Theoretical Analysis of Weak-to-Strong Generalization
Authors: Hunter Lang
David Sontag
Aravindan Vijayaraghavan
Venue: NeurIPS 2024 poster (2024)
Keywords: Weak supervision
weak-to-strong generalization
self-supervised learning
semi-supervised learning
Primary area: natural_language_processing
TL;DR: We give a theoretical account of strong student models learning to improve over their weaker teachers.

Abstract: Strong student models can learn from weaker teachers: when trained on the predictions of a weaker model, a strong pretrained student can learn to correct the weak model’s errors and generalize to examples where the teacher is not confident, even when these examples are excluded from training. This enables learning from cheap, incomplete, and possibly incorrect label information, such as coarse logical rules or the generations of a language model. We show that existing weak supervision theory results fail to account for both of these effects, which we call pseudolabel correction and coverage expansion, respectively. We give a new bound based on expansion properties of the data distribution and student hypothesis class that directly accounts for pseudolabel correction and coverage expansion. Our bound generalizes results from the co-training and self-training literature and captures the intuition that weak-to-strong generalization occurs when the mistakes of the weak model are hard for the strong model to fit without incurring additional error. We show that these expansion properties can be checked from finite data and give empirical evidence that they hold in practice.

Theoretical Analysis of Weak-to-Strong Generalization
Hunter Lang MIT CSAIL
David Sontag MIT CSAIL
Aravindan Vijayaraghavan Northwestern University
Abstract
Strong student models learn from weaker teachers: when trained on the pre- dictions of a weaker model, a strong pretrained student can learn to correct the weak model's errors and generalize to examples where the teacher is not confident, even when these examples are excluded from training. This enables learning from cheap, incomplete, and possibly incorrect label information, such as coarse logical rules or the generations of a language model. We show that existing weak supervi- sion theory fails account for both of these effects, which call pseudolabel correction and coverage expansion, respectively. We give a new bound based on expansion properties of the data distribution and student hypothesis class that directly accounts for pseudolabel correction and coverage expansion. Our bounds capture the intuition that weak-to-strong generalization occurs when the strong model is unable to fit the mistakes of the weak teacher without incurring additional error. We show that these expansion properties can be checked from finite data and give empirical evidence that they hold in practice.
Introduction
Weakly-supervised learning allows practitioners to train models with possibly-incorreet, easy-to- obtain pseudolabels instead of accurate and expensive ground-truth labels. For example, suppose the goal is to classify documents based on whether they have positive or negative sentiment. Instead of employing humans label examples æ; with positive/negative sentiment labels yı, weak supervision enables models to learn from simple rules, such as: if 'incredible' E Ei, sentiment positive. Called programmatic weak supervision, a wealth of literature has shown how aggregate rules, often called "labeling functions", into individual pseudolabels that can be used to train a model [e.g.. 51, 49, 63, 21]. In typical pipelines, this consists of fine-tuning a pre-trained neural network [71]. This method has met with a huge amount of empirical success in natural language processing [51, 28, 36, 70] computer vision [74, 62], and verticals such healthcare [25, 26, 20, 67].
Another emerging trend in weak supervision is to use the zero-shot or few-shot outputs of a large language model (LLM) as pseudolabels for training another language model-the student model often outperforms its noisy "teacher", and this technique even works when the teacher is less powerful model than the student, distinguishing it from classical knowledge distillation [65, 17, 18, 35, 2, 24, 33, 66, 39, 10]. Good data selection can be critical, and several approaches carefully choose a "confident" subset of the pseudolabels [e.g. 35, 18, 39, 66]. The increasing prevalence of LLM use in crowdwork [64] highlights the importance of better understanding this learning method.
In both cases, the pscudolabels ÿ given by the pseudolabeler teacher model, which is some function of the input example æ. The pseudolabeler may make errors (ū(x) * y(x)),, and it may some points where the teacher abstains from providing weak label.
cover every point-there In the examples above, these are points not covered by the rules and points where the teacher LLM is confident, respectively. priori, it seems that powerful enough classifier should exactly fit the pseudolabeler on the covered data and have trivial performance on the uncovered data. However, this what happens in practice-the empirical success of weak supervision is due two surprising. related phenomena that together comprise weak-to-strong generulization: (a) Pseudolabel correction:
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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The performance of the model exceeds the performance of the pseudolabels used to train it; and (b) Coverage expansion: The model performs well even on the portion of example space X' that is not covered by pseudolabels. These empirical outcomes are key to the success of weak supervision.
Surprisingly, the existing theoretical literature on programmatic weak supervision does not address either phenomenon- majority of weak supervision theory literature either focuses on how to adeptly combine the outputs of multiple "weak rules" into single pseudolabel T.. [51, 49, 50, 21], or treats learning from weak supervision as learning from noisy labels [e.g. 46, 59, 15J, but as we discuss in Section 2, this framing does not capture the setting we consider, where the pseudolabels used to train one model are the outputs of another model.
In this work, give a theoretical analysis of weakly-supervisee learning that provably accounts for the effects of pseudolabel correction and coverage expansion. Our results use a natural expansion condition on the population data distribution. Informally, expansion implies that "bad" points (points with incorrect pseudolabels, or points with no pseudolabel at all) have many "good" neighhors (points with correct pseudolabels). If the learned student model is relatively robust on the neighborhoods of interest, then making a mistake on bad point means making many mistakes on good points as well. This allows to prove relationship between the student model's error on the weak labels (the training objective) and the student model's error on the true labels (the desired objective). Our assumptions and bounds in Section formalize this intuition. Section details a procedure for checking our expansion conditions from finite data, and in Section 6 and Appendix E, we give empirical evidence that these conditions hold on real data.
To the best of our knowledge, our results provide the first error bounds for programmatic weak supervision with realistic assumptions. We show that our bounds generalize and connect several existing results from the co-training, self-training, and distribution shift literature [e.g., 6, 3, 69, 12] and adapt them the weak supervision setting. For example, we show in Appendix C.I that Theorem 4.2 generalizes the co-training results of Blum and Mitchell [6]. We discuss these generalizations in detail in Sections 2 and 4 and Appendix C. Our result in Section 5 is the first among these works to prove that the expansion assumptions can, in principle, be checked using finite data. Unlike most prior work in this space, our experiments in Section attempt check whether the expansion assumptions hold in practice. While our experiments are limited in scope, our attempt to systematically check expansion a practical scenario is a major departure from previous work.
Finally, prior work with expansion assumptions similar to ours (in the co-training [6, 3], self-training [69], and distribution shift [12] literature) requires that the classifiers are either perfectly robust [6, 3] or adversarially robust [69, 12] for their bounds to apply. Empirical results suggest that adversarial
training has fairly limited value for improving coverage expansion and pseudolabel correction [38]. and that these two effects still occur for student models that are not adversarially trained [71, 10].
To close this gap, we make a connection to the literature on robust expansion [27, 34, 42] and prove error bounds for student models that are merely "robust on average." Unlike prior work, these bounds allow for the presence of adversarial examples for every input point.
2 Related Work
Ratner al. [51, 49, 50), Fu et al. [21] focus how to combine the outputs of multiple "weak rules" into a single pseudolabel ((z) for each covered example, a problem with a long history in the crowdsourcing literature [e.g. 16. 30, 29]. However, empirical results indicate that this is not the important aspect of weak supervision: most methods for combining weak rules fail to significantly outperform majority vote once the final classifier is trained [72]. Works in this literature that do provide error bounds for the student (e.g., Fu et al. [21], Ratner et al. [51]) either fail to capture weak- assumptions-foor example, Ratner et al. [51] assumes that yx) and x are conditionally independent given f(x). If this were true, there would be no gain from training a classifier, since ğ(x) already captures all the information that x contains about (x). Work that treats learning from weak supervision as a noisy label learning problem [e.g. 45, 46, 59, 15] does not capture the types of weak supervision we consider. When the supervision comes from weaker model (be it rule-based or weaker LLM), there no exogenous noise process that corrupts the training labels. There are simply some points that deterministically get the wrong labels and some points with no label. This rules out common noise models like class-conditional noise [46] and Tsybakov noise [60], and is arguably not appropriate to model as instance-dependent noise [59, 15], since for each the noise is deterministically 0 or 1.
to-strong generalization effects, as shown in Section 3, or make

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Burns et al. [10] conducted a large empirical study showing widespread weak-to-strone generalization effects when training a strong language model on the generations of weaker model. Our results are a step toward a theoretical understanding of these effects. Several works have used expansion to give provable guarantees in other settings where models are learning from each other. Balcan et al. [3] use expansion to analyze co-trained [6] classifiers. Our expansion assumption is similar to their "left-right" expansion, but we generalize beyond the multi-view setup of co-training and account for error propagation. Wei et al. [69] give provable guarantees for self-training under expansion assumptions similar to ours. We provide a different pseudolabel correction bound, tighter guarantees for coverage expansion under weaker assumptions, and generalize both results to classifiers that are not adversarially robust. Cai et al. [12] use expansion to prove general guarantees for pseudolabel correction, semi- supervise learning, and unsupervised domain adaptation, but their results require the student to be very adversarially robust. Compared to all these expansion works, we also outline a rigorous theoretical framework for checking expansion finite data and provide more empirical evidence for our assumptions. We discuss more related work in Appendix A.
3 Setup and Shortcomings of Existing Bounds
Notation. x refers random variable with distribution 'D and italicized letters refer to realizations of x. We will assume for ease of exposition that the input space X' is a discrete' (but possibly very large) set, such all vectors in Rd up fixed numerical precision. For A C X' we A == A. We assume there is ground-truth function of interest, y X' + = =I,,... k}, and a pseudolabeler ÿ : X + y U {Ø). which assigns each point either a label in y or the special "abstention" symbol Ø. The function ÿ can also be thought of as the teacher model, but we are primarily concerned with instances where the teacher is much less capable than the "student" it will be used train.
Define S = {x (xx) * Ø} to be the covered subset of Xx i.e., the subset of X' that has pseudolabel, and let T { ((x) Ø = X' \ S be the uncovered set. This notation serves to emphasize that training occurs on a (pseudolabeled) source subset S, and then evaluation occurs on the union of S and the (uncovered) larges T. Let {Xx] be partition of X' such that within each X, the ground-truth label is constant. For example, we could set X; = {=]y((>) =] to be the set of points with ground-truth label We will use this definition of X, for convenience, but all our results hold for more general partitions. Each of S and T can further be partitioned as S; = S n X, T; T n Xr Finally, each St can be further partitioned into the correctly pseudolabeled examples Sgood = (x E S. ((x) = y(z) and the cowooreclly-ssuuolbbelleee examples Stud = S \ Soood
Let 0; = P(S (S)) be the error rate of ÿ on S,. We assume < aa < 1 for all
Problem Setup. For two classifiers f.g: : X + y' and a set U C X. we use err(f, /U)) to represent ((f(x) * /x)]|x U) their probability of disagreement conditioned on x falling in the set U. Here the probability is over X ~ DD this will often be omitted for notational convenience. We will be par- ticularly interested in classifiers obtained by minimizing the error on the non-abstaining weak labels over the strong model hypothesis class F. i.e., (approximate) solutions to arg min feF err(ʃ: 7 (S). The ultimate goal is to obtain upper bounds on the error err(f, y X') for such classifiers. That is, we want to upper bound the error of classifier f on the true labels over the entire input space X.
There are two key challenges. First, the classifier is trained using y. not y, and ÿ may have arbitrary errors that not captured by any well-studied noise model such as class-conditional or Tsybakov noise [60]- we' ve assumed ÿ and the true labels y are both deterministic functions of the input, so there is no exogenous noise process that corrupts the training labels.
Second, we care about the performance of f on the entire space X, but we only train on the covered samples from 8 C X. Again, since is an arbitrary deterministic function, our samples from S are not distributed according to D, and S and have no overlap, ruling out approaches like importance- weighting. The following example elaborates on the issues at play and illustrates the shortcomings of existing bounds in the weak supervision literature.
3.1 Shortcomings of Existing Bounds: Illustrative Example
A special case of weak-to-strom st-oon generalization is training a strong pretrained model on the outputs of very coarse rules. Following the example from Section 1, suppose our goal is to obtain sentiment
'This is done to simplify some of the arguments, as in HaoChen et al. [22]. As shown there, the results can be generalized to continuous X with additional regularity conditions. Our bounds have no dependence on |x]].
3

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classifier, so A' is the space of text documents, 3 = [-1,1 1) and i given by the following rules: if ' 'incredible' E x. ((x) +1. If 'horrible' E x. () -1. Otherwise, ((z) 2.
Assume for simplicity that "incredible" and "horrible" never co-oceur, so ÿ is well-defined. This example shows that the student model hypothesis class F and the training procedure both play vital role in weak-to-strong generalization. Suppose F is the class of bag-of-words classifiers, and we obtain a student by minimizing err(f [ S). Without modifications to the training procedure (such as L2 regularization), f may place a large positive weight on "incredible", a large negative weight on "horrible", and zero weight on all other tokens. This model has zero error on the weak labels, so it exactly minimizes the training objective. It reproduces the pseudolabels on the covered set and has trivial performance on the uncovered set, so there no weak-to- te--tt--ttrre strong generalization. On the other hand, if we were to instead train a linear probe on top of SentenceBERT [52] representation, we would obtain a model that improves over I on S (the covered set of documents containing either "horrible" or "incredible") and has reasonable performance on T (the uncovered set). Section 6 contains precise results for this example, but the critical (seemingly obvious) aspect that the student representation and the training details matter for achieving weak-to-strong generalization.
The following proposition (proven in Appendix B.3) illustrates how existing error bounds in the programmatic weak supervision literature, which do not account for training details and the student hypothesis class, are unable to capture pseudolabel correction and coverage expansion.
for y E [-1, and assume that the weak label error rates a-1 = 01 = a, and that the weak labels cover each class equally often: PØ = Øly y) = Pp = 2). Let f = minser err(f.ū) ÿ S) be the classifier minimizing the weak label error on the covered set. Then the bound from Fu et al. [21,
Proposition 3.1. Suppose the label marginals for the above example satisfy P(y = y)
Theorem 3 simplifies (in our notation) to: err(J .y) P(S). 4a(1 a) + P(T).
The first term accounts for the error of on the covered set S. The weak labels themselves have error on S, but the bound for f is 4c<1 (1-a) > whenever a 1- so Fu et al. [21]'s bound does not allow for pseudolabel correction in this example. The second term accounts for the error of f on the uncov- ered set T. A random guess achieves error & on T, but the bound charges every point in T as error so it also does not account for coverage expansion or even the performance of random guessing on T.
Expansion assumptions similar to ours have also been studied in the context of self self-training [69] and domain adaptation [12]. The following proposition shows that while the results of Wei et al. [69] can be adapted to weakly-supervised learning, our bounds capture the full weak supervision setup better, since Wei et al. [69, Theorem 4.3] was not designed to deal with partial coverage P E SS < 1, so applying it to weakly-supervinn
learning still requires fairly large coverage.
Proposition 3.2 (informal). Suppose the coverage P(x E S) in the example above is less than 2/3. Then the bound from Wei et al. [69, Theorem 4.3] does not apply since directly adapting it to the weak supervision setting requires P(x E S) 2 2/3.
Empirically, coverage expansion and pseudolabel correction can both occur in the low-coverage regime [38]. Finally, as mentioned in Section I, Wei et al. [69] assumes the classifier is adversarially robust. In contrast, we provide bounds that directly account for coverage expansion, place no restrictions on the amount of weak label coverage, and allow for the presence of many adversarial examples. As the example in this section suggests is necessary, the model hypothesis class and the training details (in particular, the robustness of the model) play a central role in our bounds. The following definitions attempt to capture these properties.
3.2 Definitions
Definition 1 (Neighborhood). Let N be a neighborhood function that maps each point æ to a set of points (xx) C X that we call the neighborhood of x. We will assume that N satisfies x e N(xx) x' E (xx)., i.e., that the neighborhoods are symmetric. We can extend N to a function of sets (AA UZEA (zz). Examples to keep in mind are N(x)) {xx [1(((x)) - -p((=') S r}
for some representation up : X Rdd or, in the case of text inputs x, the set of fluent paraphrases of x. However, our results work with any definition of N.
Definition 2 (1p-robust). For an arbitrary classifier I and point æ. define ((J,=) = =P([(x'') # f(x)|x E (x))) the probability f gives different labels to and random neighbor x' of

