A central question in computer science and statistics is whether efficient algorithms can achieve the information-theoretic limits of statistical problems. Many computational-statistical tradeoffs have been shown under average-case assumptions, but since statistical problems are average-case in nature, it has been a challenge to base them on standard worst-case assumptions. In PAC learning where such tradeoffs were first studied, the question is whether computational efficiency can come at the cost of using more samples than information-theoretically necessary. We base such tradeoffs on $\mathsf{NP}$-hardness and obtain: $\circ$ Sharp computational-statistical tradeoffs assuming $\mathsf{NP}$ requires exponential time: For every polynomial $p(n)$, there is an $n$-variate class $C$ with VC dimension $1$ such that the sample complexity of time-efficiently learning $C$ is $\Theta(p(n))$. $\circ$ A characterization of $\mathsf{RP}$ vs. $\mathsf{NP}$ in terms of learning: $\mathsf{RP} = \mathsf{NP}$ iff every $\mathsf{NP}$-enumerable class is learnable with $O(\mathrm{VCdim}(C))$ samples in polynomial time. The forward implication has been known since (Pitt and Valiant, 1988); we prove the reverse implication. Notably, all our lower bounds hold against improper learners. These are the first $\mathsf{NP}$-hardness results for improperly learning a subclass of polynomial-size circuits, circumventing formal barriers of Applebaum, Barak, and Xiao (2008).
The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and relevance. Towards addressing this, we prove a distributional-lifting theorem: This upgrades a learner that succeeds with respect to a limited distribution family $\mathcal{D}$ to one that succeeds with respect to any distribution $D^\star$, with an efficiency overhead that scales with the complexity of expressing $D^\star$ as a mixture of distributions in $\mathcal{D}$. Recent work of Blanc, Lange, Malik, and Tan considered the special case of lifting uniform-distribution learners and designed a lifter that uses a conditional sample oracle for $D^\star$, a strong form of access not afforded by the standard PAC model. Their approach, which draws on ideas from semi-supervised learning, first learns $D^\star$ and then uses this information to lift. We show that their approach is information-theoretically intractable with access only to random examples, thereby giving formal justification for their use of the conditional sample oracle. We then take a different approach that sidesteps the need to learn $D^\star$, yielding a lifter that works in the standard PAC model and enjoys additional advantages: it works for all base distribution families, preserves the noise tolerance of learners, has better sample complexity, and is simpler.
Machine learning is now ubiquitous in societal decision-making, for example in evaluating job candidates or loan applications, and it is increasingly important to take into account how classified agents will react to the learning algorithms. The majority of recent literature on strategic classification has focused on reducing and countering deceptive behaviors by the classified agents, but recent work of Attias et al. identifies surprising properties of learnability when the agents genuinely improve in order to attain the desirable classification, such as smaller generalization error than standard PAC-learning. In this paper we characterize so-called learnability with improvements across multiple new axes. We introduce an asymmetric variant of minimally consistent concept classes and use it to provide an exact characterization of proper learning with improvements in the realizable setting. While prior work studies learnability only under general, arbitrary agent improvement regions, we give positive results for more natural Euclidean ball improvement sets. In particular, we characterize improper learning under a mild generative assumption on the data distribution. We further show how to learn in more challenging settings, achieving lower generalization error under well-studied bounded noise models and obtaining mistake bounds in realizable and agnostic online learning. We resolve open questions posed by Attias et al. for both proper and improper learning.
Attribute-efficient learning of sparse halfspaces has been a fundamental problem in machine learning theory. In recent years, machine learning algorithms are faced with prevalent data corruptions or even adversarial attacks. It is of central interest to design efficient algorithms that are robust to noise corruptions. In this paper, we consider that there exists a constant amount of malicious noise in the data and the goal is to learn an underlying $s$-sparse halfspace $w^* \in \mathbb{R}^d$ with $\text{poly}(s,\log d)$ samples. Specifically, we follow a recent line of works and assume that the underlying distribution satisfies a certain concentration condition and a margin condition at the same time. Under such conditions, we show that attribute-efficiency can be achieved by simple variants to existing hinge loss minimization programs. Our key contribution includes: 1) an attribute-efficient PAC learning algorithm that works under constant malicious noise rate; 2) a new gradient analysis that carefully handles the sparsity constraint in hinge loss minimization.
