G( + )
Cay / 3
+2p(1+ +2 3 a/
I p 1/T nOz nn +
(x*
+ 2V20) +
VN
VN
+ 60 x + /20) +
Co Cov3log 3ly3log
VN
Br - 81 G
+
(31)
VN
2(c,,
2n) 6xp
Notice the following decomposition: f(Or)- - f(00) 2 f(0T) - f(0) 2 -G110- 0 6/|
Also notice
the fact (a - 2) 2 _a²²= 1/2 holding for any a > 0,b > 0, we know
(rr ( 22-2Heo)) | ol - 2 マ (r 2H 0-- - 01 - -2
Putting pieces together yields the following inequality:
a allor - 01 b ||r - 0| - c S 0,
where:
VN - 201 )
a=
1(c, + )
b
Cg 3log + VNoo +G,
GG (2 [
Φ cg + G2) + Cg
3 3log 2 (1* +2V2p) + 22p(1+ x
3log AMD
c= by
VN
A
+ 60 (1* + v20) + VN
2(c, G2
C23log _
2GC, /oCgvllog 3log z (c, bn)
Assume 6o S
so b 5
G. Solving the above quadratic inequality yields:
VN
0r 81
b+ v 12 +
Auc
Va
2a
2GC, / 3og
1G(c, + +2
VN(r- 2HEE)
VN(r - 22He))
2(r 2H
4(c, + 2))
++ +2) + (C, (X- + 2v2p) + V2p(1 + v2p+ 3log 3
VN(r 22
VN
1(c,02 + 2
Eo (X' Vep)
+
VN(r - 22
2CC, 3 log 60(c, 1
A
VN - 22
2(r 2H
(c + Go) C, 3log? 3 (X* +2/2p) /2p(1 V2p+ X*G
3 3log
+ (x' + v20)
2/N
21

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Published as a conference paper at ICLR 2026
where at the last step we use the fact Vab _ a²+b²
-Finally, note the following decomposition:
2
f(0) f(0*) f(0) f(0r) + f(0r) f((²)
s G 0r 8 + f(0r) f(0))
20 3 log o
G +
cn 0 G
|r |0r- + (x+
v20)
(Og G²) + (Cg (x* + 2v2p) + 22p(1 + 22 xx))) 3 3
VN
(X++V20)VN and plugging bound of |rr 0/ yields:
Co Coy3log, f
Recalling 60 S
f(0) - f(0")
G+ 20, v 3 blog 2h
|2GCg v 3log + 6G + 2)
A
G2 +
VN(r - cH
2(r
2Hc
Co Coy 1o81083 1og G, Gy
+ (C + G2) + 20 y log ah (A+ + p) + PG
log 3
+
+ G+
VN
Now simplify the above bound. By the definition of c, we know Cy S G have
f(0) - f(0))
2G + 20oy / /3log
2GC 3 3og + 6G G2 G2 +2 2
A
+1 C, G+
VN(r - 2/1
2(r
2Hc
+ 2G + 20g / /3og 3
e² Cr(Cg + Q2) + (C, (X* + P) + pG) log
G 11
VN
Finally, by definition of c, we know 2 2
en g'
e²
2 pOg.
2²
2 p12/6G./108 o and
og
by
G/en 2 C. G² which concludes the proof: (Hoo)
GC VoGylog// log
G A
p C
log
- f(0)) 2 G log 1
(r
2Hc
N
Plugging the bound that x s G/r (see MMahdavi et al.
2012 Remark 1)) will conclude the proof.
C.2 PROOF OF THEOREM 2
Proof. Now we proceed proving Theorem We split the proof into parts: proof of optimiza- tion guarantee (the gradient complexity) and proof of statistical rate.
Convergence rate The total gradient complexity of A be the sum of the complexi- ties of each calling of procedure cP-Solver. Theorem gives the complexity of CP-Sol ver, and
22

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Published as a conference paper at ICLR 2026
what need to compute is in each call. Recall that, ey is the enror between the true constraint and the approximate we actually used CP-Solver. Hence, we first compute those error as follows:
(0'( (', a')l - (0) - 91. (0) - (01,a-c.7)) - 1 (
(0) gå. ( {lā åsl, 1 91,r(0)) S max &T
(0) (0) max {lāa (0) ghg (0)), 19/ (0) (r)
S max Eag: t. R (0 (0) R (Or,aa)))))
< max
{
cas: TT
Now we verify that the choice of the tolerance in Algorithm 2 can ensure that
Ri (0) min
R R (0) S E1,S ES,T.
