arXiv subjects

Vladimir Braverman

Publications and source records attributed to Vladimir Braverman.

At least 55 records · Page 3Linked to original sources

arXiv2023

Private Data Stream Analysis for Universal Symmetric Norm Estimation

We study how to release summary statistics on a data stream subject to the constraint of differential privacy. In particular, we focus on releasing the family of symmetric norms, which are invariant under sign-flips and coordinate-wise permutations on an input data stream and include $L_p$ norms, $k$-support norms, top-$k$ norms, and the box norm as special cases. Although it may be possible to design and analyze a separate mechanism for each symmetric norm, we propose a general parametrizable framework that differentially privately releases a number of sufficient statistics from which the approximation of all symmetric norms can be simultaneously computed. Our framework partitions the coordinates of the underlying frequency vector into different levels based on their magnitude and releases approximate frequencies for the "heavy" coordinates in important levels and releases approximate level sizes for the "light" coordinates in important levels. Surprisingly, our mechanism allows for the release of an arbitrary number of symmetric norm approximations without any overhead or additional loss in privacy. Moreover, our mechanism permits $(1+α)$-approximation to each of the symmetric norms and can be implemented using sublinear space in the streaming model for many regimes of the accuracy and privacy parameters.

cs.DS↗

arXiv2023

Finite-Sample Analysis of Learning High-Dimensional Single ReLU Neuron

This paper considers the problem of learning a single ReLU neuron with squared loss (a.k.a., ReLU regression) in the overparameterized regime, where the input dimension can exceed the number of samples. We analyze a Perceptron-type algorithm called GLM-tron (Kakade et al., 2011) and provide its dimension-free risk upper bounds for high-dimensional ReLU regression in both well-specified and misspecified settings. Our risk bounds recover several existing results as special cases. Moreover, in the well-specified setting, we provide an instance-wise matching risk lower bound for GLM-tron. Our upper and lower risk bounds provide a sharp characterization of the high-dimensional ReLU regression problems that can be learned via GLM-tron. On the other hand, we provide some negative results for stochastic gradient descent (SGD) for ReLU regression with symmetric Bernoulli data: if the model is well-specified, the excess risk of SGD is provably no better than that of GLM-tron ignoring constant factors, for each problem instance; and in the noiseless case, GLM-tron can achieve a small risk while SGD unavoidably suffers from a constant risk in expectation. These results together suggest that GLM-tron might be preferable to SGD for high-dimensional ReLU regression.

cs.LG↗

arXiv2023

A framework for dynamically training and adapting deep reinforcement learning models to different, low-compute, and continuously changing radiology deployment environments

While Deep Reinforcement Learning has been widely researched in medical imaging, the training and deployment of these models usually require powerful GPUs. Since imaging environments evolve rapidly and can be generated by edge devices, the algorithm is required to continually learn and adapt to changing environments, and adjust to low-compute devices. To this end, we developed three image coreset algorithms to compress and denoise medical images for selective experience replayed-based lifelong reinforcement learning. We implemented neighborhood averaging coreset, neighborhood sensitivity-based sampling coreset, and maximum entropy coreset on full-body DIXON water and DIXON fat MRI images. All three coresets produced 27x compression with excellent performance in localizing five anatomical landmarks: left knee, right trochanter, left kidney, spleen, and lung across both imaging environments. Maximum entropy coreset obtained the best performance of $11.97\pm 12.02$ average distance error, compared to the conventional lifelong learning framework's $19.24\pm 50.77$.

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arXiv2023

Multi-environment lifelong deep reinforcement learning for medical imaging

Deep reinforcement learning(DRL) is increasingly being explored in medical imaging. However, the environments for medical imaging tasks are constantly evolving in terms of imaging orientations, imaging sequences, and pathologies. To that end, we developed a Lifelong DRL framework, SERIL to continually learn new tasks in changing imaging environments without catastrophic forgetting. SERIL was developed using selective experience replay based lifelong learning technique for the localization of five anatomical landmarks in brain MRI on a sequence of twenty-four different imaging environments. The performance of SERIL, when compared to two baseline setups: MERT(multi-environment-best-case) and SERT(single-environment-worst-case) demonstrated excellent performance with an average distance of $9.90\pm7.35$ pixels from the desired landmark across all 120 tasks, compared to $10.29\pm9.07$ for MERT and $36.37\pm22.41$ for SERT($p<0.05$), demonstrating the excellent potential for continuously learning multiple tasks across dynamically changing imaging environments.