We study the complexity of learning real-valued Multi-Index Models (MIMs) under the Gaussian distribution. A $K$-MIM is a function $f:\mathbb{R}^d\to \mathbb{R}$ that depends only on the projection of its input onto a $K$-dimensional subspace. We give a general algorithm for PAC learning a broad class of MIMs with respect to the square loss, even in the presence of adversarial label noise. Moreover, we establish a nearly matching Statistical Query (SQ) lower bound, providing evidence that the complexity of our algorithm is qualitatively optimal as a function of the dimension. Specifically, we consider the class of bounded variation MIMs with the property that degree at most $m$ distinguishing moments exist with respect to projections onto any subspace. In the presence of adversarial label noise, the complexity of our learning algorithm is $d^{O(m)}2^{\mathrm{poly}(K/\epsilon)}$. For the realizable and independent noise settings, our algorithm incurs complexity $d^{O(m)}2^{\mathrm{poly}(K)}(1/\epsilon)^{O(K)}$. To complement our upper bound, we show that if for some subspace degree-$m$ distinguishing moments do not exist, then any SQ learner for the corresponding class of MIMs requires complexity $d^{\Omega(m)}$. As an application, we give the first efficient learner for the class of positive-homogeneous $L$-Lipschitz $K$-MIMs. The resulting algorithm has complexity $\mathrm{poly}(d) 2^{\mathrm{poly}(KL/\epsilon)}$. This gives a new PAC learning algorithm for Lipschitz homogeneous ReLU networks with complexity independent of the network size, removing the exponential dependence incurred in prior work.
Recently, the authors introduced the theory of high-arity PAC learning, which is well-suited for learning graphs, hypergraphs and relational structures. In the same initial work, the authors proved a high-arity analogue of the Fundamental Theorem of Statistical Learning that almost completely characterizes all notions of high-arity PAC learning in terms of a combinatorial dimension, called the Vapnik--Chervonenkis--Natarajan (VCN${}_k$) $k$-dimension, leaving as an open problem only the characterization of non-partite, non-agnostic high-arity PAC learnability. In this work, we complete this characterization by proving that non-partite non-agnostic high-arity PAC learnability implies a high-arity version of the Haussler packing property, which in turn implies finiteness of VCN${}_k$-dimension. This is done by obtaining direct proofs that classic PAC learnability implies classic Haussler packing property, which in turn implies finite Natarajan dimension and noticing that these direct proofs nicely lift to high-arity.
We consider the relationship between learnability of a "base class" of functions on a set $X$, and learnability of a class of statistical functions derived from the base class. For example, we refine results showing that learnability of a family $h_p: p \in Y$ of functions implies learnability of the family of functions $h_\mu=\lambda p: Y. E_\mu(h_p)$, where $E_\mu$ is the expectation with respect to $\mu$, and $\mu$ ranges over probability distributions on $X$. We will look at both Probably Approximately Correct (PAC) learning, where example inputs and outputs are chosen at random, and online learning, where the examples are chosen adversarily. For agnostic learning, we establish improved bounds on the sample complexity of learning for statistical classes, stated in terms of combinatorial dimensions of the base class. We connect these problems to techniques introduced in model theory for "randomizing a structure". We also provide counterexamples for realizable learning, in both the PAC and online settings.
We formalize the generalization error bound using Rademacher complexity in the Lean 4 theorem prover. Generalization error quantifies the gap between a learning machine's performance on given training data versus unseen test data, and Rademacher complexity serves as an estimate of this error based on the complexity of learning machines, or hypothesis class. Unlike traditional methods such as PAC learning and VC dimension, Rademacher complexity is applicable across diverse machine learning scenarios including deep learning and kernel methods. We formalize key concepts and theorems, including the empirical and population Rademacher complexities, and establish generalization error bounds through formal proofs of McDiarmid's inequality, Hoeffding's lemma, and symmetrization arguments.
