Machine Learning (Theory)

8/6/2011

Interesting thing at UAI 2011

Tags: Conferences,Papers,Reinforcement jl@ 3:44 pm

I had a chance to attend UAI this year, where several papers interested me, including:

  1. Hoifung Poon and Pedro Domingos Sum-Product Networks: A New Deep Architecture. We’ve already discussed this one, but in a nutshell, they identify a large class of efficiently normalizable distributions and do learning with it.
  2. Yao-Liang Yu and Dale Schuurmans, Rank/norm regularization with closed-form solutions: Application to subspace clustering. This paper is about matrices, and in particular they prove that certain matrices are the solution of matrix optimizations. I’m not matrix inclined enough to fully appreciate this one, but I believe many others may be, and anytime closed form solutions come into play, you get 2 order of magnitude speedups, as they show experimentally.
  3. Laurent Charlin, Richard Zemel and Craig Boutilier, A Framework for Optimizing Paper Matching. This is about what works in matching papers to reviewers, as has been tested at several previous NIPS. We are looking into using this system for ICML 2012.

In addition I wanted to comment on Karl Friston‘s invited talk. At the outset, he made a claim that seems outlandish to me: The way the brain works is to minimize surprise as measured by a probabilistic model. The majority of the talk was not actually about this—instead it was about how probabilistic models can plausibly do things that you might not have thought possible, such as birdsong. Nevertheless, I think several of us in the room ended up stuck on the claim in questions afterward.

My personal belief is that world modeling (probabilistic or not) is a useful subroutine for intelligence, but it could not possibly be the entirety of intelligence. A key reason for this is the bandwidth of our senses—we simply take in too much information to model everything with equal attention. It seems critical for the efficient functioning of intelligence that only things which might plausibly matter are modeled, and only to the degree that matters. In other words, I do not model the precise placement of items on my desk, or even the precise content of my desk, because these details simply do not matter.

This argument can be made in another way. Suppose for the moment that all the brain does is probabilistic modeling. Then, the primary notion of failure to model is “surprise”, which is low probability events occurring. Surprises (stumbles, car wrecks, and other accidents) certainly can be unpleasant, but this could be correct if modeling is a subroutine as well. The clincher is that there are many unpleasant things which are not surprises, including keeping your head under water, fasting, and self-inflicted wounds.

Accounting for the unpleasantness of these events requires more than probabilistic modeling. In other words, it requires rewards, which is why reinforcement learning is important. As a byproduct, rewards also naturally create a focus of attention, addressing the computational efficiency issue. Believing that intelligence is just probabilistic modeling is another example of simple wrong answer.

1/24/2010

Specializations of the Master Problem

One thing which is clear on a little reflection is that there exists a single master learning problem capable of encoding essentially all learning problems. This problem is of course a very general sort of reinforcement learning where the world interacts with an agent as:

  1. The world announces an observation x.
  2. The agent makes a choice a.
  3. The world announces a reward r.

The goal here is to maximize the sum of the rewards over the time of the agent. No particular structure relating x to a or a to r is implied by this setting so we do not know effective general algorithms for the agent. It’s very easy to prove lower bounds showing that an agent cannot hope to succeed here—just consider the case where actions are unrelated to rewards. Nevertheless, there is a real sense in which essentially all forms of life are agents operating in this setting, somehow succeeding. The gap between these observations drives research—How can we find tractable specializations of the master problem general enough to provide an effective solution in real problems?

The process of specializing is a tricky business, as you want to simultaneously achieve tractable analysis, sufficient generality to be useful, and yet capture a new aspect of the master problem not otherwise addressed. Consider: How is it even possible to choose a setting where analysis is tractable before you even try to analyze it? What follows is my mental map of different specializations.

Online Learning

The online learning setting is perhaps the most satisfying specialization more general than standard batch learning at present, because it turns out to additionally provide tractable algorithms for many batch learning settings.

Standard online learning models specialize in two ways: You assume that the choice of action in step 2 does not influence future observations and rewards, and you assume additional information is available in step 3, a retrospectively available reward for each action. The algorithm for an agent in this setting typically has a given name—gradient descent, weighted majority, Winnow, etc…

The general algorithm here is a more refined version of follow-the-leader than in batch learning, with online update rules. An awesome discovery about this setting is that it’s possible to compete with a set of predictors even when the world is totally adversarial, substantially strengthening our understanding of what learning is and where it might be useful. For this adversarial setting, the algorithm alters into a form of follow-the-perturbed leader, where the learning algorithm randomizes it’s action amongst the set of plausible alternatives in order to defeat an adversary.

The standard form of argument in this setting is a potential argument, where at each step you show that if the learning algorithm performs badly, there is some finite budget from which an adversary deducts it’s ability. The form of the final theorem is that you compete with the accumulated reward of a set any one-step policies h:X – > A, with a dependence log(#policies) or weaker in regret, a measure of failure to compete.

A good basic paper to read here is:
Nick Littlestone and Manfred Warmuth, The Weighted Majority Algorithm, which shows the basic information-theoretic claim clearly. Vovk‘s page on aggregating algorithms is also relevant, although somewhat harder to read.

Provably computationally tractable special cases all have linear structure, either on rewards or policies. Good results are often observed empirically by applying backpropagation for nonlinear architectures, with the danger of local minima understood.

Bandit Analysis

In the bandit setting, step 1 is omitted, and the difficulty of the problem is weakened by assuming that action in step (2) don’t alter future rewards. The goal is generally to compete with all constant arm strategies.

Analysis in this basic setting started very specialized with Gittin’s Indicies and gradually generalized over time to include IID and fully adversarial settings, with EXP3 a canonical algorithm. If there are k strategies available, the standard theorem states that you can compete with the set of all constant strategies up to regret k. The most impressive theoretical discovery in this setting is that the dependence on T, the number of timesteps, is not substantially worse than supervised learning despite the need to explore.

Given the dependence on k all of these algorithms are computationally tractable.