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> C
A
B
B
Figure Relative expansion (Definition 3) on the sets (A, B). Expansion requires that certain
subsets U C B have neighborhoods (TT) such that ((V(U))A) 2 cP(U)B). These probabilities
are represented graphically on the right-hand-side as the fractions (W(C) n Al/IA] and [U]/|B].
A classifier f + y is said to be n-robust at point æ ifr(f. x) S n. Define (f) {x r(f,z) 2) I n) as the set of ny-roust points for f.
If = 0. f is n-robust at if and only if f is adversarially robust over (xx), this definition generalizes adversarial robustness. By Markov's 's inequality, any classifier f with:
Dx- + f(x') S
is y-robust on set of probability at least z/n (see Lemma B.1). The requirement (1) is significantly more natural than adversarial robustness: classifier satisfies (1) whenever it gives most points the same labels as most of their neighbors. We refer to (1) as "average-case robustness".
Definition 3 (Expansion). Fix sets A, B C X. We say the distribution Pxx satisfies (c, q) exppansion on (A, B) if for all sets C with PUU B) > q, P((((U))A) > BP.
Figure shows examples of Definition 3 graphically. Intuitively, a pair of sets A and B satisfy this definition if large subsets of B correspond/expand (via the neighborhood N) large subsets of A. Definition 3 requires all sets U with large enough probability in B to expand to A. However, this is unnecessarily strong. Our theorems will only need certain structured sets, corresponding to elements of the student hypothesis class, to expand. This is captured by Definition 4 and is the key to our results in Section on checking the expansion property.
Definition 4 (Expansion of set collection). Fix sets A, C X and suppose M is collection of subsets of B. Then we say M satisfies (c, -eppansinn on (A, B) if all sets U E M with B) > q satisfy ((N())) BP.
4 Error Bounds for Weakly-Supervised Classifiers
In this section, we upper bound the gold error of f on the covered and uncovered sets sets-err[J, err(sy y|S) and err(S, (7))-wtth expressions involving the weak error err(S, ū[S) of I on the covered set, expansion parameters, and robustness parameters. These bounds give theoretical justification for why empirical risk minimization using the weak labels leads to weak-to-strong generalization. We condition on subsets S, C S (for pseudolabel correction) or pairs of subsets S; S. T, C T (for coverage expansion). This subset-wise conditioning allows for different parts of the distribution to expand in different amounts, yielding tighter bounds, an approach also followed by Cai et al. [12].
4.1 Adversarially Robust Models
We begin by stating our theorems for the case when the classifier of interest is robust with y 0 on most points æ. That is, we assume that for most points x, the classifier adversarially robust over (x)). This follows the assumptions in related work from other domains [69, 12]. We describe our results for the significantly more general average-case-robus classifiers Section 4.2.
Expanding Families. We first define the families M and M' of sets that must expand according to Definition Let be the hypothesis class of the strong model and for each f E F, define R(f) Ro {x: : rrfffff x) = 0} to be the set of adversarially robust points for f. For B c X and f E F, define U(B f) {ar B f(x:) o y(x)} as the set of points B where f makes mistake on the true label y. Now define M(B, F) to be the class of robust mistakes sets: M)B =
(UB f) n R(J)= : E F). Similarly, define M' (B. = ((B (U, n R(f) : f F) as the family of robust non con-mistakes on B.
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Pseudolabel Correction. Here we relate the gold error of the student model f on covered subset, err(f. S,), to the weak error of f on that set, err(f. 7/ S.)) The goal is to allow for the correction of some of the incorrect weak labels (Stad); we want our bounds for err(f, y)S to be less than ay the error rate of the weak labels. Expansion between the points with correct pseudolabels, Spood, and points with incorrect pseudolabels, Shod implies that there are many bad points with good neighbors. If the classifier is suitably robust on the neighborhoods, pscudolabel correction can occur. The bound in this section makes this intuition quantitative.
Theorem 4.1 (Pseudolabel correction). Suppose (5, ,F) satisfies (c, exxpnssion the sets (Stad, ( for q < 2/4 (1 2a). Consider an arbitrary classifier f E such that P P(f(x) +
((x or f not robust at x x{S S 3-04 Then the true error of I on S, satisfies:
4
err(J,y(S:) S
2a, ((R())s)+++r(((
+ oci
1 (1-26) 2
3
1 1-20; 20;
The expansion condition intuitively states that "good" sets (elements of M' M' F)) must have suitably many neighbors with the wrong pseudolabel (elements of Shad). The trivial error bound obtained via the triangle inequality is ((.) I err(f, ū]S Ct.. The bound in Theorem 4.1 has almost the same form, but the multiplicative term on a; allows it to be much tighter than the trivial bound. Theorem 4.1 allows for pseudolabel correction because the right-hand-side can be much less than og (the error of the weak teacher) when is large and err(J, (S,) and P P(R(F))S) are small. While Wei et al. [69, Theorem 4.3] also gives pseudolabel correction guarantees for adversarially robust classifiers, Theorem 4.1 is different bound with several desirable properties that Wei et al. [69, Theorem 4.3] lacks-we we compare the two in detail in Appendix C.2 and also show how to generalize Wei al. [69]'s results to average-case-rouutees
The following proposition shows that the expansion conditions in Theorem 4.1 rule out strong models that can exactly fit the weak model.
Proposition 4.1. Suppose there exists E F such that for all x E S, f(x)) ū(x)) Then the (c, () expanssinn conditions of Theorem 4. I are not satisfied.
Proof. The family of sets that must satisfy (c, exxxaansio between Spod and Sqood is M' {R(f) n (Sgood mistakes(f)) : f E F}. Choose f such that f = when restricted to S, since we assumed this is a valid choice. Spood is defined as the set where ý makes no mistakes (and the true label is i), Sgood mistakes(s)
Spmnn, and thus R(J) Sgood MM.. Let N = fx E Spood (2(x) n Stadd * 0} be the subset of soood with neighbors in Stad. Consider an arbitrary point x E N. so there exists x' E N(x)) n Stod Since x and a' are both in S,, y(x) = y(0'). Then we have f(x) = j(xx) y(x) v(x') * (((z) f(x'). so f(x) # f(x').
Thus & R(f). This shows N R(f). Hence
R(f) Sqood R(f)) n (Sspood n NN U ( n N))
C CR()) n ((Sqood n R(f)) U (S 0 N))
= =R()) n (Sqood n NN).
The latter set is made up entirely of points with no neighbors in Stod, so
P(A(((() n Sgood) Spadd P(N((())) (Sqood (N))) 0. But for expansion to hold, we needed P(N(R()) n ) > cP(R(f) |
Proposition 4.1 shows the strong model hypothesis class enters our error bounds indirectly via the (data-dependent) expansion parameter. Using richer class for the strong model may decrease the amount of expansion, since it may make it easier to exactly fit the weak labels. At the same time, the error of the strong model on the weak labels also appears as a term the bounds (err(f, / ÿ St)), and richer class might decrease this term, these two terms capture a tradcoff. This makes our bounds

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flexible enough to capture empirical results showing that whether a richer class or more restricted class works best for the strong model depends on the problem [73].
Coverage Expansion. In this section, we relate the error of f on an uncovered subset, err(S. y TT), to the weak error of f on the corresponding covered subset, err(f, (..). The goal is to give nontrivial error bound on these points even though we see none of them during training. Expansion from Tj to Syood implies that subsets of Ti have enough correctly-pseuddee pseudolabeled neighbors. If the student model robust on N. this is already enough to prove an error bound for T, However, we also assume that neighbors. Intuitively, this means that subsets
subsets of T; have enough of T3 have the "correct" number of neighbors in Sqood and the "correct" number of neighbors in Stad This implies a regular structure in the S-T, neighborhood connections that allows us to prove a much tighter error bound. Our empirical results suggest that this structure is present in real-world examples. We prove a weaker bound that only assumes expansion from Ti to Syood in Appendix B.
Theorem 4.2 (Error bound for uncovered points). Suppose F) satisfies (c, expeanssoo on (Stood IT)), and M' (T,.F) satisfies (c. q) expaansion on (Stadd T.). Consider an arbitrary classifier f E F that fits the weak labels well on S; and is fairly robust on Tú: err(f, U1.)) + P(R(F)TT)) <
c(1 - q -aa) Then the true error of f on Ti satisfies:
as)
err(f. S (1 (1+
P(R(F)IT)) + max (q. err(J, coti
1-20 20x
10x
c(1 d1-22.)
To qualify for the bound, ʃ must fit the weak labels well on S,, so err(J, ū):)) is small, and be adversarially robust at most points on Ti, so IP( (((()) is small. We show in Appendix C that the original co-training setup of Blum and Mitchell [6] satisfies the assumptions of Theorem 4.2 with
O. Theorem 4.2 exactly recovers the bounds of Blum and Mitchell [6]. Lang et al. [37] in this case, so Theorem 4.2 is a direct generalization of Blum and Mitchell [6] but without the restrictive assumptions regarding multi-view data and conditional independence. In Appendix B, we prove a generalization of Theorem 4.2 that allows T; to expand to Sgood and Stod at different rates (i.e., different expansion parameters).
c= 1, = O, and P P(R(S))T))
It is not immediately clear from Theorem 4.2 how the coverage P(S)-the probability that the weak labeler does not abstain-affects the bounds. As with the role of strong model hypothesis class, this enters the picture implicitly through the parameter measuring expansion between S, and Tj. Informally, for a fixed neighborhood N,, it may be easier to have expansion from T; to S; when S larger, since for each uncovered point in Tú, there are more possible covered neighbors in S,. On the other hand, increasing the coverage by including more points in S, might also affect the weak label accuracy parameters a; which also appear in the bounds. In practice, there not a consistent tradeoff between coverage and performance, as explored recently in, e.g., Lang et al. [37], so our bounds do not prescribe a functional dependence of the error on the amount of coverage and instead allow that dependence to enter through data- deta-dependent parameters.
4.2 Relaxing Robustness Requirements
The previous bounds in this section assumed the student is adversarially robust at most points. Here, we considerably generalize this requirement so our results apply to any classifier f that satisfies (1), i.e., any classifier that gives most points the same label as most of their neighbors. This requires several additional definitions and goes beyond the assumptions made in other work with expansion- based error bounds, which assume adversarial robustness most [69, 12] or (effectively) all [6] points.
To allow for this relaxed assumption on the classifier, we assume a more robust version of expansion, aptly called robust expansion [27, 34, 42). To define robust expansion, we start by defining graph over examples with edges induced by the neighborhood A and weights given by the underlying probability measure. HaoChen et al. [22] study this graph in the context of contrastive pretraining.
Definition 5 (Example graph). Let G = (X, E)) be a graph with one node for each element of X (we assumed X is a possibly very large, but finite, set), and connect two nodes (æz, x²) if E N((a)) or, equivalently, if x' E (x)),, with edge weight of w(ax x!) == PPxx)) (ar) E N(x')).
Definition (regular, non-robust expansion) is very sensitive to removal of a few edges from the example graph. A set U C. B may have (U))A large, but only because small fraction of the
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edges (by probability mass) are connected to many a E A with P(x large. If we ignored these small- proba--robaaiiity edges, the neighborhood would be much smaller. Figure (appendix) shows an example. The robust neighhorhood tries to address this issue:
Definition 6 (17-robust neighborhood size). Let U, C X. The size of the n-robust neighborhood of U in A is: (U, A) == (PM UV, 2 (1 - ()-)))(N(U); U)}.
(U, A) is the probability of the "smallest" subset of A that still captures at least a 1 - fraction of the edge weight incident on U. When 1) = O, we have Pi (U, A) PP so this recovers the size of the non-robust neighborhood. In the pathological example described above, we would have ((A(U))A) large, but (U, AA) small for some v > 0. We are now ready give a "robustified" definition of expansion, which is identical to Definition 4 except that it requires the robust neighborhood. rather than the regular neighborhood. to be large.
Definition 7 (Robust expansion). Fix sets A, B C X and suppose M is a collection of subsets of B. .M satisfies (c. 9, -robuust expansion on (A, B) for all U E .M with P(U q. (U, A) > cP) This exactly recovers Definition 3 when 1) = 0.
The following (informal) theorem shows that Theorems 4.1 and 4.2 hold for average-case-robus, classifiers when we replace expansion with robust expansion and R(f) with (f) We state and prove formal versions of Theorems 4.1 and 4.2 for average-case-rouss classifiers in Appendix B. Theorem 4.3 (Informal). Theorems 4. I and 4.2 hold exactly with (c, (. (.).)))xxppasion instead of (c, q) expanssio and (f) instead of R(J).
By Markov's 's inequality, for any 1j > 0 classifier f with Ex~D)AX'> Nix) [f(x) + [(x)] S has ((),()))A) S 7. Theorem 4.3 shows that by assuming the data distribution follows a slightly more "regular" structure (robust expansion), we can give guarantees for average-case-robust classifiers. This generalization is important since matches with empirical results: adversarial training and adversarial robustness are not required for weak-to- most empirical work on weak supervision does not include adversarial training in the pipeline [71].
generalization to occur [73, 38, 10], and
5 Checking Expansion
We now outline a statistical theory for checking the expansion properties of the population distribution from finite data. This is possible because, as described in Section 4, our results do not actually require all sets to expand-rather, they only require expansion for a class of sets that is generated by the student hypothesis class. This means we can check expansion generalization of our estimate using the complexity of the hypothesis class. The purpose of checking expansion not (currently) algorithmic- the goal of the procedures described in this section is to give empirical evidence that our assumptions hold in the real world and that our bounds correlate with actual occurrences of pseudolabel correction and coverage expansion. Exactly checking the expansion of all subsets is coNP complette [7]; whether our notion of expansion with respect to a certain family of sets M can be checked efficiently is an interesting direction for future research. We show that expansion can at least be checked statistically (i.e., from finite data), if not efficiently.
a finite dataset and control the
For fixed choice of g. the (non-robust) expansion of set family M between sets A and B is: P(N(U))A) Suppose we have two samples SA = {(x,, y( with
c mingeM: P(U B)>4 RU x ~ P(-[A), and SB = ((x:)) ing with x P--BB). For a fixed U, the denominator is straightforward to estimate using SB as: P(UIB) RJ 1 Ex,esp 1/2, E UJ. Estimating the numerator is less straightforward: due to finite sampling, SA x Sn may contain no pairs (ær, with x E (((x).. That is, the empirical neighbor graph may be empty even when the population distribution expands (see Wei et al. [69] for more thorough discussion). This is major difference between our assumptions and similar work that uses expansion-like assumptions to analyze the performance of label-propagation algorithms that use the empirical graph, such as Pukdee et al. [48]. To overcome this, we assume we have access to a neighborhood oracle n : A - B that for each A returns point n nxx) B such that n²x) E (x)). We assume nothing about the distribution of n(x)) values (i.c., we do not assume that they are drawn from B.) merely that P(n(x)) > 0).
ng
We describe how construct n in a practical scenario in Section 6.
The neighborhood oracle makes estimating the expansion numerator more straightforward, since (xx) E U. then by construction, e (U)).. Formally, (()(())A)) 2 P(n(x) e U\A), where the
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Table 1: Measured expansion and eπror bounds for the covered sets S,. Expansion values for the family of sets M (S FF) are measured using the heuristic described in Section 5 and shown in the (Stad Spood) exp. column. This column shows our heuristic finds expansion in practice. Pseudolabel eτror a; = P # y|S:). Worst-case error of trained classifier f on the weak labels B. err(f. U)S,), across 5 independent training runs. This column shows the student can't 't exactly fit the teacher labels using this representation. Value of the error upper bound in Theorem 4.1 (specifically, the tighter version, B. computed using the numbers from the other columns (details in Appendix E). For label i = O, the bound being strictly less than the errrr a of the teacher ÿ suggests pseudolabel correction may occur. Finally, the actual worst-case error of trained classifier f on the true labels y, ar((/y/Ss). across 5 independent training runs, shows pseudolabel correction does occur for label i 0.
Model
i
(Stud, ggood) exp.
emr(J
Bound val err(f, y
SentenceBERT 0 0.848
0.11
0.12
0.05
0.04
0.497
0.33
0.29
0.37
0.35
quality of na( determines the tightness of this bound. This inequality is valid for any n: A + B as long as E (n(x))). Now we can estimate: P(((()) E U|A) x 1 na Lz, [[((2,) E
U1. Putting it all together, we can form our empirical estimate of the expansion by solving é
mingeM 27 Lencs, (M(m))e]
Ea [[ subject to: L ExiesB [[x; E UJ 2 q 4-6, E, where E is chosen
appropriately to account for empirical error in estimating the probability B() The following theorem, proven in Appendix D, shows that the expansion on the population distribution can't be too much smaller than the expansion on the empirical distribution.
Theorem 5.1 (Expansion generalization, informal). For arbitrary U E M,, define the population and empirical expansion estimates as: c(U) P(n(x) E UX A)/P(x U|x E B) and a(U) = na > 1 [n(x:)) E U// T [(x; E U.. Then for any ô E (0, with probability at least 1 - 8, supueM ((U) - c(U) S ( (VC(M))/a) where Ô hides constants and log factors in VC(M), NA nB. and 1/5.
Heuristic approximation. While Theorem 5.1 gives a rigorous statistical theory for checking expansion from finite data, it is unfortunately still intractable to compute the set with the worst expansion on the empirical data, i.e.. to solve Ú arg minge. ê(U'). Instead, our experiments in Section 6 use a simple randomized heuristic for approximating this minimization. If the learning algorithm A: 5T + F is deterministic conditioned our hypothesis class F of interest to those f E F such that there exists a training sample S C S with A(S)). Since each f E F generates set U()) E M, given training sample S, we can compute f = A(S), then use our "test" sample SB to compute (f).. Repeating this procedure for many samples S and choosing the smallest value approximates MingEM ê(U'). Table I shows the classification task
the observed training data, we can simplify
expansion measurements for different hypothesis classes on a weakly-superviet inspired by the example in Section 3. Appendix contains more results and a much more detailed description of the setup and process of checking expansion, but these results indicate that expansion present and correlates with performance.
6 Experiments
Setup. We explore training linear classifiers on top of the contrastively-fine -inett SentenceBERT embeddings' [52]. As shown in Muennighoff et al. [44], training simple classifiers on top of these com- plex pretrained representations leads to very competitive performance. We study binary sentiment pre- diction for movie reviews on the IMDb dataset [41], continuing with the example from Section 3. For the teacher model, we use a very coarse rule [49] based on the presence of the unigrams "incredible" and "horrible". Let Cuw,, æ)) be the number of times word w appears in input ac. The weak label ((x) is 1 when C(incredible, ar) > C(horrible, x=), 0 when C'horribll iile. x:) > C(incredible xx). and Ø otherwise. This counts the occurrences of "horrible" and "incredible" and assigns the binary label corresponding to the word that occurs strictly more often, and abstains otherwise.
e HuggiingFaceHuu model ID sentence- wentence-transforneee trwnsforrnrsss....... bransforrerrs/ll--pppe -pas-eeet-------rr

