arXiv subjects

Sashank J. Reddi

Publications and source records attributed to Sashank J. Reddi.

At least 37 records · Page 2Linked to original sources

arXiv2021

Distilling Double Descent

Distillation is the technique of training a "student" model based on examples that are labeled by a separate "teacher" model, which itself is trained on a labeled dataset. The most common explanations for why distillation "works" are predicated on the assumption that student is provided with \emph{soft} labels, \eg probabilities or confidences, from the teacher model. In this work, we show, that, even when the teacher model is highly overparameterized, and provides \emph{hard} labels, using a very large held-out unlabeled dataset to train the student model can result in a model that outperforms more "traditional" approaches. Our explanation for this phenomenon is based on recent work on "double descent". It has been observed that, once a model's complexity roughly exceeds the amount required to memorize the training data, increasing the complexity \emph{further} can, counterintuitively, result in \emph{better} generalization. Researchers have identified several settings in which it takes place, while others have made various attempts to explain it (thus far, with only partial success). In contrast, we avoid these questions, and instead seek to \emph{exploit} this phenomenon by demonstrating that a highly-overparameterized teacher can avoid overfitting via double descent, while a student trained on a larger independent dataset labeled by this teacher will avoid overfitting due to the size of its training set.

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arXiv2020

$O(n)$ Connections are Expressive Enough: Universal Approximability of Sparse Transformers

Recently, Transformer networks have redefined the state of the art in many NLP tasks. However, these models suffer from quadratic computational cost in the input sequence length $n$ to compute pairwise attention in each layer. This has prompted recent research into sparse Transformers that sparsify the connections in the attention layers. While empirically promising for long sequences, fundamental questions remain unanswered: Can sparse Transformers approximate any arbitrary sequence-to-sequence function, similar to their dense counterparts? How does the sparsity pattern and the sparsity level affect their performance? In this paper, we address these questions and provide a unifying framework that captures existing sparse attention models. We propose sufficient conditions under which we prove that a sparse attention model can universally approximate any sequence-to-sequence function. Surprisingly, our results show that sparse Transformers with only $O(n)$ connections per attention layer can approximate the same function class as the dense model with $n^2$ connections. Lastly, we present experiments comparing different patterns/levels of sparsity on standard NLP tasks.

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arXiv2020

Adaptive Sampling Distributed Stochastic Variance Reduced Gradient for Heterogeneous Distributed Datasets

We study distributed optimization algorithms for minimizing the average of \emph{heterogeneous} functions distributed across several machines with a focus on communication efficiency. In such settings, naively using the classical stochastic gradient descent (SGD) or its variants (e.g., SVRG) with a uniform sampling of machines typically yields poor performance. It often leads to the dependence of convergence rate on maximum Lipschitz constant of gradients across the devices. In this paper, we propose a novel \emph{adaptive} sampling of machines specially catered to these settings. Our method relies on an adaptive estimate of local Lipschitz constants base on the information of past gradients. We show that the new way improves the dependence of convergence rate from maximum Lipschitz constant to \emph{average} Lipschitz constant across machines, thereby, significantly accelerating the convergence. Our experiments demonstrate that our method indeed speeds up the convergence of the standard SVRG algorithm in heterogeneous environments.

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arXiv2020

Why distillation helps: a statistical perspective

Knowledge distillation is a technique for improving the performance of a simple "student" model by replacing its one-hot training labels with a distribution over labels obtained from a complex "teacher" model. While this simple approach has proven widely effective, a basic question remains unresolved: why does distillation help? In this paper, we present a statistical perspective on distillation which addresses this question, and provides a novel connection to extreme multiclass retrieval techniques. Our core observation is that the teacher seeks to estimate the underlying (Bayes) class-probability function. Building on this, we establish a fundamental bias-variance tradeoff in the student's objective: this quantifies how approximate knowledge of these class-probabilities can significantly aid learning. Finally, we show how distillation complements existing negative mining techniques for extreme multiclass retrieval, and propose a unified objective which combines these ideas.

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arXiv2020

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.

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arXiv2020

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.

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arXiv2020

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.

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arXiv2020

Escaping Saddle Points with Adaptive Gradient Methods

Adaptive methods such as Adam and RMSProp are widely used in deep learning but are not well understood. In this paper, we seek a crisp, clean and precise characterization of their behavior in nonconvex settings. To this end, we first provide a novel view of adaptive methods as preconditioned SGD, where the preconditioner is estimated in an online manner. By studying the preconditioner on its own, we elucidate its purpose: it rescales the stochastic gradient noise to be isotropic near stationary points, which helps escape saddle points. Furthermore, we show that adaptive methods can efficiently estimate the aforementioned preconditioner. By gluing together these two components, we provide the first (to our knowledge) second-order convergence result for any adaptive method. The key insight from our analysis is that, compared to SGD, adaptive methods escape saddle points faster, and can converge faster overall to second-order stationary points.

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arXiv2019

AdaCliP: Adaptive Clipping for Private SGD

Privacy preserving machine learning algorithms are crucial for learning models over user data to protect sensitive information. Motivated by this, differentially private stochastic gradient descent (SGD) algorithms for training machine learning models have been proposed. At each step, these algorithms modify the gradients and add noise proportional to the sensitivity of the modified gradients. Under this framework, we propose AdaCliP, a theoretically motivated differentially private SGD algorithm that provably adds less noise compared to the previous methods, by using coordinate-wise adaptive clipping of the gradient. We empirically demonstrate that AdaCliP reduces the amount of added noise and produces models with better accuracy.

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arXiv2019

On the Convergence of Adam and Beyond

Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.

