We propose a new framework of variance-reduced Hamiltonian Monte Carlo (HMC) methods for sampling from an $L$-smooth and $m$-strongly log-concave distribution, based on a unified formulation of biased and unbiased variance reduction methods. We study the convergence properties for HMC with gradient estimators which satisfy the Mean-Squared-Error-Bias (MSEB) property. We show that the unbiased gradient estimators, including SAGA and SVRG, based HMC methods achieve highest gradient efficiency with small batch size under high precision regime, and require $\tilde{O}(N + \kappa^2 d^{\frac{1}{2}} \varepsilon^{-1} + N^{\frac{2}{3}} \kappa^{\frac{4}{3}} d^{\frac{1}{3}} \varepsilon^{-\frac{2}{3}} )$ gradient complexity to achieve $\epsilon$-accuracy in 2-Wasserstein distance. Moreover, our HMC methods with biased gradient estimators, such as SARAH and SARGE, require $\tilde{O}(N+\sqrt{N} \kappa^2 d^{\frac{1}{2}} \varepsilon^{-1})$ gradient complexity, which has the same dependency on condition number $\kappa$ and dimension $d$ as full gradient method, but improves the dependency of sample size $N$ for a factor of $N^\frac{1}{2}$. Experimental results on both synthetic and real-world benchmark data show that our new framework significantly outperforms the full gradient and stochastic gradient HMC approaches. The earliest version of this paper was submitted to ICML 2020 with three weak accept but was not finally accepted.

Communication of model updates between client nodes and the central aggregating server is a major bottleneck in federated learning, especially in bandwidth-limited settings and high-dimensional models. Gradient quantization is an effective way of reducing the number of bits required to communicate each model update, albeit at the cost of having a higher error floor due to the higher variance of the stochastic gradients. In this work, we propose an adaptive quantization strategy called AdaQuantFL that aims to achieve communication efficiency as well as a low error floor by changing the number of quantization levels during the course of training. Experiments on training deep neural networks show that our method can converge in much fewer communicated bits as compared to fixed quantization level setups, with little or no impact on training and test accuracy.

The recent outbreak of the COVID-19 shocked humanity leading to the death of millions of people worldwide. To stave off the spread of the virus, the authorities in the US, employed different strategies including the mask mandate (MM) order issued by the states' governors. Although most of the previous studies pointed in the direction that MM can be effective in hindering the spread of viral infections, the effectiveness of MM in reducing the degree of exposure to the virus and, consequently, death rates remains indeterminate. Indeed, the extent to which the degree of exposure to COVID-19 takes part in the lethality of the virus remains unclear. In the current work, we defined a parameter called the average death ratio as the monthly average of the ratio of the number of daily deaths to the total number of daily cases. We utilized survey data provided by New York Times to quantify people's abidance to the MM order. Additionally, we implicitly addressed the extent to which people abide by the MM order that may depend on some parameters like population, income, and political inclination. Using different machine learning classification algorithms we investigated how the decrease or increase in death ratio for the counties in the US West Coast correlates with the input parameters. Our results showed a promising score as high as 0.94 with algorithms like XGBoost, Random Forest, and Naive Bayes. To verify the model, the best performing algorithms were then utilized to analyze other states (Arizona, New Jersey, New York and Texas) as test cases. The findings show an acceptable trend, further confirming usability of the chosen features for prediction of similar cases.

We propose a new framework, inspired by random matrix theory, for analyzing the dynamics of stochastic gradient descent (SGD) when both number of samples and dimensions are large. This framework applies to any fixed stepsize and the finite sum setting. Using this new framework, we show that the dynamics of SGD on a least squares problem with random data become deterministic in the large sample and dimensional limit. Furthermore, the limiting dynamics are governed by a Volterra integral equation. This model predicts that SGD undergoes a phase transition at an explicitly given critical stepsize that ultimately affects its convergence rate, which we also verify experimentally. Finally, when input data is isotropic, we provide explicit expressions for the dynamics and average-case convergence rates (i.e., the complexity of an algorithm averaged over all possible inputs). These rates show significant improvement over the worst-case complexities.

