Sketching is one of the most fundamental tools in large-scale machine learning. It enables runtime and memory saving via randomly compressing the original large problem into lower dimensions. In this paper, we propose a novel sketching scheme for the first order method in large-scale distributed learning setting, such that the communication costs between distributed agents are saved while the convergence of the algorithms is still guaranteed. Given gradient information in a high dimension $d$, the agent passes the compressed information processed by a sketching matrix $R\in \mathbb{R}^{s\times d}$ with $s\ll d$, and the receiver de-compressed via the de-sketching matrix $R^\top$ to ``recover'' the information in original dimension. Using such a framework, we develop algorithms for federated learning with lower communication costs. However, such random sketching does not protect the privacy of local data directly. We show that the gradient leakage problem still exists after applying the sketching technique by presenting a specific gradient attack method. As a remedy, we prove rigorously that the algorithm will be differentially private by adding additional random noises in gradient information, which results in a both communication-efficient and differentially private first order approach for federated learning tasks. Our sketching scheme can be further generalized to other learning settings and might be of independent interest itself.
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We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb{R}^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb{R}^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting.
Much like the classical Fisher linear discriminant analysis (LDA), the recently proposed Wasserstein discriminant analysis (WDA) is a linear dimensionality reduction method that seeks a projection matrix to maximize the dispersion of different data classes and minimize the dispersion of same data classes via a bi-level optimization. In contrast to LDA, WDA can account for both global and local interconnections between data classes by using the underlying principles of optimal transport. In this paper, a bi-level nonlinear eigenvector algorithm (WDA-nepv) is presented to fully exploit the structures of the bi-level optimization of WDA. The inner level of WDA-nepv for computing the optimal transport matrices is formulated as an eigenvector-dependent nonlinear eigenvalue problem (NEPv), and meanwhile, the outer level for trace ratio optimizations is formulated as another NEPv. Both NEPvs can be computed efficiently under the self-consistent field (SCF) framework. WDA-nepv is derivative-free and surrogate-model-free when compared with existing algorithms. Convergence analysis of the proposed WDA-nepv justifies the utilization of the SCF for solving the bi-level optimization of WDA. Numerical experiments with synthetic and real-life datasets demonstrate the classification accuracy and scalability of WDA-nepv.
Motivated by Carbon Emissions Trading Schemes, Treasury Auctions, and Procurement Auctions, which all involve the auctioning of homogeneous multiple units, we consider the problem of learning how to bid in repeated multi-unit pay-as-bid auctions. In each of these auctions, a large number of (identical) items are to be allocated to the largest submitted bids, where the price of each of the winning bids is equal to the bid itself. The problem of learning how to bid in pay-as-bid auctions is challenging due to the combinatorial nature of the action space. We overcome this challenge by focusing on the offline setting, where the bidder optimizes their vector of bids while only having access to the past submitted bids by other bidders. We show that the optimal solution to the offline problem can be obtained using a polynomial time dynamic programming (DP) scheme. We leverage the structure of the DP scheme to design online learning algorithms with polynomial time and space complexity under full information and bandit feedback settings. We achieve an upper bound on regret of $O(M\sqrt{T\log |\mathcal{B}|})$ and $O(M\sqrt{|\mathcal{B}|T\log |\mathcal{B}|})$ respectively, where $M$ is the number of units demanded by the bidder, $T$ is the total number of auctions, and $|\mathcal{B}|$ is the size of the discretized bid space. We accompany these results with a regret lower bound, which match the linear dependency in $M$. Our numerical results suggest that when all agents behave according to our proposed no regret learning algorithms, the resulting market dynamics mainly converge to a welfare maximizing equilibrium where bidders submit uniform bids. Lastly, our experiments demonstrate that the pay-as-bid auction consistently generates significantly higher revenue compared to its popular alternative, the uniform price auction.
Algorithmic Gaussianization is a phenomenon that can arise when using randomized sketching or sampling methods to produce smaller representations of large datasets: For certain tasks, these sketched representations have been observed to exhibit many robust performance characteristics that are known to occur when a data sample comes from a sub-gaussian random design, which is a powerful statistical model of data distributions. However, this phenomenon has only been studied for specific tasks and metrics, or by relying on computationally expensive methods. We address this by providing an algorithmic framework for gaussianizing data distributions via averaging, proving that it is possible to efficiently construct data sketches that are nearly indistinguishable (in terms of total variation distance) from sub-gaussian random designs. In particular, relying on a recently introduced sketching technique called Leverage Score Sparsified (LESS) embeddings, we show that one can construct an $n\times d$ sketch of an $N\times d$ matrix $A$, where $n\ll N$, that is nearly indistinguishable from a sub-gaussian design, in time $O(\text{nnz}(A)\log N + nd^2)$, where $\text{nnz}(A)$ is the number of non-zero entries in $A$. As a consequence, strong statistical guarantees and precise asymptotics available for the estimators produced from sub-gaussian designs (e.g., for least squares and Lasso regression, covariance estimation, low-rank approximation, etc.) can be straightforwardly adapted to our sketching framework. We illustrate this with a new approximation guarantee for sketched least squares, among other examples.
The Weighted Path Order of Yamada is a powerful technique for proving termination. It is also supported by CeTA, a certifier for checking untrusted termination proofs. To be more precise, CeTA contains a verified function that computes for two terms whether one of them is larger than the other for a given WPO, i.e., where all parameters of the WPO have been fixed. The problem of this verified function is its exponential runtime in the worst case. Therefore, in this work we develop a polynomial time implementation of WPO that is based on memoization. It also improves upon an earlier verified implementation of the Recursive Path Order: the RPO-implementation uses full terms as keys for the memory, a design which simplified the soundness proofs, but has some runtime overhead. In this work, keys are just numbers, so that the lookup in the memory is faster. Although trivial on paper, this change introduces some challenges for the verification task.