NeuralODE is one example for generative machine learning based on the push forward of a simple source measure with a bijective mapping, which in the case of NeuralODE is given by the flow of a ordinary differential equation. Using Liouville's formula, the log-density of the push forward measure is easy to compute and thus NeuralODE can be trained based on the maximum Likelihood method such that the Kulback-Leibler divergence between the push forward through the flow map and the target measure generating the data becomes small. In this work, we give a detailed account on the consistency of Maximum Likelihood based empirical risk minimization for a generic class of target measures. In contrast to prior work, we do not only consider the statistical learning theory, but also give a detailed numerical analysis of the NeuralODE algorithm based on the 2nd order Runge-Kutta (RK) time integration. Using the universal approximation theory for deep ReQU networks, the stability and convergence rated for the RK scheme as well as metric entropy and concentration inequalities, we are able to prove that NeuralODE is a probably approximately correct (PAC) learning algorithm.
Recent advancements in cognitive science and multi-round reasoning techniques for Large Language Models (LLMs) suggest that iterative thinking processes improve problem-solving performance in complex tasks. Inspired by this, approaches like Chain-of-Thought, debating, and self-refinement have been applied to auto-regressive LLMs, achieving significant successes in tasks such as mathematical reasoning, commonsense reasoning, and multi-hop question answering. Despite these successes, the theoretical basis for how multi-round reasoning enhances problem-solving abilities remains underexplored. In this work, we investigate the approximation, learnability, and generalization properties of multi-round auto-regressive models. We show that Transformers with finite context windows are universal approximators for steps of Turing-computable functions and can approximate any Turing-computable sequence-to-sequence function through multi-round reasoning. We extend PAC learning to sequence generation and demonstrate that multi-round generation is learnable even when the sequence length exceeds the model's context window. Finally, we examine how generalization error propagates across rounds, and show how the aforementioned approaches can help constrain this error, ensuring outputs stay within an expectation boundary. This work sheds light on the systemic theoretical foundations of multi-round sequence learning and reasoning, emphasizing its role in inference complexity.
One of the most basic lower bounds in machine learning is that in nearly any nontrivial setting, it takes $\textit{at least}$ $1/\epsilon$ samples to learn to error $\epsilon$ (and more, if the classifier being learned is complex). However, suppose that data points are agents who have the ability to improve by a small amount if doing so will allow them to receive a (desired) positive classification. In that case, we may actually be able to achieve $\textit{zero}$ error by just being "close enough". For example, imagine a hiring test used to measure an agent's skill at some job such that for some threshold $\theta$, agents who score above $\theta$ will be successful and those who score below $\theta$ will not (i.e., learning a threshold on the line). Suppose also that by putting in effort, agents can improve their skill level by some small amount $r$. In that case, if we learn an approximation $\hat{\theta}$ of $\theta$ such that $\theta \leq \hat{\theta} \leq \theta + r$ and use it for hiring, we can actually achieve error zero, in the sense that (a) any agent classified as positive is truly qualified, and (b) any agent who truly is qualified can be classified as positive by putting in effort. Thus, the ability for agents to improve has the potential to allow for a goal one could not hope to achieve in standard models, namely zero error. In this paper, we explore this phenomenon more broadly, giving general results and examining under what conditions the ability of agents to improve can allow for a reduction in the sample complexity of learning, or alternatively, can make learning harder. We also examine both theoretically and empirically what kinds of improvement-aware algorithms can take into account agents who have the ability to improve to a limited extent when it is in their interest to do so.
