arXiv subjects

Dimitris Papailiopoulos

Publications and source records attributed to Dimitris Papailiopoulos.

At least 73 records · Page 4Linked to original sources

arXiv2019

Convergence and Margin of Adversarial Training on Separable Data

Adversarial training is a technique for training robust machine learning models. To encourage robustness, it iteratively computes adversarial examples for the model, and then re-trains on these examples via some update rule. This work analyzes the performance of adversarial training on linearly separable data, and provides bounds on the number of iterations required for large margin. We show that when the update rule is given by an arbitrary empirical risk minimizer, adversarial training may require exponentially many iterations to obtain large margin. However, if gradient or stochastic gradient update rules are used, only polynomially many iterations are required to find a large-margin separator. By contrast, without the use of adversarial examples, gradient methods may require exponentially many iterations to achieve large margin. Our results are derived by showing that adversarial training with gradient updates minimizes a robust version of the empirical risk at a $\mathcal{O}(\ln(t)^2/t)$ rate, despite non-smoothness. We corroborate our theory empirically.

cs.LG↗

arXiv2019

Does Data Augmentation Lead to Positive Margin?

Data augmentation (DA) is commonly used during model training, as it significantly improves test error and model robustness. DA artificially expands the training set by applying random noise, rotations, crops, or even adversarial perturbations to the input data. Although DA is widely used, its capacity to provably improve robustness is not fully understood. In this work, we analyze the robustness that DA begets by quantifying the margin that DA enforces on empirical risk minimizers. We first focus on linear separators, and then a class of nonlinear models whose labeling is constant within small convex hulls of data points. We present lower bounds on the number of augmented data points required for non-zero margin, and show that commonly used DA techniques may only introduce significant margin after adding exponentially many points to the data set.

cs.LG↗

arXiv2019

ErasureHead: Distributed Gradient Descent without Delays Using Approximate Gradient Coding

We present ErasureHead, a new approach for distributed gradient descent (GD) that mitigates system delays by employing approximate gradient coding. Gradient coded distributed GD uses redundancy to exactly recover the gradient at each iteration from a subset of compute nodes. ErasureHead instead uses approximate gradient codes to recover an inexact gradient at each iteration, but with higher delay tolerance. Unlike prior work on gradient coding, we provide a performance analysis that combines both delay and convergence guarantees. We establish that down to a small noise floor, ErasureHead converges as quickly as distributed GD and has faster overall runtime under a probabilistic delay model. We conduct extensive experiments on real world datasets and distributed clusters and demonstrate that our method can lead to significant speedups over both standard and gradient coded GD.

cs.LG↗

arXiv2018

ATOMO: Communication-efficient Learning via Atomic Sparsification

Distributed model training suffers from communication overheads due to frequent gradient updates transmitted between compute nodes. To mitigate these overheads, several studies propose the use of sparsified stochastic gradients. We argue that these are facets of a general sparsification method that can operate on any possible atomic decomposition. Notable examples include element-wise, singular value, and Fourier decompositions. We present ATOMO, a general framework for atomic sparsification of stochastic gradients. Given a gradient, an atomic decomposition, and a sparsity budget, ATOMO gives a random unbiased sparsification of the atoms minimizing variance. We show that recent methods such as QSGD and TernGrad are special cases of ATOMO and that sparsifiying the singular value decomposition of neural networks gradients, rather than their coordinates, can lead to significantly faster distributed training.

stat.ML↗

arXiv2018

A Geometric Perspective on the Transferability of Adversarial Directions

State-of-the-art machine learning models frequently misclassify inputs that have been perturbed in an adversarial manner. Adversarial perturbations generated for a given input and a specific classifier often seem to be effective on other inputs and even different classifiers. In other words, adversarial perturbations seem to transfer between different inputs, models, and even different neural network architectures. In this work, we show that in the context of linear classifiers and two-layer ReLU networks, there provably exist directions that give rise to adversarial perturbations for many classifiers and data points simultaneously. We show that these "transferable adversarial directions" are guaranteed to exist for linear separators of a given set, and will exist with high probability for linear classifiers trained on independent sets drawn from the same distribution. We extend our results to large classes of two-layer ReLU networks. We further show that adversarial directions for ReLU networks transfer to linear classifiers while the reverse need not hold, suggesting that adversarial perturbations for more complex models are more likely to transfer to other classifiers. We validate our findings empirically, even for deeper ReLU networks.

