Provable Algorithms for Machine Learning Problems

Provable Algorithms for Machine Learning Problems by Rong Ge.


Modern machine learning algorithms can extract useful information from text, images and videos. All these applications involve solving NP-hard problems in average case using heuristics. What properties of the input allow it to be solved effciently? Theoretically analyzing the heuristics is very challenging. Few results were known.

This thesis takes a di fferent approach: we identify natural properties of the input, then design new algorithms that provably works assuming the input has these properties. We are able to give new, provable and sometimes practical algorithms for learning tasks related to text corpus, images and social networks.

The first part of the thesis presents new algorithms for learning thematic structure in documents. We show under a reasonable assumption, it is possible to provably learn many topic models, including the famous Latent Dirichlet Allocation. Our algorithm is the first provable algorithms for topic modeling. An implementation runs 50 times faster than latest MCMC implementation and produces comparable results.

The second part of the thesis provides ideas for provably learning deep, sparse representations. We start with sparse linear representations, and give the fi rst algorithm for dictionary learning problem with provable guarantees. Then we apply similar ideas to deep learning: under reasonable assumptions our algorithms can learn a deep network built by denoising autoencoders.

The fi nal part of the thesis develops a framework for learning latent variable models. We demonstrate how various latent variable models can be reduced to orthogonal tensor decomposition, and then be solved using tensor power method. We give a tight sample complexity analysis for tensor power method, which reduces the number of sample required for learning many latent variable models.

In theory, the assumptions in this thesis help us understand why intractable problems in machine learning can often be solved; in practice, the results suggest inherently new approaches for machine learning. We hope the assumptions and algorithms inspire new research problems and learning algorithms.

Admittedly an odd notion, starting with the data rather than an answer and working back towards data but it does happen. 😉

Given the performance improvements for LDA (50X), I anticipate this approach being applied to algorithms for “big data.”

I first saw this in a tweet by Chris Deihl.

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