We introduces a general linear framework that unifies the study of multi-winner voting rules and proportionality axioms, demonstrating that many prominent multi-winner voting rules-including Thiele methods, their sequential variants, and approval-based committee scoring rules-are linear. Similarly, key proportionality axioms such as Justified Representation (JR), Extended JR (EJR), and their strengthened variants (PJR+, EJR+), along with core stability, can fit within this linear structure as well. Leveraging PAC learning theory, we establish general and novel upper bounds on the sample complexity of learning linear mappings. Our approach yields near-optimal guarantees for diverse classes of rules, including Thiele methods and ordered weighted average rules, and can be applied to analyze the sample complexity of learning proportionality axioms such as approximate core stability. Furthermore, the linear structure allows us to leverage prior work to extend our analysis beyond worst-case scenarios to study the likelihood of various properties of linear rules and axioms. We introduce a broad class of distributions that extend Impartial Culture for approval preferences, and show that under these distributions, with high probability, any Thiele method is resolute, CORE is non-empty, and any Thiele method satisfies CORE, among other observations on the likelihood of commonly-studied properties in social choice. We believe that this linear theory offers a new perspective and powerful new tools for designing and analyzing multi-winner rules in modern social choice applications.
Contrastive learning has emerged as a powerful paradigm for self-supervised representation learning. This work analyzes the theoretical limits of contrastive learning under nasty noise, where an adversary modifies or replaces training samples. Using PAC learning and VC-dimension analysis, lower and upper bounds on sample complexity in adversarial settings are established. Additionally, data-dependent sample complexity bounds based on the l2-distance function are derived.
We study contrastive learning under the PAC learning framework. While a series of recent works have shown statistical results for learning under contrastive loss, based either on the VC-dimension or Rademacher complexity, their algorithms are inherently inefficient or not implying PAC guarantees. In this paper, we consider contrastive learning of the fundamental concept of linear representations. Surprisingly, even under such basic setting, the existence of efficient PAC learners is largely open. We first show that the problem of contrastive PAC learning of linear representations is intractable to solve in general. We then show that it can be relaxed to a semi-definite program when the distance between contrastive samples is measured by the $\ell_2$-norm. We then establish generalization guarantees based on Rademacher complexity, and connect it to PAC guarantees under certain contrastive large-margin conditions. To the best of our knowledge, this is the first efficient PAC learning algorithm for contrastive learning.
Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\mathcal{D}$ are simply more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\mathrm{err}(h^*_{\mathcal{D}})$, the error of the best hypothesis in $\mathcal{H}$ for $\mathcal{D}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\mathcal{D}$ for which $\tau = \mathrm{err}(h^*_{\mathcal{D}})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS `24) addresses this shortcoming by including $\tau$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $\tau + \Omega \left(\sqrt{\frac{\tau (d + \log(1 / \delta))}{m}} + \frac{d + \log(1 / \delta)}{m} \right)$ in a regime where $\tau > d/m$, and leave open the question of whether there may be a higher lower bound when $\tau \approx d/m$, with $d$ denoting $\mathrm{VC}(\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \cdot \tau + O \left(\sqrt{\frac{\tau (d + \log(1 / \delta))}{m}} + \frac{d + \log(1 / \delta)}{m} \right)$ for a constant $c \leq 2.1$, thus matching the lower bound when $\tau \approx d/m$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS `24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2.1 to 1, which would completely settle the complexity of agnostic learning.
We study the problem of multiclass list classification with (semi-)bandit feedback, where input examples are mapped into subsets of size $m$ of a collection of $K$ possible labels, and the feedback consists of the predicted labels which lie in the set of true labels of the given example. Our main result is for the $(\varepsilon,\delta)$-PAC variant of the problem for which we design an algorithm that returns an $\varepsilon$-optimal hypothesis with high probability using a sample complexity of $O \big( (\mathrm{poly}(K/m) + sm / \varepsilon^2) \log (|H|/\delta) \big)$ where $H$ is the underlying (finite) hypothesis class and $s$ is an upper bound on the number of true labels for a given example. This bound improves upon known bounds for combinatorial semi-bandits whenever $s \ll K$. Moreover, in the regime where $s = O(1)$ the leading terms in our bound match the corresponding full-information rates, implying that bandit feedback essentially comes at no cost. Our PAC learning algorithm is also computationally efficient given access to an ERM oracle for $H$. Additionally, we consider the regret minimization setting where data can be generated adversarially, and establish a regret bound of $\widetilde O(|H| + \sqrt{smT \log |H|})$. Our results generalize and extend those of Erez et al. (2024) who consider the simpler single-label setting corresponding to $s=m=1$, and in fact hold for the more general contextual combinatorial semi-bandit problem with $s$-sparse rewards.