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Neighborhood function and oracle. We set (x) to be the examples obtainable from x: by sampling from a pretrained paraphrase model. As described in Section 5, to measure expansion between sets A and B, we need neighborhood oracle n : A B that, given x E A, returns point x' E (a))) B. Our results require to measure expansion between (Stood T.)) (St Ti), and by rejection sampling from
). For
E & (resp. a E S we generate a target point æ' E (=x) pretrained paraphrase model. Because ÿ takes a simple form, we can efficiently approximate this step by setting the logits of tokens "horrible" and "incredible" to - -oo during decoding so they are never generated. For æ E Stad, to generate a neighbor x" E <N(x)) n Sgood we prompt GPT-4 to paraphrase randomly chosen sentence from x' (a(x)) n T; and include the correct word in its paraphrase, so that x" E Squad. We rejection sample until this constraint is satisfied. Figure (appendix) shows examples of these procedures.
Expansion results. Table I measures the expansion of the set family M' F) on the sets (Stad, Spood) ) using the procedure from Section 5. Theorem 4.1 shows this is related to pseudolabel correction. For the SentenceBERT model, the measured expansion is high and the student fits the weak labels well, but doesn't overfit the teacher labels (i.e., err(J; f )).))
F))
0). For label 0, our bound indicates that pseudolahel correction should be present, since the value for our error bound less than ag There indeed pseudolabel correction on this label: cf 3) So) < ogg For label 1, our bound indicates that pscudolabel correction may not occur-the the bound value is greater than o, since the the measured expansion is lower for this label and the error of the student on the weak labels is higher. As suggested by the bound, there is no pseudolabel correction: err(f. y|S} > an- Our expansion-bas based theory can therefore differentiate between cases where pseudolabel correction does and does not occur. Table 4 (appendix) shows the measured expansion values between the set pairs (Spood Ti and (Strad, Ti). which Theorem 4.2 shows are related to the amount of coverage expansion. For example, for label 1, T; expands to both Sgood (c = 0.75) and to Shoad (c = 0.55). The fact that both expansions are nonzero gives evidence for the structure assumed Theorem 4.2. In this case, our coverage expansion bounds show that the student model has nontrivial performance on the uncovered sets T:-for example, for label 1, the worst-case value of err(f, y (Ti) in all the training runs is 0.29, and the value of our bound is 0.33. Appendix E contains more details on how the models are trained and the bounds are computed.
7 Limitations & Conclusion
In this work, we proved error bounds based on expansion properties of the data distribution and student hypothesis class that directly allow for weak-to-strong generalization, gave statistical theory for checking these expansion properties, and gave empirical evidence that they hold in practice. Our empirical procedure for finding the worst-expanding set generated by our hypothesis class is ultimately still a heuristic, and our experiments are limited in scope. However, Sections and go beyond prior work by testing assumptions more carefully. Finally, this work contains new weak supervision algorithuss (e.g., new training methods) for improving weak-to-strong generalization. While our work does not propose new weak supervision algorithms, we believe our theory suggests a framework for encouraging weak-to-strong generalization effects: find neighborhood structure and student hypothesis class pair that expands, then find the student model f F that minimizes the error on the weak labels while staying as robust as possible on the neighborhoods. Both expansion and contrastive pre- pre-training are related spectral properties of the underlying neighborhood graph [22]. Can we improve the performance of weakly-supervised learning by imbuing the contrastive pretraining objective with knowledge of the pseudolabeler ÿ? Burns et al. [10] show that weak-to- -o- strong generalization effects are much stronger in classification problems than in other settings such as reward modeling. Adapting our analysis techniques to reward modeling and using our results to design algorithms for weak-to-stron to-strong reward model training are interesting directions for future work.
Acknowledgments
Thanks to Hussein Mozannar and Ilker Demirel for feedback on drafts of this paper. DS and HL were partially supported by NSF AitF award CCF-1723344, AV was supported by NSF grants FECS2216970 and CC1-2 and HI. was supported by the NDSEG fellowship. Finally, this project was partially supported by OpenAI "Superalignment Fast" grant.
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A Additional Related Work
Programmatic Weak Supervision. Pukdee et al. [48] study the performance of label propagation for weak supervision. Unlike our work, they assume expansion with respect to the empirical sample graph and study the performance of particular learning algorithm (label propagation). Chen et al. [13] show how to propagate weak labels using the embeddings of a strong pretrained model to provably improve performance, but they also focus on the empirical graph, and not on guarantees for the classifiers trained on the weak labels. Unlike most other work on weak supervision, which tries to give separate algorithms and guarantees for (i) the procedure for creating pseudolabel out of multiple labeling functions and (ii) the student training, Sam and Kolter [55], Rühling Cachay al. [54] consider direct end-to-end weak supervision. However, they do not give theoretical error guarantees for the student models. Cabannnes et al. [11] give theoretical guarantees for different flavor of weak supervision, when the "teacher" gives a set of labels that contains the ground-truth but may also contain other incorrect labels. Robinson et al. [53] show that when only a few gold labels are available, weak labels from a teacher model can speed up the learning process.
Domain Adaptation. What we call "coverage expansion" is very related to some work on domain adaptation that can still apply when the source and target distributions do not overlap, such as Ben- David et al. [4]. Blitzer et al. [5]. Our expansion assumptions for coverage expansion, which require the expansion of certain families of sets (generated by the student hypothesis class), are qualitatively very related to the FAF distance [4]. Assuming the FAF distance is small essentially says that the mistake set of any hypothesis in FF must have similar probability in both the source and target distribution, so the target error can't an't be much higher than the source error. Kifer et al. [31] showed that this distance can be statistically estimated from a finite sample. In an imprecise sense, our results suggest that the FAF distance is small when every hypothesis E F is robust on the neighborhoods N and the distribution has good expansion. We also have one unified set of assumptions that leads to guarantees for both coverage expansion/domain adaptation and pseudolabel correction, and our coverage expansion bounds account for the error of the teacher model on the source domain. Abbe et al. [1] also give theoretical guarantees for learning boolean functions when part of the data domain is completely unseen during training.
Knowledge Distillation, Pseudolabel correction. The observation that distilled model can outperform its teacher (i.e., what Burns et al. [10] calls weak-to-strong generalization) goes back at least to Buciluă et al. [9]. In the context of knowledge distillation, Stanton et al. [58] show that even when student models have the capacity to match the teacher, they don't match them exactly. They give empirical examples of this phenomenon and argue that it can be due to optimization issues during student training. Clearly, this effect is critical for weak-to-strong generalization, and our expansion and robustness assumptions effectively try to capture the structure (whether implicit, due to the optimization process, or explicit, due to the choice of student hypothesis class) that rules out this seaettteacher-itteee fitting behavior. In the context of self-training, Chen et al. [14), Kumar et al. [32], Oymak and Cihad Gulcu [47], Frei et al. [19] all (effectively) give pseudolabel correction guarantees under certain distributional assumptions, such as when the data distributed according to a Gaussian Mixture Model [47, 19]. In contrast, like other work with expansion assumptions [3, 69, 12, 22], results do not assume the input data follows any specific distributional form. Indeed, Wei et al. [69] showed that distributions similar to the Gaussian Mixture Models studied in these works satisfy notion of expansion very similar to the one we study here.
B Error bound proofs
We directly prove the "average-case-robuss" versions of Theorems 4.1 and 4.2. Theorems 4. 1 and 4.2 directly follow from the equivalence between expansion and robust expansion, and adversarial robustness and g-robustnesss when 17- = 0. For convenience, we reproduce the definitions of the example graph, robbbuss,ssss, and robust expansion here. Figure 2 shows examples of good and bad robust expansion.
We begin by proving some lemmas that will be useful for both the pseudolabel correction and coverage expansion bounds.
Definition (Example graph). Lei G = (X, E) be graph with one node for each element of X (we assumed X is a possibly very large, but finite, set), and connect two nodes (æz, x²) if E (a')) or equivalently, x' e (x),, with edge weight of w(x, x²) == PP()) 1/x E (x)')).
16

---

robust neighborhood
neighborhood
AKUI
AXU
robust neighborkood
neighborhood!
Figure 2: Examples of good (left) and bad (right) robust expansion. In both cases, there is a core subset C (U) that accounts most the edge weight incident on U (at least 1 fraction, for some small r)). The robust expansion is good when every such subset has large probability.
Definition (17-robust). For an arbitrary classifier I and point define r(J, P(f(x') o [(=)|xx' E (x))) as the probability gives different labels and a random neighbor x' of a. classifier f X + 3 is said to be n-robust
point if x) n, For an arbitrary classifier
let (f) = {x r(f.)) S. n) be the set of n-robust points for f.
Definition (77-robust neighborhood size). Let A, C. X. The size of the n-robust neighborhood of U in A is: (U) == miny Definition (Robust expansion). Fix sets A, C X and suppose M is collection of subsets of B. M satisfies (c. 9, 1) -robbust expansion on (A, if for all U E M with PUU B) >
(P(VIA) (PP(V
w(V U) 2 (1 - ()))) U)),
P1 (U, A) > BB) This exactly recovers Definition 4 when 1 0.
The following lemma shows that a classifier that robust on average must also 17-robust on a set of large probability.
Lemma B.1. Fix sei A C and a classifier J, and suppose
Ex.DIAX # f(x') S
for some 7 > Then for any n > 0, ((),(
Proof. Rewriting the condition on slightly,
Ex-Di ( (f(x) & f(x') N(x))) 5:
Recall that r(æ. f) = (f(x) # [xx) (x())) So we have BXND/A f)] S 7. Markov's
inequality implies that for any 17 > O,
P(r(x, f) > 7/A) EXNDIAlr(x, f)]
17
77
Since R,) = (æ: r(x /) S n) we thus have
((I,(())A) < 1j
?
which concludes the proof.
Good and bad edges. Consider an arbitrary classifier f and arbitrary set U C and fix x' E U. E (U). We say the pair (x, æ') 3') is bad if f(x) * f(z'); otherwise the pair is good. Let NUU) be the subset of NUU reachable by good edges. Formally, .N(x') = {x E <V(x') (x,x') x') good} and ((A) E (xx).. We are suppressing the dependence of Ñ on the classifier f for notational convenience. The following lemma guarantees that if f is n-robust on all points in U, bad edges do account for much of the weight between NU) and in the example graph (Definition 5). This is the key to our sarrrge-case-robustin and f is robbstt on U, the neighborhood (U) of points reachable by good edges must be large. Lemma B.2. Consider an arbitrary set U and suppose that for all æ U, f is such that rff, x) S n, Le.. U C (f) Then:
results, since it implies that if the robust expansion is large
w(N(U).U) Z (U)(N(
17

