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.
In real-world applications, the distribution of the data, and our goals, evolve over time. The prevailing theoretical framework for studying machine learning, namely probably approximately correct (PAC) learning, largely ignores time. As a consequence, existing strategies to address the dynamic nature of data and goals exhibit poor real-world performance. This paper develops a theoretical framework called "Prospective Learning" that is tailored for situations when the optimal hypothesis changes over time. In PAC learning, empirical risk minimization (ERM) is known to be consistent. We develop a learner called Prospective ERM, which returns a sequence of predictors that make predictions on future data. We prove that the risk of prospective ERM converges to the Bayes risk under certain assumptions on the stochastic process generating the data. Prospective ERM, roughly speaking, incorporates time as an input in addition to the data. We show that standard ERM as done in PAC learning, without incorporating time, can result in failure to learn when distributions are dynamic. Numerical experiments illustrate that prospective ERM can learn synthetic and visual recognition problems constructed from MNIST and CIFAR-10. Code at https://github.com/neurodata/prolearn.
We consider a weakly supervised learning problem called Learning from Label Proportions (LLP), where examples are grouped into ``bags'' and only the average label within each bag is revealed to the learner. We study various learning rules for LLP that achieve PAC learning guarantees for classification loss. We establish that the classical Empirical Proportional Risk Minimization (EPRM) learning rule (Yu et al., 2014) achieves fast rates under realizability, but EPRM and similar proportion matching learning rules can fail in the agnostic setting. We also show that (1) a debiased proportional square loss, as well as (2) a recently proposed EasyLLP learning rule (Busa-Fekete et al., 2023) both achieve ``optimistic rates'' (Panchenko, 2002); in both the realizable and agnostic settings, their sample complexity is optimal (up to log factors) in terms of $\epsilon, \delta$, and VC dimension.
Distillation is the task of replacing a complicated machine learning model with a simpler model that approximates the original [BCNM06,HVD15]. Despite many practical applications, basic questions about the extent to which models can be distilled, and the runtime and amount of data needed to distill, remain largely open. To study these questions, we initiate a general theory of distillation, defining PAC-distillation in an analogous way to PAC-learning [Val84]. As applications of this theory: (1) we propose new algorithms to extract the knowledge stored in the trained weights of neural networks -- we show how to efficiently distill neural networks into succinct, explicit decision tree representations when possible by using the ``linear representation hypothesis''; and (2) we prove that distillation can be much cheaper than learning from scratch, and make progress on characterizing its complexity.
We propose bounded fitting as a scheme for learning description logic concepts in the presence of ontologies. A main advantage is that the resulting learning algorithms come with theoretical guarantees regarding their generalization to unseen examples in the sense of PAC learning. We prove that, in contrast, several other natural learning algorithms fail to provide such guarantees. As a further contribution, we present the system SPELL which efficiently implements bounded fitting for the description logic $\mathcal{ELH}^r$ based on a SAT solver, and compare its performance to a state-of-the-art learner.
We establish a simple connection between robust and differentially-private algorithms: private mechanisms which perform well with very high probability are automatically robust in the sense that they retain accuracy even if a constant fraction of the samples they receive are adversarially corrupted. Since optimal mechanisms typically achieve these high success probabilities, our results imply that optimal private mechanisms for many basic statistics problems are robust. We investigate the consequences of this observation for both algorithms and computational complexity across different statistical problems. Assuming the Brennan-Bresler secret-leakage planted clique conjecture, we demonstrate a fundamental tradeoff between computational efficiency, privacy leakage, and success probability for sparse mean estimation. Private algorithms which match this tradeoff are not yet known -- we achieve that (up to polylogarithmic factors) in a polynomially-large range of parameters via the Sum-of-Squares method. To establish an information-computation gap for private sparse mean estimation, we also design new (exponential-time) mechanisms using fewer samples than efficient algorithms must use. Finally, we give evidence for privacy-induced information-computation gaps for several other statistics and learning problems, including PAC learning parity functions and estimation of the mean of a multivariate Gaussian.
A central goal in the cognitive sciences is the development of numerical models for mental representations of object concepts. This paper introduces Variational Interpretable Concept Embeddings (VICE), an approximate Bayesian method for embedding object concepts in a vector space using data collected from humans in a triplet odd-one-out task. VICE uses variational inference to obtain sparse, non-negative representations of object concepts with uncertainty estimates for the embedding values. These estimates are used to automatically select the dimensions that best explain the data. We derive a PAC learning bound for VICE that can be used to estimate generalization performance or determine a sufficient sample size for experimental design. VICE rivals or outperforms its predecessor, SPoSE, at predicting human behavior in the triplet odd-one-out task. Furthermore, VICE's object representations are more reproducible and consistent across random initializations, highlighting the unique advantage of using VICE for deriving interpretable embeddings from human behavior.