However, the setting is flawed, because the set of constant strategies is inevitably too weak in practice—it’s an example of optimal decision making given that you ignore almost all information. Adding back the observation in step 1 allows competing with a large set of policies, while the regret grows only as log(#policies) or weaker. Canonical algorithms here are EXP4 (computationally intractable, but information theoretically near-optimal), Epoch-Greedy (computationally tractable given an oracle optimizer), and the Offset Tree providing a reduction to supervised binary classification.

MDP analysis

A substantial fraction of reinforcement learning has specialized on the Markov Decision Process setting, where the observation x is a state s, which is a sufficient statistic for predicting all future observations. Compared to the previous settings, dealing with time dependence is explicitly required, but learning typically exists in only primitive forms.

The first work here was in the 1950′s where the actual MDP was assumed known and the problem was simply computing a good policy, typically via dynamic programming style solutions. More recently, principally in the 1990′s, the setting where the MDP was not assumed known was analyzed. A very substantial theoretical advancement was the E3 algorithm which requires only O(S2A) experience to learn a near-optimal policy where the world is an MDP with S state and A actions per state. A further improvement on this is Delayed Q-Learning, where only O(SA) experience is required. There are many variants on the model-based approach and not much for the model-free approach. Lihong Li‘s thesis probably has the best detailed discussion at present.

There are some unsatisfactory elements of the analysis here. First, I’ve suppressed the dependence on the definition of “approximate” and the typical time horizon, for which the dependence is often bad and the optimality is unclear. The second is the dependence on S, which is intuitively unremovable, with this observation formalized in the lower bound Sham and I worked on (section 8.6 of Sham’s thesis). Empirically, these and related algorithms are often finicky, because in practice the observation isn’t a sufficient statistic and the number of states isn’t small, so approximating things as such is often troublesome.

A very different variant of this setting is given by Control theory, which I know less about than I should. The canonical setting for control theory is with a known MDP having linear transition dynamics. More exciting are the system identification problems where the system must be first identified. I don’t know any good relatively assumption free results for this setting.

Oracle Advice Shortcuts

Techniques here specialize the setting to situations in which some form of oracle advice is available when a policy is being learned. A good example of this is an oracle which provides samples from the distribution of observations visited by a good policy. Using this oracle, conservative policy iteration is guaranteed to perform well, so long as a base learning algorithm can predict well. This algorithm was refined and improved a bit by PSDP, which works via dynamic programming, improving guarantees to work with regret rather than errors.

An alternative form of oracle is provide by access to a good policy at training time. In this setting, Searn has similar provable guarantees with a similar analysis.

The oracle based algorithms appear to work well anywhere these oracles are available.

Uncontrolled Delay

In the uncontrolled delay setting, step (2) is removed, and typically steps (1) and (3) are collapsed into one observation, where the goal becomes state tracking. Most of the algorithms for state tracking are heavily model dependent, implying good success within particular domains. Examples include Kalman filters, hidden markov models, and particle filters which typical operate according to an explicit probabilistic model of world dynamics.

Relatively little is known for a nonparametric version of this problem. One observation is that the process of predicting adjacent observations well forms states as a byproduct when the observations are sufficiently rich as detailed here.

A basic question is: What’s missing from the above? A good answer is worth a career.

6/3/2009

Functionally defined Nonlinear Dynamic Models

Suppose we have a set of observations over time x1,x2,…,xt and want to predict some future event yt+1. An inevitable problem arises, because learning a predictor h(x1,…,xt) of yt+1 is generically intractable due to the size of the input. To make this problem tractable, what’s necessary is a method for summarizing the relevant information in past observations for the purpose of prediction in the future. In other words, state is required.

Existing approaches for deriving state have some limitations.

  1. Hidden Markov models learned with EM suffer from local minima, use tabular learning approaches which provide dubious generalization ability, and often require substantial a.priori specification of the observations.
  2. Kalman Filters and Particle Filters are very parametric in the sense that substantial information must be specified up front.
  3. Dynamic Bayesian Networks (graphical models through time) require substantial a.priori specification and often require the solution of difficult computational problems to use. Some of these difficulties are representational rather than computational.
  4. The Subspace-ID approach from control theory uses a linear representation, with the basic claim that it works well when all transformations are linear, but not so well when things are nonlinear. (Thanks to Drew for pointing it out.) In making this post, I ran across this two day tutorial which discusses extensions of this idea to nonlinear systems. Unfortunately, I’ll miss the tutorial, and I haven’t found the related paper.

The point of this paper at ICML is that some dynamic systems (those which are “invertible”), can be decomposed into separate bounded resource prediction problems which, when solved, create an implicit definition of state. This allows us to use any general purpose supervised learning algorithm to solve the state formation problem without requiring linearity or any specific representation. When writing papers you don’t generally gush too hard, but it’s fair to say that I’m excited by this approach.

  1. It’s not a known dead end.
  2. It doesn’t require lots of prior specification & information when you have lots of data.
  3. It leverages the huge amount of work that has gone into supervised learning algorithm design.
  4. It works in controlled systems also, where the control is simply another observation.
  5. It works with generalization from the start, rather than requiring the (often awkward) addition of generalization later.
  6. It doesn’t require predicting everything in order to predict what you want.
  7. It can work with very large observation spaces, and can even work better the larger the observation space, because larger observations imply more invertibility.

I expect some people reading this paper will be disappointed that it doesn’t solve all problems. That’s good news for anyone interested in research. For those who aren’t note that this is (in some sense) a generalization of subspace ID, and hence that there are other applications of the approach known to work in practice. Furthermore, we have some sample complexity analysis in the linear case.

It’s relatively rare to have a paper about a new approach to solving a problem as intractable as nonlinear dynamics has proved to be, so if you see a flaw please speak up.

5/8/2009

Computability in Artificial Intelligence

Normally I do not blog, but John kindly invited me to do so. Since computability issues play a major role in Artificial Intelligence and Machine Learning, I would like to take the opportunity to comment on that and raise some questions.

