## Optimization and Algorithms 1

Moderator: Hossein Mobahi

Abstract:

### Chat is not available.

Tue 20 July 17:00 - 17:20 PDT
##### A Tale of Two Efficient and Informative Negative Sampling Distributions

Shabnam Daghaghi · Tharun Medini · Nicholas Meisburger · Beidi Chen · Mengnan Zhao · Anshumali Shrivastava

Softmax classifiers with a very large number of classes naturally occur in many applications such as natural language processing and information retrieval. The calculation of full softmax is costly from the computational and energy perspective. There have been various sampling approaches to overcome this challenge, popularly known as negative sampling (NS). Ideally, NS should sample negative classes from a distribution that is dependent on the input data, the current parameters, and the correct positive class. Unfortunately, due to the dynamically updated parameters and data samples, there is no sampling scheme that is provably adaptive and samples the negative classes efficiently. Therefore, alternative heuristics like random sampling, static frequency-based sampling, or learning-based biased sampling, which primarily trade either the sampling cost or the adaptivity of samples per iteration are adopted. In this paper, we show two classes of distributions where the sampling scheme is truly adaptive and provably generates negative samples in near-constant time. Our implementation in C++ on CPU is significantly superior, both in terms of wall-clock time and accuracy, compared to the most optimized TensorFlow implementations of other popular negative sampling approaches on powerful NVIDIA V100 GPU.

[ ]
Tue 20 July 17:20 - 17:25 PDT
##### TeraPipe: Token-Level Pipeline Parallelism for Training Large-Scale Language Models

Zhuohan Li · Siyuan Zhuang · Shiyuan Guo · Danyang Zhuo · Hao Zhang · Dawn Song · Ion Stoica

Model parallelism has become a necessity for training modern large-scale deep language models. In this work, we identify a new and orthogonal dimension from existing model parallel approaches: it is possible to perform pipeline parallelism within a single training sequence for Transformer-based language models thanks to its autoregressive property. This enables a more fine-grained pipeline compared with previous work. With this key idea, we design TeraPipe, a high-performance token-level pipeline parallel algorithm for synchronous model-parallel training of Transformer-based language models. We develop a novel dynamic programming-based algorithm to calculate the optimal pipelining execution scheme given a specific model and cluster configuration. We show that TeraPipe can speed up the training by 5.0x for the largest GPT-3 model with 175 billion parameters on an AWS cluster with 48 p3.16xlarge instances compared with state-of-the-art model-parallel methods. The code for reproduction can be found at https://github.com/zhuohan123/terapipe

[ ]
Tue 20 July 17:25 - 17:30 PDT
##### Quantization Algorithms for Random Fourier Features

Xiaoyun Li · Ping Li

The method of random projection (RP) is the standard technique for dimensionality reduction, approximate near neighbor search, compressed sensing, etc., which provides a simple and effective scheme for approximating pairwise inner products and Euclidean distances in massive data. Closely related to RP, the method of random Fourier features (RFF) has also become popular for approximating the (nonlinear) Gaussian kernel. RFF applies a specific nonlinear transformation on the projected data from RP. In practice, using the Gaussian kernel often leads to better performance than the linear kernel (inner product). After random projections, quantization is an important step for efficient data storage, computation and transmission. Quantization for RP has been extensively studied in the literature. In this paper, we focus on developing quantization algorithms for RFF. The task is in a sense challenging due to the tuning parameter $\gamma$ in the Gaussian kernel. For example, the quantizer and the quantized data might be tied to each specific Gaussian kernel parameter $\gamma$. Our contribution begins with the analysis on the probability distributions of RFF, and an interesting discovery that the marginal distribution of RFF is free of the parameter $\gamma$. This significantly simplifies the design of the Lloyd-Max (LM) quantization scheme for RFF in that there would be only one LM quantizer (regardless of $\gamma$). Detailed theoretical analysis is provided on the kernel estimators and approximation error, and experiments confirm the effectiveness and efficiency of the proposed method.

