Majority-of-Three: The Simplest Optimal Learner?

Developing an optimal PAC learning algorithm in the realizable setting, where empirical risk minimization (ERM) is suboptimal, was a major open problem in learning theory for decades. The problem was finally resolved by Hanneke a few years ago. Unfortunately, Hanneke's algorithm is quite complex as it returns the majority vote of many ERM classifiers that are trained on carefully selected subsets of the data. It is thus a natural goal to determine the simplest algorithm that is optimal. In this work we study the arguably simplest algorithm that could be optimal: returning the majority vote of three ERM classifiers. We show that this algorithm achieves the optimal in-expectation bound on its error which is provably unattainable by a single ERM classifier. Furthermore, we prove a near-optimal high-probability bound on this algorithm's error. We conjecture that a better analysis will prove that this algorithm is in fact optimal in the high-probability regime.

Replicable Learning of Large-Margin Halfspaces

We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first dimension-independent replicable algorithms for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. [2022] with respect to all the relevant parameters. Moreover, our first algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. We also design an SGD-based replicable algorithm that, in some parameters' regimes, achieves better sample and time complexity than our first algorithm. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun, Gaboardi, Hopkins, Impagliazzo, Lei, Pitassi, Sorrell, and Sivakumar [STOC, 2023], we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau^2$ and worse sample complexity dependence on $\epsilon$ than one of our previous algorithms. We then design an improved algorithm with better sample complexity than all three of our previous algorithms and running time exponential in $1/\tau^{2}$.

Boosting, Voting Classifiers and Randomized Sample Compression Schemes

In boosting, we aim to leverage multiple weak learners to produce a strong learner. At the center of this paradigm lies the concept of building the strong learner as a voting classifier, which outputs a weighted majority vote of the weak learners. While many successful boosting algorithms, such as the iconic AdaBoost, produce voting classifiers, their theoretical performance has long remained sub-optimal: the best known bounds on the number of training examples necessary for a voting classifier to obtain a given accuracy has so far always contained at least two logarithmic factors above what is known to be achievable by general weak-to-strong learners. In this work, we break this barrier by proposing a randomized boosting algorithm that outputs voting classifiers whose generalization error contains a single logarithmic dependency on the sample size. We obtain this result by building a general framework that extends sample compression methods to support randomized learning algorithms based on sub-sampling.

Sparse Dimensionality Reduction Revisited

The sparse Johnson-Lindenstrauss transform is one of the central techniques in dimensionality reduction. It supports embedding a set of $n$ points in $\mathbb{R}^d$ into $m=O(\varepsilon^{-2} \lg n)$ dimensions while preserving all pairwise distances to within $1 \pm \varepsilon$. Each input point $x$ is embedded to $Ax$, where $A$ is an $m \times d$ matrix having $s$ non-zeros per column, allowing for an embedding time of $O(s \|x\|_0)$. Since the sparsity of $A$ governs the embedding time, much work has gone into improving the sparsity $s$. The current state-of-the-art by Kane and Nelson (JACM'14) shows that $s = O(\varepsilon ^{-1} \lg n)$ suffices. This is almost matched by a lower bound of $s = \Omega(\varepsilon ^{-1} \lg n/\lg(1/\varepsilon))$ by Nelson and Nguyen (STOC'13). Previous work thus suggests that we have near-optimal embeddings. In this work, we revisit sparse embeddings and identify a loophole in the lower bound. Concretely, it requires $d \geq n$, which in many applications is unrealistic. We exploit this loophole to give a sparser embedding when $d = o(n)$, achieving $s = O(\varepsilon^{-1}(\lg n/\lg(1/\varepsilon)+\lg^{2/3}n \lg^{1/3} d))$. We also complement our analysis by strengthening the lower bound of Nelson and Nguyen to hold also when $d \ll n$, thereby matching the first term in our new sparsity upper bound. Finally, we also improve the sparsity of the best oblivious subspace embeddings for optimal embedding dimensionality.

AdaBoost is not an Optimal Weak to Strong Learner

AdaBoost is a classic boosting algorithm for combining multiple inaccurate classifiers produced by a weak learner, to produce a strong learner with arbitrarily high accuracy when given enough training data. Determining the optimal number of samples necessary to obtain a given accuracy of the strong learner, is a basic learning theoretic question. Larsen and Ritzert (NeurIPS'22) recently presented the first provably optimal weak-to-strong learner. However, their algorithm is somewhat complicated and it remains an intriguing question whether the prototypical boosting algorithm AdaBoost also makes optimal use of training samples. In this work, we answer this question in the negative. Concretely, we show that the sample complexity of AdaBoost, and other classic variations thereof, are sub-optimal by at least one logarithmic factor in the desired accuracy of the strong learner.