The general attitude is that AI is about finding efficient smart algorithms. For large parts of machine learning, the same attitude is not too dangerous. If you want to concentrate on conceptual problems, simply become a statistician. There is no analogous escape for modern research on AI (as opposed to GOFAI rooted in logic).

Let me show by analogy why limiting research to computational questions is bad for any field.

Except in computer science, computational aspects play little role in the development of fundamental theories: Consider e.g. set theory with axiom of choice, foundations of logic, exact/full minimax for zero-sum games, quantum (field) theory, string theory, … Indeed, at least in physics, every new fundamental theory seems to be less computable than previous ones. Of course, once a subject has been formalized, further research (a) analyzes the structure of the theory and (b) tries to compute efficient approximations. Only in (b) do computational aspects play a role.

So my question is: Why are computational questions so prevalent in AI research? Here are some unconvincing arguments I’ve heard:

A) Because AI is a subfield of computer science, and the task of computer scientists is to find (efficient) algorithms for well-defined problems?

I think it does not do any (real-world) problem any good to confine it to computer science. Of course, philosophers and cognitive scientists also care about AI, but where are the mathematicians?

B) Because formalizing AI and finding efficient smart programs goes hand-in-hand? Separating these two issues would lead to no, or at best to results which are misleading or useless in the construction of intelligent machines?

I am not aware of any convincing argument that separating the issues of “axiomatizing a field” and “finding efficient solutions” will (likely) fail for AI. The examples above of other fields actually indicate the opposite. Of course, interaction is important to avoid both sides running wild. For instance, von Neumann’s minimax solution for games, albeit infeasible for most games, is the cornerstone of most practical approximations.

C) Because there is some deep connection between intelligence and computation which can not be disentangled?

Sure, you could say that intelligence is by definition about computationally efficient decision making. This is as unconvincing as argument (A). Pointing out that the human brain is a computational device is quite useful in many ways, but doesn’t proves (C) either. Of course, ultimately we want a “fast” smart algorithm. How is AI different from wanting a fast algorithm computing primes, which you derive from a non-algorithmic definition of primes; or drawing fractals?

D) Because AI is trivial if computational issues are ignored? All conceptual problems have already been solved?

Many have expressed ideas that some form of exhaustive search over all possible solutions and picking the “best” one does the job. This works essentially for exactly those problems that are well-defined. For instance, optimal minimax play of a zero-sum game or solving NP complete problems are conceptually trivial, i.e. if computation time is ignored. But in general AI and machine learning, there is not a universally agreed-upon objective function. The Turing test is informal (involves a human judge in the loop), maximizing expected reward (the true distribution is not known, so expectation w.r.t. to what?), etc. The AIXI model, briefly discussed at this blog, is the first complete and formal such criterion, for which, let me phrase it that way, no flaw has yet been identified. Shane Legg’s award-winning thesis gives an informal introduction and contains lots of discussion.

Conceptual and computational problems in AI should be studied jointly as well as separately, but the latter is not (yet) fashionable. When AI was more logic oriented, some good logicians helped develop the foundations of “deductive” AI. Where are the researchers giving modern “inductive” AI its foundation? I am talking about generic learning agents, not classifying i.i.d. data. Reinforcement learners? Well, most of the hard results are from adaptive control theorists, but it’s reassuring to see parts of these communities merging. It’s a pity that so few mathematicians are interested in AI. A field “mathematical AI” with the prestige of “mathematical physics” would be exciting. As a start: 40% of the COLT & ALT papers on generic learning agents, 30% induction, 20% time-series forecasting, 10% i.i.d. Currently it’s reversed.

5/6/2009

Machine Learning to AI

I recently had fun discussions with both Vikash Mansinghka and Thomas Breuel about approaching AI with machine learning. The general interest in taking a crack at AI with machine learning seems to be rising on many fronts including DARPA.

As a matter of history, there was a great deal of interest in AI which died down before I began research. There remain many projects and conferences spawned in this earlier AI wave, as well as a good bit of experience about what did not work, or at least did not work yet. Here are a few examples of failure modes that people seem to run into:

  1. Supply/Product confusion. Sometimes we think “Intelligences use X, so I’ll create X and have an Intelligence.” An example of this is the Cyc Project which inspires some people as “intelligences use ontologies, so I’ll create an ontology and a system using it to have an Intelligence.” The flaw here is that Intelligences create ontologies, which they use, and without the ability to create ontologies you don’t have an Intelligence. If we are lucky, the substantial effort invested in Cyc won’t be wasted, as it has a large quantity of information stored in a plausibly useful format. If we are unlucky, it fails to even be partially useful, because the format is unnatural for the internal representations of an Intelligence.
  2. Uncertainty second. Many of the older AI programs had no role for uncertainty. If you asked the people working on them, they might agree that uncertainty was an important but secondary concern to be solved after the main problem. Unfortunately, it seems that uncertainty is a primary concern in practice. One example of this is blocks world where a system for planning how to rearrange blocks on a table might easily fail in practice because the robot fails to grab a block properly. Many people think of uncertainty as a second order concern, because they don’t experience uncertainty in their daily lives. I believe this is incorrect—a mental illusion due to the effect that focusing attention on a specific subject implies reducing uncertainty on that subject. More generally, because any Intelligence is a small part of the world, the ability of any intelligence to perceive, understand, and manipulate the world is inherently limited, requiring the ability to deal with uncertainty. For statistics & ML people, it’s important to not breath a sigh of relief too easily, as the problem is pernicious. For example many ML techniques based around conditional independence routinely suffer from excess certainty.
  3. Computation second. Some people try to create an intelligence without reference to efficient computation. AIXI is an extreme example of this sort. The algorithm is very difficult to deploy in practice because there were no computational constraints other than computability designed into it’s creation. It’s important to understand that computational constraints and uncertainty go together: because there are computational constraints, an intelligence is forced to deal with uncertainty since not everything which might follow at a mathematical level can be inferred in the available computational budget.
  4. AI-Hard problems. There was a time when some people thought, “If we could just get a program that mastered chess so well it could beat the best humans, we will learn enough about AI to create an AI.” Deep Blue put that theory to rest. Current efforts on Poker and Go seem more promising, but no one believes they are “AI-Hard” for good reason. It’s not even clear that the Turing Test is a reliable indicator, because (for example) we might imagine that there is Intelligence which can not imitate a human, or that there are programs that can imitate humans well enough to fool humans without being able to achieve everything that an Intelligence could. Perhaps the best evidence is something singularity-style: AI exists when it can substantially improve it’s own abilities.
  5. Asymptopia. In machine learning there are many theorems of the form “learning algorithm A can solve any learning problem in the limit of infinite data”. Here A might be nearest neighbors, decision trees, two-layer neural networks, support vector machines, nonparametric statistics, nonparametric Bayes, or something else. These theorem are ok, but insufficient. Often the algorithms are not computationally acceptable, and even if so, they are not sufficiently efficient with respect to the amount of experience required to learn.

