Implicit Regularization in Hierarchical Tensor Factorization and Deep Convolutional Networks
The ability of large neural networks to generalize is commonly believed to stem from an implicit regularization — a tendency... Continue
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
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