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arXiv2023

AutoCoreset: An Automatic Practical Coreset Construction Framework

A coreset is a tiny weighted subset of an input set, that closely resembles the loss function, with respect to a certain set of queries. Coresets became prevalent in machine learning as they have shown to be advantageous for many applications. While coreset research is an active research area, unfortunately, coresets are constructed in a problem-dependent manner, where for each problem, a new coreset construction algorithm is usually suggested, a process that may take time or may be hard for new researchers in the field. Even the generic frameworks require additional (problem-dependent) computations or proofs to be done by the user. Besides, many problems do not have (provable) small coresets, limiting their applicability. To this end, we suggest an automatic practical framework for constructing coresets, which requires (only) the input data and the desired cost function from the user, without the need for any other task-related computation to be done by the user. To do so, we reduce the problem of approximating a loss function to an instance of vector summation approximation, where the vectors we aim to sum are loss vectors of a specific subset of the queries, such that we aim to approximate the image of the function on this subset. We show that while this set is limited, the coreset is quite general. An extensive experimental study on various machine learning applications is also conducted. Finally, we provide a ``plug and play" style implementation, proposing a user-friendly system that can be easily used to apply coresets for many problems. Full open source code can be found at \href{https://github.com/alaamaalouf/AutoCoreset}{\text{https://github.com/alaamaalouf/AutoCoreset}}. We believe that these contributions enable future research and easier use and applications of coresets.

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arXiv2023

Lower Bounds for Pseudo-Deterministic Counting in a Stream

Many streaming algorithms provide only a high-probability relative approximation. These two relaxations, of allowing approximation and randomization, seem necessary -- for many streaming problems, both relaxations must be employed simultaneously, to avoid an exponentially larger (and often trivial) space complexity. A common drawback of these randomized approximate algorithms is that independent executions on the same input have different outputs, that depend on their random coins. Pseudo-deterministic algorithms combat this issue, and for every input, they output with high probability the same ``canonical'' solution. We consider perhaps the most basic problem in data streams, of counting the number of items in a stream of length at most $n$. Morris's counter [CACM, 1978] is a randomized approximation algorithm for this problem that uses $O(\log\log n)$ bits of space, for every fixed approximation factor (greater than $1$). Goldwasser, Grossman, Mohanty and Woodruff [ITCS 2020] asked whether pseudo-deterministic approximation algorithms can match this space complexity. Our main result answers their question negatively, and shows that such algorithms must use $Ω(\sqrt{\log n / \log\log n})$ bits of space. Our approach is based on a problem that we call Shift Finding, and may be of independent interest. In this problem, one has query access to a shifted version of a known string $F\in\{0,1\}^{3n}$, which is guaranteed to start with $n$ zeros and end with $n$ ones, and the goal is to find the unknown shift using a small number of queries. We provide for this problem an algorithm that uses $O(\sqrt{n})$ queries. It remains open whether $poly(\log n)$ queries suffice; if true, then our techniques immediately imply a nearly-tight $Ω(\log n/\log\log n)$ space bound for pseudo-deterministic approximate counting.

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arXiv2023

Nearly Optimal Distinct Elements and Heavy Hitters on Sliding Windows

We study the distinct elements and $\ell_p$-heavy hitters problems in the sliding window model, where only the most recent $n$ elements in the data stream form the underlying set. We first introduce the composable histogram, a simple twist on the exponential (Datar et al., SODA 2002) and smooth histograms (Braverman and Ostrovsky, FOCS 2007) that may be of independent interest. We then show that the composable histogram along with a careful combination of existing techniques to track either the identity or frequency of a few specific items suffices to obtain algorithms for both distinct elements and $\ell_p$-heavy hitters that are nearly optimal in both $n$ and $ε$. Applying our new composable histogram framework, we provide an algorithm that outputs a $(1+ε)$-approximation to the number of distinct elements in the sliding window model and uses $\mathcal{O}\left(\frac{1}{ε^2}\log n\log\frac{1}ε\log\log n+\frac{1}ε\log^2 n\right)$ bits of space. For $\ell_p$-heavy hitters, we provide an algorithm using space $\mathcal{O}\left(\frac{1}{ε^p}\log^3 n\left(\log\log n+\log\frac{1}ε\right)\right)$ for $0<p\le 2$, improving upon the best-known algorithm for $\ell_2$-heavy hitters (Braverman et al., COCOON 2014), which has space complexity $\mathcal{O}\left(\frac{1}{ε^4}\log^3 n\right)$. We also show lower bounds of $Ω\left(\frac{1}ε\log^2 n+\frac{1}{ε^2}\log n\right)$ for distinct elements and $Ω\left(\frac{1}{ε^p}\log^2 n\right)$ for $\ell_p$-heavy hitters.

