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We study the fundamental problem of sampling independent events, called subset sampling. Specifically, consider a set of $n$ events $S=\{x_1, \ldots, x_n\}$, where each event $x_i$ has an associated probability $p(x_i)$. The subset sampling problem aims to sample a subset $T \subseteq S$, such that every $x_i$ is independently included in $S$ with probability $p_i$. A naive solution is to flip a coin for each event, which takes $O(n)$ time. However, the specific goal is to develop data structures that allow drawing a sample in time proportional to the expected output size $\mu=\sum_{i=1}^n p(x_i)$, which can be significantly smaller than $n$ in many applications. The subset sampling problem serves as an important building block in many tasks and has been the subject of various research for more than a decade. However, most of the existing subset sampling approaches are conducted in a static setting, where the events or their associated probability in set $S$ is not allowed to be changed over time. These algorithms incur either large query time or update time in a dynamic setting despite the ubiquitous time-evolving events with changing probability in real life. Therefore, it is a pressing need, but still, an open problem, to design efficient dynamic subset sampling algorithms. In this paper, we propose ODSS, the first optimal dynamic subset sampling algorithm. The expected query time and update time of ODSS are both optimal, matching the lower bounds of the subset sampling problem. We present a nontrivial theoretical analysis to demonstrate the superiority of ODSS. We also conduct comprehensive experiments to empirically evaluate the performance of ODSS. Moreover, we apply ODSS to a concrete application: influence maximization. We empirically show that our ODSS can improve the complexities of existing influence maximization algorithms on large real-world evolving social networks.

Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic optimization problems, at least when minibatching with a sufficiently large batch size. The algorithm we study can be interpreted as an accelerated randomized Kaczmarz algorithm with minibatching and heavy ball momentum. The analysis relies on carefully decomposing the momentum transition matrix, and using new spectral norm concentration bounds for products of independent random matrices. We provide numerical illustrations demonstrating that our bounds are reasonably sharp.

We investigate the generalization error of statistical learning models in a Federated Learning (FL) setting. Specifically, we study the evolution of the generalization error with the number of communication rounds between the clients and the parameter server, i.e., the effect on the generalization error of how often the local models as computed by the clients are aggregated at the parameter server. We establish PAC-Bayes and rate-distortion theoretic bounds on the generalization error that account explicitly for the effect of the number of rounds, say $ R \in \mathbb{N}$, in addition to the number of participating devices $K$ and individual datasets size $n$. The bounds, which apply in their generality for a large class of loss functions and learning algorithms, appear to be the first of their kind for the FL setting. Furthermore, we apply our bounds to FL-type Support Vector Machines (FSVM); and we derive (more) explicit bounds on the generalization error in this case. In particular, we show that the generalization error of FSVM increases with $R$, suggesting that more frequent communication with the parameter server diminishes the generalization power of such learning algorithms. Combined with that the empirical risk generally decreases for larger values of $R$, this indicates that $R$ might be a parameter to optimize in order to minimize the population risk of FL algorithms. Moreover, specialized to the case $R=1$ (sometimes referred to as "one-shot" FL or distributed learning) our bounds suggest that the generalization error of the FL setting decreases faster than that of centralized learning by a factor of $\mathcal{O}(\sqrt{\log(K)/K})$, thereby generalizing recent findings in this direction to arbitrary loss functions and algorithms. The results of this paper are also validated on some experiments.

Simulation-based inference (SBI) methods such as approximate Bayesian computation (ABC), synthetic likelihood, and neural posterior estimation (NPE) rely on simulating statistics to infer parameters of intractable likelihood models. However, such methods are known to yield untrustworthy and misleading inference outcomes under model misspecification, thus hindering their widespread applicability. In this work, we propose the first general approach to handle model misspecification that works across different classes of SBI methods. Leveraging the fact that the choice of statistics determines the degree of misspecification in SBI, we introduce a regularized loss function that penalises those statistics that increase the mismatch between the data and the model. Taking NPE and ABC as use cases, we demonstrate the superior performance of our method on high-dimensional time-series models that are artificially misspecified. We also apply our method to real data from the field of radio propagation where the model is known to be misspecified. We show empirically that the method yields robust inference in misspecified scenarios, whilst still being accurate when the model is well-specified.

