arXiv subjects

Vladimir Braverman

Publications and source records attributed to Vladimir Braverman.

At least 91 records · Page 5Linked to original sources

arXiv2020

Obtaining Adjustable Regularization for Free via Iterate Averaging

Regularization for optimization is a crucial technique to avoid overfitting in machine learning. In order to obtain the best performance, we usually train a model by tuning the regularization parameters. It becomes costly, however, when a single round of training takes significant amount of time. Very recently, Neu and Rosasco show that if we run stochastic gradient descent (SGD) on linear regression problems, then by averaging the SGD iterates properly, we obtain a regularized solution. It left open whether the same phenomenon can be achieved for other optimization problems and algorithms. In this paper, we establish an averaging scheme that provably converts the iterates of SGD on an arbitrary strongly convex and smooth objective function to its regularized counterpart with an adjustable regularization parameter. Our approaches can be used for accelerated and preconditioned optimization methods as well. We further show that the same methods work empirically on more general optimization objectives including neural networks. In sum, we obtain adjustable regularization for free for a large class of optimization problems and resolve an open question raised by Neu and Rosasco.

cs.LG↗

arXiv2020

Coresets for Clustering in Excluded-minor Graphs and Beyond

Coresets are modern data-reduction tools that are widely used in data analysis to improve efficiency in terms of running time, space and communication complexity. Our main result is a fast algorithm to construct a small coreset for k-Median in (the shortest-path metric of) an excluded-minor graph. Specifically, we give the first coreset of size that depends only on $k$, $ε$ and the excluded-minor size, and our running time is quasi-linear (in the size of the input graph). The main innovation in our new algorithm is that is iterative; it first reduces the $n$ input points to roughly $O(\log n)$ reweighted points, then to $O(\log\log n)$, and so forth until the size is independent of $n$. Each step in this iterative size reduction is based on the importance sampling framework of Feldman and Langberg (STOC 2011), with a crucial adaptation that reduces the number of \emph{distinct points}, by employing a terminal embedding (where low distortion is guaranteed only for the distance from every terminal to all other points). Our terminal embedding is technically involved and relies on shortest-path separators, a standard tool in planar and excluded-minor graphs. Furthermore, our new algorithm is applicable also in Euclidean metrics, by simply using a recent terminal embedding result of Narayanan and Nelson, (STOC 2019), which extends the Johnson-Lindenstrauss Lemma. We thus obtain an efficient coreset construction in high-dimensional Euclidean spaces, thereby matching and simplifying state-of-the-art results (Sohler and Woodruff, FOCS 2018; Huang and Vishnoi, STOC 2020). In addition, we also employ terminal embedding with additive distortion to obtain small coresets in graphs with bounded highway dimension, and use applications of our coresets to obtain improved approximation schemes, e.g., an improved PTAS for planar k-Median via a new centroid set.

cs.DS↗

arXiv2020

On the Noisy Gradient Descent that Generalizes as SGD

The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and the covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a sampling noise that arises from the mini-batch sampling procedure. Moreover, the sampling noises unify two kinds of gradient regularizing noises that belong to the Gaussian class: the one using (scaled) Fisher as covariance and the one using the gradient covariance of SGD as covariance. Finally, thanks to the flexibility of choosing noise class, an algorithm is proposed to perform noisy gradient descent that generalizes well, the variant of which even benefits large batch SGD training without hurting generalization.

cs.LG↗

arXiv2020

Universal Streaming of Subset Norms

Most known algorithms in the streaming model of computation aim to approximate a single function such as an $\ell_p$-norm. In 2009, Nelson [\url{https://sublinear.info}, Open Problem 30] asked if it possible to design \emph{universal algorithms}, that simultaneously approximate multiple functions of the stream. In this paper we answer the question of Nelson for the class of \emph{subset $\ell_0$-norms} in the insertion-only frequency-vector model. Given a family of subsets $\mathcal{S}\subset 2^{[n]}$, we provide a single streaming algorithm that can $(1\pm ε)$-approximate the subset-norm for every $S\in\mathcal{S}$. Here, the subset-$\ell_p$-norm of $v\in \mathbb{R}^n$ with respect to set $S\subseteq [n]$ is the $\ell_p$-norm of vector $v_{|S}$ (which denotes restricting $v$ to $S$, by zeroing all other coordinates). Our main result is a near-tight characterization of the space complexity of every family $\mathcal{S}\subset 2^{[n]}$ of subset-$\ell_0$-norms in insertion-only streams, expressed in terms of the "heavy-hitter dimension" of $\mathcal{S}$, a new combinatorial quantity that is related to the VC-dimension of $\mathcal{S}$. In contrast, we show that the more general turnstile and sliding-window models require a much larger space usage. All these results easily extend to $\ell_1$. In addition, we design algorithms for two other subset-$\ell_p$-norm variants. These can be compared to the Priority Sampling algorithm of Duffield, Lund and Thorup [JACM 2007], which achieves additive approximation $ε\|{v}\|$ for all possible subsets ($\mathcal{S}=2^{[n]}$) in the entry-wise update model. One of our algorithms extends this algorithm to handle turnstile updates, and another one achieves multiplicative approximation given a family $\mathcal{S}$.

