An Exponential Improvement on the Memorization Capacity of Deep Threshold Networks

It is well known that modern deep neural networks are powerful enough to memorize datasets even when the labels have been randomized. Recently, Vershynin (2020) settled a long standing question by Baum (1988), proving that \emph{deep threshold} networks can memorize $n$ points in $d$ dimensions using $\widetilde{\mathcal{O}}(e^{1/\delta^2}+\sqrt{n})$ neurons and $\widetilde{\mathcal{O}}(e^{1/\delta^2}(d+\sqrt{n})+n)$ weights, where $\delta$ is the minimum distance between the points. In this work, we improve the dependence on $\delta$ from exponential to almost linear, proving that $\widetilde{\mathcal{O}}(\frac{1}{\delta}+\sqrt{n})$ neurons and $\widetilde{\mathcal{O}}(\frac{d}{\delta}+n)$ weights are sufficient. Our construction uses Gaussian random weights only in the first layer, while all the subsequent layers use binary or integer weights. We also prove new lower bounds by connecting memorization in neural networks to the purely geometric problem of separating $n$ points on a sphere using hyperplanes.

Pufferfish: Communication-efficient Models At No Extra Cost

To mitigate communication overheads in distributed model training, several studies propose the use of compressed stochastic gradients, usually achieved by sparsification or quantization. Such techniques achieve high compression ratios, but in many cases incur either significant computational overheads or some accuracy loss. In this work, we present Pufferfish, a communication and computation efficient distributed training framework that incorporates the gradient compression into the model training process via training low-rank, pre-factorized deep networks. Pufferfish not only reduces communication, but also completely bypasses any computation overheads related to compression, and achieves the same accuracy as state-of-the-art, off-the-shelf deep models. Pufferfish can be directly integrated into current deep learning frameworks with minimum implementation modification. Our extensive experiments over real distributed setups, across a variety of large-scale machine learning tasks, indicate that Pufferfish achieves up to 1.64x end-to-end speedup over the latest distributed training API in PyTorch without accuracy loss. Compared to the Lottery Ticket Hypothesis models, Pufferfish leads to equally accurate, small-parameter models while avoiding the burden of "winning the lottery". Pufferfish also leads to more accurate and smaller models than SOTA structured model pruning methods.

On the Utility of Gradient Compression in Distributed Training Systems

Rapid growth in data sets and the scale of neural network architectures have rendered distributed training a necessity. A rich body of prior work has highlighted the existence of communication bottlenecks in synchronous data-parallel training. To alleviate these bottlenecks, the machine learning community has largely focused on developing gradient and model compression methods. In parallel, the systems community has adopted several High Performance Computing (HPC)techniques to speed up distributed training. In this work, we evaluate the efficacy of gradient compression methods and compare their scalability with optimized implementations of synchronous data-parallel SGD. Surprisingly, we observe that due to computation overheads introduced by gradient compression, the net speedup over vanilla data-parallel training is marginal, if not negative. We conduct an extensive investigation to identify the root causes of this phenomenon, and offer a performance model that can be used to identify the benefits of gradient compression for a variety of system setups. Based on our analysis, we propose a list of desirable properties that gradient compression methods should satisfy, in order for them to provide a meaningful end-to-end speedup

Permutation-Based SGD: Is Random Optimal?

A recent line of ground-breaking results for permutation-based SGD has corroborated a widely observed phenomenon: random permutations offer faster convergence than with-replacement sampling. However, is random optimal? We show that this depends heavily on what functions we are optimizing, and the convergence gap between optimal and random permutations can vary from exponential to nonexistent. We first show that for 1-dimensional strongly convex functions, with smooth second derivatives, there exist optimal permutations that offer exponentially faster convergence compared to random. However, for general strongly convex functions, random permutations are optimal. Finally, we show that for quadratic, strongly-convex functions, there are easy-to-construct permutations that lead to accelerated convergence compared to random. Our results suggest that a general convergence characterization of optimal permutations cannot capture the nuances of individual function classes, and can mistakenly indicate that one cannot do much better than random.

