233 research outputs found
Recommended from our members
Quantum meets optimization and machine learning
With the advent of the quantum era, what role the quantum computer will play in optimization and machine learning becomes a natural and salient question. The development of novel quantum computing techniques is essential to showcase the quantum advantage in these fields. At the same time, new findings in classical optimization and machine learning algorithms also have the potential to stimulate quantum computing research. In the dissertation, we explore the fascinating connections between quantum computing, optimization, and machine learning, paving the way for transformative advances in all three fields. First, on the quantum side, we present efficient quantum algorithms for fundamental problems in sampling, optimization, and quantum physics. Our results highlight the practical advantages of quantum computing in these fields. In addition, we introduce new approaches to quantum complexity theory for characterizing the quantum hardness of optimization and machine learning problems. Second, on the optimization side, we improve the efficiency of the state-of-the-art classical algorithms for solving semi-definite programming (SDP), Fourier sensing, dynamic distance estimation, etc. Our classical results are closely intertwined with quantum computing. Some of them serve as stepping stones to new quantum algorithms, while others are motivated by quantum applications or inspired by quantum techniques. Third, on the machine learning side, we develop fast classical and quantum algorithms for training over-parameterized neural networks with provable guarantees of convergence and generalization. Furthermore, we contribute to the security aspect of machine learning by theoretically investigating some potential approaches to (classically) protect private data in collaborative machine learning and to (quantumly) protect the copyright of machine learning models. Fourth, we investigate the concentration and discrepancy properties of hyperbolic polynomials and higher-order random walks, which could potentially be applied to quantum computing, optimization, and machine learning.Computer Science
Hyperbolic Concentration, Anti-concentration, and Discrepancy
Chernoff bound is a fundamental tool in theoretical computer science. It has
been extensively used in randomized algorithm design and stochastic type
analysis. Discrepancy theory, which deals with finding a bi-coloring of a set
system such that the coloring of each set is balanced, has a huge number of
applications in approximation algorithms design. Chernoff bound [Che52] implies
that a random bi-coloring of any set system with sets and elements will
have discrepancy with high probability, while the famous
result by Spencer [Spe85] shows that there exists an discrepancy
solution.
The study of hyperbolic polynomials dates back to the early 20th century when
used to solve PDEs by G{\aa}rding [G{\aa}r59]. In recent years, more
applications are found in control theory, optimization, real algebraic
geometry, and so on. In particular, the breakthrough result by Marcus,
Spielman, and Srivastava [MSS15] uses the theory of hyperbolic polynomials to
prove the Kadison-Singer conjecture [KS59], which is closely related to
discrepancy theory.
In this paper, we present a list of new results for hyperbolic polynomials:
* We show two nearly optimal hyperbolic Chernoff bounds: one for Rademacher
sum of arbitrary vectors and another for random vectors in the hyperbolic cone.
* We show a hyperbolic anti-concentration bound.
* We generalize the hyperbolic Kadison-Singer theorem [Br\"a18] for vectors
in sub-isotropic position, and prove a hyperbolic Spencer theorem for any
constant hyperbolic rank vectors.
The classical matrix Chernoff and discrepancy results are based on
determinant polynomial. To the best of our knowledge, this paper is the first
work that shows either concentration or anti-concentration results for
hyperbolic polynomials. We hope our findings provide more insights into
hyperbolic and discrepancy theories
Evaluation of Wind Turbine Operation Status Based on ACO + FAHP
Aiming at the shortcomings of the fuzzy analytic hierarchy process (FAHP) in the comprehensive evaluation of wind power
projects, such as the diffi culty of satisfying and modifying the consistency of the judgment matrix and the high computational complexity, a
fuzzy analytic hierarchy process based on ant colony optimization (ACO+FAHP) is proposed. Firstly, the proposed fuzzy analytic hierarchy
process based on ant colony optimization algorithm overcomes the disadvantages that the weight and consistency cannot be improved once
the judgment matrix is given. The comparison chart of the consistency ratio calculated according to this method shows that the consistency
ratio B, C1-C5 all have diff erent degrees of reduction. Then, in view of the fact that various qualitative indicators cannot be accurately
calculated, the wind turbine operating status evaluation model is established by using the fuzzy comprehensive evaluation method. In this
paper, the evaluation score of a certain wind farm is 0.731, which means that the operators need to carry out high-level maintenance at this
time
Training Multi-Layer Over-Parametrized Neural Network in Subquadratic Time
We consider the problem of training a multi-layer over-parametrized neural network to minimize the empirical risk induced by a loss function. In the typical setting of over-parametrization, the network width m is much larger than the data dimension d and the number of training samples n (m = poly(n,d)), which induces a prohibitive large weight matrix W ∈ ℝ^{m× m} per layer. Naively, one has to pay O(m²) time to read the weight matrix and evaluate the neural network function in both forward and backward computation. In this work, we show how to reduce the training cost per iteration. Specifically, we propose a framework that uses m² cost only in the initialization phase and achieves a truly subquadratic cost per iteration in terms of m, i.e., m^{2-Ω(1)} per iteration. Our result has implications beyond standard over-parametrization theory, as it can be viewed as designing an efficient data structure on top of a pre-trained large model to further speed up the fine-tuning process, a core procedure to deploy large language models (LLM)
Solid Solution Strengthened Fe Alloys
Iron (Fe)-based alloys (such as steel) are widely used structural materials in industry. Numerous methods have been applied to improve their mechanical properties. In this study, we used a technique know as magnetron sputtering to deposit various Fe-based binary alloy coatings to investigate the influence of solutes on solid solution hardening. Several factors contribute to the solid solution hardening of the alloys, such as composition, atomic radius, modulus, and lattice parameter. After preliminary calculations and analysis, we selected several solutes, including molybdenum (Mo), niobium (Nb), and zirconium (Zr). The compositions of solutes were varied to be 2.5, 5, 8 atomic %. Our nanoindentation hardness measurements show that among the three solid solution alloys, Fe-Zr has the highest hardness. The influences of solutes on microstructural and hardness evolution in these solid solution alloys are discussed
QED driven QAOA for network-flow optimization
We present a general framework for modifying quantum approximate optimization
algorithms (QAOA) to solve constrained network flow problems. By exploiting an
analogy between flow constraints and Gauss's law for electromagnetism, we
design lattice quantum electrodynamics (QED) inspired mixing Hamiltonians that
preserve flow constraints throughout the QAOA process. This results in an
exponential reduction in the size of the configuration space that needs to be
explored, which we show through numerical simulations, yields higher quality
approximate solutions compared to the original QAOA routine. We outline a
specific implementation for edge-disjoint path (EDP) problems related to
traffic congestion minimization, numerically analyze the effect of initial
state choice, and explore trade-offs between circuit complexity and qubit
resources via a particle-vortex duality mapping. Comparing the effect of
initial states reveals that starting with an ergodic (unbiased) superposition
of solutions yields better performance than beginning with the mixer
ground-state, suggesting a departure from the "short-cut to adiabaticity"
mechanism often used to motivate QAOA.Comment: 14 pages, 10 figure
- …