Solving AI is undeniably hard, as evidenced by the amount of time spent on it, and the set of approaches which haven’t succeeded. There are a couple reasons for hope this time. The first is that there is, or soon will be sufficient computation available, unlike the last time. The second is that the machine learning approach fails well, because there are industrial uses for machine learning. Consequently, we can expect a lack of success to still see substantial use in practice. This might sound like “a good downside”, but it’s actually an upside, because it implies that incremental progress has the potential for ultimate success.

Restated at an abstract level: a hard problem can generally be decomposed in many ways into subproblems. Amongst all such decompositions, a good decomposition is one with the property that solutions to the subproblems are immediately useful. The machine learning approach to AI has this goodness property, unlike many other approaches, which partially explains why the ML approach is successful despite “failing” so far to achieve AI.

One reason why AI is hard, is that it turns out tackling general problems in the world undeniably requires a substantial number of different strategies, including learning, searching, and chunking (= constructing macros), all while respecting constraints of computation and robustness to uncertainty. Given this, a fair strategy seems to be first mastering one strategy, and then incorporating others, always checking that that incorporation properly addresses real world problems. In doing this, considering the constraint ignoring approaches as limiting cases of the real system may be helpful.

11/26/2008

Efficient Reinforcement Learning in MDPs

Tags: Reinforcement,Theory jl@ 7:29 am

Claude Sammut is attempting to put together an Encyclopedia of Machine Learning. I volunteered to write one article on Efficient RL in MDPs, which I would like to invite comment on. Is something critical missing?

3/23/2008

Interactive Machine Learning

A new direction of research seems to be arising in machine learning: Interactive Machine Learning. This isn’t a familiar term, although it does include some familiar subjects.

What is Interactive Machine Learning? The fundamental requirement is (a) learning algorithms which interact with the world and (b) learn.

For our purposes, let’s define learning as efficiently competing with a large set of possible predictors. Examples include:

  1. Online learning against an adversary (Avrim’s Notes). The interaction is almost trivial: the learning algorithm makes a prediction and then receives feedback. The learning is choosing based upon the advice of many experts.
  2. Active Learning. In active learning, the interaction is choosing which examples to label, and the learning is choosing from amongst a large set of hypotheses.
  3. Contextual Bandits. The interaction is choosing one of several actions and learning only the value of the chosen action (weaker than active learning feedback).

More forms of interaction will doubtless be noted and tackled as time progresses. I created a webpage for my own research on interactive learning which helps define the above subjects a bit more.

What isn’t Interactive Machine Learning?
There are several learning settings which fail either the interaction or the learning test.

  1. Supervised Learning doesn’t fit. The basic paradigm in supervised learning is that you ask experts to label examples, and then you learn a predictor based upon the predictions of these experts. This approach has essentially no interaction.
  2. Semisupervised Learning doesn’t fit. Semisupervised learning is almost the same as supervised learning, except that you also throw in many unlabeled examples.
  3. Bandit algorithms don’t fit. They have the interaction, but not much learning happens because the sample complexity results only allow you to choose from amongst a small set of strategies. (One exception is EXP4 (page 66), which can operate in the contextual bandit setting.)
  4. MDP learning doesn’t fit. The interaction is there, but the set of policies learned over is still too limited—essentially the policies just memorize what to do in each state.
  5. Reinforcement learning may or may not fit, depending on whether you think of it as MDP learning or in a much broader sense.

All of these not-quite-interactive-learning topics are of course very useful background information for interactive machine learning.

Why now? Because it’s time, of course.

  1. We know from other fields and various examples that interaction is very powerful.
    1. From online learning against an adversary, we know that independence of samples is unnecessary in an interactive setting—in fact you can even function against an adversary.
    2. From active learning, we know that interaction sometimes allows us to use exponentially fewer labeled samples than in supervised learning.
    3. From context bandits, we gain the ability to learn in settings where traditional supervised learning just doesn’t apply.
    4. From complexity theory we have “IP=PSPACE” roughly: interactive proofs are as powerful as polynomial space algorithms, which is a strong statement about the power of interaction.
  2. We know that this analysis is often tractable. For example, since Sanjoy‘s post on Active Learning, much progress has been made. Several other variations of interactive settings have been proposed and analyzed. The older online learning against an adversary work is essentially completely worked out for the simpler cases (except for computational issues).
  3. Real world problems are driving it. One of my favorite problems at the moment is the ad display problem—How do you learn which ad is most likely to be of interest? The contextual bandit problem is a big piece of this problem.
  4. It’s more fun. Interactive learning is essentially a wide-open area of research. There are plenty of kinds of natural interaction which haven’t been formalized or analyzed. This is great for beginnners, because it means the problems are simple, and their solution does not require huge prerequisites.
  5. It’s a step closer to AI. Many people doing machine learning want to reach AI, and it seems clear that any AI must engage in interactive learning. Mastering this problem is a next step.

