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1. Coreset Clustering on Small Quantum Computers

   https://doi.org/10.3390/electronics10141690  

   Tomesh, Teague; Gokhale, Pranav; Anschuetz, Eric R.; Chong, Frederic T. (July 2021, Electronics)

   null (Ed.)

   Many quantum algorithms for machine learning require access to classical data in superposition. However, for many natural data sets and algorithms, the overhead required to load the data set in superposition can erase any potential quantum speedup over classical algorithms. Recent work by Harrow introduces a new paradigm in hybrid quantum-classical computing to address this issue, relying on coresets to minimize the data loading overhead of quantum algorithms. We investigated using this paradigm to perform k-means clustering on near-term quantum computers, by casting it as a QAOA optimization instance over a small coreset. We used numerical simulations to compare the performance of this approach to classical k-means clustering. We were able to find data sets with which coresets work well relative to random sampling and where QAOA could potentially outperform standard k-means on a coreset. However, finding data sets where both coresets and QAOA work well—which is necessary for a quantum advantage over k-means on the entire data set—appears to be challenging. 

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2. Bilevel Coreset Selection in Continual Learning: A New Formulation and Algorithm

   Hao, J; Ji, K; Liu, M (February 2024, Advances in Neural Information Processing Systems)

   Coreset is a small set that provides a data summary for a large dataset, such that training solely on the small set achieves competitive performance compared with a large dataset. In rehearsal-based continual learning, the coreset is typically used in the memory replay buffer to stand for representative samples in previous tasks, and the coreset selection procedure is typically formulated as a bilevel problem. However, the typical bilevel formulation for coreset selection explicitly performs optimization over discrete decision variables with greedy search, which is computationally expensive. Several works consider other formulations to address this issue, but they ignore the nested nature of bilevel optimization problems and may not solve the bilevel coreset selection problem accurately. To address these issues, we propose a new bilevel formulation, where the inner problem tries to find a model which minimizes the expected training error sampled from a given probability distribution, and the outer problem aims to learn the probability distribution with approximately $$K$$ (coreset size) nonzero entries such that learned model in the inner problem minimizes the training error over the whole data. To ensure the learned probability has approximately $$K$$ nonzero entries, we introduce a novel regularizer based on the smoothed top-$$K$$ loss in the upper problem. We design a new optimization algorithm that provably converges to the $$\epsilon$$-stationary point with $$O(1/\epsilon^4)$$ computational complexity. We conduct extensive experiments in various settings in continual learning, including balanced data, imbalanced data, and label noise, to show that our proposed formulation and new algorithm significantly outperform competitive baselines. From bilevel optimization point of view, our algorithm significantly improves the vanilla greedy coreset selection method in terms of running time on continual learning benchmark datasets. The code is available at \url{https://github.com/MingruiLiu-ML-Lab/Bilevel-Coreset-Selection-via-Regularization}. 

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3. Provably efficient machine learning for quantum many-body problems

   https://doi.org/10.1126/science.abk3333  

   Huang, Hsin-Yuan; Kueng, Richard; Torlai, Giacomo; Albert, Victor V.; Preskill, John (September 2022, Science)