[ ]
Tue 20 July 17:30 - 17:35 PDT
##### Rethinking Neural vs. Matrix-Factorization Collaborative Filtering: the Theoretical Perspectives

Da Xu · Chuanwei Ruan · Evren Korpeoglu · Sushant Kumar · Kannan Achan

The recent work by Rendle et al. (2020), based on empirical observations, argues that matrix-factorization collaborative filtering (MCF) compares favorably to neural collaborative filtering (NCF), and conjectures the dot product's superiority over the feed-forward neural network as similarity function. In this paper, we address the comparison rigorously by answering the following questions: 1. what is the limiting expressivity of each model; 2. under the practical gradient descent, to which solution does each optimization path converge; 3. how would the models generalize under the inductive and transductive learning setting. Our results highlight the similar expressivity for the overparameterized NCF and MCF as kernelized predictors, and reveal the relation between their optimization paths. We further show their different generalization behaviors, where MCF and NCF experience specific tradeoff and comparison in the transductive and inductive collaborative filtering setting. Lastly, by showing a novel generalization result, we reveal the critical role of correcting exposure bias for model evaluation in the inductive setting. Our results explain some of the previously observed conflicts, and we provide synthetic and real-data experiments to shed further insights to this topic.

[ ]
Tue 20 July 17:35 - 17:40 PDT
##### Concentric mixtures of Mallows models for top-$k$ rankings: sampling and identifiability

Fabien Collas · Ekhine IRUROZKI

In this paper, we study mixtures of two Mallows models for top-$k$ rankings with equal location parameters but with different scale parameters (a mixture of concentric Mallows models). These models arise when we have a heterogeneous population of voters formed by two populations, one of which is a subpopulation of expert voters. We show the identifiability of both components and the learnability of their respective parameters. These results are based upon, first, bounding the sample complexity for the Borda algorithm with top-$k$ rankings. Second, we characterize the distances between rankings, showing that an off-the-shelf clustering algorithm separates the rankings by components with high probability -provided the scales are well-separated.As a by-product, we include an efficient sampling algorithm for Mallows top-$k$ rankings. Finally, since the rank aggregation will suffer from a large amount of noise introduced by the non-expert voters, we adapt the Borda algorithm to be able to recover the ground truth consensus ranking which is especially consistent with the expert rankings.

[ ]
Tue 20 July 17:40 - 17:45 PDT
##### Heterogeneity for the Win: One-Shot Federated Clustering

Don Kurian Dennis · Tian Li · Virginia Smith

In this work, we explore the unique challenges---and opportunities---of unsupervised federated learning (FL). We develop and analyze a one-shot federated clustering scheme, kfed, based on the widely-used Lloyd's method for $k$-means clustering. In contrast to many supervised problems, we show that the issue of statistical heterogeneity in federated networks can in fact benefit our analysis. We analyse kfed under a center separation assumption and compare it to the best known requirements of its centralized counterpart. Our analysis shows that in heterogeneous regimes where the number of clusters per device $(k')$ is smaller than the total number of clusters over the network $k$, $(k'\le \sqrt{k})$, we can use heterogeneity to our advantage---significantly weakening the cluster separation requirements for kfed. From a practical viewpoint, kfed also has many desirable properties: it requires only round of communication, can run asynchronously, and can handle partial participation or node/network failures. We motivate our analysis with experiments on common FL benchmarks, and highlight the practical utility of one-shot clustering through use-cases in personalized FL and device sampling.

[ ]
Tue 20 July 17:45 - 17:50 PDT
##### Cross-Gradient Aggregation for Decentralized Learning from Non-IID Data

Yasaman Esfandiari · Sin Yong Tan · Zhanhong Jiang · Aditya Balu · Ethan Herron · Chinmay Hegde · Soumik Sarkar

Decentralized learning enables a group of collaborative agents to learn models using a distributed dataset without the need for a central parameter server. Recently, decentralized learning algorithms have demonstrated state-of-the-art results on benchmark data sets, comparable with centralized algorithms. However, the key assumption to achieve competitive performance is that the data is independently and identically distributed (IID) among the agents which, in real-life applications, is often not applicable. Inspired by ideas from continual learning, we propose Cross-Gradient Aggregation (CGA), a novel decentralized learning algorithm where (i) each agent aggregates cross-gradient information, i.e., derivatives of its model with respect to its neighbors' datasets, and (ii) updates its model using a projected gradient based on quadratic programming (QP). We theoretically analyze the convergence characteristics of CGA and demonstrate its efficiency on non-IID data distributions sampled from the MNIST and CIFAR-10 datasets. Our empirical comparisons show superior learning performance of CGA over existing state-of-the-art decentralized learning algorithms, as well as maintaining the improved performance under information compression to reduce peer-to-peer communication overhead. The code is available here on GitHub.

[ ]
Tue 20 July 17:50 - 17:55 PDT

[ ]