We consider the problem of regret minimization in non-parametric stochastic bandits. When the rewards are known to be bounded from above, there exists asymptotically optimal algorithms, with asymptotic regret depending on an infimum of Kullback-Leibler divergences (KL). These algorithms are computationally expensive and require storing all past rewards, thus simpler but non-optimal algorithms are often used instead. We introduce several methods to approximate the infimum KL which reduce drastically the computational and memory costs of existing optimal algorithms, while keeping their regret guaranties. We apply our findings to design new variants of the MED and IMED algorithms, and demonstrate their interest with extensive numerical simulations.

State-of-the-art optimization is steadily shifting towards massively parallel pipelines with extremely large batch sizes. As a consequence, CPU-bound preprocessing and disk/memory/network operations have emerged as new performance bottlenecks, as opposed to hardware-accelerated gradient computations. In this regime, a recently proposed approach is data echoing (Choi et al., 2019), which takes repeated gradient steps on the same batch while waiting for fresh data to arrive from upstream. We provide the first convergence analyses of "data-echoed" extensions of common optimization methods, showing that they exhibit provable improvements over their synchronous counterparts. Specifically, we show that in convex optimization with stochastic minibatches, data echoing affords speedups on the curvature-dominated part of the convergence rate, while maintaining the optimal statistical rate.

Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the population risk to decrease (better performance) monotonically with increasing data points. It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms that minimize the empirical risk. Non-monotonic behavior of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent. It is, therefore, crucial to pursue this concern and provide a characterization of such behavior. In this paper, we derive the first consistent and risk-monotonic (in high probability) algorithms for a general statistical learning setting under weak assumptions, consequently answering some questions posed by Viering et.

Clustering is a ubiquitous task in data science. Compared to the commonly used k-means clustering, k-medoids clustering requires the cluster centers to be actual data points and supports arbitrary distance metrics, which permits greater interpretability and the clustering of structured objects. Current state-of-the-art k-medoids clustering algorithms, such as Partitioning Around Medoids (PAM), are iterative and are quadratic in the dataset size n for each iteration, being prohibitively expensive for large datasets. We propose BanditPAM, a randomized algorithm inspired by techniques from multi-armed bandits, that reduces the complexity of each PAM iteration from O(n 2) to O(nlogn) and returns the same results with high probability, under assumptions on the data that often hold in practice. As such, BanditPAM matches state-of-the-art clustering loss while reaching solutions much faster.

Dimensionality reduction plays a central role in real-world applications for Machine Learning, among many fields. In particular, metric dimensionality reduction where data from a general metric is mapped into low dimensional space, is often used as a first step before applying machine learning algorithms. In almost all these applications the quality of the embedding is measured by various average case criteria. Metric dimensionality reduction has also been studied in Math and TCS, within the extremely fruitful and influential field of metric embedding. Yet, the vast majority of theoretical research has been devoted to analyzing the worst case behavior of embeddings and therefore has little relevance to practical settings.

One of the most commonly used methods for forming confidence intervals is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role in machine learning, its theoretical properties are still not well understood. Recent developments in probability have provided new tools to study the bootstrap method. However, they have been applied only to specific applications and contexts, and it is unclear whether these techniques are applicable to the understanding of the consistency of the bootstrap in machine learning pipelines. In this paper, we derive general stability conditions under which the empirical bootstrap estimator is consistent and quantify the speed of convergence.

Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, simple to implement, and competitive with state-of-the-art algorithms.

Our general setup leads to a number of interesting findings. We outline precise conditions that decide the sign of the optimal setting \lambda_{\opt} for the ridge parameter \lambda and confirm the implicit \ell_2 regularization effect of overparameterization, which theoretically justifies the surprising empirical observation that \lambda_{\opt} can be \textit{negative} in the overparameterized regime. We also characterize the double descent phenomenon for principal component regression (PCR) when \vX and \vbeta_{\star} are both anisotropic. Finally, we determine the optimal weighting matrix \vSigma_w for both the ridgeless ( \lambda\to 0) and optimally regularized ( \lambda \lambda_{\opt}) case, and demonstrate the advantage of the weighted objective over standard ridge regression and PCR.

Graph Convolutional Networks (GCNs), aiming to obtain the representation of a node by aggregating its neighbors, have demonstrated great power in tackling various analytics tasks on graph (network) data. The remarkable performance of GCNs typically relies on the homophily assumption of networks, while such assumption cannot always be satisfied, since the heterophily or randomness are also widespread in real-world. This gives rise to one fundamental question: whether networks with different structural properties should adopt different propagation mechanisms? Surprisingly, we discover that there are actually segmentation rules for the propagation mechanism, i.e., 1-hop, 2-hop and k -nearest neighbor ( k NN) neighbors are more suitable as neighborhoods of network with complete homophily, complete heterophily and randomness, respectively. However, the real-world networks are complex, and may present diverse structural properties, e.g., the network dominated by homophily may contain a small amount of randomness.

The vast majority of successful deep neural networks are trained using variants of stochastic gradient descent (SGD) algorithms. Recent attempts to improve SGD can be broadly categorized into two approaches: (1) adaptive learning rate schemes, such as AdaGrad and Adam and (2) accelerated schemes, such as heavy-ball and Nesterov momentum. In this paper, we propose a new optimization algorithm, Lookahead, that is orthogonal to these previous approaches and iteratively updates two sets of weights. Intuitively, the algorithm chooses a search direction by looking ahead at the sequence of fast weights" generated by another optimizer. We show that Lookahead improves the learning stability and lowers the variance of its inner optimizer with negligible computation and memory cost.