Basic Questions

  1. For natural interaction form [insert yours here], how do you learn? Some of the techniques for other methods of interactive learning may be helpful.
  2. How do we blend interactive and noninteractive learning? In many applications, there is already a pool of supervised examples around.
  3. Are there general methods for reducing interactive learning problems to supervised learning problems (which we know better)?

12/10/2007

Learning Track of International Planning Competition

The International Planning Competition (IPC) is a biennial event organized in the context of the International Conference on Automated Planning and Scheduling (ICAPS). This year, for the first time, there will a learning track of the competition. For more information you can go to the competition web-site.

The competitions are typically organized around a number of planning domains that can vary from year to year, where a planning domain is simply a class of problems that share a common action schema—e.g. Blocksworld is a well-known planning domain that contains a problem instance each possible initial tower configuration and goal configuration. Some other domains have included Logistics, Airport, Freecell, PipesWorld, and many others. For each domain the competition includes a number of problems (say 40-50) and the planners are run on each problem with a time limit for each problem (around 30 minutes). The problems are hard enough that many problems are not solved within the time limit.

Given that the planners are asked to solve many problems from individual domains, and that problems within a domain generally have common solution structures, it makes sense to consider learning from previous problem-solving experience in a domain to better solve future problems in the same domain. Here “better solve” could mean either solve the problems significantly more quickly or finding better quality plans in a similar time frame. However, no planner in any of the competitions has included a learning component. Rather, to quote Foreigner, for these planners each problem “feels like the first time”.

Perhaps one reason that planners have not incorporated learning into the competition setting is that there has not been much overlap between the ICML and ICAPS communities, although that is changing. Another reason is perhaps that the structure of the competition would deduct any “learning time” from a planner’s 30mins per problem, which could reduce the potential benefits.

The learning track for the 2008 competition is being designed so that learning time is not counted against planners. Rather, there will be a distinct learning phase and a distinct evaluation phase. During the learning phase the planners will be provided with the set of domains to be used in the competition and example problems from each. The evaluation phase will be conducted like the current competition, with the exception that the learning-based planners will be allowed to read in a learned domain-specific “knowledge file” when solving the problems. This structure is designed to help answer the following question:

Do we have techniques that can leverage a learning period to outperform state-of-the-art non-learning techniques across a wide range of domains?

My current belief is that the answer is “no”. I certainly have never seen any such demonstration. This is not because of lack of work in the area of “learning to plan” as there has been a long history dating back to some of the early planners (see my horribly outdated resource page for a taste). While many of the learning approaches have shown some degree of success, the evaluations have typically been very narrow, focusing on only 2 to 3 domains and often only a few problems. My intuition, grounded in personal experience, is that most (all) of these systems would be quite brittle when taken to new domains. The hope is that the learning track of the competition will force us to take the issue of robustness seriously and soon lead to learning systems that convincingly outperform non-learning planners across a wide range of domains given proper training experience.

I hope to see a wide range of approaches entered into the competition. I’ll mention two styles of approaches that might be particular interesting to readers of this blog.

First, one might consider applying reinforcement learning to learn “generalized policies” that can be applied to any problem from a domain. Recall that here the domain model is provided to us, so applying RL would mean that the domain model is used as a sort of simulator in which an RL algorithm is run. RL is particularly difficult in these domains due to the challenges in developing an appropriate representation for learning value functions and/or policies—the states can be viewed as sets of ground relational atoms, rather than the more typical n-dimensional vectors common in RL. Another challenge is the extremely sparse reward, which is obtained only at goal states. There has been some work on applying RL to IPC-style domains (e.g. relational reinforcement learning, approximate policy iteration, policy gradient) but much improvement is needed to compete with non-learning planners.

Second, one might consider structured-classification techniques for this problem. Here one might view the planning problem as an input X and the plan as the structured output Y. Training data can be generated by solving example planning problems using state-of-the-art planners perhaps using significant resources. This approach has been studied under the name max-margin planning, but applied to a very different class of planning problems. Other work has considered applying the Learning as Search Optimization (LaSO) framework to IPC-style domains with some success. Some of the challenges here are to automatically produce an appropriate feature set given a planning domain and ambiguity in the training data. Ambiguity here refers to the fact that there are often a huge number of equally good plans for a given problem and the training data has only one or a small number of them, making the training data incomplete.

10/24/2007

Contextual Bandits

One of the fundamental underpinnings of the internet is advertising based content. This has become much more effective due to targeted advertising where ads are specifically matched to interests. Everyone is familiar with this, because everyone uses search engines and all search engines try to make money this way.

The problem of matching ads to interests is a natural machine learning problem in some ways since there is much information in who clicks on what. A fundamental problem with this information is that it is not supervised—in particular a click-or-not on one ad doesn’t generally tell you if a different ad would have been clicked on. This implies we have a fundamental exploration problem.

A standard mathematical setting for this situation is “k-Armed Bandits”, often with various relevant embellishments. The k-Armed Bandit setting works on a round-by-round basis. On each round:

  1. A policy chooses arm a from 1 of k arms (i.e. 1 of k ads).
  2. The world reveals the reward ra of the chosen arm (i.e. whether the ad is clicked on).

As information is accumulated over multiple rounds, a good policy might converge on a good choice of arm (i.e. ad).

This setting (and its variants) fails to capture a critical phenomenon: each of these displayed ads are done in the context of a search or other webpage. To model this, we might think of a different setting where on each round:

  1. The world announces some context information x (think of this as a high dimensional bit vector if that helps).
  2. A policy chooses arm a from 1 of k arms (i.e. 1 of k ads).
  3. The world reveals the reward ra of the chosen arm (i.e. whether the ad is clicked on).