   INTRODUCTION Solving quantum many-body problems, such as finding ground states of quantum systems, has far-reaching consequences for physics, materials science, and chemistry. Classical computers have facilitated many profound advances in science and technology, but they often struggle to solve such problems. Scalable, fault-tolerant quantum computers will be able to solve a broad array of quantum problems but are unlikely to be available for years to come. Meanwhile, how can we best exploit our powerful classical computers to advance our understanding of complex quantum systems? Recently, classical machine learning (ML) techniques have been adapted to investigate problems in quantum many-body physics. So far, these approaches are mostly heuristic, reflecting the general paucity of rigorous theory in ML. Although they have been shown to be effective in some intermediate-size experiments, these methods are generally not backed by convincing theoretical arguments to ensure good performance. RATIONALE A central question is whether classical ML algorithms can provably outperform non-ML algorithms in challenging quantum many-body problems. We provide a concrete answer by devising and analyzing classical ML algorithms for predicting the properties of ground states of quantum systems. We prove that these ML algorithms can efficiently and accurately predict ground-state properties of gapped local Hamiltonians, after learning from data obtained by measuring other ground states in the same quantum phase of matter. Furthermore, under a widely accepted complexity-theoretic conjecture, we prove that no efficient classical algorithm that does not learn from data can achieve the same prediction guarantee. By generalizing from experimental data, ML algorithms can solve quantum many-body problems that could not be solved efficiently without access to experimental data. RESULTS We consider a family of gapped local quantum Hamiltonians, where the Hamiltonian H ( x ) depends smoothly on m parameters (denoted by x ). The ML algorithm learns from a set of training data consisting of sampled values of x , each accompanied by a classical representation of the ground state of H ( x ). These training data could be obtained from either classical simulations or quantum experiments. During the prediction phase, the ML algorithm predicts a classical representation of ground states for Hamiltonians different from those in the training data; ground-state properties can then be estimated using the predicted classical representation. Specifically, our classical ML algorithm predicts expectation values of products of local observables in the ground state, with a small error when averaged over the value of x . The run time of the algorithm and the amount of training data required both scale polynomially in m and linearly in the size of the quantum system. Our proof of this result builds on recent developments in quantum information theory, computational learning theory, and condensed matter theory. Furthermore, under the widely accepted conjecture that nondeterministic polynomial-time (NP)–complete problems cannot be solved in randomized polynomial time, we prove that no polynomial-time classical algorithm that does not learn from data can match the prediction performance achieved by the ML algorithm. In a related contribution using similar proof techniques, we show that classical ML algorithms can efficiently learn how to classify quantum phases of matter. In this scenario, the training data consist of classical representations of quantum states, where each state carries a label indicating whether it belongs to phase A or phase B . The ML algorithm then predicts the phase label for quantum states that were not encountered during training. The classical ML algorithm not only classifies phases accurately, but also constructs an explicit classifying function. Numerical experiments verify that our proposed ML algorithms work well in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases. CONCLUSION We have rigorously established that classical ML algorithms, informed by data collected in physical experiments, can effectively address some quantum many-body problems. These rigorous results boost our hopes that classical ML trained on experimental data can solve practical problems in chemistry and materials science that would be too hard to solve using classical processing alone. Our arguments build on the concept of a succinct classical representation of quantum states derived from randomized Pauli measurements. Although some quantum devices lack the local control needed to perform such measurements, we expect that other classical representations could be exploited by classical ML with similarly powerful results. How can we make use of accessible measurement data to predict properties reliably? Answering such questions will expand the reach of near-term quantum platforms. Classical algorithms for quantum many-body problems. Classical ML algorithms learn from training data, obtained from either classical simulations or quantum experiments. Then, the ML algorithm produces a classical representation for the ground state of a physical system that was not encountered during training. Classical algorithms that do not learn from data may require substantially longer computation time to achieve the same task. 

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4. Efficient Algorithms and Implementation of a Semiparametric Joint Model for Longitudinal and Competing Risk Data: With Applications to Massive Biobank Data

   https://doi.org/10.1155/2022/1362913  

   Li, Shanpeng; Li, Ning; Wang, Hong; Zhou, Jin; Zhou, Hua; Li, Gang (February 2022, Computational and Mathematical Methods in Medicine)

   Maex, Reinoud (Ed.)

   Semiparametric joint models of longitudinal and competing risk data are computationally costly, and their current implementations do not scale well to massive biobank data. This paper identifies and addresses some key computational barriers in a semiparametric joint model for longitudinal and competing risk survival data. By developing and implementing customized linear scan algorithms, we reduce the computational complexities from O n 2 or O n 3 to O n in various steps including numerical integration, risk set calculation, and standard error estimation, where n is the number of subjects. Using both simulated and real-world biobank data, we demonstrate that these linear scan algorithms can speed up the existing methods by a factor of up to hundreds of thousands when n > 1 0 4 , often reducing the runtime from days to minutes. We have developed an R package, FastJM, based on the proposed algorithms for joint modeling of longitudinal and competing risk time-to-event data and made it publicly available on the Comprehensive R Archive Network (CRAN). 

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5. Faster Algorithms for Fair Max-Min Diversification in R ^d

   https://doi.org/10.1145/3654940  

   Kurkure, Yash; Shamo, Miles; Wiseman, Joseph; Galhotra, Sainyam; Sintos, Stavros (May 2024, Proceedings of the ACM on Management of Data)

   The task of extracting a diverse subset from a dataset, often referred to as maximum diversification, plays a pivotal role in various real-world applications that have far-reaching consequences. In this work, we delve into the realm of fairness-aware data subset selection, specifically focusing on the problem of selecting a diverse set of size k from a large collection of n data points (FairDiv). The FairDiv problem is well-studied in the data management and theory community. In this work, we develop the first constant approximation algorithm for FairDiv that runs in near-linear time using only linear space. In contrast, all previously known constant approximation algorithms run in super-linear time (with respect to n or k) and use super-linear space. Our approach achieves this efficiency by employing a novel combination of the Multiplicative Weight Update method and advanced geometric data structures to implicitly and approximately solve a linear program. Furthermore, we improve the efficiency of our techniques by constructing a coreset. Using our coreset, we also propose the first efficient streaming algorithm for the FairDiv problem whose efficiency does not depend on the distribution of data points. Empirical evaluation on million-sized datasets demonstrates that our algorithm achieves the best diversity within a minute. All prior techniques are either highly inefficient or do not generate a good solution. 

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