Learning in Quantum Control: High-Dimensional Global Optimization for Noisy Quantum Dynamics
University of Calgary（卡尔加里大学）；ICFO-The Institute of Photonic Sciences（光子科学研究所（ICFO））；Canadian Institute for Advanced Research（加拿大高等研究院）；University of Science and Technology of China（中国科学技术大学）
Pantita Palittapongarnpim, Peter Wittek, Ehsan Zahedinejad, Shakib Vedaie, Barry C. Sanders
备注：
32 pages, 4 figures, extension of proceedings in ESANN 2016 conference submitted to Neurocomputing
Quantum control is valuable for various quantum technologies such as high-fidelity gates for universal quantum computing, adaptive quantum-enhanced metrology, and ultra-cold atom manipulation. Although supervised machine learning and reinforcement learning are widely used for optimizing control parameters in classical systems, quantum control for parameter optimization is mainly pursued via gradient-based greedy algorithms. Although the quantum fitness landscape is often compatible with greedy algorithms, sometimes greedy algorithms yield poor results, especially for large-dimensional quantum systems. We employ differential evolution algorithms to circumvent the stagnation problem of non-convex optimization. We improve quantum control fidelity for noisy system by averaging over the objective function. To reduce computational cost, we introduce heuristics for early termination of runs and for adaptive selection of search subspaces. Our implementation is massively parallel and vectorized to reduce run time even further. We demonstrate our methods with two examples, namely quantum phase estimation and quantum gate design, for which we achieve superior fidelity and scalability than obtained using greedy algorithms.
UAV Trajectory Optimization via Improved Noisy Deep Q-Network
Dept. of Computer Science（计算机科学系）；Wenzhou-Kean University（温州市凯恩大学）；School of Electrical and Computer Engineering（电气与计算机工程学院）；University of Tehran（伊朗塔里班大学）；School of Computer Science and Info. Eng.（计算机科学与信息工程学院）；Qilu Institute of Technology（青岛理工学院）
Queensland University of Technology（昆士兰科技大学）；Stanford University（斯坦福大学）；Adobe Research（奥多比研究中心）；Google Research（谷歌研究中心）
We study sample complexity of optimizing "hill-climbing friendly" functions defined on a graph under noisy observations. We define a notion of convexity, and we show that a variant of best-arm identification can find a near-optimal solution after a small number of queries that is independent of the size of the graph. For functions that have local minima and are nearly convex, we show a sample complexity for the classical simulated annealing under noisy observations. We show effectiveness of the greedy algorithm with restarts and the simulated annealing on problems of graph-based nearest neighb…
Randomized experiments are the gold standard for evaluating the effects of changes to real-world systems. Data in these tests may be difficult to collect and outcomes may have high variance, resulting in potentially large measurement error. Bayesian optimization is a promising technique for efficiently optimizing multiple continuous parameters, but existing approaches degrade in performance when the noise level is high, limiting its applicability to many randomized experiments. We derive an expression for expected improvement under greedy batch optimization with noisy observations and noisy co…
Deep neural networks (DNNs) trained on large-scale datasets have exhibited significant performance in image classification. Many large-scale datasets are collected from websites, however they tend to contain inaccurate labels that are termed as noisy labels. Training on such noisy labeled datasets causes performance degradation because DNNs easily overfit to noisy labels. To overcome this problem, we propose a joint optimization framework of learning DNN parameters and estimating true labels. Our framework can correct labels during training by alternating update of network parameters and label…
Bosch Center for Artificial Intelligence（博世人工智能中心）；ETH Zürich（苏黎世联邦理工学院）
We consider the problem of robust optimization within the well-established Bayesian optimization (BO) framework. While BO is intrinsically robust to noisy evaluations of the objective function, standard approaches do not consider the case of uncertainty about the input parameters. In this paper, we propose Noisy-Input Entropy Search (NES), a novel information-theoretic acquisition function that is designed to find robust optima for problems with both input and measurement noise. NES is based on the key insight that the robust objective in many cases can be modeled as a Gaussian process, howeve…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Parallel surrogate optimization algorithms have proven to be efficient methods for solving expensive noisy optimization problems. In this work we develop a new parallel surrogate optimization algorithm (ProSRS), using a novel tree-based "zoom strategy" to improve the efficiency of the algorithm. We prove that if ProSRS is run for sufficiently long, with probability converging to one there will be at least one point among all the evaluations that will be arbitrarily close to the global minimum. We compare our algorithm to several state-of-the-art Bayesian optimization algorithms on a suite of s…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Carnegie Mellon University（卡内基梅隆大学）
