Machine Learning (Theory)


Compositional Machine Learning Algorithm Design

There were two papers at ICML presenting learning algorithms for a contextual bandit-style setting, where the loss for all labels is not known, but the loss for one label is known. (The first might require a exploration scavenging viewpoint to understand if the experimental assignment was nonrandom.) I strongly approve of these papers and further work in this setting and its variants, because I expect it to become more important than supervised learning. As a quick review, we are thinking about situations where repeatedly:

  1. The world reveals feature values (aka context information).
  2. A policy chooses an action.
  3. The world provides a reward.

Sometimes this is done in an online fashion where the policy can change based on immediate feedback and sometimes it’s done in a batch setting where many samples are collected before the policy can change. If you haven’t spent time thinking about the setting, you might want to because there are many natural applications.

I’m going to pick on the Banditron paper (second one), which attacks the special case of the contextual bandit setting where exactly one of the rewards is 1 and all other actions result in reward 0, and show that (a) similar performance is achievable via a simple combination of existing modular technologies and (b) superior performance is achievable by optimizing some existing modular technologies for the realizable case. This algorithm is the hardest of the two to compete with, because it explicitly deals with the explore/exploit tradeoff. Note that I’m definitely not trying to minimize the paper—there is analysis in that paper which remains interesting to me and isn’t covered by what follows. I am happy that it was published. The point of this post is showing that a modular approach to building learning algorithms is a strong contender when we encounter new learning problems.

Given the problem statement, my approach to solving the problem would be to compose some modular technologies I know.

  1. Perceptron learning algorithm We chose a Binary Perceptron as a classification algorithm. This choice is intrinsically motivated by the great computational performance of a Perceptron. It’s also the closest binary supervised learning algorithm to the Banditron, which eliminates a source of variation in comparison. We could have easily chosen a different base learning algorithm, and in many applications this is highly desirable.
  2. Offset Tree Reduction The offset tree is a newer machine learning reduction from the contextual bandit setting to the standard supervised learning setting. It more robustly transforms a supervised learner’s performance into good policy performance than any other reduction. The offset tree also has good computational properties, since it produces at most log2 k binary examples per train or test event, where k is the number of actions. In some sense it’s unfair to include the offset tree because it hasn’t yet been formally published. In another sense, that’s what this post is about.
  3. Epoch-Greedy exploration. The Epoch-greedy approach shows how to handle the explore/exploit tradeoff for learning in a contextual bandit setting as a function of a sample complexity bound. For common sample complexity bounds, we get an O(T2/3) online regret where T is the total number of timesteps.
  4. Occam’s Razor Bound The Occam’s Razor bound limits the regret of an empirical error minimizing perceptron as function of the number of examples. The bound (and it’s many cousins) are often loose, so the only thing we’ll really use is the denominator which says that regret scales as 1/sqrt(number of training examples) in the worst case. Applying Epoch-Greedy to the Occam’s Razor bound gives you an exploration probability of about C/t1/3 where t is the round number.

Each component above has been analyzed in isolation and is at least a reasonable approach (some are the best possible). Each of these components is also composable. Fitting these pieces together, we get an online learning algorithm (agnostic offset-tree perceptron) that chooses to explore uniformly at random amongst the actions with probability about 1/t1/3. How well does it perform? On the 4 class reuters based dataset used in the Banditron paper, we get the following accumulated average error rates with some code.:

The right plot is from the Banditron paper. The Perceptron line in both plots is for an algorithm which learns knowing the full loss function of each example, so it represents an ideal we don’t expect to achieve here. There are three results in the left plot:

  1. The blue line is a version of the component set where the Occam’s Razor bound and the Offset Tree reduction have been optimized for the realizable case. This was the first thing we tested (and it’s the result I mentioned at Shai‘s ICML talk). It turns out this approach works substantially better than the Banditron, achieving an error rate about halfway between the Banditron and the Multiclass Perceptron. The two components that we tweaked are:
    1. Realizable case Bound It’s well known that in the realizable case the regret of a chosen classifier should scale as 1/t rather than 1/t0.5. Plugging this into epoch-greedy, we get that the probability of exploration should be about 1/t0.5.
    2. Realizable case Offset Tree A basic observation is that in the realizable setting, every observation should create an example to tune the learning algorithm. In the context of the perceptron, this implies every error creates an example. The offset tree reduction can be altered to take this into account by eliminating the importance weight from all updates, and updating even for exploitation examples which are not drawn uniform randomly.
  2. The red line is what you get with exactly the component set stated above. We were curious about the degree to which a general purpose algorithm can perform well on this application as the realizable case algorithm is definitely broken when the problem is inherently noisy. The performance is somewhat worse than Banditron. I believe this is because it explores only about 1% of the time while the Banditron plot comes from exploring about 5% of the time.
  3. The green line is from a component set where epoch greedy and the offset tree have been tweaked to keep track of and use the distribution over actions at every timestep. This tweaks allows the amount of exploration as measured by the sum of importance weights of training examples to almost double. As we see, this approach improves performance, almost as if we doubled the number of examples, giving it similar performance to the Banditron. The tweaks used for the component set are:
    1. Stochastic Epoch-Greedy Instead of deterministically exploring every 1/(bound_gap) times, choose to explore with probability 1/(bound_gap), and pass this probability to the offset tree reduction.
    2. Nonuniform Offset Tree Tweak the Offset Tree in the obvious way to take into account nonuniform exploration. In particular, 1/2 is replaced with K/p(a) where p(a) is the probability of the action taken conditioned on one of two actions being taken. We set K so that this value is 1 when a nonexploit action is taken, which implies the importance weight is p(a)/(2-p(a)) when the exploit action is taken.
    3. Importance Weighted Perceptron We dealt with importance weights generated by the offset tree reduction by scaling any update by the importance weight.

People may be dissatisfied with the component assembly approach to learning algorithm design, because all of the pieces are not analyzed together. For example, the Banditron paper essentially proves that certain conditions are sufficient for the Banditron algorithm to perform well. This is a more complete guarantee than “all the pieces we stuck together are known to work well when analyzed in isolation”, but this style of guarantee has limitations which are both obvious and often overlooked. These guarantees do not show necessity of these conditions, and hence characterize only a subset of settings where the Banditron works. Unless you are lucky enough to know that your application satisfies the sufficient conditions, you’ll basically have to try the Banditron and see if it works for any particular application. Another drawback is that the complexity of analyzing all the different pieces simultaneously means that it’s difficult to use the best elements together. This last point is what leaves me dissatisfied with the complete analysis approach—it produces algorithms which are simple to analyze rather than optimized for performance.

If you are thinking “I have a better algorithm for solving one problem”, then you’ve missed the point of this post. The point of this post is that compositional design is good for solving many learning problems. This post is about one example of that approach in action. We start by assembling a set of components which we know work well from individual component analysis. Then, we try to optimize the performance of the assembly by swapping components or improving individual components to address known properties of the problem or observed deficiencies. In this particular case, we end up with a better performing algorithm and better components (such as stochastic epoch greedy) which are directly reusable in other settings. The essence of this approach is understanding that there is a real vocabulary of interchangeable components and actively using it in designing learning algorithms.

I would like to thank Alina who helped substantially with this post. I would also like to thank Shai for providing data and helping setup a clean comparison and Sham for helpful discussion.


Interesting papers at COLT (and a bit of UAI & workshops)

Here are a few papers from COLT 2008 that I found interesting.

  1. Maria-Florina Balcan, Steve Hanneke, and Jenn Wortman, The True Sample Complexity of Active Learning. This paper shows that in an asymptotic setting, active learning is always better than supervised learning (although the gap may be small). This is evidence that the only thing in the way of universal active learning is us knowing how to do it properly.
  2. Nir Ailon and Mehryar Mohri, An Efficient Reduction of Ranking to Classification. This paper shows how to robustly rank n objects with n log(n) classifications using a quicksort based algorithm. The result is applicable to many ranking loss functions and has implications for others.
  3. Michael Kearns and Jennifer Wortman. Learning from Collective Behavior. This is about learning in a new model, where the goal is to predict how a collection of interacting agents behave. One claim is that learning in this setting can be reduced to IID learning.

Due to the relation with Metric-E3, I was particularly interested in a couple other papers on reinforcement learning in navigation-like spaces.
I also particularly enjoyed Dan Klein‘s talk, which was the most impressive application of graphical model technology I’ve seen.

I also attended the large scale learning challenge workshop and enjoyed Antoine Bordes talk about a fast primal space algorithm that won by a hair over other methods in the wild track. Ronan Collobert‘s talk was also notable in that they are doing relatively featuritis-free NLP.


Interesting papers, ICML 2008

Here are some papers from ICML 2008 that I found interesting.

