Understanding Aggregations of Proper Learners in Multiclass Classification

Multiclass learnability is known to exhibit a properness barrier: there are learnable classes which cannot be learned by any proper learner. Binary classification faces no such barrier for learnability, but a similar one for optimal learning, which can in general only be achieved by improper learners. Fortunately, recent advances in binary classification have demonstrated that this requirement can be satisfied using aggregations of proper learners, some of which are strikingly simple. This raises a natural question: to what extent can simple aggregations of proper learners overcome the properness barrier in multiclass classification? We give a positive answer to this question for classes which have finite Graph dimension, $d_G$. Namely, we demonstrate that the optimal binary learners of Hanneke, Larsen, and Aden-Ali et al. (appropriately generalized to the multiclass setting) achieve sample complexity $O\left(\frac{d_G + \ln(1 / \delta)}{\epsilon}\right)$. This forms a strict improvement upon the sample complexity of ERM. We complement this with a lower bound demonstrating that for certain classes of Graph dimension $d_G$, majorities of ERM learners require $\Omega \left( \frac{d_G + \ln(1 / \delta)}{\epsilon}\right)$ samples. Furthermore, we show that a single ERM requires $\Omega \left(\frac{d_G \ln(1 / \epsilon) + \ln(1 / \delta)}{\epsilon}\right)$ samples on such classes, exceeding the lower bound of Daniely et al. (2015) by a factor of $\ln(1 / \epsilon)$. For multiclass learning in full generality -- i.e., for classes of finite DS dimension but possibly infinite Graph dimension -- we give a strong refutation to these learning strategies, by exhibiting a learnable class which cannot be learned to constant error by any aggregation of a finite number of proper learners.

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.

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.

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 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.