Multiple methods of finding the vertices belonging to a planted dense subgraph in a random dense $G(n, p)$ graph have been proposed, with an emphasis on planted cliques. Such methods can identify the planted subgraph in polynomial time, but are all limited to several subgraph structures. Here, we present PYGON, a graph neural network-based algorithm, which is insensitive to the structure of the planted subgraph. This is the first algorithm that uses advanced learning tools for recovering dense subgraphs. We show that PYGON can recover cliques of sizes $\Theta\left(\sqrt{n}\right)$, where $n$ is the size of the background graph, comparable with the state of the art. We also show that the same algorithm can recover multiple other planted subgraphs of size $\Theta\left(\sqrt{n}\right)$, in both directed and undirected graphs. We suggest a conjecture that no polynomial time PAC-learning algorithm can detect planted dense subgraphs with size smaller than $O\left(\sqrt{n}\right)$, even if in principle one could find dense subgraphs of logarithmic size.
Boosting is an ensemble learning method that converts a weak learner into a strong learner in the PAC learning framework. Freund and Schapire designed the Godel prize-winning algorithm named AdaBoost that can boost learners, which output binary hypotheses. Recently, Arunachalam and Maity presented the first quantum boosting algorithm with similar theoretical guarantees. Their algorithm, which we refer to as QAdaBoost henceforth, is a quantum adaptation of AdaBoost and only works for the binary hypothesis case. QAdaBoost is quadratically faster than AdaBoost in terms of the VC-dimension of the hypothesis class of the weak learner but polynomially worse in the bias of the weak learner. Izdebski et al. posed an open question on whether we can boost quantum weak learners that output non-binary hypothesis. In this work, we address this open question by developing the QRealBoost algorithm which was motivated by the classical RealBoost algorithm. The main technical challenge was to provide provable guarantees for convergence, generalization bounds, and quantum speedup, given that quantum subroutines are noisy and probabilistic. We prove that QRealBoost retains the quadratic speedup of QAdaBoost over AdaBoost and further achieves a polynomial speedup over QAdaBoost in terms of both the bias of the learner and the time taken by the learner to learn the target concept class. Finally, we perform empirical evaluations on QRealBoost and report encouraging observations on quantum simulators by benchmarking the convergence performance of QRealBoost against QAdaBoost, AdaBoost, and RealBoost on a subset of the MNIST dataset and Breast Cancer Wisconsin dataset.
Computational learning theory states that many classes of boolean formulas are learnable in polynomial time. This paper addresses the understudied subject of how, in practice, such formulas can be learned by deep neural networks. Specifically, we analyze boolean formulas associated with model-sampling benchmarks, combinatorial optimization problems, and random 3-CNFs with varying degrees of constrainedness. Our experiments indicate that: (i) neural learning generalizes better than pure rule-based systems and pure symbolic approach; (ii) relatively small and shallow neural networks are very good approximators of formulas associated with combinatorial optimization problems; (iii) smaller formulas seem harder to learn, possibly due to the fewer positive (satisfying) examples available; and (iv) interestingly, underconstrained 3-CNF formulas are more challenging to learn than overconstrained ones. Such findings pave the way for a better understanding, construction, and use of interpretable neurosymbolic AI methods.
Prediction is a well-studied machine learning task, and prediction algorithms are core ingredients in online products and services. Despite their centrality in the competition between online companies who offer prediction-based products, the \textit{strategic} use of prediction algorithms remains unexplored. The goal of this paper is to examine strategic use of prediction algorithms. We introduce a novel game-theoretic setting that is based on the PAC learning framework, where each player (aka a prediction algorithm aimed at competition) seeks to maximize the sum of points for which it produces an accurate prediction and the others do not. We show that algorithms aiming at generalization may wittingly mispredict some points to perform better than others on expectation. We analyze the empirical game, i.e., the game induced on a given sample, prove that it always possesses a pure Nash equilibrium, and show that every better-response learning process converges. Moreover, our learning-theoretic analysis suggests that players can, with high probability, learn an approximate pure Nash equilibrium for the whole population using a small number of samples.