We study the problem of black-box optimization of a noisy function in the presence of low-cost approximations or fidelities, which is motivated by problems like hyper-parameter tuning. In hyper-parameter tuning evaluating the black-box function at a point involves training a learning algorithm on a large data-set at a particular hyper-parameter and evaluating the validation error. Even a single such evaluation can be prohibitively expensive. Therefore, it is beneficial to use low-cost approximations, like training the learning algorithm on a sub-sampled version of the whole data-set. These low…
Several data analysis techniques employ similarity relationships between data points to uncover the intrinsic dimension and geometric structure of the underlying data-generating mechanism. In this paper we work under the model assumption that the data is made of random perturbations of feature vectors lying on a low-dimensional manifold. We study two questions: how to define the similarity relationship over noisy data points, and what is the resulting impact of the choice of similarity in the extraction of global geometric information from the underlying manifold. We provide concrete mathemati…
Many existing fairness criteria for machine learning involve equalizing some metric across protected groups such as race or gender. However, practitioners trying to audit or enforce such group-based criteria can easily face the problem of noisy or biased protected group information. First, we study the consequences of naively relying on noisy protected group labels: we provide an upper bound on the fairness violations on the true groups G when the fairness criteria are satisfied on noisy groups $\hat{G}$. Second, we introduce two new approaches using robust optimization that, unlike the naive …
This paper analyzes DONE, an online optimization algorithm that iteratively minimizes an unknown function based on costly and noisy measurements. The algorithm maintains a surrogate of the unknown function in the form of a random Fourier expansion (RFE). The surrogate is updated whenever a new measurement is available, and then used to determine the next measurement point. The algorithm is comparable to Bayesian optimization algorithms, but its computational complexity per iteration does not depend on the number of measurements. We derive several theoretical results that provide insight on how…
University of Alberta（阿尔伯塔大学）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Imperial College London（伦敦帝国理工学院）
Algorithms for bandit convex optimization and online learning often rely on constructing noisy gradient estimates, which are then used in appropriately adjusted first-order algorithms, replacing actual gradients. Depending on the properties of the function to be optimized and the nature of ``noise'' in the bandit feedback, the bias and variance of gradient estimates exhibit various tradeoffs. In this paper we propose a novel framework that replaces the specific gradient estimation methods with an abstract oracle. With the help of the new framework we unify previous works, reproducing their res…
University of California, Berkeley（加州大学伯克利分校）；Columbia University（哥伦比亚大学）；University of Michigan（密歇根大学）
Causal inference from observational datasets often relies on measuring and adjusting for covariates. In practice, measurements of the covariates can often be noisy and/or biased, or only measurements of their proxies may be available. Directly adjusting for these imperfect measurements of the covariates can lead to biased causal estimates. Moreover, without additional assumptions, the causal effects are not point-identifiable due to the noise in these measurements. To this end, we study the partial identification of causal effects given noisy covariates, under a user-specified assumption on th…
This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework…
SGD does not produce robust results on datasets with label noise. Because the gradients calculated according to the losses of the noisy samples cause the optimization process to go in the wrong direction. In this paper, as an alternative to SGD, we recommend using samples with loss less than a threshold value determined during the optimization process, instead of using all samples in the mini-batch. Our proposed method, Adaptive-k, aims to exclude label noise samples from the optimization process and make the process robust. On noisy datasets, we found that using a threshold-based approach, su…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error that can be controlled through a batch size parameter. Our proposed oracles are appealing in several practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed (i.i.d.) samples, or simulation optimization, where the function measurements are `biased' due to computational constraints. In either case, increasing the batch size reduces the estimation error. We highlight the applicability of our biased gradient oracles in a risk-s…