Internet-of-Things (IoT) networks are expected to support the wireless connection of massive energy limited IoT nodes. The emerging wireless powered backscatter communications (WPBC) enable IoT nodes to harvest energy from the incident radio frequency signals transmitted by a power beacon (PB) to support their circuit operation, but the energy consumption of the PB (a potentially high cost borne by the network operator) has not been sufficiently studied for WPBC. In this paper, we aim to minimize the energy consumption of the PB while satisfying the throughput requirement per IoT node by jointly optimizing the time division multiple access (TDMA) time slot duration and backscatter reflection coefficient of each IoT node and the PB transmit power per time slot. As the formulated joint optimization problem is non-convex, we transform it into a convex problem by using auxiliary variables, then employ the Lagrange dual method to obtain the optimal solutions. To reduce the implementation complexity required for adjusting the PB's transmit power every time slot, we keep the PB transmit power constant in each time block and solve the corresponding PB energy consumption minimization problem by using auxiliary variables, the block coordinated decent method and the successive convex approximation technique. Based on the above solutions, two iterative algorithms are proposed for the dynamic PB transmit power scheme and the static PB transmit power scheme. The simulation results show that the dynamic PB transmit power scheme and the static PB transmit power scheme both achieve a lower PB energy consumption than the benchmark schemes, and the former achieves the lowest PB energy consumption.

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.

Private Information Retrieval (PIR) schemes allow a client to retrieve any file of interest, while hiding the file identity from the database servers. In contrast to most existing PIR schemes that assume honest-but-curious servers, we study the case of dishonest servers. The latter provide incorrect answers and try to persuade the client to output the wrong result. We introduce several PIR schemes with information-theoretic privacy and result verification for the case of two servers. Security guarantees can be information-theoretical or computational, and the verification keys can be public or private. In this work, our main performance metric is the download rate.

In many industrial applications, obtaining labeled observations is not straightforward as it often requires the intervention of human experts or the use of expensive testing equipment. In these circumstances, active learning can be highly beneficial in suggesting the most informative data points to be used when fitting a model. Reducing the number of observations needed for model development alleviates both the computational burden required for training and the operational expenses related to labeling. Online active learning, in particular, is useful in high-volume production processes where the decision about the acquisition of the label for a data point needs to be taken within an extremely short time frame. However, despite the recent efforts to develop online active learning strategies, the behavior of these methods in the presence of outliers has not been thoroughly examined. In this work, we investigate the performance of online active linear regression in contaminated data streams. Our study shows that the currently available query strategies are prone to sample outliers, whose inclusion in the training set eventually degrades the predictive performance of the models. To address this issue, we propose a solution that bounds the search area of a conditional D-optimal algorithm and uses a robust estimator. Our approach strikes a balance between exploring unseen regions of the input space and protecting against outliers. Through numerical simulations, we show that the proposed method is effective in improving the performance of online active learning in the presence of outliers, thus expanding the potential applications of this powerful tool.

Given a traversal algorithm, cover time is the expected number of steps needed to visit all nodes in a given graph. A smaller cover time means a higher exploration efficiency of traversal algorithm. Although random walk algorithms have been studied extensively in the existing literature, there has been no cover time result for any non-Markovian method. In this work, we stand on a theoretical perspective and show that the negative feedback strategy (a count-based exploration method) is better than the naive random walk search. In particular, the former strategy can locally improve the search efficiency for an arbitrary graph. It also achieves smaller cover times for special but important graphs, including clique graphs, tree graphs, etc. Moreover, we make connections between our results and reinforcement learning literature to give new insights on why classical UCB and MCTS algorithms are so useful. Various numerical results corroborate our theoretical findings.

Quantum data access and quantum processing can make certain classically intractable learning tasks feasible. However, quantum capabilities will only be available to a select few in the near future. Thus, reliable schemes that allow classical clients to delegate learning to untrusted quantum servers are required to facilitate widespread access to quantum learning advantages. Building on a recently introduced framework of interactive proof systems for classical machine learning, we develop a framework for classical verification of quantum learning. We exhibit learning problems that a classical learner cannot efficiently solve on their own, but that they can efficiently and reliably solve when interacting with an untrusted quantum prover. Concretely, we consider the problems of agnostic learning parities and Fourier-sparse functions with respect to distributions with uniform input marginal. We propose a new quantum data access model that we call "mixture-of-superpositions" quantum examples, based on which we give efficient quantum learning algorithms for these tasks. Moreover, we prove that agnostic quantum parity and Fourier-sparse learning can be efficiently verified by a classical verifier with only random example or statistical query access. Finally, we showcase two general scenarios in learning and verification in which quantum mixture-of-superpositions examples do not lead to sample complexity improvements over classical data. Our results demonstrate that the potential power of quantum data for learning tasks, while not unlimited, can be utilized by classical agents through interaction with untrusted quantum entities.