Doubly-stochastic mining for heterogeneous retrieval

Modern retrieval problems are characterised by training sets with potentially billions of labels, and heterogeneous data distributions across subpopulations (e.g., users of a retrieval system may be from different countries), each of which poses a challenge. The first challenge concerns scalability: with a large number of labels, standard losses are difficult to optimise even on a single example. The second challenge concerns uniformity: one ideally wants good performance on each subpopulation. While several solutions have been proposed to address the first challenge, the second challenge has received relatively less attention. In this paper, we propose doubly-stochastic mining (S2M ), a stochastic optimization technique that addresses both challenges. In each iteration of S2M, we compute a per-example loss based on a subset of hardest labels, and then compute the minibatch loss based on the hardest examples. We show theoretically and empirically that by focusing on the hardest examples, S2M ensures that all data subpopulations are modelled well.

Federated Learning with Only Positive Labels

We consider learning a multi-class classification model in the federated setting, where each user has access to the positive data associated with only a single class. As a result, during each federated learning round, the users need to locally update the classifier without having access to the features and the model parameters for the negative classes. Thus, naively employing conventional decentralized learning such as the distributed SGD or Federated Averaging may lead to trivial or extremely poor classifiers. In particular, for the embedding based classifiers, all the class embeddings might collapse to a single point. To address this problem, we propose a generic framework for training with only positive labels, namely Federated Averaging with Spreadout (FedAwS), where the server imposes a geometric regularizer after each round to encourage classes to be spreadout in the embedding space. We show, both theoretically and empirically, that FedAwS can almost match the performance of conventional learning where users have access to negative labels. We further extend the proposed method to the settings with large output spaces.

Robust Large-Margin Learning in Hyperbolic Space

Recently, there has been a surge of interest in representation learning in hyperbolic spaces, driven by their ability to represent hierarchical data with significantly fewer dimensions than standard Euclidean spaces. However, the viability and benefits of hyperbolic spaces for downstream machine learning tasks have received less attention. In this paper, we present, to our knowledge, the first theoretical guarantees for learning a classifier in hyperbolic rather than Euclidean space. Specifically, we consider the problem of learning a large-margin classifier for data possessing a hierarchical structure. Our first contribution is a hyperbolic perceptron algorithm, which provably converges to a separating hyperplane. We then provide an algorithm to efficiently learn a large-margin hyperplane, relying on the careful injection of adversarial examples. Finally, we prove that for hierarchical data that embeds well into hyperbolic space, the low embedding dimension ensures superior guarantees when learning the classifier directly in hyperbolic space.

Reliable Distributed Clustering with Redundant Data Assignment

In this paper, we present distributed generalized clustering algorithms that can handle large scale data across multiple machines in spite of straggling or unreliable machines. We propose a novel data assignment scheme that enables us to obtain global information about the entire data even when some machines fail to respond with the results of the assigned local computations. The assignment scheme leads to distributed algorithms with good approximation guarantees for a variety of clustering and dimensionality reduction problems.

Low-Rank Bottleneck in Multi-head Attention Models

Attention based Transformer architecture has enabled significant advances in the field of natural language processing. In addition to new pre-training techniques, recent improvements crucially rely on working with a relatively larger embedding dimension for tokens. Unfortunately, this leads to models that are prohibitively large to be employed in the downstream tasks. In this paper we identify one of the important factors contributing to the large embedding size requirement. In particular, our analysis highlights that the scaling between the number of heads and the size of each head in the current architecture gives rise to a low-rank bottleneck in attention heads, causing this limitation. We further validate this in our experiments. As a solution we propose to set the head size of an attention unit to input sequence length, and independent of the number of heads, resulting in multi-head attention layers with provably more expressive power. We empirically show that this allows us to train models with a relatively smaller embedding dimension and with better performance scaling.

Are Transformers universal approximators of sequence-to-sequence functions?

Despite the widespread adoption of Transformer models for NLP tasks, the expressive power of these models is not well-understood. In this paper, we establish that Transformer models are universal approximators of continuous permutation equivariant sequence-to-sequence functions with compact support, which is quite surprising given the amount of shared parameters in these models. Furthermore, using positional encodings, we circumvent the restriction of permutation equivariance, and show that Transformer models can universally approximate arbitrary continuous sequence-to-sequence functions on a compact domain. Interestingly, our proof techniques clearly highlight the different roles of the self-attention and the feed-forward layers in Transformers. In particular, we prove that fixed width self-attention layers can compute contextual mappings of the input sequences, playing a key role in the universal approximation property of Transformers. Based on this insight from our analysis, we consider other simpler alternatives to self-attention layers and empirically evaluate them.

