Machine Learning (Theory)


ICML papers

Tags: Machine Learning,Papers jl@ 7:25 am

Here are some ICML papers which interested me.

  1. Arindam Banerjee had a paper which notes that PAC-Bayes bounds, a core theorem in online learning, and the optimality of Bayesian learning statements share a core inequality in their proof.
  2. Pieter Abbeel, Morgan Quigley and Andrew Y. Ng have a paper discussing RL techniques for learning given a bad (but not too bad) model of the world.
  3. Nina Balcan and Avrim Blum have a paper which discusses how to learn given a similarity function rather than a kernel. A similarity function requires less structure than a kernel, implying that a learning algorithm using a similarity function might be applied in situations where no effective kernel is evident.
  4. Nathan Ratliff, Drew Bagnell, and Marty Zinkevich have a paper describing an algorithm which attempts to fuse A* path planning with learning of transition costs based on human demonstration.

Papers (2), (3), and (4), all seem like an initial pass at solving interesting problems which push the domain in which learning is applicable.

I’d like to encourage discussion of what papers interested you and why. Maybe we’ll all learn a little bit, and it’s very likely that we all missed interesting papers in a multitrack conference.


Presentation of Proofs is Hard.

Tags: Research,Teaching jl@ 7:38 pm

When presenting part of the Reinforcement Learning theory tutorial at ICML 2006, I was forcibly reminded of this.

There are several difficulties.

  1. When creating the presentation, the correct level of detail is tricky. With too much detail, the proof takes too much time and people may be lost to boredom. With too little detail, the steps of the proof involve too-great a jump. This is very difficult to judge.

    1. What may be an easy step in the careful thought of a quiet room is not so easy when you are occupied by the process of presentation.
    2. What may be easy after having gone over this (and other) proofs is not so easy to follow in the first pass by a viewer.

    These problems seem only correctable by process of repeated test-and-revise.

  2. When presenting the proof, simply speaking with sufficient precision is substantially harder than in normal conversation (where precision is not so critical). Practice can help here.
  3. When presenting the proof, going at the right pace for understanding is difficult. When we use a blackboard/whiteboard, a natural reasonable pace is imposed by the process of writing. Unfortunately, writing doesn’t scale well to large audiences for vision reasons, losing this natural pacing mechanism.
  4. It is difficult to entertain with a proof—there is nothing particularly funny about it. This particularly matters for a large audience which tends to naturally develop an expectation of being entertained.

Given all these difficulties, it is very tempting to avoid presenting proofs. Avoiding the proof in any serious detail is fairly reasonable in a conference presentation—the time is too short and the people viewing are too heavily overloaded to follow the logic well. The “right” level of detail is often the theorem statement.

Nevertheless, avoidance is not always possible because the proof is one of the more powerful mechanisms we have for doing research.


Online convex optimization at COLT

Tags: Machine Learning,Online,Papers jl@ 2:07 pm

At ICML 2003, Marty Zinkevich proposed the online convex optimization setting and showed that a particular gradient descent algorithm has regret O(T0.5) with respect to the best predictor where T is the number of rounds. This seems to be a nice model for online learning, and there has been some significant follow-up work.

At COLT 2006 Elad Hazan, Adam Kalai, Satyen Kale, and Amit Agarwal presented a modification which takes a Newton step guaranteeing O(log T) regret when the first and second derivatives are bounded. Then they applied these algorithms to portfolio management at ICML 2006 (with Robert Schapire) yielding some very fun graphs.


Regularization = Robustness

The Gibbs-Jaynes theorem is a classical result that tells us that the highest entropy distribution (most uncertain, least committed, etc.) subject to expectation constraints on a set of features is an exponential family distribution with the features as sufficient statistics. In math,

argmax_p H(p)
s.t. E_p[f_i] = c_i

is given by e^{\sum \lambda_i f_i}/Z. (Z here is the necessary normalization constraint, and the lambdas are free parameters we set to meet the expectation constraints).

A great deal of statistical mechanics flows from this result, and it has proven very fruitful in learning as well. (Motivating work in models in text learning and Conditional Random Fields, for instance. ) The result has been demonstrated a number of ways. One of the most elegant is the “geometric” version here.

In the case when the expectation constraints come from data, this tells us that the maximum entropy distribution is exactly the maximum likelihood distribution in the exponential family. It’s a surprising connection and the duality it flows from appears in a wide variety of work. (For instance, Martin Wainwright’s approximate inference techniques rely (in essence) on this result.)

In practice, we know that Maximum Likelihood with a lot of features is bound to overfit. The traditional trick is to pull a sleight of hand in the derivation. We start with the primal entropy problem, move to the dual, and in the dual add a “prior” that penalizes the lambdas. (Typically an l_1 or l_2 penalty or constraint.) This game is played in a variety of papers, and it’s a sleight of hand because the penalties don’t come from the motivating problem (the primal) but rather get tacked on at the end. In short: it’s a hack.

So I realized a few months back, that the primal (entropy) problem that regularization relates to is remarkably natural. Basically, it tells us that regularization in the dual corresponds directly to uncertainty (mini-max) about the constraints in the primal. What we end up with is a distribution p that is robust in the sense that it maximizes the entropy subject to a large set of potential constraints. More recently, I realized that I’m not even close to having been the first to figure that out. Miroslav Dudík, Steven J. Phillips and Robert E. Schapire, have a paper that derives this relation and then goes a step further to show what performance guarantees the method provides. It’s a great paper and I hope you get a chance to check it out:

Performance guarantees for regularized maximum entropy density estimation.

(Even better: if you’re attending ICML this year, I believe you will see Rob Schapire talk about some of this and related material as an invited speaker.)

It turns out the idea generalizes quite a bit. In Robust design of biological experiments. P. Flaherty, M. I. Jordan and A. P. Arkin show a related result where regularization directly follows from a robustness or uncertainty guarantee. And if you want the whole, beautiful framework you’re in luck. Yasemin Altun and Alex Smola have a paper (that I haven’t yet finished, but at least begins very well) that generalizes the regularized maximum entropy duality to a whole class of statistical inference procedures. If you’re at COLT, you can check this out as well.

Unifying Divergence Minimization and Statistical Inference via Convex Duality

The deep, unifying result seems to be what the title of the post says: robustness = regularization. This viewpoint makes regularization seem like much less of a hack, and goes further in suggesting just what range of constants might be reasonable. The work is very relevant to learning, but the general idea goes beyond to various problems where we only approximately know constraints.


IJCAI is out of season

IJCAI is running January 6-12 in Hyderabad India rather than a more traditional summer date. (Presumably, this is to avoid melting people in the Indian summer.)

The paper deadline(June 23 abstract / June 30 submission) are particularly inconvenient if you attend COLT or ICML. But on the other hand, it’s a good excuse to visit India.

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