Accordion: Adaptive Gradient Communication via Critical Learning Regime Identification

Distributed model training suffers from communication bottlenecks due to frequent model updates transmitted across compute nodes. To alleviate these bottlenecks, practitioners use gradient compression techniques like sparsification, quantization, or low-rank updates. The techniques usually require choosing a static compression ratio, often requiring users to balance the trade-off between model accuracy and per-iteration speedup. In this work, we show that such performance degradation due to choosing a high compression ratio is not fundamental. An adaptive compression strategy can reduce communication while maintaining final test accuracy. Inspired by recent findings on critical learning regimes, in which small gradient errors can have irrecoverable impact on model performance, we propose Accordion a simple yet effective adaptive compression algorithm. While Accordion maintains a high enough compression rate on average, it avoids over-compressing gradients whenever in critical learning regimes, detected by a simple gradient-norm based criterion. Our extensive experimental study over a number of machine learning tasks in distributed environments indicates that Accordion, maintains similar model accuracy to uncompressed training, yet achieves up to 5.5x better compression and up to 4.1x end-to-end speedup over static approaches. We show that Accordion also works for adjusting the batch size, another popular strategy for alleviating communication bottlenecks.

Attack of the Tails: Yes, You Really Can Backdoor Federated Learning

Due to its decentralized nature, Federated Learning (FL) lends itself to adversarial attacks in the form of backdoors during training. The goal of a backdoor is to corrupt the performance of the trained model on specific sub-tasks (e.g., by classifying green cars as frogs). A range of FL backdoor attacks have been introduced in the literature, but also methods to defend against them, and it is currently an open question whether FL systems can be tailored to be robust against backdoors. In this work, we provide evidence to the contrary. We first establish that, in the general case, robustness to backdoors implies model robustness to adversarial examples, a major open problem in itself. Furthermore, detecting the presence of a backdoor in a FL model is unlikely assuming first order oracles or polynomial time. We couple our theoretical results with a new family of backdoor attacks, which we refer to as edge-case backdoors. An edge-case backdoor forces a model to misclassify on seemingly easy inputs that are however unlikely to be part of the training, or test data, i.e., they live on the tail of the input distribution. We explain how these edge-case backdoors can lead to unsavory failures and may have serious repercussions on fairness, and exhibit that with careful tuning at the side of the adversary, one can insert them across a range of machine learning tasks (e.g., image classification, OCR, text prediction, sentiment analysis).

Optimal Lottery Tickets via SubsetSum: Logarithmic Over-Parameterization is Sufficient

The strong {\it lottery ticket hypothesis} (LTH) postulates that one can approximate any target neural network by only pruning the weights of a sufficiently over-parameterized random network. A recent work by Malach et al.~\cite{MalachEtAl20} establishes the first theoretical analysis for the strong LTH: one can provably approximate a neural network of width $d$ and depth $l$, by pruning a random one that is a factor $O(d^4l^2)$ wider and twice as deep. This polynomial over-parameterization requirement is at odds with recent experimental research that achieves good approximation with networks that are a small factor wider than the target. In this work, we close the gap and offer an exponential improvement to the over-parameterization requirement for the existence of lottery tickets. We show that any target network of width $d$ and depth $l$ can be approximated by pruning a random network that is a factor $O(\log(dl))$ wider and twice as deep. Our analysis heavily relies on connecting pruning random ReLU networks to random instances of the \textsc{SubsetSum} problem. We then show that this logarithmic over-parameterization is essentially optimal for constant depth networks. Finally, we verify several of our theoretical insights with experiments.

Closing the convergence gap of SGD without replacement

Stochastic gradient descent without replacement sampling is widely used in practice for model training. However, the vast majority of SGD analyses assumes data sampled with replacement, and when the function minimized is strongly convex, an $\mathcal{O}\left(\frac{1}{T}\right)$ rate can be established when SGD is run for $T$ iterations. A recent line of breakthrough work on SGD without replacement (SGDo) established an $\mathcal{O}\left(\frac{n}{T^2}\right)$ convergence rate when the function minimized is strongly convex and is a sum of $n$ smooth functions, and an $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^3}{T^3}\right)$ rate for sums of quadratics. On the other hand, the tightest known lower bound postulates an $\Omega\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ rate, leaving open the possibility of better SGDo convergence rates in the general case. In this paper, we close this gap and show that SGD without replacement achieves a rate of $\mathcal{O}\left(\frac{1}{T^2}+\frac{n^2}{T^3}\right)$ when the sum of the functions is a quadratic, and offer a new lower bound of $\Omega\left(\frac{n}{T^2}\right)$ for strongly convex functions that are sums of smooth functions.