cs.DS↗

arXiv2020

Schatten Norms in Matrix Streams: Hello Sparsity, Goodbye Dimension

Spectral functions of large matrices contains important structural information about the underlying data, and is thus becoming increasingly important. Many times, large matrices representing real-world data are \emph{sparse} or \emph{doubly sparse} (i.e., sparse in both rows and columns), and are accessed as a \emph{stream} of updates, typically organized in \emph{row-order}. In this setting, where space (memory) is the limiting resource, all known algorithms require space that is polynomial in the dimension of the matrix, even for sparse matrices. We address this challenge by providing the first algorithms whose space requirement is \emph{independent of the matrix dimension}, assuming the matrix is doubly-sparse and presented in row-order. Our algorithms approximate the Schatten $p$-norms, which we use in turn to approximate other spectral functions, such as logarithm of the determinant, trace of matrix inverse, and Estrada index. We validate these theoretical performance bounds by numerical experiments on real-world matrices representing social networks. We further prove that multiple passes are unavoidable in this setting, and show extensions of our primary technique, including a trade-off between space requirements and number of passes.

cs.DS↗

arXiv2020

Communication-efficient distributed SGD with Sketching

Large-scale distributed training of neural networks is often limited by network bandwidth, wherein the communication time overwhelms the local computation time. Motivated by the success of sketching methods in sub-linear/streaming algorithms, we introduce Sketched SGD, an algorithm for carrying out distributed SGD by communicating sketches instead of full gradients. We show that Sketched SGD has favorable convergence rates on several classes of functions. When considering all communication -- both of gradients and of updated model weights -- Sketched SGD reduces the amount of communication required compared to other gradient compression methods from $\mathcal{O}(d)$ or $\mathcal{O}(W)$ to $\mathcal{O}(\log d)$, where $d$ is the number of model parameters and $W$ is the number of workers participating in training. We run experiments on a transformer model, an LSTM, and a residual network, demonstrating up to a 40x reduction in total communication cost with no loss in final model performance. We also show experimentally that Sketched SGD scales to at least 256 workers without increasing communication cost or degrading model performance.

cs.LG↗

arXiv2020

Data-Independent Neural Pruning via Coresets

Previous work showed empirically that large neural networks can be significantly reduced in size while preserving their accuracy. Model compression became a central research topic, as it is crucial for deployment of neural networks on devices with limited computational and memory resources. The majority of the compression methods are based on heuristics and offer no worst-case guarantees on the trade-off between the compression rate and the approximation error for an arbitrarily new sample. We propose the first efficient, data-independent neural pruning algorithm with a provable trade-off between its compression rate and the approximation error for any future test sample. Our method is based on the coreset framework, which finds a small weighted subset of points that provably approximates the original inputs. Specifically, we approximate the output of a layer of neurons by a coreset of neurons in the previous layer and discard the rest. We apply this framework in a layer-by-layer fashion from the top to the bottom. Unlike previous works, our coreset is data independent, meaning that it provably guarantees the accuracy of the function for any input $x\in \mathbb{R}^d$, including an adversarial one. We demonstrate the effectiveness of our method on popular network architectures. In particular, our coresets yield 90\% compression of the LeNet-300-100 architecture on MNIST while improving the accuracy.

cs.LG↗

arXiv2019

Six Dimensional Streaming Algorithm for Cluster Finding in N-Body Simulations

Cosmological N-body simulations are crucial for understanding how the Universe evolves. Studying large-scale distributions of matter in these simulations and comparing them to observations usually involves detecting dense clusters of particles called "halos,'' which are gravitationally bound and expected to form galaxies. However, traditional cluster finders are computationally expensive and use massive amounts of memory. Recent work by Liu et al (Liu et al. (2015)) showed the connection between cluster detection and memory-efficient streaming algorithms and presented a halo finder based on heavy hitter algorithm. Later, Ivkin et al. (Ivkin et al. (2018)) improved the scalability of suggested streaming halo finder with efficient GPU implementation. Both works map particles' positions onto a discrete grid, and therefore lose the rest of the information, such as their velocities. Therefore, two halos travelling through each other are indistinguishable in positional space, while the velocity distribution of those halos can help to identify this process which is worth further studying. In this project we analyze data from the Millennium Simulation Project (Springel et al. (2005)) to motivate the inclusion of the velocity into streaming method we introduce. We then demonstrate a use of suggested method, which allows one to find the same halos as before, while also detecting those which were indistinguishable in prior methods.