---

Proof. Since every x' E U is in (f), we have:
2 -f, = (x(f(x) # f E (ax())
[x(f(x) & E N(x'))
P(
ExEx (l(x)) # (x E E
ExEx 1/2 E ((a)))
Enex 1[f(x) # f(x')]11z E ((2
Enex 1/x E (()P)) EBEN 1[f(x) + f(x''(()) (1x)) Euexw(0 w(2x
so for all x' E U we have:
Σ
1(x æ²) bad (ad] x')Sn) Sn
Σ
w(xx2')
DEX
#EX
Because w(x, x²) over M(x'), So:
>
a EEW (x'),, we can replace the summations over all of X with the sum
Σ
[[xx, x') bad (2, x') In
Σ
w(x, x').
(2)
mEN(x()
Now we can simplify:
w (U())) Σ Σ
w(xx
r'EU (EN(w)
Σ
Σ
11(x, x') good]
E'U (EN(2²)
where the inequality is because some elements of Ñ(U) might be reachable by mixture of good and bad edges. The seft-hand-side counts both, and the right-hand-side only counts the contribution of the good edges. Note that [(xx, x²) good] = 1 [ x²) bad]. Plugging this in gives:
w w(N(U).U)
Σ
Σ
(1 1[(x, x') (2d))
E'EU xEN(x²)
Σ
Σ
w(x x') - Σ
Σ
1[(x, x²) bad (ax)
o' EU (EN(2
x'EU wEN((")
Σ
Σ
w(x - 1 Σ
Σ
w(x, x')
EU (EN(2))
x wEN(w")
(1 1))
Σ w(x
E'EU (EN((=)
(1 (1- (I-))w(V(U))UU)
where we used (2) in the second inequality.
Expanding set family. Now construct the set families that must expand for our results in this section. Let F be the hypothesis class for the strong model, y the ground-truth function, and B c X an arbitrary set. For each f E F let (B, f) (a B f(x) + y(*)} be the set of f's mistakes on in B. Then we define the family:
M,( (B FEF).
That is, (BFFF) is the family of rj-rouust (Definition 2) mistake sets of F the set B. This exactly agrees with the definition of M(B.F) from Section when 7) = 0. Similarly, we define:
MB = (R,(F) n (B (U, f
as the family of robust non- non-mistake sets. Again, this exactly agrees with the definition of MB) from Section 4 when n 0.
18

---

B.1
Pseudolabel correction
We directly prove the "average-case cosrage-case-rbusst version of Theorem 4.1. Theorem 4.1 then follows from the equivalence between expansion and robust expansion, and adversarial robustness and n-robusttesss when 7) = 0.
Theorem B.1 (Pseudolabel correction). Suppose (S FF) satisfies (c, q, roobuusstt expansion on (Stad (good) )or some and 2 0. Consider arbitrary classifier I such that P(f(x) + x(x) or f not q-robust x x{S:) - Osy- Then f satisfies the following error bound:
err(f, ((.)) ax, - -1) 2d' P, (R)
err(J.y(S:) <
22
where d' = cc((1 Ci + cas).
Corollary B.1 (Simplified bound, average-case-robus version of Theorem 4.1). Suppose the condi- tions of Theorem B.1 hold. For any A such that err(S. S ca, (1 a1 A), the error of f satisfies:
20;
err(fyy 1 - -2a,
((R,(()))))++ + a,,(1 (c.)).
Moreover, if satisfies
FX--DS + [(x)) s
then
20; 7 e ta,(1 2cA).
err(J,y(8.) < 1 20; !]
Proof. The first part directly follows from Theorem B.1 by substituting the value of c', using S 1, and using the form of A to simplify the bound. Theorem 4.1 follows from taking A 3/4, which was small enough for the error condition hold empirically. The condition on in Theorem 4.1 implies that if (x) + (xx not ro-rotttt at S() s +3a, then
(x * (x) or f not n--obust x(S S - ais
so the conditions of Theorem B.1 hold. This upper bound on could be replaced by instead assuming
PP A 5(x) or f not ry-roustt at x(S.) min q az 1 a; 3ca;
The second part follows directly from Lemma B.I.
Proof of Theorem B. Let M; = {a S. f(x) + y(x)} be the set of mistakes of f on the true labels in S,. Similarly, let D; = (r S f(x) * [(x)) the set of mistakes of f on the weak labels in Sg. Define U, = S; M; and let V; = (f) n U, O Sgoud, Note that V; E M' FF) so if it's large enough, it expands according to expansion assumption. The following lemma shows that V. is large enough to expand.
Lemma B.3. PVVV]S >
Proof. Suppose for a contradiction that (V, Sgood
S q. Then by definition of V,,
P(V.
= P V. Sgood
1 P(V. Sgood Sgoood
PP(((S, \ Ma) R,, O ggood Sgoody
1 1-P(((M O Sgood) U R2 [f] ggood
19

---

Fix an arbitrary E M, n Sqood By definition of M,, f(x) F v(x). By definition of Spood
((x) = v()). Hence f(x) + ū(z),
D,, and therefore M, O Syood
D,. Then:
q 2 (V:
2 1 P(D; (f
= 1 - (1 P,(,P( U R (f) n Spood
2 1 (1 ()) (f)( (S.)
= 1 - (1 0: ox)))((((x) + j(x) or f not 17-robust at x|S.)
2 1 (1 0:) -
= q + a2 1------00 -3)))
>
The second-to-lase inequality used the assumption on P(f(x) + j(x) or f not 1--roust at x|S.) and the final inequality used < 1 and 0 < < 1/2. Assuming (VVisSpood) S thus leads to q > 9, contradiction. So P(V, (Sgood) q-
Recall that for a set A X, we define (AA) C (A) as the subset of points reachable from A by good edge f()) f(x'). ). By Lemma B.2, since V; C R (f),
wVV(V,), V.) 2 (1 - (1-n)w(V(V3) V.). Then by Lemma B.3, since (Stad syood) satisfy (c, q, -oobust expansion,
P(N())) 2 PV, Stmd) 2 cP((V)Sspod)
Fix an arbitrary a E (V.). By defimition of N, there exists x' E V; such that f(x) = f(x'). Since x' V2 Ug, we have that f(x) = f(x') y(z') (since x' & M.), and g(x') = =y(x). since æ and x' are both in S,. This shows f(x) = v and therefore æ & Mi, so a E U, = SS M,. Since æ was arbitrary, (VV) U. Thus,
P(U 2 cIP(Vi)s;,0dd))
Points E U, are correctly labeled by S. and points E Shed are incorrectly labeled by the pseudolabeler ÿ. This inequality thus guarantees that there must be some points that gets correct and I gets incorrect. It also gives quantitative lower bound on how much probability is assigned to these points. Intuitively, (3) is already a "pseudolabel correction" result, and the remainder of the proof is deriving final bound on the error from (3).
The following lemma converts (3) into relationship between P(U,]Shud) and P(U, n (f)(ss).
Lemma B.4. StU 2 CP(UU n (, + c'ouP((U n (n( where e' = c/(1 - 0ş co.).
Proof.
P(U,)Stadd 2 cPP(V.lS Sgood
(by (3))
P(V n WWnnddoddsss SSnod(s.)
1- ai
P(U. n (J) Spood S.)
(P(U.n ((IIS)) - IPU n (2) I.ad)s
Rearranging, 04
P(U n (f)S)) < (PUU,)
P(U, n CIU,,B,, (f)
Shad
1-a; a;
= PPU,
+
Co:
P(U, n R. (f))tt(d)))
1 1 as
=PUUU (ffdd +
Cas
(P(U,Is
P(U, (f)
1 l-a; a;
1 00:
P(U, n R,, (f) 004
az
1 - a;
20

---

And finally,
P(U, n (()(s) +
Co2
P(U n (,)( hod S ((
04 co2
05 00
The following lemma decomposes the disagreement set D; into its components in Syood and Sped
Lemma B.5. D; D (M, n (MMnsoo U (U; S
Proof. First, fix a Mj n Sqaad Since a E M1. f(x) * y(z). Since a E Sgood 7(x) = y Thus f(x) + ((x), so x E Dr. Second, fix a E U, n.n. Stud Since a U, = Si Mr f(x) = (()). Since Stnd, ((x) + y(z). Thus f(x) & ((x), so Dz.
By Lemma B.5 and using P (Spood (S:) (S) P(D.(S) ((M (1 a;) + PPUil
(S (S:)
aii
Applying Lemma B.4,
P(D.(S,) 2 (1 az (MM coc (P(UInR,,()IS + a.U(U ( (((()R)((())((
(4)
Applying Lemma B.4 again,
P(U,(S:) a,P(U + - (1U11))(U31)) ス c'a (PUU (. aPP(U) (, (T) + (
so we have
1 (P(U.)8.) d'a (P(U, (.U (f))s
S 1 aj
Combining this with U, = S \ Mr
P(M goood = PUU soood
1
(P(U.[S:) - d'as (P(U. + (R, ))
> 1 aq
Plugging this into (4) gives:
P(D.(S.) 2 (1 a:) - P(U.)S)) + 2e'a; (P(U: n R2(())))) =.PU))
2 (1 a.) - (1- (MM((())))) 26 (1 1 UI-(MUR, + (R, spand(s))
P(M()S) a2 1) 2d'o; (P(U, n R2, S 11nd PPMMU -PM (()Ss))) 2 S.M + (2c' - 2d',, P U (f(())
Letp = (S.))) Then, using the union bound for the last term,
P(D, 2 p a; - 2d (Rr,
= -1122'a;)p + a - 1) - 2c' 2c a (f)(s)
Since P(D,(S.) = err(f. (.S.), rearranging terms we get:
er(f.y/S) P(MIS)) < err(s.ūSs) az(2c' - 1) + 2d a, (PR((())))))))
1-2c' 2c'a
21

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B.2 Coverage expansion
We directly prove the "average-case rase-robust" version of Theorem 4.2. Theorem 4.2 then follows from the equivalence between expansion and robust expansion, and adversarial robustness and n-robustness, when 0. The following theorem also allows for T; to expand to S and Sped in different amounts, controlled by the two expansion parameters ej and cg. This allows for empirically tighter bounds and is used in our experiments in Section 6. Theorem 4.2 is a special case of Theorem B.2 taking 7 = 0 and 61 = C2.
Theorem B.2 (Average-case- robuust version of Theorem 4.2). Suppose there exists C1,02, 9.7 > O such that (T, F, v) satisfies (c1, 9, -xxeaansoo on (S,o0d TT) and M (T,FFF satisfies (02,9: q. (.77)xxxaassion on (Stad, T.). Fix an arbitrary classifier f:X + y that satisfies:
err(f.ū[s) + ((),( < (1(1 - q - -
Let max/ci, Lete=max(cı,og ) and min(c1, c2). Then the error of f on T is most:
((RI()))T)) + max (1. ? err(y, U)S:) c204
err(S.u(::)= (1 c/ē 20
(cz + c2)as)
Moreover, if satisfies
EX-DTT.X-DWWI//(( # f(x') Sy
then the error of J on Tr is bounded by:
err(f.y(Ti)) S 1
04
7 + max 9. c1 (ei + c2)a:)
errig, ū|5,) ) c204
c/C - 20%
17
Proof. Let M, = (0 E T; : f(x) + y(0)} and =
= T, \ Mj = fo E T f(x) = y(x)}
= M, E (f) and V C, (f). We have U. E (T, Define their robust subsets U,=
and V; E MT,
Recall that for a set A, (AA) is the subset of (A) consisting of neighbors reachable by good edges (edges where f assigns the same label to both endpoints). Since U₃ C (f)
Lemma B.2 implies that (U,) has enough weight to qualify as one of the sets in the definition of the robust neighborhood size Pu- Sgood). In particular, it implies that ((N( (Spood 2 (U, sgood Similarly, (V4) satisfies IP (V 2 Pi-n(Vi, Spad). There are now three cases to consider:
Case 1: (PU))))) > q, P(VIT) > q. In this case, the expansion assumptions imply:
P(N(U))) 2 (TU)
(VV(())) 2 (VV.TT)
For all x E (UU,) n Sgood there exists x' E U2 with f(x) = f(x'). Then:
Def. of S,T.,
f(x) f(x)) + y(x')
y(x) ((x)
x'El.
xESpood
so f(x) # ((z). Similarly, for all x E (VVV n Shad there exists x' E V/ with f(z) f(x'). Then
Def. of S.,
f(x) = f(x') = u(x')
vx)) + ((x).
æVEV
so f(x) + 5(xx).
Let D. (x E S: f(x) + )(()}} be the set of points in S₁ where I and the teacher ÿ disagree. These arguments showed that ((U))) n Soood c D, and N(V)) n Stad c Dr. Since these sets are disjoint,
P(D.(S.) Z F(N(U.) n Sgood (S:) + P(N(U))) n Studd (S:)) -P(N(U)) 5.000d )1 (1-aa) as) + (U.) (stadday
2
22

---

Since U, UV; = (2) and U, and V; are disjoint subsets of T.
P(VV[T)) P(R, P(U))T))
Plugging this into the previous inequality and simplifying gives:
P(UIIT)) < P(D:[S:) e2a,P (2,))( PPD S.) c20; + (2a,
6 (c1 + c2)a;
01 -- + +22
((DD))) C204 + g/c -2a; (()
03
CI (ci 02)0:
We can then bound I(M = P(U,T))) + P(M, n (,(TT) S P(U,IT)) - ((),)) so all combined, we have:
((R,(T)TT)) + P(D.(S.)- - 0204
P(MIT)) S (1+
a
c/c 201
01 (a1 + t2)aa
Case 2: P(U. & q. Here we directly upper-bound P(U; with q and P(MIT.) S P(U.)T)) + ((),,()) q (f),())
Case P(U.)T.) > q, P(V.T) S q. Since (U) + P(V[T)) ((),()) in this case we have P(U 2 ((ITT)
q. Since P(U, (T) > q, U, expands, so as in Case we have:
(S 2 P(U:)T).
and therefore:
((D()))) 2 (NN(U (U,) (Sp00) (1 as) 2 c1 a11-a,)P(U,11)) 2 a1 (F)((((((() 9).
But F(D:(S,) = egi(g. (.. (.)) and we assumed:
q a.) S c 90( a:), err(g, أ) S + (())(f) < (1-
since e S 1, this implies
err(g, (S)) + ci o:) < ci () - as);
so:
(1s.) < < - q) as)
-
So Case leads contradiction. Combining Cases and 2 yields the final bound. The second part follows directly from Lemma B.1.
If = = c, we have c/é and C (c. + c2)a; = c(1 20x), which gives simpler bound for coverage expansion of weeaae-ccsse-oouse classifiers. When 7 = O. ((),()) == (()()) by definition, and (c. q, -xpaansin is equivalent to (c, -)-xxannsioon These two simplifications directly yield Theorem 4.2 for coverage expansion of adversarially robust classifiers.
The following theorem gives a more loose bound that only assumes expansion from T; to Sgood Theorem B.3 (Coverage expansion, weaker assumptions), Suppose MT, F) satisfies (c, 9: n)- robust expansion (Sgood TT:) for some f on T, is bounded by:
- y. The error of 0. Fix arbitrary classifier X
err(f. S ((R((() + max (g. enr(f.0(s))
c(1-aa)
Proof. Define M2 = fx: (xx * y(x)) } n Tj as the set of mistakes of f in T;, and let U,
M; n (J). Let Dj = (a E S, f(x)) * ](x)) be the set of points in S where f disagrees with the weak labels. Note that err (f. = (()) and err( v[))) P(M( & (PU,)TT) - ((R... So the goal is to bound P(U,)T.). Recall that ((U) is the subset of NUU) consisting of neighbors reachable by good edges (edges where I assigns the same label to both endpoints). Since U, C (f), Lemma B.2 implies that (U,)) has enough weight to qualify as one of the sets V in the definition of the robust neighborhood size PU- Spond)) ). In particular, it implies that (UA Sppood 2 (U, Sgood). ). Note that U, E MT, FF) since it is a mistake set intersected with robust set. Then, if TT)) > 9. (c, -robuuu expansion implies that P(N(U))) Soood 2 (U. Sfo00 > (T.
We now proceed in two cases.
23