(0)50
To ensure CP-Sol ver Rimiss ÚS,T, E(ES,T, GS.T) outputs as,r-accurate solution, by the condition Theorem 3 we need (0) - g (0) S ((,., hence we require
6b S ((ES)
(32)
To S r(,T,
(33)
4 S E(ES,T,
(34)
We also require E S EL,T, so we know choosing & = min (E(ES.T,TS.T), e1.7) suffices.
To ensure CP-Solver RRg.ir da. 5(G,r)), c+) (0) gäg (0)) s E(c r4), hence we require
output ac-arraate solution, we need
Cas S E(TT, r'T,;
(35)
TTa-aar S E(6,, r'7)
(36)
To ensure CP-Solver (Rp.m.s. Da, f(és.T's. és) output e's é1.s-accurate solution, need lāa(0) Säg (0) s r')s hence we require
Cas S E(cs, r's),
(37)
CT < E('s, r's)
(38)
To ensure CP-Sol cP-solver(a', (0(,)'', Eleag, ras) (as) output eas -acurrae solution, we need (0).((0) - 91.r < É(Eăg, Tās ), hence we require
TT.--a.. S Eleag,
(39)
From (2).. (35) and 37) suffices and from (33
we know that choosing eās
S min [E(é's I's (())), r4)), Eles,
('s)
(36.
37 and (39 we know that choosing CT.a-eor
min (E(e's.r r. (( 1/)) E(e's E(E,) r suffices.
The complexity immediately follows by plugging enror tolerances into Theorem 3
Statistical Rate If R.p.m, Rip. (60) 261,s, then know
min Rep (0) (0)S0
min
Rep. (0) 2 Rip (0) Rip.n, (0's
(8's)-(a)) - 61.8 2 C1S
8:na S61.9
Hence the set 10: Dās (0) S 50,9 (0) < 0} does not intersect with {0: 9as (0) < 00)'s(0) < {19)
which is (h H' R. (h) S Re (7l') 3E1,s} slightly inflated version of Hy,s) .) so we output 0,.
23

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Published as a conference paper at ICLR 2026
Otherwise, sn. s. 20,,s. then
min R ( min Rp (0) _ s s.ppi,s('3) - - 61.8 (0)
Bggg S0,,
gās (0) < 0, 9/ 50} (Hlp) intersects with which implies that the set gas
{0: gās (0) < 0,0's(0 S +1,a)} which is {h E H' s. (h) S pm (H') + ei,s} (a
slightly shrinked version of Ml's ). so we output model θ in their intersection.
Hence the output of our optimization procedure is consistent with the learning algorithm ( and the same generalization analysis applies here.
D ADDITIONAL EXPERIMENTS
In this section, we present additional experiments on synthetic Gaussian data in dimension 10. The source class-0 distribution is (2-2, 1). the target class-O distribution N (0, (0, I). and both source
and target class-1 distributions are [11, T). In Figure 4 we fix no,r = n1, = 40 and vary the number of source samples Ro,s n1.s from 50 to 3000, with a 0.1. In Figure []] we fix No,s 11S 1000 and vary the number of target samples 110,7 = RIT from 50 to 1000, again with a = 0.1. Each experiment is repeated over independent trials.
The results in Figure show that the source-only method exhibits very large Type-I error, while the proposed method interpolates toward the target-only method as expected. is important to note that, in the implementation, several constants appear in the quantities E,D for i {0, 1} and D {5,T). These c-terms arise from uniform concentration inequalities controlling the gap between empirical and population errors. In theory, such inequalities known to be conservative, and the associated constants are not tight; in practice, however, these parameters can be tuned more carefully. Consequently, while the proposed method provides practical insight into how design an adaptive transfer-learning procedure, further tuning and experimental investigation are needed to select these constants appropriately. as they directly influence the final error performance.
In Figure S as the number of target samples increases, the type-I errors of both TLA and the target- only method converge a 0.1. The slight increase in the type-II error of TLA reflects the inherent trade-off between type-I and type-Il errors: reducing one typically leads to an increase in the other.