astro-ph.GA↗

arXiv2019

Memory-Efficient Performance Monitoring on Programmable Switches with Lean Algorithms

Network performance problems are notoriously difficult to diagnose. Prior profiling systems collect performance statistics by keeping information about each network flow, but maintaining per-flow state is not scalable on resource-constrained NIC and switch hardware. Instead, we propose sketch-based performance monitoring using memory that is sublinear in the number of flows. Existing sketches estimate flow monitoring metrics based on flow sizes. In contrast, performance monitoring typically requires combining information across pairs of packets, such as matching a data packet with its acknowledgment to compute a round-trip time. We define a new class of \emph{lean} algorithms that use memory sublinear in both the size of input data and the number of flows. We then introduce lean algorithms for a set of important statistics, such as identifying flows with high latency, loss, out-of-order, or retransmitted packets. We implement prototypes of our lean algorithms on a commodity programmable switch using the P4 language. Our experiments show that lean algorithms detect $\sim$82\% of top 100 problematic flows among real-world packet traces using just 40KB memory.

cs.DS↗

arXiv2019

Multiparametric Deep Learning Tissue Signatures for Muscular Dystrophy: Preliminary Results

A current clinical challenge is identifying limb girdle muscular dystrophy 2I(LGMD2I)tissue changes in the thighs, in particular, separating fat, fat-infiltrated muscle, and muscle tissue. Deep learning algorithms have the ability to learn different features by using the inherent tissue contrasts from multiparametric magnetic resonance imaging (mpMRI). To that end, we developed a novel multiparametric deep learning network (MPDL) tissue signature model based on mpMRI and applied it to LGMD2I. We demonstrate a new tissue signature model of muscular dystrophy with the MPDL algorithm segments different tissue types with excellent results.

eess.IV↗

arXiv2019

Improved Algorithms for Time Decay Streams

In the time-decay model for data streams, elements of an underlying data set arrive sequentially with the recently arrived elements being more important. A common approach for handling large data sets is to maintain a \emph{coreset}, a succinct summary of the processed data that allows approximate recovery of a predetermined query. We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for $k$-median clustering, where we provide a constant factor approximation algorithm that utilizes the online facility location algorithm. Our algorithm stores $\mathcal{O}(k\log(hΔ)+h)$ points where $h$ is the half-life of the decay function and $Δ$ is the aspect ratio of the dataset. Our techniques extend to $k$-means clustering and $M$-estimators as well.

cs.DS↗

arXiv2019

Streaming Quantiles Algorithms with Small Space and Update Time

Approximating quantiles and distributions over streaming data has been studied for roughly two decades now. Recently, Karnin, Lang, and Liberty proposed the first asymptotically optimal algorithm for doing so. This manuscript complements their theoretical result by providing a practical variants of their algorithm with improved constants. For a given sketch size, our techniques provably reduce the upper bound on the sketch error by a factor of two. These improvements are verified experimentally. Our modified quantile sketch improves the latency as well by reducing the worst case update time from $O(1/\varepsilon)$ down to $O(\log (1/\varepsilon))$. We also suggest two algorithms for weighted item streams which offer improved asymptotic update times compared to naïve extensions. Finally, we provide a specialized data structure for these sketches which reduces both their memory footprints and update times.

cs.DS↗

arXiv2019

Coresets for Ordered Weighted Clustering

We design coresets for Ordered k-Median, a generalization of classical clustering problems such as k-Median and k-Center, that offers a more flexible data analysis, like easily combining multiple objectives (e.g., to increase fairness or for Pareto optimization). Its objective function is defined via the Ordered Weighted Averaging (OWA) paradigm of Yager (1988), where data points are weighted according to a predefined weight vector, but in order of their contribution to the objective (distance from the centers). A powerful data-reduction technique, called a coreset, is to summarize a point set $X$ in $\mathbb{R}^d$ into a small (weighted) point set $X'$, such that for every set of $k$ potential centers, the objective value of the coreset $X'$ approximates that of $X$ within factor $1\pm ε$. When there are multiple objectives (weights), the above standard coreset might have limited usefulness, whereas in a \emph{simultaneous} coreset, which was introduced recently by Bachem and Lucic and Lattanzi (2018), the above approximation holds for all weights (in addition to all centers). Our main result is a construction of a simultaneous coreset of size $O_{ε, d}(k^2 \log^2 |X|)$ for Ordered k-Median. To validate the efficacy of our coreset construction we ran experiments on a real geographical data set. We find that our algorithm produces a small coreset, which translates to a massive speedup of clustering computations, while maintaining high accuracy for a range of weights.

