Saturday, October 29, 2016

Unsupervised Learning II: the power of improper convex relaxations

In this continuation post I'd like to bring out a critical component of learning theory that is essentially missing in today's approaches for unsupervised learning: improper learning by convex relaxations.

Consider the task of learning a sparse linear separator. The sparsity, or $\ell_0$, constraint is non-convex and computationally hard. Here comes the Lasso - replace the $\ell_0$ constraint with $\ell_1$ convex relaxation --- et voila --- you've enabled polynomial-time solvable convex optimization.

Another more timely example: latent variable models for recommender systems, a.k.a. the matrix completion problem. Think of a huge matrix, with one dimension corresponding to users and the other corresponding to media items (e.g. movies as in the Netflix problem). Given a set of entries in the matrix, corresponding to ranking of movies that the users already gave feedback on, the problem is to complete the entire matrix and figure out the preferences of people on movies they haven't seen.
This is of course ill posed without further assumptions. The low-rank assumption intuitively states that people's preferences are governed by few factors (say genre, director, etc.).  This corresponds to the user-movie matrix having low algebraic rank.

Completing a low-rank matrix is NP-hard in general. However, stemming from the compressed-sensing literature, the statistical-reconstruction approach is to assume additional statistical and structural properties over the user-movie matrix. For example, if this matrix is "incoherent", and the observations of the entires are obtained via a uniform distribution, then this paper by Emmanuel Candes and Ben Recht shows efficient reconstruction is still possible.

But is there an alternative to incoherent assumptions such as incoherence and i.i.d. uniform observations?

A long line of research has taken the "improper learning by convex relaxation approach" and applied it to matrix completion in works such as Srebro and Shraibman, considering convex relaxations to rank such as the trace norm and the max norm. Finally, in  joint work with Satyen Kale and Shai Shalev-Shwartz, we take this approach to the extreme by not requiring any distribution at all, giving regret bounds in the online convex optimization model (see previous post).

By the exposition above one may think that this technique of improper convex relaxations applies only to problems whose hardness comes from "sparsity".  This is far from the truth, and in the very same paper referenced above, we show how the technique can be applied to combinatorial problems such as predicting cuts in graphs, and outcomes of matches in tournaments.

In fact, improper learning is such a powerful concept, that problems such that the complexity of problems such as learning DNF formulas has remained open for quite a long time, despite the fact that showing proper learnability was long known to be NP-hard.

On the other hand, improper convex relaxation is not an all powerful magic pill. When designing convex relaxations to learning problems, special care is required so not to increase sample complexity. Consider for example the question of predicting tournaments, as given in this COLT open problem formulation by Jake Abernethy. The problem, loosely stated, is to iteratively predict the outcome of a match between two competing teams from a league of N teams. The goal is to compete, or make as few mistakes, as the best ranking of teams in hindsight. Since the number of rankings scales as $N!$, the multiplicative weights update method can be used to guarantee regret that scales as $\sqrt{T \log N!} = O(\sqrt{T N \log N})$. However, the latter, naively implemented, runs in time proportional to $N!$. Is there an efficient algorithm for the problem attaining similar regret bounds?

A naive improper learning relaxation would treat each pair of competing teams as an instance of binary prediction, for which we have efficient algorithms. The resulting regret bound, however, would scale as the number of pairs of teams over $N$ candidate teams, or as $\sqrt{T N^2}$, essentially removing any meaningful prediction property. What has gone wrong?

By removing the restriction of focus from rankings to pairs of teams, we have enlarged the decision set significantly, and increased the number of examples needed to learn. This is a general and important concern for improper convex relaxations: one needs to relax the problem in such a way that sample complexity (usually measured in terms of Rademacher complexity) doesn't explode. For the aforementioned problem of predicting tournaments, a convex relaxation that preserves sample complexity up to logarithmic factors is indeed possible, and described in the same paper.

In the coming posts we'll describe a framework for unsupervised learning that allows us to use the power of improper convex relaxations.

Thursday, October 6, 2016

A mathematical definition of unsupervised learning?