---

Case 1: P(U,TT)) > q. In this case, because (Stood T2) satisfy (c, r...uust expansion, as discussed above, we know that (U) z cl (U:(i.) and thus
(PN(( n Sgood S.) 2 cl1 a:)
Fix (U,)) n S By the definition of (U)),, there must be a point x' E U3 reachable from by good edge. That is, there is point x' U, such that f(x) f(x'). Then the following set of incqualities holds:
SE
at J'E.M.
((x)
u(a) y(x')
+ f(x') f(x).
Defn of S., T.
Defn of N
This shows f(x) # 5(xx), so x E D,. Hence ((U),)
Sgood C Dr and we have:
err(f, uT(f. = P(D.IS)) 2 P(N(U,) n goood
S)) 2 c(1 P(U,,
so
err(Juu(T) S ((),(())TTT)) + err(J, S(:) c(1 a:))
Case 2: P(U, & g. In this case we directly use the bound T.) err(g, ()) S ((I,())):) + Combining the two cases yields the theorem.
q, so we get S
B.3
Proposition 3.1
To prove Proposition 3.1, reproduce (a tighter, intermediate version of) Theorem 3 from Fu et al. [21] and translate to our notation, then simplify the bound in the setting of the example from Section 3.
The goal Fu et al. [21] is to estimate set of graphical model parameters μe such that, given a vector of labeling function outputs X(x), the graphical model P, (y)X(())) approximates Pyy|)). A classifier then trained on weak labels sampled from P (x)))(())). Define err(f) = P(x,y)= [xx & y] and err,, B(xx)=D(P (Pyp) Acx) ( + 7]]
Theorem B.4. Fu et al. [21. Theorem 3, intermediate step, p.39] Let f x + y be an arbitrary classifier. Then:
Σ = - - Pu = ylA(x) = x(X>))
err(f) S errp(f) + Bxx
(5)
The remainder of the theorem accounts for finite-sample issues and estimation error for the optimal graphical model parameters but this is the fundamental relationship proven between classifier's s clean error and weak-label error. The second term in the RHS of (5) is exactly twice the total variation distance between P(y[x) and I (y (x)). For the final theorem statement, Pinsker's 's inequality is used upper bound this term using the KL divergence between the true probability Py x) and the
estimated graphical model probability P (y (y)A(x)), but the TV version is tighter, so we proceed by analyzing the intermediate bound (5).
In the example from Section 3, there is only one labeling function:
i good E æ
A(x) = ū(()) = 0
bad E
Ø
otherwise.
and we have assumed that the true label is deterministic function of x, i.e., y v(x). We can lower-bound (5) by using (fr 2 and exactly computing the total-variation distance term assuming the optimal graphical model parameters a are known, in which case the joint distribution
24

---

Py
X(x)) Fxx)))(x)) exactly matches the true joint distribution P(y, A(x)):
Pa(y - 0,A(x) - 0) - P(y -
A(x) -
0,
1 - 0) = Fly 0)P(A( - = (1 -a)
Pa = O, 0,4(x) = 1) = P(y - 0, A(x) = 1) = P(y = 0)((((()))))))))))))))))))())))_ = 11y = 0) = a
(y - 1, 1()(x) 1) P(y 1, A(x) - 1) - Ply 1))(((()) 11y = 1)
2 (1 0)
1
(y - I, 1x) 0) = P(y = 1, M(x) = 0) = P(y - 1))P((()) Oly = 2
(y = O, A(x) = Ø) = P(y = 0, xxx) = = = P(y = 0)P(A(X) = Ø,y = 0) = ((( = Ø)
2
Pn(y = 1,x(x) = Ø) = P(y = 1, X(x) = =0) P(y = 1)P(A(X) = Øly = 1) 2
P(A(x) = Ø),
where we used the assumptions from Proposition 3.1 that PÚù = Oly = 1) = PO 1,y = 0) = a
(the error rates of the weak labels are equal for both classes) and (P((x) Øly = y) = P(X(x) = Ø) (the weak labels cover each class equally often).
Now can this joint distribution to compute the value of the total variation distance term in (5). There are six cases.
Case x E Sg Sgood. Here æ is a true negative (æ Sp) so y(œ) 0. Since æ E Sq Sgood (i.c., the pseudolabel agrees with the true label), X(x) O. Here we have:
F(y = 1|x = x) =
F(y 0|x x)
(n = 0, M(x) = 0)
21
(y = 0/A(x) =) = Py = 0, A(x) = 0) + P, - 1, 1xx) =
1 - a
2(1 0)
20
(y = 1|A(x) = 0) Py = 0, A(x) = 0) Pu = L, x(x) = =)
P = 1, = 0)
70
= 0
[11 0)
2
And so the (doubled) total variation distance for this is
11 - (y 0/A(x) 0)l 10 - (y = 1|A(x) = 0)1 = 11111-110 a) + 1 20.
Case x E ES Syood Here a true positive (a: 81) so y(x) 1. Since x E Sgood (i.e., the pseudolabel agrees with the true label), X(x) 1. By the analogous argument to Case I, the doubled TV distance term for this 2ax
Case 3:
Shhd, Here is true negative (æ: So) y(æ) = 0. Since E Shed (i.e., the pseudolabel disagrees with the true label), A(x) = 1. Here we have:
Ply = 1|x = x)
P(y 0|x x)) 1
Fu O, 0,A(x) 1)
( = 0/{(x) = =1)= Py = 0, A(x) = 1) + Paly= = 1,) =1)
1
2 2 (1 a)
a
2
P = 1, 1,x()) =1)
Py = 1|A(x) = =1) Py 0() 1) - Paly = 1,x()) =1
=1- -1-a
And so the (doubled) total variation distance for this æ is
1 - (y 0|A(x) = 1) + 0 (y = 1/x(x) = =1]= | + -1 2(1 -a).
Case 4: E Stud Here is true positive (x E S1) y(z) = 1. Since E Shud (i.e., the pseudolabel disagrees with the true label), X(x)) = 0. By symmetry with Case 3, the (doubled) TV distance for this x is 2(1 a).
25

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Case 5: a E Tu. Here æ is true negative by definition of To, so y(a) = 0. Thus P(y = 0|x
2) 1. Since x T. the pseudolabeler abstains, i.c. \(x) 2. The (doubled) total variation distance for this x. is thus:
1-PP,y = 0|A(x) = Ø)) + (y = 1J{(x) = の)ه =
1 2
+ 10 2
Case 6: a E Ti. By symmetry with Case 5, in this case the doubled TV term is 1.
Finishing the proof. Finally, we can simplify (5). For any partition (V)) of X,
Σ P(y = y/x xx)=xxx-- - Pu(y = vl>(x) - X6X1011
)'X
= P (x)
Σ [ply x|x= P sl((x) X(XI)XXE
Applying this with our 6 cases above, obtain:
(x' w'>
Σ IP = y/x x)) Puly = 8/A(x)xx---)>)))))(())) = X(x))1
= P(Sgood) -2a
+ P(5 -2x
+ P(Sbadd
2(1 a)
+ P(Stad)
2(1 a))
+ P(TO)
+ P(Ti).
Now we can group terms together using
P(Syood) P(S,, ((SS)) (Sy) P(S a) y
P(Shad) = (P Syhad (S,) = P(S,)a
P(So) + P(Si) = P(S)
P(Tb) P(Ti) = P(T)
for y E {0, 1} obtain:
Σ IPly y/x y/x=
Py vlA(x) =(x()> P(S))4a( a) P(T).
Since ignored the nonnegative (fr term in (5), the following bound is tighter than Fu et al. [21, Theorem 3]:
err(f) < 4a - a) P(T).
However, this is looser than on S (since < 3/4), so it can never account for pscudolabel correction, and charges every point T as an error, so can't account for coverage expansion.
B.4 Proposition 3.2
Here we reproduce Theorem 4.3 from Wei et al. [69] in our notation and show how to apply their bound (initially designed for the full-coverage P(S) 1 pseudolabel correction setting) to the weak supervision setup. The essential idea the reduction to treat points x E T, where x(x) = Ø, as mistakes from the teacher. Unfortunately, this effectively limits the application of the bound cases where (P)) is large.
For classifier g, let Mg(g) = fa E X; g(x) + y(x)) be the mistakes of g X; (with respect to the true labels y(x)). In particular, M,(g) = fa E X, : g(xx) + i). Recall that the pseudolabeler is classifier ÿ - (e)). our notation, Mz(r) = T, U Mj( contains T, because for all E T, j(x) = Ø (()).
Definitions.
26

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Definition (Expansion, Wei et al. [69]). F())X) satisfies (c, (c.aa)xxxpasion for all U C. M.) with P(U)X;) I a, P(N(U)))) 2 min(cP(U|x)).1)
Theorem (Wei al. [69. Theorem 4.3), our notation). Let max; (())) and suppose that a < 1/3 and for all i, P--)X) satisfies (c. -xppaniinnn for > 3. Suppose the classifier minimizes:
+ L(g.0) = c c err(g, 9)
2c FlEe's e NVxx) g(x) + g(z')}
c
Then و satisfies the following error bound:
2
2c c- P/ax' E EN(x) : v(a) + y(x')).
err(g. - c err(ū.y)
This bound shows that when the expansion (according to their definition) is large and the ground- truth classifier ÿ is udversarially robust over / on most points, classifer 9 that minimizes L(g. ÿ) enjoys good upper bound on it crror. In particular, this bound can be lower than the error of the pscudolabels err(ū, y)- However, the assumptions require that ((M( < 1/3 for all i. Since T₁ M.(). this requires that P(T; < 1/3 for all i.
In addition to requiring high coverage, this application of Wei et al. [69, Theorem 4.3] also gives one unified bound for the classifier error on the full input space X' instead of dealing with coverage expansion and pseudolabel correction effects separately. Empirically, in real weak supervision settings, the two effects don't always occur together, so ideally a theory for weak supervision would treat the two effects separately. We take this approach with our bounds.
B.5 Note on finite-sample error bounds
Frei et al. [19] give a finite-sample analysis that shows self-training can give weak-to-strong general- ization effects under more strict distributional assumptions than what we consider here. Wei et al. [69] and Cai et al. [12] both provide mamte error bounds for the student model in addition to using expansion to prove relationships between the weak label eπror and true label error the population distribution. They follow a modular analysis framework that is also followed (for example) by Blum and Mitchell [6], Balcan et al. [3], Lang et al. [37]: first, the population error the clean labels is related via structural assumptions to the population error on the weak labels. Second, and almost orthogonally, off-the-shelf concentration//iiie--eee sample arguments are applied the weak label population error. This approach is modular in the sense that once the population error relation established, the finite-sample issues can be dealt with by purely considering err( S)) and the remainder of the analysis is almost unrelated to each paper's 's specific setup of self seff-training. co-training, domain adaptation, or weak supervision. For example, Wei et al. [69]. Cai al. [12] both apply an off-the off-the-shelf generalization result for deep networks from Wei and Ma [68] to err (f, 7 S) in this step. The key novelty each paper thus to establish reasonable structural assumptions that relate the clean and weak errors on the population data, so this is also the focus of our results. Other work with expansion assumptions, such as HaoChen al. [23], which gives error bounds for linear classifiers trained on top of contrastive representations, also focuses exclusively on the population case because of the same argument.
Connections and comparisons to existing bounds
C.1 Co-training with conditionally independent views
Suppose that X' = X2 x X2, so each example æ = (æ1, xz2) has two "views". Assume that the weak labels ğ can be written as a function of 2; alone, and the hypothesis class F is such that each f E F is a function of æy alone. Finally, suppose the distribution of x is such that for all A C X1. B C X2. and E
PP A, Bly(x) = i = P(xx E Aly(x) i|p[x2 E Bly(x) i].
That is, the two views are conditionally independent given the (unobserved) true label.
This is precisely the conditionally independent view setup of Blum and Mitchell [6]. They prove the following relationship between the clean error and weak label error of any classifier.
27