Quantile regression is increasingly encountered in modern big data applications due to its robustness and flexibility. We consider the scenario of learning the conditional quantiles of a specific target population when the available data may go beyond the target and be supplemented from other sources that possibly share similarities with the target. A crucial question is how to properly distinguish and utilize useful information from other sources to improve the quantile estimation and inference at the target. We develop transfer learning methods for high-dimensional quantile regression by detecting informative sources whose models are similar to the target and utilizing them to improve the target model. We show that under reasonable conditions, the detection of the informative sources based on sample splitting is consistent. Compared to the naive estimator with only the target data, the transfer learning estimator achieves a much lower error rate as a function of the sample sizes, the signal-to-noise ratios, and the similarity measures among the target and the source models. Extensive simulation studies demonstrate the superiority of our proposed approach. We apply our methods to tackle the problem of detecting hard-landing risk for flight safety and show the benefits and insights gained from transfer learning of three different types of airplanes: Boeing 737, Airbus A320, and Airbus A380.
We consider the vulnerability of fairness-constrained learning to small amounts of malicious noise in the training data. Konstantinov and Lampert (2021) initiated the study of this question and presented negative results showing there exist data distributions where for several fairness constraints, any proper learner will exhibit high vulnerability when group sizes are imbalanced. Here, we present a more optimistic view, showing that if we allow randomized classifiers, then the landscape is much more nuanced. For example, for Demographic Parity we show we can incur only a $\Theta(\alpha)$ loss in accuracy, where $\alpha$ is the malicious noise rate, matching the best possible even without fairness constraints. For Equal Opportunity, we show we can incur an $O(\sqrt{\alpha})$ loss, and give a matching $\Omega(\sqrt{\alpha})$lower bound. In contrast, Konstantinov and Lampert (2021) showed for proper learners the loss in accuracy for both notions is $\Omega(1)$. The key technical novelty of our work is how randomization can bypass simple "tricks" an adversary can use to amplify his power. We also consider additional fairness notions including Equalized Odds and Calibration. For these fairness notions, the excess accuracy clusters into three natural regimes $O(\alpha)$,$O(\sqrt{\alpha})$ and $O(1)$. These results provide a more fine-grained view of the sensitivity of fairness-constrained learning to adversarial noise in training data.
We study the consistent k-center clustering problem. In this problem, the goal is to maintain a constant factor approximate $k$-center solution during a sequence of $n$ point insertions and deletions while minimizing the recourse, i.e., the number of changes made to the set of centers after each point insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that in the incremental setting, where deletions are not allowed, one can obtain $k \cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and $k$-median, and demonstrated a matching lower bound. However, no algorithm for the fully dynamic setting achieves less than the trivial $O(k)$ changes per update, which can be obtained by simply reclustering the full dataset after every update. In this work, we give the first algorithm for consistent $k$-center clustering for the fully dynamic setting, i.e., when both point insertions and deletions are allowed, and improves upon a trivial $O(k)$ recourse bound. Specifically, our algorithm maintains a constant factor approximate solution while ensuring worst-case constant recourse per update, which is optimal in the fully dynamic setting. Moreover, our algorithm is deterministic and is therefore correct even if an adaptive adversary chooses the insertions and deletions.
As soon as abstract mathematical computations were adapted to computation on digital computers, the problem of efficient representation, manipulation, and communication of the numerical values in those computations arose. Strongly related to the problem of numerical representation is the problem of quantization: in what manner should a set of continuous real-valued numbers be distributed over a fixed discrete set of numbers to minimize the number of bits required and also to maximize the accuracy of the attendant computations? This perennial problem of quantization is particularly relevant whenever memory and/or computational resources are severely restricted, and it has come to the forefront in recent years due to the remarkable performance of Neural Network models in computer vision, natural language processing, and related areas. Moving from floating-point representations to low-precision fixed integer values represented in four bits or less holds the potential to reduce the memory footprint and latency by a factor of 16x; and, in fact, reductions of 4x to 8x are often realized in practice in these applications. Thus, it is not surprising that quantization has emerged recently as an important and very active sub-area of research in the efficient implementation of computations associated with Neural Networks. In this article, we survey approaches to the problem of quantizing the numerical values in deep Neural Network computations, covering the advantages/disadvantages of current methods. With this survey and its organization, we hope to have presented a useful snapshot of the current research in quantization for Neural Networks and to have given an intelligent organization to ease the evaluation of future research in this area.
Learning from a few examples remains a key challenge in machine learning. Despite recent advances in important domains such as vision and language, the standard supervised deep learning paradigm does not offer a satisfactory solution for learning new concepts rapidly from little data. In this work, we employ ideas from metric learning based on deep neural features and from recent advances that augment neural networks with external memories. Our framework learns a network that maps a small labelled support set and an unlabelled example to its label, obviating the need for fine-tuning to adapt to new class types. We then define one-shot learning problems on vision (using Omniglot, ImageNet) and language tasks. Our algorithm improves one-shot accuracy on ImageNet from 87.6% to 93.2% and from 88.0% to 93.8% on Omniglot compared to competing approaches. We also demonstrate the usefulness of the same model on language modeling by introducing a one-shot task on the Penn Treebank.
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