Extracting structure from data is considered by many to be the frontier of machine learning. Yet even defining "structure", or the very goal of learning structure from unlabelled data, is not well understood or rigorously defined.

In this post we'll give a little background to unsupervised learning and argue that, compared to supervised learning, we lack a well-founded theory to tackle even the most basic questions. In the next post, we'll introduce a new candidate framework.

Unsupervised learning is a commonplace component of many undergraduate machine learning courses and books. At the core, the treatment of unsupervised learning is a collection of methods for analyzing data and extracting hidden structure from this data.  Take for example John Duchi's "Machine Learning" course at Stanford. Under "Unsupervised learning", we have the topics: Clustering, K-means, EM. Mixture of Gaussians, Factor analysis, PCA (Principal components analysis), ICA (Independent components analysis). This is an excellent representation of the current state-of-the-art:

  • Clustering and K-means:  by grouping "similar" elements together, usually according to some underlying metric, we can create artificial "labels" for data elements in hope that this rough labelling will be of future use, either in a supervised learning task on the same dataset, or in "understanding" the data. K-means is a widely used algorithm for finding k representative centers of the data, allowing for k "labels".
  • Mixture of Gaussians: an exemplary and widely used method stemming from the generative approach to unsupervised learning. This approach stipulates that the data is generated from a distribution, in this case a mixture of normal random variables. There are numerous algorithms for finding the centers of these Gaussians, thereby learning the structure of the data.
  • Factor analysis, PCA and ICA: the spectral approach to unsupervised learning is perhaps the most widely used, in which the covariance matrix of the data is approximated by the largest few singular vectors. This type of clustering is widely successful in text (word embeddings) and many other forms of data. 

The above techniques are widely used and successful in practice, and under suitable conditions polynomial-time algorithms are known. The common thread amongst them is finding structure in the data, which turns out for many practical applications to be useful in a variety of supervised learning tasks.

Notable unsupervised learning techniques that are missing from the above list are Restricted Boltzman machines (RBMs), Autoencoders and Dictionary Learning (which was conceived in the context of deep neural networks). The latter techniques attempt to find a succinct representation of the data. Various algorithms and heuristics for RBMs, Autotencoders and Dictionary Learning exist, but these satisfy at least one of the following:

1) come without rigorous performance guarantees.

2) run in exponential time in the worst case.

3) assume strong generative assumptions about the input.

Recent breakthroughs in polynomial time unsupervised learning due to Sanjeev and his group address points (1) & (2) above and require only (3). Independently the same is also achievable by the method of moments, see e.g. this paper, originally from Sham's group @ MSR New England, and many more recent advances. What's the downside of this approach? The Arora-Hazan debate over this point, which the theory-ML students are exposed to in our weekly seminar, is subject for a longer post...

What is missing then? Compare the above syllabus with that of supervised learning, in most undergrad courses and books, the difference in terms of theoretical foundations is stark. For supervised learning  we have Vapnik's statistical learning theory -  a deep and elegant mathematical theory that classifies exactly the learnable from labeled real-valued examples. Valiant's computational learning theory adds the crucial element of computation, and over the combined result we have hundreds of scholarly articles describing in detail problems that are learnable efficiently / learnable but inefficiently /  improperly learnable / various other notions.

Is there meaning, in pure mathematical sense such as Vapnik and Valiant instilled into supervised learning, to the unsupervised world?

I like to point out in my ML courses that computational learning theory say something philosophical about science: the concept of "learning", or a "theory" such as a physical theory of the world, has a precise mathematical meaning independent of humans. While many of us scientists seek a "meaningful" theory in the human sense, there need not be one! It could very well be that a physical theory, for example that of condensed matter, has a "theory" in the Vapnik-Valiant sense, but not one that would be as "meaningful" and "elegant" in the Feynman sense.

How can we then, give mathematical meaning to unsupervised learning in a way that:

  1. Allows natural notions of generalization from examples
  2. Improves future supervised learning tasks regardless of their nature
  3. Allows for natural family of algorithms (hopefully but not necessarily - based on convex optimization)

This will be the starting point for our next post...