Predicting Generalization using GANs

A central problem of generalization theory is the following: Given a training dataset and a deep net trained with that... Continue

Does Gradient Flow Over Neural Networks Really Represent Gradient Descent?

TL;DR A lot was said in this blog (cf. post by Sanjeev) about the importance of studying trajectories of gradient... Continue

Implicit Regularization in Tensor Factorization: Can Tensor Rank Shed Light on Generalization in Deep Learning?

In effort to understand implicit regularization in deep learning, a lot of theoretical focus is being directed at matrix factorization,... Continue

Rip van Winkle's Razor, a Simple New Estimate for Adaptive Data Analysis

Can you trust a model whose designer had access to the test/holdout set? This implicit question in Dwork et al... Continue

When are Neural Networks more powerful than Neural Tangent Kernels?

The empirical success of deep learning has posed significant challenges to machine learning theory: Why can we efficiently train neural... Continue

Beyond log-concave sampling (Part 3)

In the first post of this series, we introduced the challenges of sampling distributions beyond log-concavity. In Part 2 we... Continue

Beyond log-concave sampling (Part 2)

In our previous blog post, we introduced the challenges of sampling distributions beyond log-concavity. We first introduced the problem of... Continue

Can implicit regularization in deep learning be explained by norms?

This post is based on my recent paper with Noam Razin (to appear at NeurIPS 2020), studying the question of... Continue

How to allow deep learning on your data without revealing the data

Today’s online world and the emerging internet of things is built around a Faustian bargain: consumers (and their internet of... Continue

Mismatches between Traditional Optimization Analyses and Modern Deep Learning

You may remember our previous blog post showing that it is possible to do state-of-the-art deep learning with learning rate... Continue

Beyond log-concave sampling

As the growing number of posts on this blog would suggest, recent years have seen a lot of progress in... Continue

Training GANs - From Theory to Practice

GANs, originally discovered in the context of unsupervised learning, have had far reaching implications to science, engineering, and society. However,... Continue

An equilibrium in nonconvex-nonconcave min-max optimization

While there has been incredible progress in convex and nonconvex minimization, a multitude of problems in ML today are in... Continue

Exponential Learning Rate Schedules for Deep Learning (Part 1)

This blog post concerns our ICLR20 paper on a surprising discovery about learning rate (LR), the most basic hyperparameter in... Continue

Ultra-Wide Deep Nets and Neural Tangent Kernel (NTK)

(Crossposted at CMU ML.) Traditional wisdom in machine learning holds that there is a careful trade-off between training error and... Continue

Understanding implicit regularization in deep learning by analyzing trajectories of gradient descent

Sanjeev’s recent blog post suggested that the conventional view of optimization is insufficient for understanding deep learning, as the value... Continue

Landscape Connectivity of Low Cost Solutions for Multilayer Nets

A big mystery about deep learning is how, in a highly nonconvex loss landscape, gradient descent often finds near-optimal solutions... Continue

Is Optimization a Sufficient Language for Understanding Deep Learning?

In this Deep Learning era, machine learning usually boils down to defining a suitable objective/cost function for the learning task... Continue

Contrastive Unsupervised Learning of Semantic Representations: A Theoretical Framework

Semantic representations (aka semantic embeddings) of complicated data types (e.g. images, text, video) have become central in machine learning, and... Continue

The search for biologically plausible neural computation: A similarity-based approach

This is the second post in a series reviewing recent progress in designing artificial neural networks (NNs) that resemble natural... Continue

Understanding optimization in deep learning by analyzing trajectories of gradient descent

Neural network optimization is fundamentally non-convex, and yet simple gradient-based algorithms seem to consistently solve such problems. This phenomenon is... Continue

Simple and efficient semantic embeddings for rare words, n-grams, and language features

Distributional methods for capturing meaning, such as word embeddings, often require observing many examples of words in context. But most... Continue

When Recurrent Models Don't Need to be Recurrent

In the last few years, deep learning practitioners have proposed a litany of different sequence models. Although recurrent neural networks... Continue

Deep-learning-free Text and Sentence Embedding, Part 2

This post continues Sanjeev’s post and describes further attempts to construct elementary and interpretable text embeddings. The previous post described... Continue

Deep-learning-free Text and Sentence Embedding, Part 1

Word embeddings (see my old post1 and post2) capture the idea that one can express “meaning” of words using a... Continue

Limitations of Encoder-Decoder GAN architectures

This is yet another post about Generative Adversarial Nets (GANs), and based upon our new ICLR’18 paper with Yi Zhang.... Continue

Can increasing depth serve to accelerate optimization?