We study the problem of computable multiclass learnability within the Probably Approximately Correct (PAC) learning framework of Valiant (1984). In the recently introduced computable PAC (CPAC) learning framework of Agarwal et al. (2020), both learners and the functions they output are required to be computable. We focus on the case of finite label space and start by proposing a computable version of the Natarajan dimension and showing that it characterizes CPAC learnability in this setting. We further generalize this result by establishing a meta-characterization of CPAC learnability for a certain family of dimensions: computable distinguishers. Distinguishers were defined by Ben-David et al. (1992) as a certain family of embeddings of the label space, with each embedding giving rise to a dimension. It was shown that the finiteness of each such dimension characterizes multiclass PAC learnability for finite label space in the non-computable setting. We show that the corresponding computable dimensions for distinguishers characterize CPAC learning. We conclude our analysis by proving that the DS dimension, which characterizes PAC learnability for infinite label space, cannot be expressed as a distinguisher (even in the case of finite label space).
Adversarially robust PAC learning has proved to be challenging, with the currently best known learners [Montasser et al., 2021a] relying on improper methods based on intricate compression schemes, resulting in sample complexity exponential in the VC-dimension. A series of follow up work considered a slightly relaxed version of the problem called adversarially robust learning with tolerance [Ashtiani et al., 2023, Bhattacharjee et al., 2023, Raman et al., 2024] and achieved better sample complexity in terms of the VC-dimension. However, those algorithms were either improper and complex, or required additional assumptions on the hypothesis class H. We prove, for the first time, the existence of a simpler learner that achieves a sample complexity linear in the VC-dimension without requiring additional assumptions on H. Even though our learner is improper, it is "almost proper" in the sense that it outputs a hypothesis that is "similar" to a hypothesis in H. We also use the ideas from our algorithm to construct a semi-supervised learner in the tolerant setting. This simple algorithm achieves comparable bounds to the previous (non-tolerant) semi-supervised algorithm of Attias et al. [2022a], but avoids the use of intricate subroutines from previous works, and is "almost proper."
We investigate the problem of verifying different properties of discrete time dynamical systems, namely, reachability, safety and reach-while-avoid. To achieve this, we adopt a data driven perspective and using past systems' trajectories as data, we aim at learning a specific function termed \emph{certificate} for each property we wish to verify. The certificate construction problem is treated as a safety informed neural network training process, where we use a neural network to learn the parameterization of each certificate, while the loss function we seek to minimize is designed to encompass conditions on the certificate to be learned that encode the satisfaction of the associated property. Besides learning a certificate, we quantify probabilistically its generalization properties, namely, how likely it is for a certificate to be valid (and hence for the associated property to be satisfied) when it comes to a new system trajectory not included in the training data set. We view this problem under the realm of probably approximately correct (PAC) learning under the notion of compression, and use recent advancements of the so-called scenario approach to obtain scalable generalization bounds on the learned certificates. To achieve this, we design a novel algorithm that minimizes the loss function and hence constructs a certificate, and at the same time determines a quantity termed compression, which is instrumental in obtaining meaningful probabilistic guarantees. This process is novel per se and provides a constructive mechanism for compression set calculation, thus opening the road for its use to more general non-convex optimization problems. We verify the efficacy of our methodology on several numerical case studies, and compare it (both theoretically and numerically) with closely related results on data-driven property verification.
Recent advances in the binary classification setting by Hanneke [2016b] and Larsen [2023] have resulted in optimal PAC learners. These learners leverage, respectively, a clever deterministic subsampling scheme and the classic heuristic of bagging Breiman [1996]. Both optimal PAC learners use, as a subroutine, the natural algorithm of empirical risk minimization. Consequently, the computational cost of these optimal PAC learners is tied to that of the empirical risk minimizer algorithm. In this work, we seek to provide an alternative perspective on the computational cost imposed by the link to the empirical risk minimizer algorithm. To this end, we show the existence of an optimal PAC learner, which offers a different tradeoff in terms of the computational cost induced by the empirical risk minimizer.