We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations under two different designs (i.e.$~$a sort of random Fourier design and Gaussian design). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of random noise. The current paper makes two contributions by demonstrating that: (1) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations. (2) convex relaxation also achieves minimax-optimal statistical…
In black-box optimization problems, we aim to maximize an unknown objective function, where the function is only accessible through feedbacks of an evaluation or simulation oracle. In real-life, the feedbacks of such oracles are often noisy and available after some unknown delay that may depend on the computation time of the oracle. Additionally, if the exact evaluations are expensive but coarse approximations are available at a lower cost, the feedbacks can have multi-fidelity. In order to address this problem, we propose a generic extension of hierarchical optimistic tree search (HOO), calle…
Australian Artificial Intelligence Institute, University of Technology Sydney（悉尼科技大学澳大利亚人工智能研究所）；University of Washington, Seattle（西雅图华盛顿大学）；University of Maryland, College Park（马里兰大学帕克分校）；Hong Kong Baptist University（香港浸会大学）
Federated learning (FL) aims at training a global model on the server side while the training data are collected and located at the local devices. Hence, the labels in practice are usually annotated by clients of varying expertise or criteria and thus contain different amounts of noises. Local training on noisy labels can easily result in overfitting to noisy labels, which is devastating to the global model through aggregation. Although recent robust FL methods take malicious clients into account, they have not addressed local noisy labels on each device and the impact to the global model. In …
University of Macau（澳门大学）；Hong Kong Baptist University（香港浸会大学）；Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）
Federated learning (FL) collaboratively trains a shared global model depending on multiple local clients, while keeping the training data decentralized in order to preserve data privacy. However, standard FL methods ignore the noisy client issue, which may harm the overall performance of the shared model. We first investigate critical issue caused by noisy clients in FL and quantify the negative impact of the noisy clients in terms of the representations learned by different layers. We have the following two key observations: (1) the noisy clients can severely impact the convergence and perfor…
Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）
We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has access to zeroth-order information). Optimization performance is evaluated by the expected difference of function values at the estimated optimum and the true optimum. In contrast to the classical optimization setup, first-order information like gradients are not directly accessibl…
Learning Locally, Revising Globally: Global Reviser for Federated Learning with Noisy Labels
College of Computer Science, Sichuan University, Chengdu, China（四川大学计算机学院，中国成都）；Engineering Research Center of Machine Learning（机器学习工程研究中心）；University of Sydney, Sydney, Australia（悉尼大学，澳大利亚悉尼）；Southeast University, Nanjing, China（东南大学，中国南京）
Imperial College London（帝国理工学院）；National University of Singapore（新加坡国立大学）
We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as $Ω(σ\sqrt{T / \log (T)}) \cap \mathcal{O}(σ\sqrt{T} \cdot \log T)$ and $Ω(σ/ \sqrt{T \log (T)}) \cap \mathcal{O}(σ\log T / \sqrt{T})$ respectively, where $σ^2$ is the noise variance. Thus, our upper and lower bounds are tight up to a factor of $\mathcal{O}( (\log T)^{1.5} )$. The upper bound uses an algorithm…
Optimization plays a key role in the training of deep neural networks. Deciding when to stop training can have a substantial impact on the performance of the network during inference. Under certain conditions, the generalization error can display a double descent pattern during training: the learning curve is non-monotonic and seemingly diverges before converging again after additional epochs. This optimization pattern can lead to early stopping procedures to stop training before the second convergence and consequently select a suboptimal set of parameters for the network, with worse performan…
Chung-Ang University（中央大学）；University of Copenhagen（哥本哈根大学）
Regularization is essential for avoiding over-fitting to training data in network optimization, leading to better generalization of the trained networks. The label noise provides a strong implicit regularization by replacing the target ground truth labels of training examples by uniform random labels. However, it can cause undesirable misleading gradients due to the large loss associated with incorrect labels. We propose a first-order optimization method (Label-Noised Trim-SGD) that uses the label noise with the example trimming in order to remove the outliers based on the loss. The proposed a…
