The Sample Complexity of Replicable Realizable PAC Learning

In this paper, we consider the problem of replicable realizable PAC learning. We construct a particularly hard learning problem and show a sample complexity lower bound with a close to $(\log|H|)^{3/2}$ dependence on the size of the hypothesis class $H$. Our proof uses several novel techniques and works by defining a particular Cayley graph associated with $H$ and analyzing a suitable random walk on this graph by examining the spectral properties of its adjacency matrix. Furthermore, we show an almost matching upper bound for the lower bound instance, meaning if a stronger lower bound exists, one would have to consider a different instance of the problem.

Learning with Monotone Adversarial Corruptions

We study the extent to which standard machine learning algorithms rely on exchangeability and independence of data by introducing a monotone adversarial corruption model. In this model, an adversary, upon looking at a "clean" i.i.d. dataset, inserts additional "corrupted" points of their choice into the dataset. These added points are constrained to be monotone corruptions, in that they get labeled according to the ground-truth target function. Perhaps surprisingly, we demonstrate that in this setting, all known optimal learning algorithms for binary classification can be made to achieve suboptimal expected error on a new independent test point drawn from the same distribution as the clean dataset. On the other hand, we show that uniform convergence-based algorithms do not degrade in their guarantees. Our results showcase how optimal learning algorithms break down in the face of seemingly helpful monotone corruptions, exposing their overreliance on exchangeability.

Improved Margin Generalization Bounds for Voting Classifiers

In this paper we establish a new margin-based generalization bound for voting classifiers, refining existing results and yielding tighter generalization guarantees for widely used boosting algorithms such as AdaBoost (Freund and Schapire, 1997). Furthermore, the new margin-based generalization bound enables the derivation of an optimal weak-to-strong learner: a Majority-of-3 large-margin classifiers with an expected error matching the theoretical lower bound. This result provides a more natural alternative to the Majority-of-5 algorithm by (H\o gsgaard et al. 2024) , and matches the Majority-of-3 result by (Aden-Ali et al. 2024) for the realizable prediction model.

Tight Generalization Bounds for Large-Margin Halfspaces

We prove the first generalization bound for large-margin halfspaces that is asymptotically tight in the tradeoff between the margin, the fraction of training points with the given margin, the failure probability and the number of training points.

Improved Replicable Boosting with Majority-of-Majorities

We introduce a new replicable boosting algorithm which significantly improves the sample complexity compared to previous algorithms. The algorithm works by doing two layers of majority voting, using an improved version of the replicable boosting algorithm introduced by Impagliazzo et al. [2022] in the bottom layer.

An Exponential Separation Between Quantum and Quantum-Inspired Classical Algorithms for Machine Learning

Achieving a provable exponential quantum speedup for an important machine learning task has been a central research goal since the seminal HHL quantum algorithm for solving linear systems and the subsequent quantum recommender systems algorithm by Kerenidis and Prakash. These algorithms were initially believed to be strong candidates for exponential speedups, but a lower bound ruling out similar classical improvements remained absent. In breakthrough work by Tang, it was demonstrated that this lack of progress in classical lower bounds was for good reasons. Concretely, she gave a classical counterpart of the quantum recommender systems algorithm, reducing the quantum advantage to a mere polynomial. Her approach is quite general and was named quantum-inspired classical algorithms. Since then, almost all the initially exponential quantum machine learning speedups have been reduced to polynomial via new quantum-inspired classical algorithms. From the current state-of-affairs, it is unclear whether we can hope for exponential quantum speedups for any natural machine learning task. In this work, we present the first such provable exponential separation between quantum and quantum-inspired classical algorithms. We prove the separation for the basic problem of solving a linear system when the input matrix is well-conditioned and has sparse rows and columns.

Derandomizing Multi-Distribution Learning

Multi-distribution or collaborative learning involves learning a single predictor that works well across multiple data distributions, using samples from each during training. Recent research on multi-distribution learning, focusing on binary loss and finite VC dimension classes, has shown near-optimal sample complexity that is achieved with oracle efficient algorithms. That is, these algorithms are computationally efficient given an efficient ERM for the class. Unlike in classical PAC learning, where the optimal sample complexity is achieved with deterministic predictors, current multi-distribution learning algorithms output randomized predictors. This raises the question: can these algorithms be derandomized to produce a deterministic predictor for multiple distributions? Through a reduction to discrepancy minimization, we show that derandomizing multi-distribution learning is computationally hard, even when ERM is computationally efficient. On the positive side, we identify a structural condition enabling an efficient black-box reduction, converting existing randomized multi-distribution predictors into deterministic ones.

The Many Faces of Optimal Weak-to-Strong Learning

Boosting is an extremely successful idea, allowing one to combine multiple low accuracy classifiers into a much more accurate voting classifier. In this work, we present a new and surprisingly simple Boosting algorithm that obtains a provably optimal sample complexity. Sample optimal Boosting algorithms have only recently been developed, and our new algorithm has the fastest runtime among all such algorithms and is the simplest to describe: Partition your training data into 5 disjoint pieces of equal size, run AdaBoost on each, and combine the resulting classifiers via a majority vote. In addition to this theoretical contribution, we also perform the first empirical comparison of the proposed sample optimal Boosting algorithms. Our pilot empirical study suggests that our new algorithm might outperform previous algorithms on large data sets.

Optimal Parallelization of Boosting

Recent works on the parallel complexity of Boosting have established strong lower bounds on the tradeoff between the number of training rounds $p$ and the total parallel work per round $t$. These works have also presented highly non-trivial parallel algorithms that shed light on different regions of this tradeoff. Despite these advancements, a significant gap persists between the theoretical lower bounds and the performance of these algorithms across much of the tradeoff space. In this work, we essentially close this gap by providing both improved lower bounds on the parallel complexity of weak-to-strong learners, and a parallel Boosting algorithm whose performance matches these bounds across the entire $p$ vs.~$t$ compromise spectrum, up to logarithmic factors. Ultimately, this work settles the true parallel complexity of Boosting algorithms that are nearly sample-optimal.

Revisiting Agnostic PAC Learning

PAC learning, dating back to Valiant'84 and Vapnik and Chervonenkis'64,'74, is a classic model for studying supervised learning. In the agnostic setting, we have access to a hypothesis set $\mathcal{H}$ and a training set of labeled samples $(x_1,y_1),\dots,(x_n,y_n) \in \mathcal{X} \times \{-1,1\}$ drawn i.i.d. from an unknown distribution $\mathcal{D}$. The goal is to produce a classifier $h : \mathcal{X} \to \{-1,1\}$ that is competitive with the hypothesis $h^\star_{\mathcal{D}} \in \mathcal{H}$ having the least probability of mispredicting the label $y$ of a new sample $(x,y)\sim \mathcal{D}$. Empirical Risk Minimization (ERM) is a natural learning algorithm, where one simply outputs the hypothesis from $\mathcal{H}$ making the fewest mistakes on the training data. This simple algorithm is known to have an optimal error in terms of the VC-dimension of $\mathcal{H}$ and the number of samples $n$. In this work, we revisit agnostic PAC learning and first show that ERM is in fact sub-optimal if we treat the performance of the best hypothesis, denoted $\tau:=\Pr_{\mathcal{D}}[h^\star_{\mathcal{D}}(x) \neq y]$, as a parameter. Concretely we show that ERM, and any other proper learning algorithm, is sub-optimal by a $\sqrt{\ln(1/\tau)}$ factor. We then complement this lower bound with the first learning algorithm achieving an optimal error for nearly the full range of $\tau$. Our algorithm introduces several new ideas that we hope may find further applications in learning theory.