---

Theorem C.1 (Blum and Mitchell [6], rephrased), Fix f : 22 + y. Let a, = P(S (i). In the conditionally independent view setting, the errors of f satisfy:
err(f.uls:) err(S.0|S) az (1- - 20:)
enr(JuuT.) S err(S, (V[[W -a;
(1 - 20:)
This exactly matches our pseudolabel correction result in Theorem B.1 and our coverage expansion result from Theorem 4.2 when c = 1, O, and PP P(R(T))T)) = O. We now show how to set N such that the conditionally independent view distribution provably expands with these parameters and any classifier has P P()(f)]T,) = O. Our results thus directly generalize those of Blum and Mitchell [6] in three directions, allowing for the views to be dependent, allowing small sets to not have good expansion, and allowing for the case when the classifier is not perfectly robust over N.
Lemma C.I. Suppose the hypothesis class G is such that g X2 Y for all g G. That is, the hypotheses are functions of 22 alone. In the appropriate families of sets M( and M'(-.) G) satisfy (c, -)xxxanssion (Sgood, Ti). (Stad, T.), (Sqood Shadd), and (Shod, Sq Sqood). all with c = 1 and q 0. Additionally, any classifier & G is adversarially robust at all x E Xx so [() = 0 for all E 9 and all A C X. Plugging these coefficients into our Theorems B.I and 4.2 yields Theorem C.I. Our bounds therefore exactly recover the Blum and Mitchell [6]'s s bounds in the setting.
view setup, the ieniitionnall-iddeeeneteee the pairs of sets
Proof. Let ((x)) (2))) (2) be the set of all points in X' that share the same value of ø2 as æ. That is, x' E Nxx) iff x = x2. By construction, any g X22 - y has (g)) = all points in the same neighborhood have the same value of æ2, so must be constant on every neighborhood. This proves r(f, x) for all x.
We will show c = for the pair (Sqood T:)) Completely analogous arguments hold for the three other pairs. The expansion amount e between sgood and Ti for set size 2 0 is given by:
P(A(U))S) (Soood IPN((U) Spoodly ly i) P(T,ly i) P(U]T:)
min U.P(UTT
P(S y= PIU, T i)
Fix arbitrary g E 9 and let set U C T, be the set of x T. such that g(x) * y(z). A point Sqood is in (U) iff there exists x' U with N²x'). By our neighborhood construction. this is equivalent to 22 = wzz. Let U22 = {x² : x E U} and consider a point æ E Sgood, Then spood satisfies æ E (U)) iff æ2 E U2. Likewise, a point æ E T, is in U iff a E U2. since the errors of g (and thus, membership in U) only depend on Ø2. Now consider membership in Spood. x E sgood iff ((11) = i. Conditioned on y = i, x E T. iff ū(x1) = Ø. We can thus rewrite the expression above as:
((N(U), Sqmady y =i) P(T.ly i) P(S y i) PlU,Tlly=1) =) P(i(x1) ily = F( E U2, 0j (x11) Oly i)
P(æz E U2.0(m)) = = i) P(((r)) = Øly =
By the conditional independence assumption the two views, the numerator of the first term and the denominator of the second term factor and cancel out their denominator and numerator, respectively. This yields:
P(22 E U2ly = i)
1
P(1. E U2
Since U was arbitrary, this shows that in the munditionlllyindeeendee view setup has (c, q)- expansion for the pair (5good .T,) with c = 1 and q = 0. The argument for the remaining three pairs of sets is completely analogous.
C.2 Wei et al. [69]'s pseudolabel correction bound
Proposition 3.2 shows that Wei et al. [69]'s error bound can't easily be ported to the full weak supervision setting (i.e., pseudolabel correction and coverage expansion) because it requires high coverage (P()). In this section, we show that even for pseudolabel correction, our bound is signifi- cantly different from Wei et al.'s 'ss bound and has three desirable properties that Wei et al. [69, Theorem
28

---

A.2] (the more general version of Wei et al. [69, Theorem 4.3]) lacks: (a) our bound more directly generalizes the Blum and Mitchell [6] bounds in the caaitiionlllyindeeeeeeeen view setting from Section C.1, (b) our expansion parameter naturally appears in our bound, making it clear that more expansion yields a tighter bound, and (c) our bound empirically applies to more classifiers since places less strict conditions on the student model.
First, we prove a direct analogue of Wei al. [69, Theorem A.2] using our multiplicative expansion assumptions, since the original uses slightly different additive expansion assumption. The proof' are almost identical.
Theorem C.2 (Wei et al. [69] Theorem A.2, restated). Suppose M(S F) satisfies (c, -)-xxnsssionn on (Sgond Shed) for some c > 0s/(( 03). Let ē c- and suppose that the classifier I satisfies:
(([(x)) + j(x)) or f not robust al arS:: S azi(1 + 1g(2-11
Then:
err(f,y(S.) 2 qa: + +P(((F)))))
aq.
Note:
The following result shows our robust expansion technique can also be used to directly
generalize this result average-case-roban classifiers, whereas Wei al. [69, Theorem A.2] only applies to adversarially robust classifiers. Theorem C.2 is a special case of Theorem C.3 with vj 0.
Theorem C.3. Suppose Mn , F) satisfies (c, (e,g,n)-eea -exp)anixnsion on (Sgood : Shadd for some a:/(1 az). Let ē c²-a; and additionally suppose: as
P(f((x)) + ((x)) or f not 17-rouust at x=Ss)) S a, + g(c- - 1))
Then:
arr(fyySS) s ((R,(()()))) + err(s, eer(f. 04-
Furthermore, if satisfies:
Ex Ex f(x)] 5n
for some y 2 O, then P(R, s ?
Proof. We directly prove Theorem C.3 since Theorem C.2 is a special case with 1) = 0. The proof is largely similar to Wei et al. [69, Theorem A.2] but with our multiplicative version of the expansion assumptions and our generalization to robust expansion. Let M, = {x E (z) + y(x)) be the set of mistakes of f in S;; note that err (ʃ. y Ss)) (S) Let D: = far E fz)) A j(x)) be the set of disagreements between / and ÿ in S note that err(J, (()) ((D)
We can partition M, into three sets:
St A (S; \ D Stod
A2 = D, n M n Shod
A3 D, Sg0oo
Points æ E Ai must be in M, since f(z) ;j(x) # y(z) by the definitions of St \ D, and Shad
respectively, so A₁ A n M. Points æ E A3 must be in M; since f(x) * 5(xx) = y(x) by the definitions of D; and ggoon respectiively, be in Dj since otherwise we'd have (x) j(x)) y(x) and thus æ g M, so M, O Sgood C Ag-
As M, M. Finally, any point in M, M& must
Together, this indicates that {A1 A2, Aa} is partition of M, since we' ve shown:
A U Ag U
(S, (S,\D,) (
n Stod
U U n M, n Shod
UD n M: n Sgood
M, n Shad M2 0
M ggood
M,
29

---

Now define Uj = M; n Squd n (f) Note that A1 n (f) and A2 n (f, are disjoint subsets of Ug. Since U; (f), Lemma B.2 implies that w(N(U.), U.) 2 (1 n)w(((UU.) U.), so P(N(U.)) Sqood) 2 Pi (S U.). Since Uj is mistake set intersected with a robust set, Ur E M(S FF
Suppose for the sake of contradiction that P(U,|S!ad) q. By the expansion assumption, have:
P(N(U)) (S.) (1 (.) 2 (1 P1 (S, U4)
c(1 - )(UU((1))))) (1 (1- PUU,)S)).
04
Since we assumed > 0g a-a;
for some è we have
P(N(U)) n n Sqood (S.) è PPU.)S.).
Now we decompose S; into three disjoint sets (note the differences between Vj's and Aj's):
V1 S, \ D;
1/2 D: Stad
V/3 = Dj aons S
Consider a point a E (U,,) n 5 By definition of N, there exists x' E U, reachable from ac by good edge, so f(x)= = f(x'). Since æ and x' are both in S,, g(x) y(x'). Since U3 M,, f(x') + v(x'), so we must have f(x) + v = 7(x)). Hence x E D; n sgood This shows that (U,))) n Spood V3. Using the decomposition above, we have:
1 = P(S))S;) = P
2 (V) + + n n. Spood (S:)
> + PPV (FPU))
(6)
Lemma C.2.
a, S ((V))).) + P(U.))))++ had n n R. (f)(())
Proof.
0g = PP(ttad (S:) = IP n D:(.)) + P(Staad n (S,) D.)(S:)
(.) P(S n. n (T)()).)) + P(S n Vi n (()))))
Consider x E Shad n V/1 n (f) Since x E Shad g(x) + f(z). Since x E V1 = S \ D,, f(x) = (()) Thus f(x) * y(x) and hence x E M,. Because U = M, n Shad n R,((), this shows that Stod n Vi n (f) C U.. The lemma follows.
Plugging Lemma C.2 into (6) yields:
> P(V) + ai + (ī -8- 1)P(U))) PVV n Stad n (()S))
PVV n (Shadd n RR a; + (2 1))(()))
P(V n (Syo0d +
(ā ((- I)I((U
2 P(V n R2 (f) + a, + +
(7)
Now recall that V1 = S; \ Di. So:
P(V n ())
P(D, n (,()))s))
= -1)((f()) * ((x) or f not r7-roustt at x:S,)
30

---

We assumed in the theorem statement that: (()(a)) * ū(z) or not ry-roust at a/S:) - a; + q(ē 1060-10 -1)
a + (ē - P(;
a; (c 1),().)).).
And hence (7) gives 1, contradiction. must therefore be the case that P(U; (UUdd < q. Since A1 n (f) and Ag n (J) are disjoint subsets of Ug we have that P(A1 n R2 ()) + P(A21 n
(1)()) S gaz-
We can now finish the bound.
err(f, v|S))
P(MIS)) = P(A U2
U RRAA))))) R()1ss)
s S(((A)UA Aa) +
+
Since (Ag (S, \ DJ)) n (2) = (A1 U Sq00d) n (f, we have:
(Aa ( (S, D.) n (J) A₁ Ru (f) Sqood (T)
All sets that are arguments to the unions above are disjoint, so:
P(As n (,)))) - P(D, (n(()) P(A R, (f)1s:) + P(Sqood c (f)(())
Using F(D; (S)S,) 1 P(D, U (f))s)), P(Syood R2( I 1 as. and (PA (s)())s)) S goz, we obtain:
P(As n (S)()) < IDD; U R. (f) (S)) qa: a.s.
Plugging this to the earlier bound gives:
err(g, vlS,) - P( R, (f)).) + 2qa, + P(D; U R2(())s)) ai
which we can leave as-is:
erT(f. v|s,) < P ((R,())) + 2qa; + P(f(x)) + y(Φ) or f not p-robust at æ|S.)
or union-bound obtain:
(|S:) < qa: + (F(R(())))) +er(f.g]S.) a..
The second part of the theorem follows directly from Lemma B.I.
Comparing Theorems 4.1 and C.2 the Co- Co-Training setting. In Section C.1 we argued that the view setup of Blum and Mitchell [6] satisfies our expansion assumptions with c and q = O, and any model inherently perfectly robust because it's only trained on one of the views, so P()(f))S,) = for all f E F. We can plug these values into the conditions of Theorem C.2. particular, Theorem C.2 only applies to classifiers that satisfy:
er(S,W|S) S as (1 + q(č - 1)) ai
But in the conditionally independent view setting, minseF S.) ag this case, the error bound in Theorem C.2 simplifies to:
err(f S eer(f. a; =0
Thus, in this setting, Theorem C.2 only applies to classifiers that minimize the weak error as much as possible, and therefore have error on the true labels. contrast, our Theorem B. applies to any with frr.f < -- 1---ar a = 1 az When = 1, the value of c' in Theorem B. reduces to c' = 1, and our bound exactly recovers the Blum and Mitchell bound for any classifier with error at most 1 ag This relaxation of which classifiers qualify for the bound would be important for finite-sample
31

---

Table 2: Comparison of measured values of the amount of expansion e ("exp. e val.") on the data described in Section 6. Theorem B. (our pseudolabel correction result) requires (S F) to expand on the pair (Stadd _S90od) for some e > 0. That is, requires robust non non-mistakes to expand to points with the wrong pseudolabels. We call this "good-to-bad" or G2B expansion. On the other hand, Theorem C.2 requires M(SSud, F)) expand on the pair (Strad Shad) for some c > a:/(1 ax;). In other words, requires robust mistakes to expand to points with the correct pseudolabel. We call this "bad-to- good" or B2G expansion. These results show that empirically, we may have the G2B e > O, so our bounds apply, but the B2G c < as/(1 - as). so Theorem C.2 does not apply. This is the case for the values.
(Stad, Spond) (G2B) exp. val.
(Spood , ,Shad) (B2G) exp. val.
03/(1 0;)
0
0.11
0.12
0.85
0.17
0.33
0.49
0.50
0.32
guarantees, since Theorem C.2 only applies to I for which (r[[j minger err(f'; S.) It is not possible to guarantee that we can obtain such an f from finite sample using standard improper PAC learning techniques. In contrast, our bound still applies to student models f with enrf, S.) fin e S.) > O.
Our result can also recover the same bound as Theorem C.2 in this setting: when e 1, we can take A = 1 in Corollary B.1. The condition on the classifier then becomes err(f. ūl S.) S ag and in that case the bound in Corollary B.1 simplifies to
err(f. S)) I err(f ((fff(j)))) + az - 2cA) -er(f ff ū1S a. =
so Theorem B.1 can exactly match Theorem C.2 in this case. This discussion shows that our result can match the bound from Theorem C.2 in this setting, but is more stable, since it applies to classifiers that do not attain the exact optimal weak label error on the population.
coefficient. The bound in Theorem B.1 has a coefficient of 2a. on the probability
1-2a;
of non-robust points TP (R())).)). The bound in Theorem C.3 has a coefficient of Our dependence on the probability of non-robust points is better when a < 1/3.
Expansion term not in the bound. Theorem C.2 does not have the amount of expansion in the еттor bound-instead, the expansion is only present in the pre-conditions that the classifier needs to meet for the bound to apply. In Wei et al. [69], the authors avoid this by introducing loss function that contains the expansion and assuming the classifier I minimizes that loss, essentially re-introducing a dependence on e. However, achieving this bound (Wei et al. [69, Theorem 4.3)) therefore requires knowing the expansion up-front in order minimize the suitably-scabe loss function. In contrast, our pseudolabel correction bound directly has the amount of expansion c on the right-hand-side.
Empirically less sensitive. Theorem C.2 requires c > a;/(1 a;)) for it to apply. This is not an artifact of our conversion from Wei et al. [69]'s additive expansion to our multiplicative expansion, as shown by the following lemma.
Lemma C.3. Suppose that the distribution P(-)S)) satisfies (q, -addiiivee expansion on Spod 169, Definition A.1] for some d 0. Then IP(((S) satisfies (c, 2. -xxpansion on (Sgood Shaud with e > as/(1 az).
Proof. (g. adddiivve expansion implies that for any U Shad with P(U(S.) > q.
P(V(U) Shoud S.)) 2 P(U (S.) + 8
(UN((U) n S ggood
Ss) 2 PUU n Sbad (S,) + 5
P(N(U)) (ggood
ggood )1 a;) 2 P(U Shad aa; + 5,
32

---

so ((S (UU|Stad) Jas > 9, so ((U)s > g/a;, Hence this example satisfies (c. g/a;) exxpansion on the pair of sets (Spood Stad).
Poul) We assumed U C Stnud was an arbitrary set with (U)) =
I-a:
Lemma C.3 shows that if Wei et al. [69]'s additive expansion assumptions hold, our (Sgord, Sphad expansion assumption holds with > a4/(1 (.,)). However, our experiments suggest that this amount of expansion may not be present empirically for the error sets of actual trained classifiers. In contrast, our pseudolabel correction result only requires c > and the bounds computed using our empirical measurements are close to the true error of the classifiers. Table 2 shows example of this on the data from Section For label I, a3/(1 a;) = 0.49 but the measured value of the expansion between (Sqood Shad) is only 0.32, so Theorem C.2 does not apply. Of course, this does not rule out the possibility that C.2 may apply with different choice of the neighborhood N, but for least one natural setup (N the set of paraphrases of the text input æ), there is not enough measured expansion for it to apply.
D Checking expansion: Statistical theory
Theorem D.1 (Theorem 5.1, formal), For U E M, define:
<(U) = P(n(x) UUEE
P(xEUXX B)
and its empirical version:
E(U) = nA E 1 [n();;) U]
ーー 12, U]
Suppose there is a lower bound q > 0 such that q(U) 2 q for all U E M. Define 7 n and let m A + n.. Then for any d (0, 1], with probability at least 1- ô over the sampling of SA and SB.
(VCMM log VC[M) + log 8
sup a(U) c(U) S 4(4+ vny
UEM
nA
Proof of Theorem D.I. We start off with a lemma providing concentation results for fixed set U E M.
Lemma D.1 (Concentration for fixed U). Fix an arbitrary set U M. Define:
p(U)= = P(((() E A) q(U) = P)
And their empirical versions:
P(U)
1
Σ
[[((x;)) E U]
@(U)
1
Σ 1|x E U..
TEA
Ti
B.ESA
w.ESn
Suppose there is a lower bound q > 0 such that q(U) 2 q for all U e M. Define y = ngq/n nA Then for any 8 E (0, 1].
((U) p(U) 8 a(U)
7G2
P
2exp
-2n
q(U)
(4+ v7)2
where the probability is with respect to the sampling of SA and Sp.
Proof. Fix e > and let E be the event that (1 +)g(U) < @(U). Let W refer the event that
B(U) P(U)
r(U) > 5. Then
Q(U5
P[W] = P(W APP + EIPE
< P(W E P
33