Moderate-size quantum computers are now publicly accessible over the cloud, opening the exciting possibility of performing dynamical simulations of quantum systems. However, while rapidly improving, these devices have short coherence times, limiting the depth of algorithms that may be successfully implemented. Here we demonstrate that, despite these limitations, it is possible to implement long-time, high fidelity simulations on current hardware. Specifically, we simulate an XY-model spin chain on the Rigetti and IBM quantum computers, maintaining a fidelity of at least 0.9 for over 600 time steps. This is a factor of 150 longer than is possible using the iterated Trotter method. Our simulations are performed using a new algorithm that we call the fixed state Variational Fast Forwarding (fsVFF) algorithm. This algorithm decreases the circuit depth and width required for a quantum simulation by finding an approximate diagonalization of a short time evolution unitary. Crucially, fsVFF only requires finding a diagonalization on the subspace spanned by the initial state, rather than on the total Hilbert space as with previous methods, substantially reducing the required resources.

A recently introduced technique for a sparse optimization problem called "safe screening" allows us to identify irrelevant variables in the early stage of optimization. In this paper, we first propose a flexible framework for safe screening based on the Fenchel-Rockafellar duality and then derive a strong safe screening rule for norm-regularized least squares by the framework. We call the proposed screening rule for norm-regularized least squares "dynamic Sasvi" because it can be interpreted as a generalization of Sasvi. Unlike the original Sasvi, it does not require the exact solution of a more strongly regularized problem; hence, it works safely in practice. We show that our screening rule can eliminate more features and increase the speed of the solver in comparison with other screening rules both theoretically and experimentally.

Clustering is a fundamental unsupervised learning task used for various applications, such as anomaly detection (Leung and Leckie, 2005), recommender systems (Shepitsen et al., 2008) and cancer diagnosis (Zheng et al., 2014). In recent years, research on sequential clustering has been actively studied, motivated by applications in which data arrives sequentially, such as online recommender systems (Nasraoui et al., 2007) and online community detection (Aggarwal, 2003). In this work, we study k-median clustering in the sequential no-substitution setting, a term first introduced in Hess and Sabato (2020). In this setting, a stream of data points is sequentially observed, and some of these points are selected by the algorithm as cluster centers. However, a point can be selected as a center only immediately after it is observed, before observing the next point. In addition, a selected center cannot be substituted later. This setting is motivated by applications in which center selection is mapped to a real-world irreversible action, such as providing users with promotional gifts or recruiting participants to a clinical trial. The goal in the no-substitution k-median setting is to obtain a near-optimal k-median risk value, while selecting a number of centers that is as close as possible to k.

We study infinite-horizon discounted two-player zero-sum Markov games, and develop a decentralized algorithm that provably converges to the set of Nash equilibria under self-play. Our algorithm is based on running an Optimistic Gradient Descent Ascent algorithm on each state to learn the policies, with a critic that slowly learns the value of each state. To the best of our knowledge, this is the first algorithm in this setting that is simultaneously rational (converging to the opponent's best response when it uses a stationary policy), convergent (converging to the set of Nash equilibria under self-play), agnostic (no need to know the actions played by the opponent), symmetric (players taking symmetric roles in the algorithm), and enjoying a finite-time last-iterate convergence guarantee, all of which are desirable properties of decentralized algorithms.

There has recently been significant interest in training reinforcement learning (RL) agents in vision-based environments. This poses many challenges, such as high dimensionality and potential for observational overfitting through spurious correlations. A promising approach to solve both of these problems is a self-attention bottleneck, which provides a simple and effective framework for learning high performing policies, even in the presence of distractions. However, due to poor scalability of attention architectures, these methods do not scale beyond low resolution visual inputs, using large patches (thus small attention matrices). In this paper we make use of new efficient attention algorithms, recently shown to be highly effective for Transformers, and demonstrate that these new techniques can be applied in the RL setting. This allows our attention-based controllers to scale to larger visual inputs, and facilitate the use of smaller patches, even individual pixels, improving generalization. In addition, we propose a new efficient algorithm approximating softmax attention with what we call hybrid random features, leveraging the theory of angular kernels. We show theoretically and empirically that hybrid random features is a promising approach when using attention for vision-based RL.

We present a novel approach for time series classification where we represent time series data as plot images and feed them to a simple CNN, outperforming several state-of-the-art methods. We propose a simple and highly replicable way of plotting the time series, and feed these images as input to a non-optimized shallow CNN, without any normalization or residual connections. These representations are no more than default line plots using the time series data, where the only pre-processing applied is to reduce the number of white pixels in the image. We compare our method with different state-of-the-art methods specialized in time series classification on two real-world non public datasets, as well as 98 datasets of the UCR dataset collection. The results show that our approach is very promising, achieving the best results on both real-world datasets and matching / beating the best state-of-the-art methods in six UCR datasets. We argue that, if a simple naive design like ours can obtain such good results, it is worth further exploring the capabilities of using image representation of time series data, along with more powerful CNNs, for classification and other related tasks.