cs.LG↗

arXiv2018

DRACO: Byzantine-resilient Distributed Training via Redundant Gradients

Distributed model training is vulnerable to byzantine system failures and adversarial compute nodes, i.e., nodes that use malicious updates to corrupt the global model stored at a parameter server (PS). To guarantee some form of robustness, recent work suggests using variants of the geometric median as an aggregation rule, in place of gradient averaging. Unfortunately, median-based rules can incur a prohibitive computational overhead in large-scale settings, and their convergence guarantees often require strong assumptions. In this work, we present DRACO, a scalable framework for robust distributed training that uses ideas from coding theory. In DRACO, each compute node evaluates redundant gradients that are used by the parameter server to eliminate the effects of adversarial updates. DRACO comes with problem-independent robustness guarantees, and the model that it trains is identical to the one trained in the adversary-free setup. We provide extensive experiments on real datasets and distributed setups across a variety of large-scale models, where we show that DRACO is several times, to orders of magnitude faster than median-based approaches.

stat.ML↗

arXiv2018

The Effect of Network Width on the Performance of Large-batch Training

Distributed implementations of mini-batch stochastic gradient descent (SGD) suffer from communication overheads, attributed to the high frequency of gradient updates inherent in small-batch training. Training with large batches can reduce these overheads; however, large batches can affect the convergence properties and generalization performance of SGD. In this work, we take a first step towards analyzing how the structure (width and depth) of a neural network affects the performance of large-batch training. We present new theoretical results which suggest that--for a fixed number of parameters--wider networks are more amenable to fast large-batch training compared to deeper ones. We provide extensive experiments on residual and fully-connected neural networks which suggest that wider networks can be trained using larger batches without incurring a convergence slow-down, unlike their deeper variants.

stat.ML↗

arXiv2018

Gradient Coding via the Stochastic Block Model

Gradient descent and its many variants, including mini-batch stochastic gradient descent, form the algorithmic foundation of modern large-scale machine learning. Due to the size and scale of modern data, gradient computations are often distributed across multiple compute nodes. Unfortunately, such distributed implementations can face significant delays caused by straggler nodes, i.e., nodes that are much slower than average. Gradient coding is a new technique for mitigating the effect of stragglers via algorithmic redundancy. While effective, previously proposed gradient codes can be computationally expensive to construct, inaccurate, or susceptible to adversarial stragglers. In this work, we present the stochastic block code (SBC), a gradient code based on the stochastic block model. We show that SBCs are efficient, accurate, and that under certain settings, adversarial straggler selection becomes as hard as detecting a community structure in the multiple community, block stochastic graph model.

stat.ML↗

arXiv2018

Speeding Up Distributed Machine Learning Using Codes

Codes are widely used in many engineering applications to offer robustness against noise. In large-scale systems there are several types of noise that can affect the performance of distributed machine learning algorithms -- straggler nodes, system failures, or communication bottlenecks -- but there has been little interaction cutting across codes, machine learning, and distributed systems. In this work, we provide theoretical insights on how coded solutions can achieve significant gains compared to uncoded ones. We focus on two of the most basic building blocks of distributed learning algorithms: matrix multiplication and data shuffling. For matrix multiplication, we use codes to alleviate the effect of stragglers, and show that if the number of homogeneous workers is $n$, and the runtime of each subtask has an exponential tail, coded computation can speed up distributed matrix multiplication by a factor of $\log n$. For data shuffling, we use codes to reduce communication bottlenecks, exploiting the excess in storage. We show that when a constant fraction $α$ of the data matrix can be cached at each worker, and $n$ is the number of workers, \emph{coded shuffling} reduces the communication cost by a factor of $(α+ \frac{1}{n})γ(n)$ compared to uncoded shuffling, where $γ(n)$ is the ratio of the cost of unicasting $n$ messages to $n$ users to multicasting a common message (of the same size) to $n$ users. For instance, $γ(n) \simeq n$ if multicasting a message to $n$ users is as cheap as unicasting a message to one user. We also provide experiment results, corroborating our theoretical gains of the coded algorithms.

