26 pages, 23 figures, expanded version of a paper accepted for publication in Pattern Recognition. This revision fixed minor typos in the two algorithms
Dynamic Time Warping (DTW) is a popular similarity measure for aligning and comparing time series. Due to DTW's high computation time, lower bounds are often employed to screen poor matches. Many alternative lower bounds have been proposed, providing a range of different trade-offs between tightness and computational efficiency. LB Keogh provides a useful trade-off in many applications. Two recent lower bounds, LB Improved and LB Enhanced, are substantially tighter than LB Keogh. All three have the same worst case computational complexity - linear with respect to series length and constant with respect to window size. We present four new DTW lower bounds in the same complexity class. LB Petitjean is substantially tighter than LB Improved, with only modest additional computational overhead. LB Webb is more efficient than LB Improved, while often providing a tighter bound. LB Webb is always tighter than LB Keogh. The parameter free LB Webb is usually tighter than LB Enhanced. A parameterized variant, LB Webb Enhanced, is always tighter than LB Enhanced. A further variant, LB Webb*, is useful for some constrained distance functions. In extensive experiments, LB Webb proves to be very effective for nearest neighbor search.
There has been renewed recent interest in developing effective lower bounds for Dynamic Time Warping (DTW) distance between time series. These have many applications in time series indexing, clustering, forecasting, regression and classification. One of the key time series classification algorithms, the nearest neighbor algorithm with DTW distance (NN-DTW) is very expensive to compute, due to the quadratic complexity of DTW. Lower bound search can speed up NN-DTW substantially. An effective and tight lower bound quickly prunes off unpromising nearest neighbor candidates from the search space a…
College of Equipment Management and UAV Engineering, Air Force Engineering University（空军工程大学装备管理与无人机工程学院）；Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University（西北工业大学光学影像分析与学习中心（OPTIMAL））
Dynamic time warping (DTW) is a robust similarity measure of time series. However, it does not satisfy triangular inequality and has high computational complexity, severely limiting its applications in similarity search on large-scale datasets. Usually, we resort to lower bounding distances to speed up similarity search under DTW. Unfortunately, there is still a lack of an effective lower bounding distance that can measure unequal-length time series and has desirable tightness. In the paper, we propose a novel lower bounding distance LB_Keogh+, which is a seamless combination of sequence exten…
Dynamic Time Warping (DTW) is a popular time series distance measure that aligns the points in two series with one another. These alignments support warping of the time dimension to allow for processes that unfold at differing rates. The distance is the minimum sum of costs of the resulting alignments over any allowable warping of the time dimension. The cost of an alignment of two points is a function of the difference in the values of those points. The original cost function was the absolute value of this difference. Other cost functions have been proposed. A popular alternative is the squar…
Nearest neighbor search under elastic distances is a key tool for time series analysis, supporting many applications. However, straightforward implementations of distances require $O(n^2)$ space and time complexities, preventing these applications from scaling to long series. Much work has been devoted to speeding up the NN search process, mostly with the development of lower bounds, allowing to avoid costly distance computations when a given threshold is exceeded. This threshold, provided by the similarity search process, also allows to early abandon the computation of a distance itself. Anot…
Temporal data are naturally everywhere, especially in the digital era that sees the advent of big data and internet of things. One major challenge that arises during temporal data analysis and mining is the comparison of time series or sequences, which requires to determine a proper distance or (dis)similarity measure. In this context, the Dynamic Time Warping (DTW) has enjoyed success in many domains, due to its 'temporal elasticity', a property particularly useful when matching temporal data. Unfortunately this dissimilarity measure suffers from a quadratic computational cost, which prohibit…
William G. Enloe High School（威廉·G·恩洛高中）；North Carolina State University（北卡罗来纳州立大学）
Dynamic time warping (DTW) plays an important role in analytics on time series. Despite the large body of research on speeding up univariate DTW, the method for multivariate DTW has not been improved much in the last two decades. The most popular algorithm used today is still the one developed seventeen years ago. This paper presents a solution that, as far as we know, for the first time consistently outperforms the classic multivariate DTW algorithm across dataset sizes, series lengths, data dimensions, temporal window sizes, and machines. The new solution, named TC-DTW, introduces Triangle I…