We can check that this is a critical distinction in 2 ways. First, note that policies using x can encode much more rich decisions than a policy not using x. Just think about: “if a search has the word flowers display a flower advertisement”. Second, we can try to reduce this setting to the k-Armed Bandit setting, and note that it can not be done well. There are two methods that I know of:

  1. Run a different k-Armed Bandit for every value of x. The amount of information required to do well scales linearly in the number of contexts. In contrast, good supervised learning algorithms often require information which is (essentially) independent of the number of contexts.
  2. Take some set of policies and treat every policy h(x) as a different arm. This removes an explicit dependence on the number of contexts, but it creates a linear dependence on the number of policies. Via Occam’s razor/VC dimension/Margin bounds, we already know that supervised learning requires experience much smaller than the number of policies.

We know these are bad reductions by contrast to direct methods for solving the problem. The first algorithm for solving this problem is EXP4 (page 19 = 66) which has a regret with respect to the best policy in a set of O( T0.5 (ln |H|)0.5) where T is the number of rounds and |H| is the number of policies. (Dividing by T gives error-rate like quantities.) This result is independent of the number of contexts x and only weakly dependent (similar to supervised learning) on the number of policies.

EXP4 has a number of drawbacks—it has severe computational requirements and doesn’t work for continuously parameterized policies (*). Tong and I worked out a reasonably simple meta-algorithm Epoch-Greedy which addresses these drawbacks (**), at the cost of sometimes worsening the regret bound to O(T2/3S1/3) where S is related to the representational complexity of supervised learning on the set of policies.

This T dependence is of great concern to people who have worked on bandit problems in the past (where, basically, only the dependence on T could be optimized). In many applications, the S dependence is more important. However, this does leave an important open question: Is it possible to get the best properties of EXP4 and Epoch-Greedy?

Reasonable people could argue about which setting is more important: k-Armed Bandits or Contextual Bandits. I favor Contextual Bandits, even though there has been far more work in the k-Armed Bandit setting. There are several reasons:

  1. I’m having difficulty finding interesting real-world k-Armed Bandit settings which aren’t better thought of as Contextual Bandits in practice. For myself, bandit algorithms are (at best) motivational because they can not be applied to real-world problems without altering them to take context into account.
  2. Doing things in context is one of the underlying (and very successful) tenets of machine learning. Applying this tenet here seems wise.
  3. If we want to eventually solve big problems, we must have composable subelements. Composition doesn’t work without context, because there is no “input” for an I/O diagram.

Any insights into the open question above or Contextual Bandits in general are of great interest to me.

(*) There are some simple modifications to deal with the second issue but not the first.
(**) You have to read between the lines a little bit to see this in the paper. The ERM-style algorithm in the paper could be replaced with an efficient approximate ERM algorithm which is often possible in practice.

10/19/2007

Second Annual Reinforcement Learning Competition

The Second Annual Reinforcement Learning Competition is about to get started. The aim of the competition is to facilitate direct comparisons between various learning methods on important and realistic domains. This year’s event will feature well-known benchmark domains as well as more challenging problems of real-world complexity, such as helicopter control and robot soccer keepaway.

The competition begins on November 1st, 2007 when training software is released. Results must be submitted by July 1st, 2008. The competition will culminate in an event at ICML-08 in Helsinki, Finland, at which the winners will be announced.

For more information, visit the competition website.

3/3/2007

All Models of Learning have Flaws

Attempts to abstract and study machine learning are within some given framework or mathematical model. It turns out that all of these models are significantly flawed for the purpose of studying machine learning. I’ve created a table (below) outlining the major flaws in some common models of machine learning.

The point here is not simply “woe unto us”. There are several implications which seem important.

  1. The multitude of models is a point of continuing confusion. It is common for people to learn about machine learning within one framework which often becomes there “home framework” through which they attempt to filter all machine learning. (Have you met people who can only think in terms of kernels? Only via Bayes Law? Only via PAC Learning?) Explicitly understanding the existence of these other frameworks can help resolve the confusion. This is particularly important when reviewing and particularly important for students.
  2. Algorithms which conform to multiple approaches can have substantial value. “I don’t really understand it yet, because I only understand it one way”. Reinterpretation alone is not the goal—we want algorithmic guidance.
  3. We need to remain constantly open to new mathematical models of machine learning. It’s common to forget the flaws of the model that you are most familiar with in evaluating other models while the flaws of new models get exaggerated. The best way to avoid this is simply education.
  4. The value of theory alone is more limited than many theoreticians may be aware. Theories need to be tested to see if they correctly predict the underlying phenomena.

Here is a summary what is wrong with various frameworks for learning. To avoid being entirely negative, I added a column about what’s right as well.