---

By the multiplicative Chernoff bound.
P|E] = P(((UU) < (1-c)q(U)) & expl 2ng q(U)/2).
Now the goal is to bound P P[W[E].
((U)
p(U)
M(U)
P(U)
P(W F =
> 8 E SP
8 E
((U) ((U)
c)g())
Q(U)
B(U)
P(U)
= P
6
(1 c)g(U) (UU)
F [ > + qÒ c(p + gó)l
S Fb > p qō 2e].
where we used in the second line that the samples from A and B are independent (and thus, ((U) and ((U) are independent), and in the last line that q q. 0 5 p 1,00 q 1, and 6 (0, 1]. Setting theorem, we obtain:
qó - 2e, and using the
IP [W]F] S Pl/ Ely] + + & (xp((2((6)) nA)
Combining results, we have that for any > 0:
P[W]= exp( 2 2(e))² πA) + exp(-e 2 q/2).
Let y = ngg/g// nA. Setting c = 1+ v
248
(this is valid since > 0) yields = 96.5
so we obtain
1+
(22na0²
7a2
P[ S 2xxp
(4 (4+
Corollary D.1. An almost exactly symmetric argument yields:
P(U)
p(U)
792
P
$ 2 2exp
-nA62.
((U)
Q(U)
(2+ v7)
and thus:
p(U)
((U)
b1 I| 44exp
702
P
4exp -n1102
((U)
1(U)
(4+ v?)
Now we show how apply this deviation bound for p(U)/q(U) in place of Hoeffding's inequality in the symmetrization argument from Bousquet et al. [8].
Lemma D.2 (Symmetrization). Suppose we have a ghost sample of n additional points æf drawn i.i.d. from P(- A)) and ng additional points 2 drawn i.i.d. from P(-IB). Let è' (U)= - (U) (U)
denote the empirical expansion of set U on the ghost sample, and let ē(U) Q(U) denote
the empirical expansion of set U on the original sample. Then for any t >> 0 such that n t² 2
(7) 16
79
P sup é(U) c(U) >t s 2P
sup a(U) E'(U) t/2
UEM
UEE
Proof of Lemma D.2. This follows Bousquet et al. [8] exactly, except we replace the application of one inequality with the deviation bound derived above. Let Un be the set achieving the supremum on the left-hand-side. This depends on the sample (æ1,.... #nn+
<1a(,))-(U,)>>11
Taking the expectation over the ghost sample (x', x'n )
L=c_,)-a(t.)sxP((((( c(Un) < t/2] P'(C(Un)) - (UU)) 1/2)
34

---

From Lemma D.1,
nal?
7ap
P'(((Un) cUU) 2 2/2 < 2exp
(4 + v5)²
-
by the condition on nA P. Hence:
La(vw)-((UU,) S 2P'(o(h,) - 2 (hn) > t/2],
and taking the expectation over the original sample (If In +nn) finishes the proof.
Recall that the (deterministic) neighborhood oracle n is fixed ahead of training and thus does not depend on the random sample(s). For æ; SA. and x S'A. we can redefine æ; f n(x;;) that is, we apply the neighborhood oracle as part of the sampling process for points in A. Let m = n2A + n.B. and define Ma,
{(I=:eU,- .... lu) U E .M). Recall that the growth function of class M is defined as (m = sup (11.......... [M (Mai........ Now to finish the proof, observe that the sup in the right-hand-side of the Lemma D.2 result only depends on the finile set of vectors M Man..... Fm Pi due to our simplification æ; < n(x;) for Φ, SA. S That is,
P
sup a(U) c(U) > t WEM
S 2P
sup
aU) (U) > t/2
UEM
Σ
P(e(U) (U) > t/2]
U.M
S 2
Σ
P((C(U) c(U) > 1/4)
UEM 21 Dr of Si
74
I (885((m) exp
nA
16(4 + v7)2
The first line simply applies the union bound over Ma Mm.......m... you. The second line uses the fact that if a - > e then for any real x, either |a - æ| or bb x or both are larger than c/2, applies a union bound, and uses that e(U) and è' (U) are identically distributed. The last line applies Corollary D.1 and uses the definition of the growth function. The Sauer-Shelah lemma [61, 56, 57] implies that for any class H with VC(H) d. Sxx (m) 1 $(O)) (em)" Then setting:
(cm)
d
VC(M) log VC(M]
20m
+ log 8
12 4(4+ viV
nAT²
guarantees that
P sup é(U) - c(U) > t 58
LUEM
and completes the proof.
The following proposition shows the complexity of the class of mistake sets generated by f E F is bounded by complexity of the hypothesis class F itself.
Proposition D.1. Suppose F is a binary hypothesis class and X - (0,1) is the ground-truth classifier (we do not assume y E F). Fix a set C X and for each f f F (A, = (x E A
f(x) # v(z)} be f mistakes on A. Define M (U(A, f Then VC(M) < VC(F).
Proof. Fix a finite set of points C with [V = =d, so V (0),... ... r.). Suppose that M shatters so:
.......,.....
KUNV
2d
Fix an arbitrary labeling v x 10. Let W = (a, E EV (x,) + y(z))) Since W E M. by definition of M there exists f f F such that f(z;) + y(m;) if and only if x, E W. If æ; E W. then f(x:) # y(a,) and yax + y(x:); since there are only two labels, it must be the case that f(x:) 5/ (.:)). If n, & W, then (ax) = v(æ:) = y y'(::) This shows that for any labeling y' of the points V, there exists / e that has zero error on that labeling. Hence any set shattered by
M is shattered by F. so VC(M) < VC(F).
D
35

---

E Experiment Details
Dataset details. We train on the IMDb dataset of movie reviews [41] (HuggingFaceHub ID stanfordnlp/imdee which has 25000 training examples and 25000 test examples, each with exactly 50/50 positive/negative split, and an unspecified license. For the teacher model described in Section 6, based on the presence of unigrams 'horrible' and 'incredible', the coverage rate is P(x E 5) 0.06. The fully-supervised (gold) error of linear classifier trained on top of the SentenceBERT representation we use is 10.6% (when trained using the optimization procedure and hyperparameters described below; we did not perform hyperparameter tuning, since this number is merely used as an ideal lower bound for the performance of weakly-supervised classifier). For i E {0, 1 and points
Sgund we obtained neighbors in T; and Spnd using the oracle procedure described in Section 6 and shown graphically in Figure 3. In each neighbor sampling step, we enforce the constraint y(ar") = i using the model trained the gold labels to perform rejection sampling. This ensures (as much as possible) that all neighbors have the same ground-truth label, as required by the definitions. We include these sampled neighbors in the supplementary material for reproducability.
Model training. We train the linear classifiers using the Adam optimizer [40] with global learning rate 0.01 and a weight decay of 0.1, and linear learning rate decay over 500 optimizer steps. For the heuristic in Section 5, retrain model 5 times on 5 different random subsets of the covered training samples, each 80% of the original, and use the other 20% of covered samples as a validation set to perform early stopping with the weak label. We do not perform early stopping with the true gold label, which is common practice in many works on programmatic weak supervision and gives a large performance gain [73, 10]. Since the goal of our experiments is to give evidence that expansion holds in practice, we do not tune the learning rate or weight decay parameters at all, since the initial values we chose already led to weak-to-strong generalization effects.
Compute. We used an internal machine with 4xA 100 80GB GPUs to extract all deep network embeddings and to train the linear classifiers top of those embeddings. Embeddings were only extracted once. Classifiers were trained 5 times with different random seeds and data subsets to perform the heuristic procedure in Section 5. This step repeated several times developed different versions of our bounds and recomputed different notions of expansion w.r.t. the mistake sets of the trained classifiers.
Results. Table 3 shows the results of training using the SentenceBERT representations on the pseudolabels described in Section 6. The table also shows the accuracy of the weak labels themselves in the ū(x) row. Pscudolabel correction and coverage expansion occur in different amounts depending on the class. For example, the student consistently improves over ÿ on So but not on St. This justifies our choice to analyze these effects separately for different classes. As discussed in Section 6, Tables 1 and 4 show the measured values of expansion, the worst-case weak label error of the student across the 5 training runs, and the values of our bounds from Theorems B. and B.2, respectively. Since the goal is to give evidence for the presence of expansion, we ignore IP P(RR,, (f)l.) when computing the values of the bounds and focus on the terms involving amount of expansion. The bounds in Table 1 show that our results can effectively distinguish between cases where pseudolabel correction does (on So) and does not (on 51) occur.
Table 4 measures the expansion of the set families M(T,, F) and MT) on the set pairs (Spood TT.) and (Shad, T.), respectively. First, the fact that both expansion numbers are nonzero gives evidence for the extra structure described in Section 4.1 and assumed in Theorem 4.2/Theorem B.2. As mentioned in Section 4.1, we could have also just assumed expansion on (S T2), but this extra structure gives tighter bound, and these results suggest that it is present empirically. Table 4 shows that even though the coverage F(x E S) 0.06 is very low in this example, good representations (in this case, representations such that MM, G o u) expands for G linear), can have good coverage expansion, We use the tighter bound from Theorem B.2, which allows for different amounts of expansion between (Sgood, TT) and (Shad T.).
36

---

Table 3: Test accuracy breakdown for linear probe trained with true (gold) and weak labels the IMDb data. Performance the weakly-trained model is broken down across the covered sets (Sp. Si) and the uncovered sets (To. Ti). One standard deviation across five training folds shown in parentheses. Pseudolabel correction and coverage expansion occur in different amounts depending on the class. For example, the student consistently improves over y on Sg but on Sj. This justifies our choice to analyze these effects separately for different classes.
Test Acc. (gold train)
Test Acc.
Model
So (4%)
S1 (2%)
To (46%)
Ti (48%)
(weak train)
(ax)
89.5
67.1
n/a
n/a
84.9 SentenceBERT
BoW
50.1
89.5
67.1
100.0
0.0
89.4
81.8 (1.4)
95.5 (1.4)
69.1 (4.6)
85.8 (3.1)
74.9 (4.8)
Table Measured expansion values and error bounds for the uncovered sets T, Expansion values
for the families M(Shuud on (Sgood .hha) and M FF) on (Shad, sgood), are measured using the heuristic described in Section 5. The detection of both types of expansion (expansion from T, to Soood and to Shad) gives evidence for the extra structure we described in Section 4.1 and justifies our use of Theorem B.2, which uses this structure, instead of Theorem B.3, which only uses expansion from T; to Syoud and gives looser bound. Worst-case value of the error bound in Theorem B.2, computed using the smallest expansion values and largest weak errors err(f. 7 S.) from the 5 training runs. The err(J, S:) and a; values used in the bound computation are identical to the values Table Unlike Table where the "baseline" for pseudolabel correction effects to have error bounds strictly better than ag for coverage expansion, the more relevant comparison is against random/arbitrary guessing. The actual worst-case error the student on each T, is shown as err(f. (..). As suggested by our bound values, the errors on each T; are non-trivial (much better than random or arbitrary guessing).
Model
(Sgood Tτ)
(Stad, T:)
Bd. val or([sy[T)
0
0.16
0.98
0.12
0.37
0.16
SentenceBERT
1
0.75
0.55
0.29
0.33
0.29
Paraphrase this teat: Don't change the meaning and don't include sords "hurrthla" your paraphrase.
this the asaning, but include the mard "horrible" in your paraphpase.
enjoy This wavte all. Watching it. was on inceestble burder.
del not enjoy this masie at Watccing Hs was burden.
not enjoy this mavie att Watcching it was horrible burden..
rRjectson Somple
rejaction
Figure 3: Example of our neighborhood oracle n., constructed using targeted paraphrase procedure. For covered point x E S (in this case, E Shod, since it is example weak rules 8), we first generate an uncovered point æ' E T, using constrained paraphrase model and rejection sampling to ensure the ground-truth -tuth label remains negative (we use model trained on the gold labels as a stand-in for the ground truth y). Next, we use GPT-4 to rewrite x' using the opposite word from [horrible, incredible) than the one that originally appeared. This generates another point x" E S: Since we enforce that and x" are covered by different words, we know that if æ E Sgoood (resp. Sbod), x² E Stad (resp. Sgood)
negative point mislabeled by our
37

---

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Justification: the paper does not involve crowdsourcing nor research with human subjects.
43