Sampled Softmax with Random Fourier Features

The computational cost of training with softmax cross entropy loss grows linearly with the number of classes. For the settings where a large number of classes are involved, a common method to speed up training is to sample a subset of classes and utilize an estimate of the gradient based on these classes, known as the sampled softmax method. However, the sampled softmax provides a biased estimate of the gradient unless the samples are drawn from the exact softmax distribution, which is again expensive to compute. Therefore, a widely employed practical approach (without theoretical justification) involves sampling from a simpler distribution in the hope of approximating the exact softmax distribution. In this paper, we develop the first theoretical understanding of the role that different sampling distributions play in determining the quality of sampled softmax. Motivated by our analysis and the work on kernel-based sampling, we propose the Random Fourier Softmax (RF-softmax) method that utilizes the powerful Random Fourier features to enable more efficient and accurate sampling from the (approximate) softmax distribution. We show that RF-softmax leads to low bias in estimation in terms of both the full softmax distribution and the full softmax gradient. Furthermore, the cost of RF-softmax scales only logarithmically with the number of classes.

Lifting high-dimensional nonlinear models with Gaussian regressors

We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional data $\mathbf{y}_i=f(\mathbf{a}_i^T\mathbf{x}_0)$ for some nonlinear (and potentially unknown) link function $f$, when the regressors $\mathbf{a}_i$ are iid Gaussian. Brillinger (1982) showed that ordinary least-squares estimates $\mathbf{x}_0$ up to a constant of proportionality $\mu_\ell$, which depends on $f$. Recently, Plan & Vershynin (2015) extended this result to the high-dimensional setting deriving sharp error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $\mu_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an alternative convex recovery method. In a nutshell, our method treats such link functions as if they were linear in a lifted space of higher-dimension. Interestingly, our error analysis captures the effect of both the nonlinearity and the problem's geometry in a few simple summary parameters.

Robust Gradient Descent via Moment Encoding with LDPC Codes

This paper considers the problem of implementing large-scale gradient descent algorithms in a distributed computing setting in the presence of {\em straggling} processors. To mitigate the effect of the stragglers, it has been previously proposed to encode the data with an erasure-correcting code and decode at the master server at the end of the computation. We, instead, propose to encode the second-moment of the data with a low density parity-check (LDPC) code. The iterative decoding algorithms for LDPC codes have very low computational overhead and the number of decoding iterations can be made to automatically adjust with the number of stragglers in the system. We show that for a random model for stragglers, the proposed moment encoding based gradient descent method can be viewed as the stochastic gradient descent method. This allows us to obtain convergence guarantees for the proposed solution. Furthermore, the proposed moment encoding based method is shown to outperform the existing schemes in a real distributed computing setup.

Representation Learning and Recovery in the ReLU Model

Rectified linear units, or ReLUs, have become the preferred activation function for artificial neural networks. In this paper we consider two basic learning problems assuming that the underlying data follow a generative model based on a ReLU-network -- a neural network with ReLU activations. As a primarily theoretical study, we limit ourselves to a single-layer network. The first problem we study corresponds to dictionary-learning in the presence of nonlinearity (modeled by the ReLU functions). Given a set of observation vectors $\mathbf{y}^i \in \mathbb{R}^d, i =1, 2, \dots , n$, we aim to recover $d\times k$ matrix $A$ and the latent vectors $\{\mathbf{c}^i\} \subset \mathbb{R}^k$ under the model $\mathbf{y}^i = \mathrm{ReLU}(A\mathbf{c}^i +\mathbf{b})$, where $\mathbf{b}\in \mathbb{R}^d$ is a random bias. We show that it is possible to recover the column space of $A$ within an error of $O(d)$ (in Frobenius norm) under certain conditions on the probability distribution of $\mathbf{b}$. The second problem we consider is that of robust recovery of the signal in the presence of outliers, i.e., large but sparse noise. In this setting we are interested in recovering the latent vector $\mathbf{c}$ from its noisy nonlinear sketches of the form $\mathbf{v} = \mathrm{ReLU}(A\mathbf{c}) + \mathbf{e}+\mathbf{w}$, where $\mathbf{e} \in \mathbb{R}^d$ denotes the outliers with sparsity $s$ and $\mathbf{w} \in \mathbb{R}^d$ denote the dense but small noise. This line of work has recently been studied (Soltanolkotabi, 2017) without the presence of outliers. For this problem, we show that a generalized LASSO algorithm is able to recover the signal $\mathbf{c} \in \mathbb{R}^k$ within an $\ell_2$ error of $O(\sqrt{\frac{(k+s)\log d}{d}})$ when $A$ is a random Gaussian matrix.