“How does depth help?” is a fundamental question in the theory of deep learning. Conventional wisdom, backed by theoretical studies... Continue

Proving generalization of deep nets via compression

This post is about my new paper with Rong Ge, Behnam Neyshabur, and Yi Zhang which offers some new perspective... Continue

Generalization Theory and Deep Nets, An introduction

Deep learning holds many mysteries for theory, as we have discussed on this blog. Lately many ML theorists have become... Continue

How to Escape Saddle Points Efficiently

A core, emerging problem in nonconvex optimization involves the escape of saddle points. While recent research has shown that gradient... Continue

Do GANs actually do distribution learning?

This post is about our new paper, which presents empirical evidence that current GANs (Generative Adversarial Nets) are quite far... Continue

Unsupervised learning, one notion or many?

Unsupervised learning, as the name suggests, is the science of learning from unlabeled data. A look at the wikipedia page... Continue

Generalization and Equilibrium in Generative Adversarial Networks (GANs)

The previous post described Generative Adversarial Networks (GANs), a technique for training generative models for image distributions (and other complicated... Continue

Generative Adversarial Networks (GANs), Some Open Questions

Since ability to generate “realistic-looking” data may be a step towards understanding its structure and exploiting it, generative models are... Continue

Back-propagation, an introduction

Given the sheer number of backpropagation tutorials on the internet, is there really need for another? One of us (Sanjeev)... Continue

The search for biologically plausible neural computation: The conventional approach

Inventors of the original artificial neural networks (NNs) derived their inspiration from biology. However, as artificial NNs progressed, their design... Continue

Gradient Descent Learns Linear Dynamical Systems

From text translation to video captioning, learning to map one sequence to another is an increasingly active research area in... Continue

Linear algebraic structure of word meanings

Word embeddings capture the meaning of a word using a low-dimensional vector and are ubiquitous in natural language processing (NLP).... Continue

A Framework for analysing Non-Convex Optimization

Previously Rong’s post and Ben’s post show that (noisy) gradient descent can converge to local minimum of a non-convex function,... Continue

Markov Chains Through the Lens of Dynamical Systems: The Case of Evolution

In this post, we will see the main technical ideas in the analysis of the mixing time of evolutionary Markov... Continue

Saddles Again

Thanks to Rong for the very nice blog post describing critical points of nonconvex functions and how to avoid them.... Continue

Escaping from Saddle Points

Convex functions are simple — they usually have only one local minimum. Non-convex functions can be much more complicated. In... Continue

Stability as a foundation of machine learning

Central to machine learning is our ability to relate how a learning algorithm fares on a sample to its performance... Continue

Evolution, Dynamical Systems and Markov Chains

In this post we present a high level introduction to evolution and to how we can use mathematical tools such... Continue

Word Embeddings: Explaining their properties

This is a followup to an earlier post about word embeddings, which capture the meaning of a word using a... Continue

NIPS 2015 workshop on non-convex optimization

While convex analysis has received much attention by the machine learning community, theoretical analysis of non-convex optimization is still nascent.... Continue

Nature, Dynamical Systems and Optimization

The language of dynamical systems is the preferred choice of scientists to model a wide variety of phenomena in nature.... Continue

Tensor Methods in Machine Learning

Tensors are high dimensional generalizations of matrices. In recent years tensor decompositions were used to design learning algorithms for estimating... Continue

Semantic Word Embeddings

This post can be seen as an introduction to how nonconvex problems arise naturally in practice, and also the relative... Continue

Why go off the convex path?

The notion of convexity underlies a lot of beautiful mathematics. When combined with computation, it gives rise to the area... Continue