Lower Bounds for Greedy Teaching Set Constructions

A fundamental open problem in learning theory is to characterize the best-case teaching dimension $\operatorname{TS}_{\min}$ of a concept class $\mathcal{C}$ with finite VC dimension $d$. Resolving this problem will, in particular, settle the conjectured upper bound on Recursive Teaching Dimension posed by [Simon and Zilles; COLT 2015]. Prior work used a natural greedy algorithm to construct teaching sets recursively, thereby proving upper bounds on $\operatorname{TS}_{\min}$, with the best known bound being $O(d^2)$ [Hu, Wu, Li, and Wang; COLT 2017]. In each iteration, this greedy algorithm chooses to add to the teaching set the $k$ labeled points that restrict the concept class the most. In this work, we prove lower bounds on the performance of this greedy approach for small $k$. Specifically, we show that for $k = 1$, the algorithm does not improve upon the halving-based bound of $O(\log(|\mathcal{C}|))$. Furthermore, for $k = 2$, we complement the upper bound of $O\left(\log(\log(|\mathcal{C}|))\right)$ from [Moran, Shpilka, Wigderson, and Yuhudayoff; FOCS 2015] with a matching lower bound. Most consequentially, our lower bound extends up to $k \le \lceil c d \rceil$ for small constant $c>0$: suggesting that studying higher-order interactions may be necessary to resolve the conjecture that $\operatorname{TS}_{\min} = O(d)$.

Proofs as Explanations: Short Certificates for Reliable Predictions

We consider a model for explainable AI in which an explanation for a prediction $h(x)=y$ consists of a subset $S'$ of the training data (if it exists) such that all classifiers $h' \in H$ that make at most $b$ mistakes on $S'$ predict $h'(x)=y$. Such a set $S'$ serves as a proof that $x$ indeed has label $y$ under the assumption that (1) the target function $h^\star$ belongs to $H$, and (2) the set $S$ contains at most $b$ corrupted points. For example, if $b=0$ and $H$ is the family of linear classifiers in $\mathbb{R}^d$, and if $x$ lies inside the convex hull of the positive data points in $S$ (and hence every consistent linear classifier labels $x$ as positive), then Carath\'eodory's theorem states that $x$ lies inside the convex hull of $d+1$ of those points. So, a set $S'$ of size $d+1$ could be released as an explanation for a positive prediction, and would serve as a short proof of correctness of the prediction under the assumption of realizability. In this work, we consider this problem more generally, for general hypothesis classes $H$ and general values $b\geq 0$. We define the notion of the robust hollow star number of $H$ (which generalizes the standard hollow star number), and show that it precisely characterizes the worst-case size of the smallest certificate achievable, and analyze its size for natural classes. We also consider worst-case distributional bounds on certificate size, as well as distribution-dependent bounds that we show tightly control the sample size needed to get a certificate for any given test example. In particular, we define a notion of the certificate coefficient $\varepsilon_x$ of an example $x$ with respect to a data distribution $D$ and target function $h^\star$, and prove matching upper and lower bounds on sample size as a function of $\varepsilon_x$, $b$, and the VC dimension $d$ of $H$.

Relating Misfit to Gain in Weak-to-Strong Generalization Beyond the Squared Loss

The paradigm of weak-to-strong generalization constitutes the training of a strong AI model on data labeled by a weak AI model, with the goal that the strong model nevertheless outperforms its weak supervisor on the target task of interest. For the setting of real-valued regression with the squared loss, recent work quantitatively characterizes the gain in performance of the strong model over the weak model in terms of the misfit between the strong and weak model. We generalize such a characterization to learning tasks whose loss functions correspond to arbitrary Bregman divergences when the strong class is convex. This extends the misfit-based characterization of performance gain in weak-to-strong generalization to classification tasks, as the cross-entropy loss can be expressed in terms of a Bregman divergence. In most practical scenarios, however, the strong model class may not be convex. We therefore weaken this assumption and study weak-to-strong generalization for convex combinations of $k$ strong models in the strong class, in the concrete setting of classification. This allows us to obtain a similar misfit-based characterization of performance gain, upto an additional error term that vanishes as $k$ gets large. Our theoretical findings are supported by thorough experiments on synthetic as well as real-world datasets.