Name Methodology What’s right What’s wrong
Bayesian Learning You specify a prior probability distribution over data-makers, P(datamaker) then use Bayes law to find a posterior P(datamaker|x). True Bayesians integrate over the posterior to make predictions while many simply use the world with largest posterior directly. Handles the small data limit. Very flexible. Interpolates to engineering.
  1. Information theoretically problematic. Explicitly specifying a reasonable prior is often hard.
  2. Computationally difficult problems are commonly encountered.
  3. Human intensive. Partly due to the difficulties above and partly because “first specify a prior” is built into framework this approach is not very automatable.
Graphical/generative Models Sometimes Bayesian and sometimes not. Data-makers are typically assumed to be IID samples of fixed or varying length data. Data-makers are represented graphically with conditional independencies encoded in the graph. For some graphs, fast algorithms for making (or approximately making) predictions exist. Relative to pure Bayesian systems, this approach is sometimes computationally tractable. More importantly, the graph language is natural, which aids prior elicitation.
  1. Often (still) fails to fix problems with the Bayesian approach.
  2. In real world applications, true conditional independence is rare, and results degrade rapidly with systematic misspecification of conditional independence.
Convex Loss Optimization Specify a loss function related to the world-imposed loss fucntion which is convex on some parametric predictive system. Optimize the parametric predictive system to find the global optima. Mathematically clean solutions where computational tractability is partly taken into account. Relatively automatable.
  1. The temptation to forget that the world imposes nonconvex loss functions is sometimes overwhelming, and the mismatch is always dangerous.
  2. Limited models. Although switching to a convex loss means that some optimizations become convex, optimization on representations which aren’t single layer linear combinations is often difficult.
Gradient Descent Specify an architecture with free parameters and use gradient descent with respect to data to tune the parameters. Relatively computationally tractable due to (a) modularity of gradient descent (b) directly optimizing the quantity you want to predict.
  1. Finicky. There are issues with paremeter initialization, step size, and representation. It helps a great deal to have accumulated experience using this sort of system and there is little theoretical guidance.
  2. Overfitting is a significant issue.
Kernel-based learning You chose a kernel K(x,x’) between datapoints that satisfies certain conditions, and then use it as a measure of similarity when learning. People often find the specification of a similarity function between objects a natural way to incorporate prior information for machine learning problems. Algorithms (like SVMs) for training are reasonably practical—O(n2) for instance. Specification of the kernel is not easy for some applications (this is another example of prior elicitation). O(n2) is not efficient enough when there is much data.
Boosting You create a learning algorithm that may be imperfect but which has some predictive edge, then apply it repeatedly in various ways to make a final predictor. A focus on getting something that works quickly is natural. This approach is relatively automated and (hence) easy to apply for beginners. The boosting framework tells you nothing about how to build that initial algorithm. The weak learning assumption becomes violated at some point in the iterative process.
Online Learning with Experts You make many base predictors and then a master algorithm automatically switches between the use of these predictors so as to minimize regret. This is an effective automated method to extract performance from a pool of predictors. Computational intractability can be a problem. This approach lives and dies on the effectiveness of the experts, but it provides little or no guidance in their construction.
Learning Reductions You solve complex machine learning problems by reducing them to well-studied base problems in a robust manner. The reductions approach can yield highly automated learning algorithms. The existence of an algorithm satisfying reduction guarantees is not sufficient to guarantee success. Reductions tell you little or nothing about the design of the base learning algorithm.
PAC Learning You assume that samples are drawn IID from an unknown distribution D. You think of learning as finding a near-best hypothesis amongst a given set of hypotheses in a computationally tractable manner. The focus on computation is pretty right-headed, because we are ultimately limited by what we can compute. There are not many substantial positive results, particularly when D is noisy. Data isn’t IID in practice anyways.
Statistical Learning Theory You assume that samples are drawn IID from an unknown distribution D. You think of learning as figuring out the number of samples required to distinguish a near-best hypothesis from a set of hypotheses. There are substantially more positive results than for PAC Learning, and there are a few examples of practical algorithms directly motivated by this analysis. The data is not IID. Ignorance of computational difficulties often results in difficulty of application. More importantly, the bounds are often loose (sometimes to the point of vacuousness).
Decision tree learning Learning is a process of cutting up the input space and assigning predictions to pieces of the space. Decision tree algorithms are well automated and can be quite fast. There are learning problems which can not be solved by decision trees, but which are solvable. It’s common to find that other approaches give you a bit more performance. A theoretical grounding for many choices in these algorithms is lacking.
Algorithmic complexity Learning is about finding a program which correctly predicts the outputs given the inputs. Any reasonable problem is learnable with a number of samples related to the description length of the program. The theory literally suggests solving halting problems to solve machine learning.
RL, MDP learning Learning is about finding and acting according to a near optimal policy in an unknown Markov Decision Process. We can learn and act with an amount of summed regret related to O(SA) where S is the number of states and A is the number of actions per state. Has anyone counted the number of states in real world problems? We can’t afford to wait that long. Discretizing the states creates a POMDP (see below). In the real world, we often have to deal with a POMDP anyways.
RL, POMDP learning Learning is about finding and acting according to a near optimaly policy in a Partially Observed Markov Decision Process In a sense, we’ve made no assumptions, so algorithms have wide applicability. All known algorithms scale badly with the number of hidden states.

This set is incomplete of course, but it forms a starting point for understanding what’s out there. (Please fill in the what/pro/con of anything I missed.)

11/27/2006

Continuizing Solutions

This post is about a general technique for problem solving which I’ve never seen taught (in full generality), but which I’ve found very useful.

Many problems in computer science turn out to be discretely difficult. The best known version of such problems are NP-hard problems, but I mean ‘discretely difficult’ in a much more general way, which I only know how to capture by examples.

  1. ERM In empirical risk minimization, you choose a minimum error rate classifier from a set of classifiers. This is NP hard for common sets, but it can be much harder, depending on the set.
  2. Experts In the online learning with experts setting, you try to predict well so as to compete with a set of (adversarial) experts. Here the alternating quantifiers of you and an adversary playing out a game can yield a dynamic programming problem that grows exponentially.
  3. Policy Iteration The problem with policy iteration is that you learn a new policy with respect to an old policy, which implies that simply adopting the new policy can go very wrong.

For each of these problems, there are “continuized” solutions which can yield smaller computation, more elegant mathematics, or both.

  1. ERM By shifting from choosing a single classifier to choosing a stochastic classifier we can prove a new style of bound which is significantly tighter, easier to state, and easier to understand than traditional bounds in the traditional setting. This is the PAC-Bayes bound idea.
  2. Experts By giving the adversary slightly more power—the ability to split experts and have them fractionally predict one way vs. another, the optimal policy becomes much easier to compute (quadratic in the horizon, or maybe less). This is the continuous experts idea.
  3. Policy Iteration For policy iteration, by stochastically mixing the old and the new policy, we can find a new policy better than the old policy. This is the conservative policy iteration idea.

There is some danger to continuizing. The first and second examples both involve a setting shift, which may not be valid—in general your setting should reflect your real problem rather than the thing which is easy to solve. However, even with the setting shift, the solutions appear so compellingly more elegant that it is hard to not hope to use them in a solution to the original setting.

I have not seen a good formulation of the general approach of continuizing. Nevertheless, I expect to see continuizing in more places and to use it in the future. By making it explicit, perhaps this can be made eaesier.