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arXiv2018

Stochastic Negative Mining for Learning with Large Output Spaces

We consider the problem of retrieving the most relevant labels for a given input when the size of the output space is very large. Retrieval methods are modeled as set-valued classifiers which output a small set of classes for each input, and a mistake is made if the label is not in the output set. Despite its practical importance, a statistically principled, yet practical solution to this problem is largely missing. To this end, we first define a family of surrogate losses and show that they are calibrated and convex under certain conditions on the loss parameters and data distribution, thereby establishing a statistical and analytical basis for using these losses. Furthermore, we identify a particularly intuitive class of loss functions in the aforementioned family and show that they are amenable to practical implementation in the large output space setting (i.e. computation is possible without evaluating scores of all labels) by developing a technique called Stochastic Negative Mining. We also provide generalization error bounds for the losses in the family. Finally, we conduct experiments which demonstrate that Stochastic Negative Mining yields benefits over commonly used negative sampling approaches.

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arXiv2017

Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifolds

We study optimization of finite sums of geodesically smooth functions on Riemannian manifolds. Although variance reduction techniques for optimizing finite-sums have witnessed tremendous attention in the recent years, existing work is limited to vector space problems. We introduce Riemannian SVRG (RSVRG), a new variance reduced Riemannian optimization method. We analyze RSVRG for both geodesically convex and nonconvex (smooth) functions. Our analysis reveals that RSVRG inherits advantages of the usual SVRG method, but with factors depending on curvature of the manifold that influence its convergence. To our knowledge, RSVRG is the first provably fast stochastic Riemannian method. Moreover, our paper presents the first non-asymptotic complexity analysis (novel even for the batch setting) for nonconvex Riemannian optimization. Our results have several implications; for instance, they offer a Riemannian perspective on variance reduced PCA, which promises a short, transparent convergence analysis.

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arXiv2016

AIDE: Fast and Communication Efficient Distributed Optimization

In this paper, we present two new communication-efficient methods for distributed minimization of an average of functions. The first algorithm is an inexact variant of the DANE algorithm that allows any local algorithm to return an approximate solution to a local subproblem. We show that such a strategy does not affect the theoretical guarantees of DANE significantly. In fact, our approach can be viewed as a robustification strategy since the method is substantially better behaved than DANE on data partition arising in practice. It is well known that DANE algorithm does not match the communication complexity lower bounds. To bridge this gap, we propose an accelerated variant of the first method, called AIDE, that not only matches the communication lower bounds but can also be implemented using a purely first-order oracle. Our empirical results show that AIDE is superior to other communication efficient algorithms in settings that naturally arise in machine learning applications.

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arXiv2016

Stochastic Frank-Wolfe Methods for Nonconvex Optimization

We study Frank-Wolfe methods for nonconvex stochastic and finite-sum optimization problems. Frank-Wolfe methods (in the convex case) have gained tremendous recent interest in machine learning and optimization communities due to their projection-free property and their ability to exploit structured constraints. However, our understanding of these algorithms in the nonconvex setting is fairly limited. In this paper, we propose nonconvex stochastic Frank-Wolfe methods and analyze their convergence properties. For objective functions that decompose into a finite-sum, we leverage ideas from variance reduction techniques for convex optimization to obtain new variance reduced nonconvex Frank-Wolfe methods that have provably faster convergence than the classical Frank-Wolfe method. Finally, we show that the faster convergence rates of our variance reduced methods also translate into improved convergence rates for the stochastic setting.

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arXiv2016

Fast Stochastic Methods for Nonsmooth Nonconvex Optimization

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we show provably faster convergence than batch proximal gradient descent. Finally, we prove global linear convergence rate for an interesting subclass of nonsmooth nonconvex functions, that subsumes several recent works. This paper builds upon our recent series of papers on fast stochastic methods for smooth nonconvex optimization [22, 23], with a novel analysis for nonconvex and nonsmooth functions.

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arXiv2016

Stochastic Variance Reduction for Nonconvex Optimization

We study nonconvex finite-sum problems and analyze stochastic variance reduced gradient (SVRG) methods for them. SVRG and related methods have recently surged into prominence for convex optimization given their edge over stochastic gradient descent (SGD); but their theoretical analysis almost exclusively assumes convexity. In contrast, we prove non-asymptotic rates of convergence (to stationary points) of SVRG for nonconvex optimization, and show that it is provably faster than SGD and gradient descent. We also analyze a subclass of nonconvex problems on which SVRG attains linear convergence to the global optimum. We extend our analysis to mini-batch variants of SVRG, showing (theoretical) linear speedup due to mini-batching in parallel settings.

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arXiv2016

Fast Incremental Method for Nonconvex Optimization

We analyze a fast incremental aggregated gradient method for optimizing nonconvex problems of the form $\min_x \sum_i f_i(x)$. Specifically, we analyze the SAGA algorithm within an Incremental First-order Oracle framework, and show that it converges to a stationary point provably faster than both gradient descent and stochastic gradient descent. We also discuss a Polyak's special class of nonconvex problems for which SAGA converges at a linear rate to the global optimum. Finally, we analyze the practically valuable regularized and minibatch variants of SAGA. To our knowledge, this paper presents the first analysis of fast convergence for an incremental aggregated gradient method for nonconvex problems.

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arXiv2016

On Variance Reduction in Stochastic Gradient Descent and its Asynchronous Variants

We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchronous versions of these algorithms---a crucial requirement for modern large-scale applications---have not been studied. We bridge this gap by presenting a unifying framework for many variance reduction techniques. Subsequently, we propose an asynchronous algorithm grounded in our framework, and prove its fast convergence. An important consequence of our general approach is that it yields asynchronous versions of variance reduction algorithms such as SVRG and SAGA as a byproduct. Our method achieves near linear speedup in sparse settings common to machine learning. We demonstrate the empirical performance of our method through a concrete realization of asynchronous SVRG.

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