cs.DS↗

arXiv2019

The One-Way Communication Complexity of Dynamic Time Warping Distance

We resolve the randomized one-way communication complexity of Dynamic Time Warping (DTW) distance. We show that there is an efficient one-way communication protocol using $\widetilde{O}(n/α)$ bits for the problem of computing an $α$-approximation for DTW between strings $x$ and $y$ of length $n$, and we prove a lower bound of $Ω(n / α)$ bits for the same problem. Our communication protocol works for strings over an arbitrary metric of polynomial size and aspect ratio, and we optimize the logarithmic factors depending on properties of the underlying metric, such as when the points are low-dimensional integer vectors equipped with various metrics or have bounded doubling dimension. We also consider linear sketches of DTW, showing that such sketches must have size $Ω(n)$.

cs.DS↗

arXiv2019

DistCache: Provable Load Balancing for Large-Scale Storage Systems with Distributed Caching

Load balancing is critical for distributed storage to meet strict service-level objectives (SLOs). It has been shown that a fast cache can guarantee load balancing for a clustered storage system. However, when the system scales out to multiple clusters, the fast cache itself would become the bottleneck. Traditional mechanisms like cache partition and cache replication either result in load imbalance between cache nodes or have high overhead for cache coherence. We present DistCache, a new distributed caching mechanism that provides provable load balancing for large-scale storage systems. DistCache co-designs cache allocation with cache topology and query routing. The key idea is to partition the hot objects with independent hash functions between cache nodes in different layers, and to adaptively route queries with the power-of-two-choices. We prove that DistCache enables the cache throughput to increase linearly with the number of cache nodes, by unifying techniques from expander graphs, network flows, and queuing theory. DistCache is a general solution that can be applied to many storage systems. We demonstrate the benefits of DistCache by providing the design, implementation, and evaluation of the use case for emerging switch-based caching.

cs.DC↗

arXiv2018

Matrix Norms in Data Streams: Faster, Multi-Pass and Row-Order

A central problem in data streams is to characterize which functions of an underlying frequency vector can be approximated efficiently. Recently there has been considerable effort in extending this problem to that of estimating functions of a matrix that is presented as a data-stream. This setting generalizes classical problems to the analogous ones for matrices. For example, instead of estimating frequent-item counts, we now wish to estimate "frequent-direction" counts. A related example is to estimate norms, which now correspond to estimating a vector norm on the singular values of the matrix. Despite recent efforts, the current understanding for such matrix problems is considerably weaker than that for vector problems. We study a number of aspects of estimating matrix norms in a stream that have not previously been considered: (1) multi-pass algorithms, (2) algorithms that see the underlying matrix one row at a time, and (3) time-efficient algorithms. Our multi-pass and row-order algorithms use less memory than what is provably required in the single-pass and entrywise-update models, and thus give separations between these models (in terms of memory). Moreover, all of our algorithms are considerably faster than previous ones. We also prove a number of lower bounds, and obtain for instance, a near-complete characterization of the memory required of row-order algorithms for estimating Schatten $p$-norms of sparse matrices.

cs.DS↗

arXiv2018

DreamNLP: Novel NLP System for Clinical Report Metadata Extraction using Count Sketch Data Streaming Algorithm: Preliminary Results

Extracting information from electronic health records (EHR) is a challenging task since it requires prior knowledge of the reports and some natural language processing algorithm (NLP). With the growing number of EHR implementations, such knowledge is increasingly challenging to obtain in an efficient manner. We address this challenge by proposing a novel methodology to analyze large sets of EHRs using a modified Count Sketch data streaming algorithm termed DreamNLP. By using DreamNLP, we generate a dictionary of frequently occurring terms or heavy hitters in the EHRs using low computational memory compared to conventional counting approach other NLP programs use. We demonstrate the extraction of the most important breast diagnosis features from the EHRs in a set of patients that underwent breast imaging. Based on the analysis, extraction of these terms would be useful for defining important features for downstream tasks such as machine learning for precision medicine.

cs.LG↗

arXiv2018

Approximations of Schatten Norms via Taylor Expansions

In this paper we consider symmetric, positive semidefinite (SPSD) matrix $A$ and present two algorithms for computing the $p$-Schatten norm $\|A\|_p$. The first algorithm works for any SPSD matrix $A$. The second algorithm works for non-singular SPSD matrices and runs in time that depends on $κ= {λ_1(A)\over λ_n(A)}$, where $λ_i(A)$ is the $i$-th eigenvalue of $A$. Our methods are simple and easy to implement and can be extended to general matrices. Our algorithms improve, for a range of parameters, recent results of Musco, Netrapalli, Sidford, Ubaru and Woodruff (ITCS 2018) and match the running time of the methods by Han, Malioutov, Avron, and Shin (SISC 2017) while avoiding computations of coefficients of Chebyshev polynomials.

cs.DS↗