cs.DC↗

arXiv2018

Gradient Diversity: a Key Ingredient for Scalable Distributed Learning

It has been experimentally observed that distributed implementations of mini-batch stochastic gradient descent (SGD) algorithms exhibit speedup saturation and decaying generalization ability beyond a particular batch-size. In this work, we present an analysis hinting that high similarity between concurrently processed gradients may be a cause of this performance degradation. We introduce the notion of gradient diversity that measures the dissimilarity between concurrent gradient updates, and show its key role in the performance of mini-batch SGD. We prove that on problems with high gradient diversity, mini-batch SGD is amenable to better speedups, while maintaining the generalization performance of serial (one sample) SGD. We further establish lower bounds on convergence where mini-batch SGD slows down beyond a particular batch-size, solely due to the lack of gradient diversity. We provide experimental evidence indicating the key role of gradient diversity in distributed learning, and discuss how heuristics like dropout, Langevin dynamics, and quantization can improve it.

cs.LG↗

arXiv2017

Approximate Gradient Coding via Sparse Random Graphs

Distributed algorithms are often beset by the straggler effect, where the slowest compute nodes in the system dictate the overall running time. Coding-theoretic techniques have been recently proposed to mitigate stragglers via algorithmic redundancy. Prior work in coded computation and gradient coding has mainly focused on exact recovery of the desired output. However, slightly inexact solutions can be acceptable in applications that are robust to noise, such as model training via gradient-based algorithms. In this work, we present computationally simple gradient codes based on sparse graphs that guarantee fast and approximately accurate distributed computation. We demonstrate that sacrificing a small amount of accuracy can significantly increase algorithmic robustness to stragglers.

stat.ML↗

arXiv2017

Stability and Generalization of Learning Algorithms that Converge to Global Optima

We establish novel generalization bounds for learning algorithms that converge to global minima. We do so by deriving black-box stability results that only depend on the convergence of a learning algorithm and the geometry around the minimizers of the loss function. The results are shown for nonconvex loss functions satisfying the Polyak-Łojasiewicz (PL) and the quadratic growth (QG) conditions. We further show that these conditions arise for some neural networks with linear activations. We use our black-box results to establish the stability of optimization algorithms such as stochastic gradient descent (SGD), gradient descent (GD), randomized coordinate descent (RCD), and the stochastic variance reduced gradient method (SVRG), in both the PL and the strongly convex setting. Our results match or improve state-of-the-art generalization bounds and can easily be extended to similar optimization algorithms. Finally, we show that although our results imply comparable stability for SGD and GD in the PL setting, there exist simple neural networks with multiple local minima where SGD is stable but GD is not.

stat.ML↗

arXiv2016

CYCLADES: Conflict-free Asynchronous Machine Learning

We present CYCLADES, a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. CYCLADES is asynchronous during shared model updates, and requires no memory locking mechanisms, similar to HOGWILD!-type algorithms. Unlike HOGWILD!, CYCLADES introduces no conflicts during the parallel execution, and offers a black-box analysis for provable speedups across a large family of algorithms. Due to its inherent conflict-free nature and cache locality, our multi-core implementation of CYCLADES consistently outperforms HOGWILD!-type algorithms on sufficiently sparse datasets, leading to up to 40% speedup gains compared to the HOGWILD! implementation of SGD, and up to 5x gains over asynchronous implementations of variance reduction algorithms.

stat.ML↗

arXiv2016

Perturbed Iterate Analysis for Asynchronous Stochastic Optimization

We introduce and analyze stochastic optimization methods where the input to each gradient update is perturbed by bounded noise. We show that this framework forms the basis of a unified approach to analyze asynchronous implementations of stochastic optimization algorithms.In this framework, asynchronous stochastic optimization algorithms can be thought of as serial methods operating on noisy inputs. Using our perturbed iterate framework, we provide new analyses of the Hogwild! algorithm and asynchronous stochastic coordinate descent, that are simpler than earlier analyses, remove many assumptions of previous models, and in some cases yield improved upper bounds on the convergence rates. We proceed to apply our framework to develop and analyze KroMagnon: a novel, parallel, sparse stochastic variance-reduced gradient (SVRG) algorithm. We demonstrate experimentally on a 16-core machine that the sparse and parallel version of SVRG is in some cases more than four orders of magnitude faster than the standard SVRG algorithm.