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arXiv2023

Near Optimal Linear Algebra in the Online and Sliding Window Models

We initiate the study of numerical linear algebra in the sliding window model, where only the most recent $W$ updates in a stream form the underlying data set. We first introduce a unified row-sampling based framework that gives randomized algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model, which often use nearly optimal space and achieve nearly input sparsity runtime. Our algorithms are based on "reverse online" versions of offline sampling distributions such as (ridge) leverage scores, $\ell_1$ sensitivities, and Lewis weights to quantify both the importance and the recency of a row. Our row-sampling framework rather surprisingly implies connections to the well-studied online model; our structural results also give the first sample optimal (up to lower order terms) online algorithm for low-rank approximation/projection-cost preservation. Using this powerful primitive, we give online algorithms for column/row subset selection and principal component analysis that resolves the main open question of Bhaskara et. al.,(FOCS 2019). We also give the first online algorithm for $\ell_1$-subspace embeddings. We further formalize the connection between the online model and the sliding window model by introducing an additional unified framework for deterministic algorithms using a merge and reduce paradigm and the concept of online coresets. Our sampling based algorithms in the row-arrival online model yield online coresets, giving deterministic algorithms for spectral approximation, low-rank approximation/projection-cost preservation, and $\ell_1$-subspace embeddings in the sliding window model that use nearly optimal space.

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arXiv2023

Provable Data Subset Selection For Efficient Neural Network Training

Radial basis function neural networks (\emph{RBFNN}) are {well-known} for their capability to approximate any continuous function on a closed bounded set with arbitrary precision given enough hidden neurons. In this paper, we introduce the first algorithm to construct coresets for \emph{RBFNNs}, i.e., small weighted subsets that approximate the loss of the input data on any radial basis function network and thus approximate any function defined by an \emph{RBFNN} on the larger input data. In particular, we construct coresets for radial basis and Laplacian loss functions. We then use our coresets to obtain a provable data subset selection algorithm for training deep neural networks. Since our coresets approximate every function, they also approximate the gradient of each weight in a neural network, which is a particular function on the input. We then perform empirical evaluations on function approximation and dataset subset selection on popular network architectures and data sets, demonstrating the efficacy and accuracy of our coreset construction.

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arXiv2022

Coresets for Clustering in Graphs of Bounded Treewidth

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set, and it offers significant efficiency improvements in terms of running time, storage, and communication, including in streaming and distributed settings. Our main result is a near-linear time construction of a coreset for k-Median in a general graph $G$, with size $O_{ε, k}(\mathrm{tw}(G))$ where $\mathrm{tw}(G)$ is the treewidth of $G$, and we complement the construction with a nearly-tight size lower bound. The construction is based on the framework of Feldman and Langberg [STOC 2011], and our main technical contribution, as required by this framework, is a uniform bound of $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. We validate our coreset on real-world road networks, and our scalable algorithm constructs tiny coresets with high accuracy, which translates to a massive speedup of existing approximation algorithms such as local search for graph k-Median.

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arXiv2022

From Local to Global: Spectral-Inspired Graph Neural Networks

Graph Neural Networks (GNNs) are powerful deep learning methods for Non-Euclidean data. Popular GNNs are message-passing algorithms (MPNNs) that aggregate and combine signals in a local graph neighborhood. However, shallow MPNNs tend to miss long-range signals and perform poorly on some heterophilous graphs, while deep MPNNs can suffer from issues like over-smoothing or over-squashing. To mitigate such issues, existing works typically borrow normalization techniques from training neural networks on Euclidean data or modify the graph structures. Yet these approaches are not well-understood theoretically and could increase the overall computational complexity. In this work, we draw inspirations from spectral graph embedding and propose $\texttt{PowerEmbed}$ -- a simple layer-wise normalization technique to boost MPNNs. We show $\texttt{PowerEmbed}$ can provably express the top-$k$ leading eigenvectors of the graph operator, which prevents over-smoothing and is agnostic to the graph topology; meanwhile, it produces a list of representations ranging from local features to global signals, which avoids over-squashing. We apply $\texttt{PowerEmbed}$ in a wide range of simulated and real graphs and demonstrate its competitive performance, particularly for heterophilous graphs.