We present a new synthesis algorithm to solve program synthesis over noisy datasets, i.e., data that may contain incorrect/corrupted input-output examples. Our algorithm uses an abstraction refinement based optimization process to synthesize programs which optimize the tradeoff between the loss over the noisy dataset and the complexity of the synthesized program. The algorithm uses abstractions to divide the search space of programs into subspaces by computing an abstract value that represents outputs for all programs in a subspace. The abstract value allows our algorithm to compute, for each …
Harvard University（哈佛大学）；Vector Institute for Artificial Intelligence（矢量人工智能研究所）；University of Toronto（多伦多大学）；Atinary Technologies Sàrl（阿蒂纳里技术有限公司）
Research challenges encountered across science, engineering, and economics can frequently be formulated as optimization tasks. In chemistry and materials science, recent growth in laboratory digitization and automation has sparked interest in optimization-guided autonomous discovery and closed-loop experimentation. Experiment planning strategies based on off-the-shelf optimization algorithms can be employed in fully autonomous research platforms to achieve desired experimentation goals with the minimum number of trials. However, the experiment planning strategy that is most suitable to a scien…
This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-à-vis random noise. When the rank and the …
Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this …
Robust Federated Learning against Noisy Clients via Masked Optimization
Institute of Computing Technology, Chinese Academy of Sciences & University of Chinese Academy of Sciences（计算技术研究所，中国科学院及中国科学院大学）；Department of Electronic Engineering, The Chinese University of Hong Kong（电子工程系，香港中文大学）；Institute of Computing Technology, Chinese Academy of Sciences（计算技术研究所，中国科学院）
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Google Research（谷歌研究院）；DeepMind（深度思维）
Increasing the batch size is a popular way to speed up neural network training, but beyond some critical batch size, larger batch sizes yield diminishing returns. In this work, we study how the critical batch size changes based on properties of the optimization algorithm, including acceleration and preconditioning, through two different lenses: large scale experiments, and analysis of a simple noisy quadratic model (NQM). We experimentally demonstrate that optimization algorithms that employ preconditioning, specifically Adam and K-FAC, result in much larger critical batch sizes than stochasti…
In this paper, we consider the problem of sequentially optimizing a black-box function $f$ based on noisy samples and bandit feedback. We assume that $f$ is smooth in the sense of having a bounded norm in some reproducing kernel Hilbert space (RKHS), yielding a commonly-considered non-Bayesian form of Gaussian process bandit optimization. We provide algorithm-independent lower bounds on the simple regret, measuring the suboptimality of a single point reported after $T$ rounds, and on the cumulative regret, measuring the sum of regrets over the $T$ chosen points. For the isotropic squared-expon…
University of Southern California（南加州大学）；University of California, Irvine（加州大学欧文分校）
We propose an efficient ADMM method with guarantees for high-dimensional problems. We provide explicit bounds for the sparse optimization problem and the noisy matrix decomposition problem. For sparse optimization, we establish that the modified ADMM method has an optimal convergence rate of $\mathcal{O}(s\log d/T)$, where $s$ is the sparsity level, $d$ is the data dimension and $T$ is the number of steps. This matches with the minimax lower bounds for sparse estimation. For matrix decomposition into sparse and low rank components, we provide the first guarantees for any online method, and pro…
University of Minnesota, Twin Cities（明尼苏达大学双城分校）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recent empirical work on stochastic gradient descent (SGD) applied to over-parameterized deep learning has shown that most gradient components over epochs are quite small. Inspired by such observations, we rigorously study properties of Truncated SGD (T-SGD), that truncates the majority of small gradient components to zeros. Considering non-convex optimization problems, we show that the convergence rate of T-SGD matches the order of vanilla SGD. We also establish the generalization error bound for T-SGD. Further, we propose Noisy Truncated SGD (NT-SGD), which adds Gaussian noise to the truncat…
School of Industrial and Systems Engineering at Georgia Tech（佐治亚理工学院工业与系统工程学院）
Numerous empirical evidences have corroborated the importance of noise in nonconvex optimization problems. The theory behind such empirical observations, however, is still largely unknown. This paper studies this fundamental problem through investigating the nonconvex rectangular matrix factorization problem, which has infinitely many global minima due to rotation and scaling invariance. Hence, gradient descent (GD) can converge to any optimum, depending on the initialization. In contrast, we show that a perturbed form of GD with an arbitrary initialization converges to a global optimum that i…