We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information…
Nanyang Technological University（南洋理工大学）；South China University of Technology（华南理工大学）
In this paper, for the purpose of data centre energy consumption monitoring and analysis, we propose to detect the running programs in a server by classifying the observed power consumption series. Time series classification problem has been extensively studied with various distance measurements developed; also recently the deep learning based sequence models have been proved to be promising. In this paper, we propose a novel distance measurement and build a time series classification algorithm hybridizing nearest neighbour and long short term memory (LSTM) neural network. More specifically, f…
Massachusetts Institute of Technology（麻省理工学院）；Microsoft Research（微软研究院）；Google Research（谷歌研究院）
We study the fundamental problem of prediction with expert advice and develop regret lower bounds for a large family of algorithms for this problem. We develop simple adversarial primitives, that lend themselves to various combinations leading to sharp lower bounds for many algorithmic families. We use these primitives to show that the classic Multiplicative Weights Algorithm (MWA) has a regret of $\sqrt{\frac{T \ln k}{2}}$, there by completely closing the gap between upper and lower bounds. We further show a regret lower bound of $\frac{2}{3}\sqrt{\frac{T\ln k}{2}}$ for a much more general fa…
Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Google Research（谷歌研究院）
We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in learning theory and generalization bounds for unbounded loss functions.
Leonard N. Stern School of Business, New York University（纽约大学伦纳德·N·斯特恩商学院）；School of Computer Science, Carnegie Mellon University（卡内基梅隆大学计算机学院）
In this short note we consider a dynamic assortment planning problem under the capacitated multinomial logit (MNL) bandit model. We prove a tight lower bound on the accumulated regret that matches existing regret upper bounds for all parameters (time horizon $T$, number of items $N$ and maximum assortment capacity $K$) up to logarithmic factors. Our results close an $O(\sqrt{K})$ gap between upper and lower regret bounds from existing works.
Technion, Israel Institute of Technology（以色列理工学院）；Nvidia Research（英伟达研究院）
The Combinatorial Multi-Armed Bandit problem is a sequential decision-making problem in which an agent selects a set of arms on each round, observes feedback for each of these arms and aims to maximize a known reward function of the arms it chose. While previous work proved regret upper bounds in this setting for general reward functions, only a few works provided matching lower bounds, all for specific reward functions. In this work, we prove regret lower bounds for combinatorial bandits that hold under mild assumptions for all smooth reward functions. We derive both problem-dependent and pro…
In the Bayesian approach to sequential decision making, exact calculation of the (subjective) utility is intractable. This extends to most special cases of interest, such as reinforcement learning problems. While utility bounds are known to exist for this problem, so far none of them were particularly tight. In this paper, we show how to efficiently calculate a lower bound, which corresponds to the utility of a near-optimal memoryless policy for the decision problem, which is generally different from both the Bayes-optimal policy and the policy which is optimal for the expected MDP under the c…
We prove a tight lower bound (up to constant factors) on the sample complexity of any non-interactive local differentially private protocol for optimizing a linear function over the simplex. This lower bound also implies a tight lower bound (again, up to constant factors) on the sample complexity of any non-interactive local differentially private protocol implementing the exponential mechanism. These results reveal that any local protocol for these problems has exponentially worse dependence on the dimension than corresponding algorithms in the central model. Previously, Kasiviswanathan et al…
Huazhong University of Science and Technology（华中科技大学）；National University of Singapore（新加坡国立大学）
Finite-sum optimization plays an important role in the area of machine learning, and hence has triggered a surge of interest in recent years. To address this optimization problem, various randomized incremental gradient methods have been proposed with guaranteed upper and lower complexity bounds for their convergence. Nonetheless, these lower bounds rely on certain conditions: deterministic optimization algorithm, or fixed probability distribution for the selection of component functions. Meanwhile, some lower bounds even do not match the upper bounds of the best known methods in certain cases…