Exploring Facets of Language Generation in the Limit

The recent work of Kleinberg and Mullainathan [KM24] provides a concrete model for language generation in the limit: given a sequence of examples from an unknown target language, the goal is to generate new examples from the target language such that no incorrect examples are generated beyond some point. In sharp contrast to strong negative results for the closely related problem of language identification, they establish positive results for language generation in the limit for all countable collections of languages. Follow-up work by Raman and Tewari [RT24] studies bounds on the number of distinct inputs required by an algorithm before correct language generation is achieved -- namely, whether this is a constant for all languages in the collection (uniform generation) or a language-dependent constant (non-uniform generation). We show that every countable language collection has a generator which has the stronger property of non-uniform generation in the limit. However, while the generation algorithm of [KM24] can be implemented using membership queries, we show that any algorithm cannot non-uniformly generate even for collections of just two languages, using only membership queries. We also formalize the tension between validity and breadth in the generation algorithm of [KM24] by introducing a definition of exhaustive generation, and show a strong negative result for exhaustive generation. Our result shows that a tradeoff between validity and breadth is inherent for generation in the limit. Finally, inspired by algorithms that can choose to obtain feedback, we consider a model of uniform generation with feedback, completely characterizing language collections for which such uniform generation with feedback is possible in terms of a complexity measure of the collection.

A Characterization of List Regression

There has been a recent interest in understanding and characterizing the sample complexity of list learning tasks, where the learning algorithm is allowed to make a short list of $k$ predictions, and we simply require one of the predictions to be correct. This includes recent works characterizing the PAC sample complexity of standard list classification and online list classification. Adding to this theme, in this work, we provide a complete characterization of list PAC regression. We propose two combinatorial dimensions, namely the $k$-OIG dimension and the $k$-fat-shattering dimension, and show that they optimally characterize realizable and agnostic $k$-list regression respectively. These quantities generalize known dimensions for standard regression. Our work thus extends existing list learning characterizations from classification to regression.

Credit Attribution and Stable Compression

Credit attribution is crucial across various fields. In academic research, proper citation acknowledges prior work and establishes original contributions. Similarly, in generative models, such as those trained on existing artworks or music, it is important to ensure that any generated content influenced by these works appropriately credits the original creators. We study credit attribution by machine learning algorithms. We propose new definitions--relaxations of Differential Privacy--that weaken the stability guarantees for a designated subset of $k$ datapoints. These $k$ datapoints can be used non-stably with permission from their owners, potentially in exchange for compensation. Meanwhile, the remaining datapoints are guaranteed to have no significant influence on the algorithm's output. Our framework extends well-studied notions of stability, including Differential Privacy ($k = 0$), differentially private learning with public data (where the $k$ public datapoints are fixed in advance), and stable sample compression (where the $k$ datapoints are selected adaptively by the algorithm). We examine the expressive power of these stability notions within the PAC learning framework, provide a comprehensive characterization of learnability for algorithms adhering to these principles, and propose directions and questions for future research.

Quantifying the Gain in Weak-to-Strong Generalization

Recent advances in large language models have shown capabilities that are extraordinary and near-superhuman. These models operate with such complexity that reliably evaluating and aligning them proves challenging for humans. This leads to the natural question: can guidance from weak models (like humans) adequately direct the capabilities of strong models? In a recent and somewhat surprising work, Burns et al. (2023) empirically demonstrated that when strong models (like GPT-4) are finetuned using labels generated by weak supervisors (like GPT-2), the strong models outperform their weaker counterparts -- a phenomenon they term weak-to-strong generalization. In this work, we present a theoretical framework for understanding weak-to-strong generalization. Specifically, we show that the improvement in performance achieved by strong models over their weaker counterparts is quantified by the misfit error incurred by the strong model on labels generated by the weaker model. Our theory reveals several curious algorithmic insights. For instance, we can predict the amount by which the strong model will improve over the weak model, and also choose among different weak models to train the strong model, based on its misfit error. We validate our theoretical findings through various empirical assessments.