stat.ML↗

arXiv2022

New Frameworks for Offline and Streaming Coreset Constructions

A coreset for a set of points is a small subset of weighted points that approximately preserves important properties of the original set. Specifically, if $P$ is a set of points, $Q$ is a set of queries, and $f:P\times Q\to\mathbb{R}$ is a cost function, then a set $S\subseteq P$ with weights $w:P\to[0,\infty)$ is an $ε$-coreset for some parameter $ε>0$ if $\sum_{s\in S}w(s)f(s,q)$ is a $(1+ε)$ multiplicative approximation to $\sum_{p\in P}f(p,q)$ for all $q\in Q$. Coresets are used to solve fundamental problems in machine learning under various big data models of computation. Many of the suggested coresets in the recent decade used, or could have used a general framework for constructing coresets whose size depends quadratically on what is known as total sensitivity $t$. In this paper we improve this bound from $O(t^2)$ to $O(t\log t)$. Thus our results imply more space efficient solutions to a number of problems, including projective clustering, $k$-line clustering, and subspace approximation. Moreover, we generalize the notion of sensitivity sampling for sup-sampling that supports non-multiplicative approximations, negative cost functions and more. The main technical result is a generic reduction to the sample complexity of learning a class of functions with bounded VC dimension. We show that obtaining an $(ν,α)$-sample for this class of functions with appropriate parameters $ν$ and $α$ suffices to achieve space efficient $ε$-coresets. Our result implies more efficient coreset constructions for a number of interesting problems in machine learning; we show applications to $k$-median/$k$-means, $k$-line clustering, $j$-subspace approximation, and the integer $(j,k)$-projective clustering problem.

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arXiv2022

The Power of Uniform Sampling for Coresets

Motivated by practical generalizations of the classic $k$-median and $k$-means objectives, such as clustering with size constraints, fair clustering, and Wasserstein barycenter, we introduce a meta-theorem for designing coresets for constrained-clustering problems. The meta-theorem reduces the task of coreset construction to one on a bounded number of ring instances with a much-relaxed additive error. This reduction enables us to construct coresets using uniform sampling, in contrast to the widely-used importance sampling, and consequently we can easily handle constrained objectives. Notably and perhaps surprisingly, this simpler sampling scheme can yield coresets whose size is independent of $n$, the number of input points. Our technique yields smaller coresets, and sometimes the first coresets, for a large number of constrained clustering problems, including capacitated clustering, fair clustering, Euclidean Wasserstein barycenter, clustering in minor-excluded graph, and polygon clustering under Fréchet and Hausdorff distance. Finally, our technique yields also smaller coresets for $1$-median in low-dimensional Euclidean spaces, specifically of size $\tilde{O}(\varepsilon^{-1.5})$ in $\mathbb{R}^2$ and $\tilde{O}(\varepsilon^{-1.6})$ in $\mathbb{R}^3$.

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arXiv2022

The Power and Limitation of Pretraining-Finetuning for Linear Regression under Covariate Shift

We study linear regression under covariate shift, where the marginal distribution over the input covariates differs in the source and the target domains, while the conditional distribution of the output given the input covariates is similar across the two domains. We investigate a transfer learning approach with pretraining on the source data and finetuning based on the target data (both conducted by online SGD) for this problem. We establish sharp instance-dependent excess risk upper and lower bounds for this approach. Our bounds suggest that for a large class of linear regression instances, transfer learning with $O(N^2)$ source data (and scarce or no target data) is as effective as supervised learning with $N$ target data. In addition, we show that finetuning, even with only a small amount of target data, could drastically reduce the amount of source data required by pretraining. Our theory sheds light on the effectiveness and limitation of pretraining as well as the benefits of finetuning for tackling covariate shift problems.