TLA - Only Target
Only Source
TLA -
Only Target
Only Source
Turget Type-11 emur
Target Type-I error
0.80
0.15
0.70
0.60
0.1
0.50
0.40
5. 10-2
0.30
2-10 10
0.20
0.10
.50
1,000
2,000
3,000
50
1,000
2,000
3,000
1o.55 131,5
noo.
11,5
Figure 4: The performance of our algorithm (TLA), along with two baselines using only source data and only
target data- is evaluated under a Type-1 error threshold of = 0.1. The experiment is conducted on synthetic Gaussian data in dimension 10, where the source class-O distribution is (-22, 7), the target class-O distribution N (0, /), and both source and target class-1 distributions are N T). We fix no. fu. 40 and vary the number source samples no,5 = n1,s.
24

---

Published as a conference paper at ICLR 2026
TLA
Only Target
Only Source
TLA
Only Target
Only Source
-10
Target Type-II ppe-IIenme croo
Turget Type-I error
0.80
0.70
0.60
0.50
2
0.40
0.30
0.20
0.10
50
250
450
650
850
1,000
50
250
450
650
850
1,000
RR,T7 11,TT
no. = 11,7
Figure 5: The performance of our algorithm (TLA), along with two baselines- only source data and only target data is evaluated under Type-I error threshold of 0.1. The experiment is conducted synthetic Gaussian data dimension 10, where source class distribution N (-2, I), the target class distribution (0, I). and both source and target class-1 distributions are [1, 1). We fix Re.s = n1S 1000 and vary the number of Target samples no,T 11,T-
25


Reviews:
[{"review_id": "CjpzGP57Lc", "reviewer": "Reviewer_Wj1g", "summary": "The paper studies transfer learning for Neyman–Pearson (NP) classification when both class-conditional distributions may shift between source and target domains. It proposes a two-stage adaptive procedure: (i) calibrate a source-side Type-I threshold `α̂_S` so that the source constraint aligns with the target NP constraint, thereby pruning hypotheses that would violate the target Type-I bound; (ii) within this restricted set, leverage source class-1 data (and target data) to further reduce Type-II risk. The paper derives generalization bounds stated via a *transfer modulus*—functions `ϕ₀` and `ϕ₁` translating source performance to target—and shows recovery of prior results when `μ₀,S = μ₀,T`. It also provides a computational oracle via a sequence of convex programs (SGDA-style) with gradient-complexity guarantees, and evaluates the method on two climate datasets where locations define domain shifts. :contentReference[oaicite:0]{index=0}", "questions": "1. The theory relies on a convex hypothesis class $H$ and convex, Lipschitz, bounded surrogate losses, whereas the experiments use an MLP. Do MLPs constitute a convex hypothesis class?\n2. Can we derive similar theoretical guarantees for transfer learning under an overlapping-support assumption between the source and target datasets?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I was not previously familiar with the NP classification setting, but I find both the problem and the authors’ extensions interesting. The authors also provide sufficient theoretical guarantees for their proposed algorithm, along with supportive simulation studies. In terms of novelty and substance, I believe the paper is worthy of publication at ICLR, although this is not my primary area of expertise.", "weaknesses": "1. It is somewhat difficult to discern a strong novelty in the transfer-learning extension, even though the authors present solid theoretical results and simulations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "d8VMDiLxpC", "reviewer": "Reviewer_oLLY", "summary": "The authors study Neyman-Pearson classification in the context of transfer learning, where both source and target domains provide positive and negative samples. The objective is to minimize the false negative (Type-II) rate while ensuring the false positive (Type-I) rate remains below a specified threshold on the target distribution. While prior work has mainly focused on the case where the negative distributions are identical between source and target, this paper explores the more general scenario where they differ. The primary contribution is an adaptive error bound on the target false negative rate, which interpolates between using only target samples and leveraging additional source samples. The authors further propose a computational algorithm with matching error guarantees, assuming convexity for both the loss function and classifier class. Experimental results validate the algorithm's adaptive control of the Type-II error while adhering to the target Type-I error constraint.", "questions": "- In what concrete scenarios is the excess term $R_{\\varphi,\\mu_1,T}(h^\\ast_{S,T,\\alpha}) - R_{\\varphi,\\mu_1,T}(h^\\ast_{T,\\alpha})$ provably small? Can you characterize such cases via properties of the source/target distributions?