  1. Risi Kondor and Karsten Borgwardt, The Skew Spectrum of Graphs. This paper is about a new family of functions on graphs which is invariant under node label permutation. They show that these quantities appear to yield good features for learning.
  2. Sanjoy Dasgupta and Daniel Hsu. Hierarchical sampling for active learning. This is the first published practical consistent active learning algorithm. The abstract is also pretty impressive.
  3. Lihong Li, Michael Littman, and Thomas Walsh Knows What It Knows: A Framework For Self-Aware Learning. This is an attempt to create learning algorithms that know when they err, (other work includes Vovk). It’s not yet clear to me what the right model for feature-dependent confidence intervals is.
  4. Novi Quadrianto, Alex Smola, TIberio Caetano, and Quoc Viet Le Estimating Labels from Label Proportions. This is an example of learning in a specialization of the offline contextual bandit setting.
  5. Filip Radlinski, Robert Kleinberg and Thorsten Joachims
    Learning Diverse Rankings with Multi-Armed Bandits. Learning should be used to solve the diversity problem, and doing it in an online bandit-like setting is quite natural. I believe the setting can be generalized to a setting with features without too much work.
  6. Rich Caruana, Nikos Karampatziakis, Ainur Yessenalina An Empirical Evaluation of Supervised Learning in High Dimensions. This paper doesn’t need an abstract given the title :). I hadn’t previously appreciated how well a random forest works in high dimensions.
  7. Sham M. Kakade, Shai Shalev-Shwartz, and Ambuj Tewari. Efficient Bandit Algorithms for Online Multiclass Prediction. A paper about an online contextual bandit setting specialized to a multiclass realizable yet otherwise adversarial setting that yields a practical algorithm.

I’d like to add that I thought the conference organization (and the colocation with COLT and UAI) are particularly well done, the best I’ve seen. The key seems to be tight integration of the colocating conference programs and hordes of local volunteers making sure everything is working. I was also happy to see a 10 years award for best paper 10 years ago.


To Dual or Not

Tags: Machine Learning,Online jl@ 11:26 pm

Yoram and Shai‘s online learning tutorial at ICML brings up a question for me, “Why use the dual?”

The basic setting is learning a weight vector wi so that the function f(x)= sumi wi xi optimizes some convex loss function.

The functional view of the dual is that instead of (or in addition to) keeping track of wi over the feature space, you keep track of a vector aj over the examples and define wi = sumj aj xji.

The above view of duality makes operating in the dual appear unnecessary, because in the end a weight vector is always used. The tutorial suggests that thinking about the dual gives a unified algorithmic font for deriving online learning algorithms. I haven’t worked with the dual representation much myself, but I have seen a few examples where it appears helpful.

  1. Noise When doing online optimization (i.e. online learning where you are allowed to look at individual examples multiple times), the dual representation may be helpful in dealing with noisy labels.
  2. Rates One annoyance of working in the primal space with a gradient descent style algorithm is that finding the right learning rate can be a bit tricky.
  3. Kernelization Instead of explicitly computing the weight vector, the representation of the solution can be left in terms of ai, which is handy when the weight vector is only implicitly defined.

A substantial drawback of dual analysis is that it doesn’t seem to apply well in nonconvex situations. There is a reasonable (but not bullet-proof) argument that learning over nonconvex functions is unavoidable in solving some problems.


More Presentation Preparation

Tags: Machine Learning,Research jl@ 8:01 am

We’ve discussed presentation preparation before, but I have one more thing to add: transitioning. For a research presentation, it is substantially helpful for the audience if transitions are clear. A common outline for a research presentation in machine leanring is:

  1. The problem. Presentations which don’t describe the problem almost immediately lose people, because the context is missing to understand the detail.
  2. Prior relevant work. In many cases, a paper builds on some previous bit of work which must be understood in order to understand what the paper does. A common failure mode seems to be spending too much time on prior work. Discuss just the relevant aspects of prior work in the language of your work. Sometimes this is missing when unneeded.
  3. What we did. For theory papers in particular, it is often not possible to really cover the details. Prioritizing what you present can be very important.
  4. How it worked. Many papers in Machine Learning have some sort of experimental test of the algorithm. Sometimes this is missing when the work is theoretical.

What seems to often happen, is that there is no transitioning in the presentation. This can happen in one of two ways:

  1. Content Confusion. Sometimes the problem description is merged into (2), and (3). Sometimes (2) and (3) are merged. When this happens, it can be very difficult to follow. The solution is to rewrite to isolate the presentation components.
  2. Untransition. Sometimes the presentation does have a reasonable structure as above, but there are just no transitions in the delivery, creating apparent content confusion. This is easy to fix. An approach I often use is to just have an outline slide with the next subject highlighted between pieces of the transition. The delivery of the presentation can also handle this well. For example, have an extra long pause after stating the problem and check to see if the audience has questions.
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