Testing with Non-identically Distributed Samples

We examine the extent to which sublinear-sample property testing and estimation applies to settings where samples are independently but not identically distributed. Specifically, we consider the following distributional property testing framework: Suppose there is a set of distributions over a discrete support of size $k$, $\textbf{p}_1, \textbf{p}_2,\ldots,\textbf{p}_T$, and we obtain $c$ independent draws from each distribution. Suppose the goal is to learn or test a property of the average distribution, $\textbf{p}_{\mathrm{avg}}$. This setup models a number of important practical settings where the individual distributions correspond to heterogeneous entities -- either individuals, chronologically distinct time periods, spatially separated data sources, etc. From a learning standpoint, even with $c=1$ samples from each distribution, $\Theta(k/\varepsilon^2)$ samples are necessary and sufficient to learn $\textbf{p}_{\mathrm{avg}}$ to within error $\varepsilon$ in TV distance. To test uniformity or identity -- distinguishing the case that $\textbf{p}_{\mathrm{avg}}$ is equal to some reference distribution, versus has $\ell_1$ distance at least $\varepsilon$ from the reference distribution, we show that a linear number of samples in $k$ is necessary given $c=1$ samples from each distribution. In contrast, for $c \ge 2$, we recover the usual sublinear sample testing of the i.i.d. setting: we show that $O(\sqrt{k}/\varepsilon^2 + 1/\varepsilon^4)$ samples are sufficient, matching the optimal sample complexity in the i.i.d. case in the regime where $\varepsilon \ge k^{-1/4}$. Additionally, we show that in the $c=2$ case, there is a constant $\rho > 0$ such that even in the linear regime with $\rho k$ samples, no tester that considers the multiset of samples (ignoring which samples were drawn from the same $\textbf{p}_i$) can perform uniformity testing.

Harnessing the Power of Choices in Decision Tree Learning

We propose a simple generalization of standard and empirically successful decision tree learning algorithms such as ID3, C4.5, and CART. These algorithms, which have been central to machine learning for decades, are greedy in nature: they grow a decision tree by iteratively splitting on the best attribute. Our algorithm, Top-$k$, considers the $k$ best attributes as possible splits instead of just the single best attribute. We demonstrate, theoretically and empirically, the power of this simple generalization. We first prove a {\sl greediness hierarchy theorem} showing that for every $k \in \mathbb{N}$, Top-$(k+1)$ can be dramatically more powerful than Top-$k$: there are data distributions for which the former achieves accuracy $1-\varepsilon$, whereas the latter only achieves accuracy $\frac1{2}+\varepsilon$. We then show, through extensive experiments, that Top-$k$ outperforms the two main approaches to decision tree learning: classic greedy algorithms and more recent "optimal decision tree" algorithms. On one hand, Top-$k$ consistently enjoys significant accuracy gains over greedy algorithms across a wide range of benchmarks. On the other hand, Top-$k$ is markedly more scalable than optimal decision tree algorithms and is able to handle dataset and feature set sizes that remain far beyond the reach of these algorithms.

Multiclass Learnability Does Not Imply Sample Compression

A hypothesis class admits a sample compression scheme, if for every sample labeled by a hypothesis from the class, it is possible to retain only a small subsample, using which the labels on the entire sample can be inferred. The size of the compression scheme is an upper bound on the size of the subsample produced. Every learnable binary hypothesis class (which must necessarily have finite VC dimension) admits a sample compression scheme of size only a finite function of its VC dimension, independent of the sample size. For multiclass hypothesis classes, the analog of VC dimension is the DS dimension. We show that the analogous statement pertaining to sample compression is not true for multiclass hypothesis classes: every learnable multiclass hypothesis class, which must necessarily have finite DS dimension, does not admit a sample compression scheme of size only a finite function of its DS dimension.