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Reviews:
[{"review_id": "pMENAXJHD7", "reviewer": "Reviewer_KtpZ", "summary": "This paper proposes a theoretical framework to interpret the weak-to-strong generalization phenomenon. It shows that strong student models trained on noisy labels from weak teacher models can outperform the weak teacher models, correcting their errors and generalizing well to examples where the weak teacher models are not confident or abstain. The paper demonstrates that existing bounds for learning from noisy labels do not explain this phenomenon and derives new bounds based on assumptions about the strong model’s robustness in its neighborhood and its expansion property. An empirical study validates the generalization bound in a practical setting.", "questions": "- Adding comparisons with recent concurrent works ([1, 2, 3]) would be beneficial. I am curious about the authors' views on these papers and whether there are any conflicting points in theory or if the conclusions are well-aligned.\n- Including simpler or synthetic experiments could enhance the paper. The (c, q) expansion property in a practical setting is not intuitively straightforward.\n\n[1] Somerstep, Seamus, et al. \"A statistical framework for weak-to-strong generalization.\" *arXiv preprint arXiv:2405.16236* (2024).\n\n[2] Charikar, Moses, Chirag Pabbaraju, and Kirankumar Shiragur. \"Quantifying the Gain in Weak-to-Strong Generalization.\" *arXiv preprint arXiv:2405.15116* (2024).\n\n[3] Zhang, Edwin, et al. \"Transcendence: Generative Models Can Outperform The Experts That Train Them.\" *arXiv preprint arXiv:2406.11741* (2024).", "limitations": "Authors adequately addressed the limitations in Conclusion.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- Weak-to-strong generalization has gained renewed attention despite being well-known in weak supervision. However, there has been no theoretical analysis of why a model trained with noisy labels can be more accurate than the labels themselves. This paper’s contribution is crucial as it provides a theoretical framework for this problem.\n- The theory explains under what conditions weak-to-strong generalization can occur, which has practical implications.\n- The paper is technically solid, with reasonable assumptions for derivation. Sections 4.2 and 5 add to its practicality.\n- The writing is clear and the main ideas are easy to follow.", "weaknesses": "- The experiment is limited to only one setting, though it is already mentioned in the limitations section. This is a minor concern since the paper is primarily theoretical.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> __Adding comparisons with recent concurrent works ([1, 2, 3]) would be beneficial.__\n\nThanks for pointing these out! First, we just want to note that all 3 of these works appeared on arXiv after the NeurIPS submission deadline, so we were not aware of them at the time of submission. \n\nBroadly, [1] and [2] are similar to our paper in that they make some structural assumptions under which weak-to-strong generalization can provably occur. [1] uses a framing of weak supervision as transfer learning to give pseudolabel correction results when the data is distributed according to a Gaussian Mixture Model, whereas we do not make specific distributional assumptions and also focus on coverage expansion in addition to pseudolabel correction. As discussed in our response to Reviewer 3HaP, Gaussian Mixture Models can be shown to satisfy notions of expansion. [2] is the most similar to our paper, but focuses on pseudolabel correction for regression problems, whereas our work tries to explain both coverage expansion and pseudolabel correction for classification problems—both phenomena are important in weak supervision settings that we are interested in. Additionally, the underlying assumptions and conceptual ideas are very different. In our work, we use structural notions of expansion to explain these two phenomena, and our bounds directly generalize existing results from other literature (co-training, self-training), and extend this theory to more general settings. In [2] they assume convexity of the hypothesis space for fine-tuning, and show that projections onto this space can explain pseudolabel correction effects. We feel these two explanations are very different but potentially complementary, and it would be very interesting future work to develop a common framework that incorporates both. [3] considers a qualitatively very different setting, showing that when the training data consists of a mixture of data generated by different experts, a strong student model can be better than the best single expert that generated the data. Their results are more related to classical work on crowdsourcing, where by observing generations from many experts the learner can outperform the best single expert. Our paper and [1] and [2] consider the case where there is a single weak teacher model (which itself may be an ensemble of many models, but this is not exposed to the learner). \n\n\n> __Including simpler or synthetic experiments could enhance the paper. The (c, q) expansion property in a practical setting is not intuitively straightforward.__\n\nThanks for the suggestion! Per our responses to the other reviewers, we will include a worked example that shows (c,q)-expansion does not hold when the strong model can exactly fit the weak model’s errors. Subject to space limitations, we can also include a version of the content in Appendix C.1, which gives a very straightforward setting (“conditionally independent views”) where the expansion assumptions hold. These examples should give more intuition for how the assumption works.", "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": "Pd3QcZmGvb", "reviewer": "Reviewer_3HaP", "summary": "The paper provides a theoretical explanation for weak-to-strong generalization. A weak model produces pseudolabels that can have errors and may not cover the entire input space. The paper argues current weak supervision theory fails to explain how and when the strong model can correct the psuedolabels (pseudolabel correction) and expand the coverage of pseudolabels (coverage expansion). The authors derive a new bound based on the expansion properties of the data distribution  and the hypothesis class of the strong model. Their bound suggests that generalization occurs when the strong model is unable to fit the mistakes of the weak model without incurring additional errors. They provide experiments to show the expansion properties are verifiable in practice.", "questions": "1. How do the bounds depend on the size of the covered points and by how much does the coverage improve?\n\n2. Where does the complexity of the strong model's function class come into play, how do the bounds depend on it? What happens in the case of self-training when the function class of strong and weak models are the same.", "limitations": "Yes.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The paper shows the gaps in the existing theory of learning from weak labels ( potentially erroneous labels available for part of the input space). In programmatic weak supervision (PWS) the focus has been on aggregating several weak labeling sources and showing that learning with aggregated labels is as good as learning from clean labels when the weak labeling sources satisfy certain conditions. The paper shows shortcomings of the previous results on PWS – failing to explain pseudolabel correction and coverage expansion. \n\n2. It provides upper bounds on the expected error of the learned strong model on the part of the space that is covered by the psueodlabels and the part that is not covered. The first bounds suggests the error decreases in the covered part implying pseudolabel correction. A non-trivial error bound is provided on the error of the strong model in the uncovered set by utilizing the expansion property i.e. many points in the uncovered set have many correctly pseudo-labeled points. \n\n3. The authors also provide a theoretical result suggesting that the expansion property is statistically checkable with finite samples but is computationally hard. They provide a heuristic approximation to verify the property in practice.", "weaknesses": "1. It is not clear from the bounds how the size of covered sets (pseudolabel coverage) affects the results and why are the results in terms of subsets of the covered set. Is it possible to provide a final result conditioned on the entire covered set? \n\n2. The improvements in terms of coverage and error correction are not immediately clear. For instance, the following paper [1] explains the improvement in coverage with self-training in a specific setting. It would be useful to have a discussion and instantiation to some specific settings like in [1] to understand the results better.\n \n[1] https://arxiv.org/abs/2006.11006", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> __Is it possible to provide a final result conditioned on the entire covered set?__\n\nYes, we just left it out for space and because getting a combined bound with a simple functional form requires more definitions (max/min of the weak label errors $\\alpha_i$, the minimum expansion parameter across all the $S_i$'s, etc). Given these definitions, the combined bound essentially follows from averaging the bounds on each $S_i$. Similarly, we can give a combined coverage expansion bound conditioned on the entire uncovered set $T$ instead of the individual covered sets $T_i$. These two combined bounds can also be averaged together to get a final combined bound on the unconditioned error $err(f,y)$. We will include combined bounds in the Appendix.\n\n> __It is not clear from the bounds how the size of covered sets (pseudolabel coverage) affects the results__\n\nThanks, this is a great point. We had a discussion on the role of coverage and had to cut it for space since it affects our bounds in a somewhat subtle way. The most direct way in which coverage enters the bounds would be in a combined error bound that uses\n$err(f,y) = err(f,y|S) P(coverage) + err(f,y | T) (1-P(coverage))$ and then applies the combined-source and combined-target bounds discussed in our reply to your previous question to $err(f,y|S)$ and $err(f,y|T)$, respectively. However, this doesn’t capture the full story.\n\nThe “coverage rate” $P(S)$ also enters the picture implicitly in the $S$–$T$ expansion parameter. For a fixed neighborhood $\\mathcal{N}$ (e.g., all points with similar embedding), it will be qualitatively easier to have expansion from $T_i$ to $S_i$ when $S_i$ is larger, since (informally) for each uncovered point in $T_i$, there are more possible covered neighbors in $S_i$. But increasing the coverage by including more points in $S_i$ might also affect the weak label accuracy parameters $\\alpha_i$, which also affect the bounds.\n\nIn practice, there is not a clear tradeoff between coverage and performance, as explored recently in, e.g., [37], so it qualitatively makes sense that our bounds do not prescribe a functional dependence of the error on the amount of coverage and instead allow that dependence to enter through data-dependent parameters (expansion $c$ and weak error rate conditioned on coverage $\\alpha_i$). We hope this at least partially answers your question and we will include a more detailed discussion of the role of coverage in the final draft.\n\n> __[How do the bounds depend on] the complexity of the strong model's function class?__\n\nGreat question! Following related work on expansion-based bounds (e.g., [23]), our error bounds are expressed as relationships between population quantities (e.g., expansion, population error on the weak/ground-truth labels). There is no statistical estimation aspect, which is where the complexity of the strong model's function class directly enters the picture. We focus on population quantities because relating the population error on the weak and ground-truth labels is the key problem-specific component. Once these quantities are related, the sample complexity aspect is more standard and can be dealt with by applying existing generalization bounds. We discuss this topic in more detail in Appendix B.5.\n\nThe strong model hypothesis class can also enter the bounds indirectly via the expansion parameter, since the amount of expansion depends on that class. A richer class for the strong model may decrease the amount of expansion. For example, if the strong model class is rich enough to exactly fit the weak labels, there is zero expansion, as worked out in our response to Reviewer cokD. At the same time, the error of the strong model on the weak labels appears as a term in the bounds ($err(f,\\tilde{y} | S_i)$), and a richer class might decrease this term. So these two terms capture a potential tradeoff—a stronger hypothesis class may decrease the expansion, but it may also decrease the error on the weak labels. Whether this makes the bounds tighter or looser depends on the sizes of these decreases. As with the coverage $P(S)$, our bounds do not prescribe a functional form for this dependence and instead allow it to enter via data-dependent parameters. This should make the bounds flexible enough to capture seemingly conflicting empirical results, where sometimes a stronger hypothesis class works better and sometimes a weaker one works better, as seen for example in the WRENCH weak supervision benchmarks [73].\n\nWe will highlight this implicit dependence in Section 4. An interesting direction for future work is to see whether these tradeoffs can reveal how to pick a strong model hypothesis class for a given weak labeler, or pick a weak labeler for a given strong model hypothesis class. Thanks for the insightful question.\n\n> __It would be useful to have a discussion and instantiation to some specific settings like in [1] to understand the results better.__\n\nWe wanted to include an instantiation of the results for special cases in the main text, but were limited by space. Appendix C.1 has a worked example under the distributional assumptions of co-training (where each data point consists of two conditionally independent views). We can mention this as a simple example where the expansion assumptions are satisfied with good values for $c$. There are other distributional assumptions that lead to good expansion, such as the Gaussian Mixture Model style distributions studied in Oymak and Cihad Gulcu---Wei et al. [69,  Example 3.4] showed that GMMs satisfy their expansion assumption, which is very related to ours. We will extend our existing discussion of Oymak and Cihad Gulcu in Appendix A to comment on its relationship to our assumptions.\n\n> __What happens in the case of self-training when the function class of strong and weak models are the same?__\n\nPlease see our response to Reviewer cokD! We will include this in the paper as a worked example of a case where expansion does not hold.", "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": "pBOqs3XFBW", "reviewer": "Reviewer_cokD", "summary": "The paper provides a theoretical analysis of weak-to-strong generalization, a phenomenon where strong models can learn from weak supervisors and outperform them [[Burns et al 2023](https://arxiv.org/abs/2312.09390)].\n\nThe authors make precise assumptions about the nature of the strong student model family and the weak supervisor mistakes. Specifically, (1) near any datapoint (where neighborhoods are defined according to the strong student model class) that is incorrectly labeled by the weak supervisor, there should be many correctly labeled datapoints and (2) datapoints that are unlabeled (i.e. in the test set) also have a correct number of correctly and incorrectly labeled datapoints in the neighborhood. The authors make these ideas mathematically precise, and derive generalization bounds for empirical risk minimization for the strong student model. \n\nImportantly these bounds describe two phenomena: (1) pseudolabel correction, i.e. correcting for the mistakes of the weak supervisor and (2) coverage expansion, i.e. generalization to unlabeled datapoints. \n\nFinally, the authors provide a way of statistically testing their assumptions empirically, and show that they are applicable in one setting: sentiment analysis with a bag-of-words supervisor.", "questions": "Q1. See questions in W3.\n\nQ2. Which of the assumptions specifically captures the intuitive notion that the strong student should not be able to fit the mistakes of the weak model too well?", "limitations": "The limitations are adequately discussed.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1: The theoretical approach used by the authors makes sense intuitively. The authors often provide an informal intuition for the definitions and results that they derive.\n\nS2: The derived generalization bounds are to the best of my understanding novel and non-trivial. The authors provide detailed comparisons to existing generalization bounds for weak supervision.\n\nS3: The authors provide a rigorous statistical method, as well as a heuristic for testing their assumptions in practice.\n\nS4: The authors evaluate their assumptions and bounds in a simple empirical setting.", "weaknesses": "W1: While the core of the paper makes sense intuitively, the paper becomes more mathematically dense and hard to follow towards the end.\n\nW2: It is not obvious to me that the paper provides a novel empirical insight. As authors admit in the limitations section, they do not provide a new training method with improved weak-to-strong generalization or generally make practical recommendations. However, the paper makes a valuable contribution in formally describing conditions under which we can provably get weak-to-strong generalization.\n\nW3: The empirical evaluation is limited, and only covers one simple case. It would be very interesting to apply the bound to several settings, with varying weak-to-strong generalization. For example, can the bound predict the result in [[Burns et al 2023](https://arxiv.org/abs/2312.09390)] that weak-to-strong generalization doesn't work well on the reward modeling task? Does the bound provide an intuition for why that would be the case?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> __For example, can the bound predict the result in [Burns et al 2023] that weak-to-strong generalization doesn't work well on the reward modeling task?__\n\nGreat question! Our analysis is limited to classification for now, since we essentially assume that the class-conditional sets $\\mathcal{X}_i$ have some neighborhood structure that is respected by the hypothesis class of the strong model. This type of assumption makes the most sense when $\\mathcal{X}_i$ has some consistent semantic meaning. In the reward modeling case, the labels (chosen/rejected) are not semantically meaningful in the same way, so different structural assumptions are likely required. While our bounds don't explain the lack of weak-to-strong generalization in reward modeling, we are encouraged that they seem to plausibly capture these effects in classification problems. We will add reward modeling / ranking losses as an interesting direction for future work!\n\n> __Which of the assumptions specifically captures the intuitive notion that the strong student should not be able to fit the mistakes of the weak model too well?__\n> \n> [Related question from Reviewer 3HaP] __What happens in the case of self-training when the function class of strong and weak models are the same?__\n\nIf the strong model class can exactly fit the weak model, the expansion assumption is not satisfied, as we now show. \n\nSuppose we have a weak model $\\tilde{y} \\in \\mathcal{F}$ and aim to fit a strong model $f \\in \\mathcal{F}$. In this case we can have $f = \\tilde{y}$ (i.e., the strong model exactly fits the weak model), so suppose that is the case. The class of sets we need to expand for pseudolabel correction is $\\mathcal{M}’ = \\\\{R(g) \\cap S_i^{good} \\setminus \\text{mistakes}(g) : g \\in \\mathcal{F}\\\\}.$ \n\nSince $f = \\tilde{y}$, $S_i^{good} \\setminus \\text{mistakes}(f) = S_i^{good}$. Now consider a pair $(x,x')$ with $x\\in S_i^{good}$, $x' \\in \\mathcal{N}(x)\\cap S_i^{bad}$. Since $x$ and $x'$ are both in $S_i$, $y(x) = y(x')$. Then we have $f(x) = \\tilde{y}(x) = y(x) = y(x') \\ne \\tilde{y}(x') = f(x')$, so $f(x) \\ne f(x')$. Thus $x$ is not in $R(f)$. Since $x$ was an arbitrary point in $S_i^{good}$, this shows $R(f) \\cap S_i^{good} = \\emptyset$, so $\\mathcal{M}’$ contains $\\emptyset$. In this case, expansion requires: $0 = P(\\mathcal{N}(\\emptyset) | S_i^{bad}) > c P(\\emptyset | S_i^{good}) = 0,$ which is not possible.\n\nWorking out this example hopefully gives more intuition for the expansion assumption, and we will include it as an example in the paper. Thanks for the question!", "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:
Overall comments:
-- 

Thanks to all the reviewers for their time, effort, and helpful feedback. We are encouraged that all the reviewers found our work technically sound, novel, and potentially impactful. We have replied to individual points below. We hope the reviewers will consider raising their scores if our replies address their questions and concerns!

Meta-review: This paper addresses some of the failures of current weak-to-strong generalization results to account for phenomena that occur in practice, namely correction of the teacher’s errors and generalization where the teacher is not confident. The reviewers found the writing clear and the contributions valuable. However, they called for illustrating the results with basic experiments, even if synthetic. Also, several concurrent manuscripts were pointed out, and comparison with them would help the community better appreciate the contributions.
Decision: Accept (poster)