The Impossibility of Parallelizing Boosting

The aim of boosting is to convert a sequence of weak learners into a strong learner. At their heart, these methods are fully sequential. In this paper, we investigate the possibility of parallelizing boosting. Our main contribution is a strong negative result, implying that significant parallelization of boosting requires an exponential blow-up in the total computing resources needed for training.

Bagging is an Optimal PAC Learner

Determining the optimal sample complexity of PAC learning in the realizable setting was a central open problem in learning theory for decades. Finally, the seminal work by Hanneke (2016) gave an algorithm with a provably optimal sample complexity. His algorithm is based on a careful and structured sub-sampling of the training data and then returning a majority vote among hypotheses trained on each of the sub-samples. While being a very exciting theoretical result, it has not had much impact in practice, in part due to inefficiency, since it constructs a polynomial number of sub-samples of the training data, each of linear size. In this work, we prove the surprising result that the practical and classic heuristic bagging (a.k.a. bootstrap aggregation), due to Breiman (1996), is in fact also an optimal PAC learner. Bagging pre-dates Hanneke's algorithm by twenty years and is taught in most undergraduate machine learning courses. Moreover, we show that it only requires a logarithmic number of sub-samples to reach optimality.

Improved Coresets for Euclidean $k$-Means

Given a set of $n$ points in $d$ dimensions, the Euclidean $k$-means problem (resp. the Euclidean $k$-median problem) consists of finding $k$ centers such that the sum of squared distances (resp. sum of distances) from every point to its closest center is minimized. The arguably most popular way of dealing with this problem in the big data setting is to first compress the data by computing a weighted subset known as a coreset and then run any algorithm on this subset. The guarantee of the coreset is that for any candidate solution, the ratio between coreset cost and the cost of the original instance is less than a $(1\pm \varepsilon)$ factor. The current state of the art coreset size is $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for Euclidean $k$-means and $\tilde O(\min(k^{2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for Euclidean $k$-median. The best known lower bound for both problems is $\Omega(k \varepsilon^{-2})$. In this paper, we improve the upper bounds $\tilde O(\min(k^{3/2} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-4}))$ for $k$-means and $\tilde O(\min(k^{4/3} \cdot \varepsilon^{-2},k\cdot \varepsilon^{-3}))$ for $k$-median. In particular, ours is the first provable bound that breaks through the $k^2$ barrier while retaining an optimal dependency on $\varepsilon$.

Optimal Weak to Strong Learning

The classic algorithm AdaBoost allows to convert a weak learner, that is an algorithm that produces a hypothesis which is slightly better than chance, into a strong learner, achieving arbitrarily high accuracy when given enough training data. We present a new algorithm that constructs a strong learner from a weak learner but uses less training data than AdaBoost and all other weak to strong learners to achieve the same generalization bounds. A sample complexity lower bound shows that our new algorithm uses the minimum possible amount of training data and is thus optimal. Hence, this work settles the sample complexity of the classic problem of constructing a strong learner from a weak learner.

The Fast Johnson-Lindenstrauss Transform is Even Faster

The seminal Fast Johnson-Lindenstrauss (Fast JL) transform by Ailon and Chazelle (SICOMP'09) embeds a set of $n$ points in $d$-dimensional Euclidean space into optimal $k=O(\varepsilon^{-2} \ln n)$ dimensions, while preserving all pairwise distances to within a factor $(1 \pm \varepsilon)$. The Fast JL transform supports computing the embedding of a data point in $O(d \ln d +k \ln^2 n)$ time, where the $d \ln d$ term comes from multiplication with a $d \times d$ Hadamard matrix and the $k \ln^2 n$ term comes from multiplication with a sparse $k \times d$ matrix. Despite the Fast JL transform being more than a decade old, it is one of the fastest dimensionality reduction techniques for many tradeoffs between $\varepsilon, d$ and $n$. In this work, we give a surprising new analysis of the Fast JL transform, showing that the $k \ln^2 n$ term in the embedding time can be improved to $(k \ln^2 n)/\alpha$ for an $\alpha = \Omega(\min\{\varepsilon^{-1}\ln(1/\varepsilon), \ln n\})$. The improvement follows by using an even sparser matrix. We also complement our improved analysis with a lower bound showing that our new analysis is in fact tight.