University of Connecticut（康涅狄格大学）；IBM Research（IBM研究院）
We study the convergence of Stochastic Gradient Descent (SGD) for strongly convex objective functions. We prove for all $t$ a lower bound on the expected convergence rate after the $t$-th SGD iteration; the lower bound is over all possible sequences of diminishing step sizes. It implies that recently proposed sequences of step sizes at ICML 2018 and ICML 2019 are {\em universally} close to optimal in that the expected convergence rate after {\em each} iteration is within a factor $32$ of our lower bound. This factor is independent of dimension $d$. We offer a framework for comparing with lower…
We consider the problem of \textit{best arm identification} with a \textit{fixed budget $T$}, in the $K$-armed stochastic bandit setting, with arms distribution defined on $[0,1]$. We prove that any bandit strategy, for at least one bandit problem characterized by a complexity $H$, will misidentify the best arm with probability lower bounded by $$\exp\Big(-\frac{T}{\log(K)H}\Big),$$ where $H$ is the sum for all sub-optimal arms of the inverse of the squared gaps. Our result disproves formally the general belief - coming from results in the fixed confidence setting - that there must exist an al…
The CUR matrix decomposition is an important extension of Nyström approximation to a general matrix. It approximates any data matrix in terms of a small number of its columns and rows. In this paper we propose a novel randomized CUR algorithm with an expected relative-error bound. The proposed algorithm has the advantages over the existing relative-error CUR algorithms that it possesses tighter theoretical bound and lower time complexity, and that it can avoid maintaining the whole data matrix in main memory. Finally, experiments on several real-world datasets demonstrate significant improveme…
We present tight lower bounds on the number of kernel evaluations required to approximately solve kernel ridge regression (KRR) and kernel $k$-means clustering (KKMC) on $n$ input points. For KRR, our bound for relative error approximation to the minimizer of the objective function is $Ω(nd_{\mathrm{eff}}^λ/\varepsilon)$ where $d_{\mathrm{eff}}^λ$ is the effective statistical dimension, which is tight up to a $\log(d_{\mathrm{eff}}^λ/\varepsilon)$ factor. For KKMC, our bound for finding a $k$-clustering achieving a relative error approximation of the objective function is $Ω(nk/\varepsilon)$, …
Imperial College London（帝国理工学院）；University of Cambridge（剑桥大学）
We propose a lower bound on the log marginal likelihood of Gaussian process regression models that can be computed without matrix factorisation of the full kernel matrix. We show that approximate maximum likelihood learning of model parameters by maximising our lower bound retains many of the sparse variational approach benefits while reducing the bias introduced into parameter learning. The basis of our bound is a more careful analysis of the log-determinant term appearing in the log marginal likelihood, as well as using the method of conjugate gradients to derive tight lower bounds on the te…
Leveraging algorithmic stability to derive sharp generalization bounds is a classic and powerful approach in learning theory. Since Vapnik and Chervonenkis [1974] first formalized the idea for analyzing SVMs, it has been utilized to study many fundamental learning algorithms (e.g., $k$-nearest neighbors [Rogers and Wagner, 1978], stochastic gradient method [Hardt et al., 2016], linear regression [Maurer, 2017], etc). In a recent line of great works by Feldman and Vondrak [2018, 2019] as well as Bousquet et al. [2020b], they prove a high probability generalization upper bound of order $\tilde{\…
Technion - Israel Institute of Technology（以色列理工学院）
We consider the Max $K$-Armed Bandit problem, where a learning agent is faced with several sources (arms) of items (rewards), and interested in finding the best item overall. At each time step the agent chooses an arm, and obtains a random real valued reward. The rewards of each arm are assumed to be i.i.d., with an unknown probability distribution that generally differs among the arms. Under the PAC framework, we provide lower bounds on the sample complexity of any $(ε,δ)$-correct algorithm, and propose algorithms that attain this bound up to logarithmic factors. We compare the performance of…
We consider a standard distributed optimisation setting where $N$ machines, each holding a $d$-dimensional function $f_i$, aim to jointly minimise the sum of the functions $\sum_{i = 1}^N f_i (x)$. This problem arises naturally in large-scale distributed optimisation, where a standard solution is to apply variants of (stochastic) gradient descent. We focus on the communication complexity of this problem: our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines to solve this problem under point-to-point communic…