\n- Why is $h^\\ast_{S,T,\\alpha}$ the right pivot for the analysis? Could similar or tighter bounds be derived by pivoting around alternatives such as a target‑only optimizer on a mixed/importance‑reweighted distribution, and how would that affect the rates?\n- Theorem 1 suggests adaptive control of the false positive rate (Type‑I), yet Figure 3 appears comparable to the target‑only method. What explains this discrepancy, and under what conditions should we expect adaptivity to be visible empirically?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "This paper addresses Neyman-Pearson classification within the transfer learning framework, which is highly relevant to the conference. The Neyman-Pearson classification setup is practical due to its importance in high-stakes real-world applications, and transfer learning is motivated by the scarcity of labeled samples in the target domain.\n\nThe authors present an algorithm with error bound guarantees that avoids negative transfer, even when both the negative and positive source and target distributions differ. As previous works are limited to the setting where the negative source and target distributions are identical, this represents a theoretical contribution to a more general scenario.\n\nThe resulting error bound is characterized by a newly proposed quantity assessing the transferability of source samples, namely, the transfer modulus. This measure has the potential to capture non-polynomial associations between the source and target distributions, whereas previous transfer exponents are restricted to polynomial relationships.\n\nThe authors also provide a computational algorithm achieving the same error bound guarantee. Even under convexity assumptions, they demonstrate a polynomial runtime algorithm for transfer Neyman-Pearson classification.", "weaknesses": "The practical scenarios in which $R_{\\varphi,\\mu_1,T}(h^\\ast_{S,T,\\alpha}) - R_{\\varphi,\\mu_1,T}(h^\\ast_{T,\\alpha})$ is small are unclear. If this term is not small, using only target samples dominates the rate. In such cases, the benefit of transfer learning is not adequately demonstrated. Since $h^\\ast_{S,T,\\alpha}$ is defined as the maximizer of $R_{\\varphi,\\mu_1,S}$, the excess error between $h^\\ast_{S,T,\\alpha}$ and $h^\\ast_{T,\\alpha}$ could remain large even when the source and target distributions are similar. The justification for using $h^\\ast_{S,T,\\alpha}$ as the pivot in the error bound should be explained more clearly.\n\nThe experimental results suggest that false positive rate control is not adaptive. Theorem 1 asserts that the additive error in false positive rate control is adaptive, meaning it is bounded by the minimum of the error rates for using only target samples and the transfer method. However, Figure 3 shows that the false positive rate error is comparable to that of the method using only target samples. This discrepancy between theory and experiment should be addressed.\n\nThe paper's structure could be improved. For example, the setup section (Section 3) combines the problem formulation of Neyman-Pearson classification under the transfer learning framework, discussion of prior results, and challenges in constructing the proposed algorithm. A clearer separation of these components would be beneficial.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qqN2B681WL", "reviewer": "Reviewer_r8Y4", "summary": "Abstract: This paper investigates transfer learning within the Neyman–Pearson (NP) classification framework.", "questions": "see weaknesses", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "great theoretical results", "weaknesses": "**Weaknesses.**\nThe paper does not adequately discuss its limitations. In particular, the theoretical generalization analysis appears to rely on boundedness assumptions on the loss or surrogate loss, which is a common but restrictive condition.\n\nMoreover, there are alternative transfer learning settings—such as fine-tuning and related approaches discussed in [1] and [2]—that are not compared against the proposed setup. Including such a comparison would clarify where the NP transfer setting stands relative to more standard transfer pipelines.\n\nFinally, the motivation for studying the Neyman–Pearson formulation in the transfer learning context is not fully articulated. It would help to explain why NP constraints are especially relevant here, and to describe how competing approaches (e.g. $\\alpha$-ERM or fine-tuning) would perform or be adapted in this setting.\n\n[1] Aminian, Gholamali, Łukasz Szpruch, and Samuel N. Cohen. “Understanding Transfer Learning via Mean-field Analysis.” arXiv:2410.17128 (2024).\n[2] Bu, Y., Aminian, G., Toni, L., Wornell, G. W., & Rodrigues, M. (2022). “Characterizing and understanding the generalization error of transfer learning with Gibbs algorithm.” AISTATS 2022, pp. 8673–8699.