cs.LG↗

arXiv2022

The White-Box Adversarial Data Stream Model

We study streaming algorithms in the white-box adversarial model, where the stream is chosen adaptively by an adversary who observes the entire internal state of the algorithm at each time step. We show that nontrivial algorithms are still possible. We first give a randomized algorithm for the $L_1$-heavy hitters problem that outperforms the optimal deterministic Misra-Gries algorithm on long streams. If the white-box adversary is computationally bounded, we use cryptographic techniques to reduce the memory of our $L_1$-heavy hitters algorithm even further and to design a number of additional algorithms for graph, string, and linear algebra problems. The existence of such algorithms is surprising, as the streaming algorithm does not even have a secret key in this model, i.e., its state is entirely known to the adversary. One algorithm we design is for estimating the number of distinct elements in a stream with insertions and deletions achieving a multiplicative approximation and sublinear space; such an algorithm is impossible for deterministic algorithms. We also give a general technique that translates any two-player deterministic communication lower bound to a lower bound for {\it randomized} algorithms robust to a white-box adversary. In particular, our results show that for all $p\ge 0$, there exists a constant $C_p>1$ such that any $C_p$-approximation algorithm for $F_p$ moment estimation in insertion-only streams with a white-box adversary requires $Ω(n)$ space for a universe of size $n$. Similarly, there is a constant $C>1$ such that any $C$-approximation algorithm in an insertion-only stream for matrix rank requires $Ω(n)$ space with a white-box adversary. Our algorithmic results based on cryptography thus show a separation between computationally bounded and unbounded adversaries. (Abstract shortened to meet arXiv limits.)

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arXiv2022

The Benefits of Implicit Regularization from SGD in Least Squares Problems

Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice, which has been hypothesized to play an important role in the generalization of modern machine learning approaches. In this work, we seek to understand these issues in the simpler setting of linear regression (including both underparameterized and overparameterized regimes), where our goal is to make sharp instance-based comparisons of the implicit regularization afforded by (unregularized) average SGD with the explicit regularization of ridge regression. For a broad class of least squares problem instances (that are natural in high-dimensional settings), we show: (1) for every problem instance and for every ridge parameter, (unregularized) SGD, when provided with logarithmically more samples than that provided to the ridge algorithm, generalizes no worse than the ridge solution (provided SGD uses a tuned constant stepsize); (2) conversely, there exist instances (in this wide problem class) where optimally-tuned ridge regression requires quadratically more samples than SGD in order to have the same generalization performance. Taken together, our results show that, up to the logarithmic factors, the generalization performance of SGD is always no worse than that of ridge regression in a wide range of overparameterized problems, and, in fact, could be much better for some problem instances. More generally, our results show how algorithmic regularization has important consequences even in simpler (overparameterized) convex settings.

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arXiv2022

Last Iterate Risk Bounds of SGD with Decaying Stepsize for Overparameterized Linear Regression

Stochastic gradient descent (SGD) has been shown to generalize well in many deep learning applications. In practice, one often runs SGD with a geometrically decaying stepsize, i.e., a constant initial stepsize followed by multiple geometric stepsize decay, and uses the last iterate as the output. This kind of SGD is known to be nearly minimax optimal for classical finite-dimensional linear regression problems (Ge et al., 2019). However, a sharp analysis for the last iterate of SGD in the overparameterized setting is still open. In this paper, we provide a problem-dependent analysis on the last iterate risk bounds of SGD with decaying stepsize, for (overparameterized) linear regression problems. In particular, for last iterate SGD with (tail) geometrically decaying stepsize, we prove nearly matching upper and lower bounds on the excess risk. Moreover, we provide an excess risk lower bound for last iterate SGD with polynomially decaying stepsize and demonstrate the advantage of geometrically decaying stepsize in an instance-wise manner, which complements the minimax rate comparison made in prior works.

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arXiv2022

Sparsity and Heterogeneous Dropout for Continual Learning in the Null Space of Neural Activations

Continual/lifelong learning from a non-stationary input data stream is a cornerstone of intelligence. Despite their phenomenal performance in a wide variety of applications, deep neural networks are prone to forgetting their previously learned information upon learning new ones. This phenomenon is called "catastrophic forgetting" and is deeply rooted in the stability-plasticity dilemma. Overcoming catastrophic forgetting in deep neural networks has become an active field of research in recent years. In particular, gradient projection-based methods have recently shown exceptional performance at overcoming catastrophic forgetting. This paper proposes two biologically-inspired mechanisms based on sparsity and heterogeneous dropout that significantly increase a continual learner's performance over a long sequence of tasks. Our proposed approach builds on the Gradient Projection Memory (GPM) framework. We leverage k-winner activations in each layer of a neural network to enforce layer-wise sparse activations for each task, together with a between-task heterogeneous dropout that encourages the network to use non-overlapping activation patterns between different tasks. In addition, we introduce two new benchmarks for continual learning under distributional shift, namely Continual Swiss Roll and ImageNet SuperDog-40. Lastly, we provide an in-depth analysis of our proposed method and demonstrate a significant performance boost on various benchmark continual learning problems.

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