stat.ML↗

arXiv2016

Bipartite Correlation Clustering -- Maximizing Agreements

In Bipartite Correlation Clustering (BCC) we are given a complete bipartite graph $G$ with `+' and `-' edges, and we seek a vertex clustering that maximizes the number of agreements: the number of all `+' edges within clusters plus all `-' edges cut across clusters. BCC is known to be NP-hard. We present a novel approximation algorithm for $k$-BCC, a variant of BCC with an upper bound $k$ on the number of clusters. Our algorithm outputs a $k$-clustering that provably achieves a number of agreements within a multiplicative ${(1-δ)}$-factor from the optimal, for any desired accuracy $δ$. It relies on solving a combinatorially constrained bilinear maximization on the bi-adjacency matrix of $G$. It runs in time exponential in $k$ and $δ^{-1}$, but linear in the size of the input. Further, we show that, in the (unconstrained) BCC setting, an ${(1-δ)}$-approximation can be achieved by $O(δ^{-1})$ clusters regardless of the size of the graph. In turn, our $k$-BCC algorithm implies an Efficient PTAS for the BCC objective of maximizing agreements.

cs.DS↗

arXiv2015

Sparse PCA via Bipartite Matchings

We consider the following multi-component sparse PCA problem: given a set of data points, we seek to extract a small number of sparse components with disjoint supports that jointly capture the maximum possible variance. These components can be computed one by one, repeatedly solving the single-component problem and deflating the input data matrix, but as we show this greedy procedure is suboptimal. We present a novel algorithm for sparse PCA that jointly optimizes multiple disjoint components. The extracted features capture variance that lies within a multiplicative factor arbitrarily close to 1 from the optimal. Our algorithm is combinatorial and computes the desired components by solving multiple instances of the bipartite maximum weight matching problem. Its complexity grows as a low order polynomial in the ambient dimension of the input data matrix, but exponentially in its rank. However, it can be effectively applied on a low-dimensional sketch of the data; this allows us to obtain polynomial-time approximation guarantees via spectral bounds. We evaluate our algorithm on real data-sets and empirically demonstrate that in many cases it outperforms existing, deflation-based approaches.

stat.ML↗

arXiv2015

On the Worst-Case Approximability of Sparse PCA

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an $n^{-1/3}$-approximation; 2) NP-hardness of approximation to within $(1-\varepsilon)$, for some small constant $\varepsilon > 0$; 3) SSE-hardness of approximation to within any constant factor; and 4) an $\exp\exp\left(Ω\left(\sqrt{\log \log n}\right)\right)$ ("quasi-quasi-polynomial") gap for the standard semidefinite program.

stat.ML↗

arXiv2015

Parallel Correlation Clustering on Big Graphs

Given a similarity graph between items, correlation clustering (CC) groups similar items together and dissimilar ones apart. One of the most popular CC algorithms is KwikCluster: an algorithm that serially clusters neighborhoods of vertices, and obtains a 3-approximation ratio. Unfortunately, KwikCluster in practice requires a large number of clustering rounds, a potential bottleneck for large graphs. We present C4 and ClusterWild!, two algorithms for parallel correlation clustering that run in a polylogarithmic number of rounds and achieve nearly linear speedups, provably. C4 uses concurrency control to enforce serializability of a parallel clustering process, and guarantees a 3-approximation ratio. ClusterWild! is a coordination free algorithm that abandons consistency for the benefit of better scaling; this leads to a provably small loss in the 3-approximation ratio. We provide extensive experimental results for both algorithms, where we outperform the state of the art, both in terms of clustering accuracy and running time. We show that our algorithms can cluster billion-edge graphs in under 5 seconds on 32 cores, while achieving a 15x speedup.

cs.DC↗