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vkFyNPCglH", "reviewer": "Reviewer_dxrU", "summary": "This paper provides an algorithm, to perform NP classification under distribution in the case of transfer learning where for both class 0 and 1 data there is a small amount of target data from our distribution of interest and a larger amount of source data from an unknown but potentially similar distribution. The NP classifier is shown to satisfy the Type I error constraints with high probability up to some error term $\\epsilon$ (while at least matching the error bound given by just using Type I error) as well at least matching the Type II error given by a classifier just produced on the target data and improving upon it when the source and target are similar.\n\nAt a high level this procedure works by finding optimal classifiers on both the source and target data (with a relaxation of the Type I error on the source data to ensure some intersection exists) then takes the final classifier to be the intersection of the near optimal classifiers on both source and target data within this set (if such a classifier exists.)", "questions": "Can this be extended to the case where some relationship between the target and source distributions are known and can be leveraged for example the density ratio or some other relationship?\n\nAre there illustrative examples of how the transfer modulus or transfer exponent vary as the source and target distributions differ from one another?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "* This work effectively extends the work of Kalan \\& Kpotufe (2024); Kalan et al. (2025) to the case of distribution shift on the class 0 data. Furthermore the theoretical results illustrate meaningful improvement from the inclusion of source the data from both classes provided the transfer moduli is sufficiently small.\n\n* Strong theoretical results are provided both for the optimal classifier under the schema as well as a classifier given by a practical algorithm solving the optimisation problem.\n\n* The relative error induced by the difference in source and target distributions is nicely related to the transfer exponent, a pre-existing measure of distribution shift.\n\n* Experimental results on real world data illustrate improvement when the source and target data overlap and minimal downside when the target and source data do not.", "weaknesses": "* The additional error on the target Type I error $\\epsilon_{0,T}$ depends upon the Rademacher complexity $B_{\\mathcal{H}}$ a quantity which while bounded for may function classes is not often practically known. This makes it difficult to use these results to give meaningful finite sample, high probability bounds on the Type I error for a specific case. This is in contrast to works such as Tong (2013) and Tong et al. (2018) which use a sample splitting procedure to project the problem down to learning the threshold of a score function, effectively making $\\mathcal{H}$ have VC dimension 1.\n\n* The paper lacks a proper conclusion or discussion of future directions/limitations of the work which I feel would enhance the paper, especially is it is a very dense technical work.\n\n* The experimental results do not illustrate a case where improper handling of the source data harms Type I error, in both cases the lowest Type I error (theoretically the value we are most keen to control) are mostly given by exclusively working with the source data.\n\n* While I appreciate that it is just a very technical work. I feel it could be improved by trying to streamline notation. E.g. later in the paper, the sets $\\mathcal{H}', \\mathcal{H}'_1, \\mathcal{H}'_0$ are re-written as constraints on the functions $g$. Could these functions instead be introduced earlier therefore requiring fewer subsets of $\\mathcal{H}$ to be defined?\n\n* Some small errata (I think):\n    * Line 343 -- Definition of $\\hat{\\theta}_{T,\\alpha}$ not defined and I believe it should be $-6\\epsilon_{1,T}$ is the definition of $g_{1,T}$. \n    * Line 361 -- Typo, ``uniformly'' written twice.\n    * Line 356 -- Slackness should be $\\xi$? $\\delta$ not used and neither is $\\epsilon$ (though by CP-Solver's later use $\\epsilon$ should appear.)\n    * Line 420 -- Technically, (12) only finds $\\hat{h}$ when $\\mathcal{H}'_1\\cap\\mathcal{H}'_0\\neq\\varnothing$ and I feel this would be clearer if it was clarified (I appreciate that this is clarified in the algorithm but would benefit from clarification here as well.)\n\n## References\nXin Tong. A plug-in approach to neyman-pearson classification. *The Journal of Machine Learning\nResearch*, 14(1):3011–3040, 2013.\n\nXin Tong, Yang Feng, and Jingyi Jessica Li. “Neyman-Pearson Classification\nAlgorithms and NP Receiver Operating Characteristics”. In: *Science Advances 4.2*\n(Feb. 2, 2018),", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)