11/22/2006

Explicit Randomization in Learning algorithms

There are a number of learning algorithms which explicitly incorporate randomness into their execution. This includes at amongst others:

  1. Neural Networks. Neural networks use randomization to assign initial weights.
  2. Boltzmann Machines/Deep Belief Networks. Boltzmann machines are something like a stochastic version of multinode logistic regression. The use of randomness is more essential in Boltzmann machines, because the predicted value at test time also uses randomness.
  3. Bagging. Bagging is a process where a learning algorithm is run several different times on several different datasets, creating a final predictor which makes a majority vote.
  4. Policy descent. Several algorithms in reinforcement learning such as Conservative Policy Iteration use random bits to create stochastic policies.
  5. Experts algorithms. Randomized weighted majority use random bits as a part of the prediction process to achieve better theoretical guarantees.

A basic question is: “Should there be explicit randomization in learning algorithms?” It seems perverse to feed extra random bits into your prediction process since they don’t contain any information about the problem itself. Can we avoid using random numbers? This question is not just philosophy—we might hope that deterministic version of learning algorithms are both more accurate and faster.

There seem to be several distinct uses for randomization.

  1. Symmetry breaking. In the case of a neural network, if every weight started as 0, the gradient of the loss with respect to every weight would be the same, implying that after updating, all weights remain the same. Using random numbers to initialize weights breaks this symmetry. It is easy to believe that there are good deterministic methods for symmetry breaking.
  2. Overfit avoidance. A basic observation is that deterministic learning algorithms tend to overfit. Bagging avoids this by randomizing the input of these learning algorithms in the hope that directions of overfit for individual predictions cancel out. Similarly, using random bits internally as in a deep belief network avoids overfitting by forcing the algorithm to learn a robust-to-noise set of internal weights, which are then robust-to-overfit. Large margin learning algorithms and maximum entropy learning algorithms can be understood as deterministic operations attempting to achieve the same goal. A significant gap remains between randomized and deterministic learning algorithms: the deterministic versions just deal with linear predictions while the randomized techniques seem to yield improvements in general.
  3. Continuizing. In reinforcement learning, it’s hard to optimize a policy over multiple timesteps because the optimal decision at timestep 2 is dependent on the decision at timestep 1 and vice versa. Randomized interpolation of policies offers a method to remove this cyclic dependency. PSDP can be understood as a derandomization of CPI which trades off increased computation (learning a new predictor for each timestep individually). Whether or not we can avoid a tradeoff in general is unclear.
  4. Adversary defeating. Some algorithms, such as randomized weighted majority are designed to work against adversaries who know your algorithm, except for random bits. The randomization here is provably essential, but the setting is often far more adversarial than the real world.

The current state-of-the-art is that random bits provide performance (computational and predictive) which we don’t know (or at least can’t prove we know) how to achieve without randomization. Can randomization be removed or is it essential to good learning algorithms?

10/8/2006

Incompatibilities between classical confidence intervals and learning.

Classical confidence intervals satisfy a theorem of the form: For some data sources D,

PrS ~ D(f(D) > g(S)) > 1-d

where f is some function of the distribution (such as the mean) and g is some function of the observed sample S. The constraints on D can vary between “Independent and identically distributed (IID) samples from a gaussian with an unknown mean” to “IID samples from an arbitrary distribution D“. There are even some confidence intervals which do not require IID samples.

Classical confidence intervals often confuse people. They do not say “with high probability, for my observed sample, the bounds holds”. Instead, they tell you that if you reason according to the confidence interval in the future (and the constraints on D are satisfied), then you are not often wrong. Restated, they tell you something about what a safe procedure is in a stochastic world where d is the safety parameter.

There are a number of results in theoretical machine learning which use confidence intervals. For example,

  1. The E3 algorithm uses confidence intervals to learn a near optimal policy for any MDP with high probability.
  2. Set Covering Machines minimize a confidence interval upper bound on the true error rate of a learned classifier.
  3. The A2 uses confidence intervals to safely deal with arbitrary noise while taking advantage of active learning.

Suppose that we want to generalize thse algorithms in a reductive style. The goal would be to train a regressor to predict the output of g(S) for new situations. For example, a good regression prediction of g(S) might allow E3 to be applied to much larger state spaces. Unfortunately, this approach seems to fail badly due to a mismatch between the semantics of learning and the semantics of a classical confidence interval.

  1. It’s difficult to imagine a constructive sampling mechanism. In a large state space, we may never encounter the same state twice, so we can not form meaningful examples of the form “for this state-action, the correct confidence interval is y“.
  2. When we think of succesful learning, we typically think of it in an l1 sense—the expected error rate over the data generating distribution is small. Confidence intervals have a much stronger meaning as we would like to apply them: with high probability, in all applications, the confidence interval holds. This mismatch appears unaddressable.

It is tempting to start plugging in other notions such as Bayesian confidence intervals or quantile regression systems. Making these approaches work at a theoretical level on even simple systems is an open problem, but there is plenty of motivation to do so.

6/14/2006

Explorations of Exploration

Exploration is one of the big unsolved problems in machine learning. This isn’t for lack of trying—there are many models of exploration which have been analyzed in many different ways by many different groups of people. At some point, it is worthwhile to sit back and see what has been done across these many models.

  • Reinforcement Learning (1). Reinforcement learning has traditionally focused on Markov Decision Processes where the next state s’ is given by a conditional distribution P(s’|s,a) given the current state s and action a. The typical result here is that certain specific algorithms controlling an agent can behave within e of optimal for horizon T except for poly(1/e,T,S,A) “wasted” experiences (with high probability). This started with E3 by Satinder Singh and Michael Kearns. Sham Kakade’s thesis has significant discussion. Extensions have typically been of the form “under extra assumptions, we can prove more”, for example Factored-E3 and Metric-E3. (It turns out that the number of wasted samples can be less than the number of bits required to describe an MDP.) A weakness of all these results is that they rely upon (a) assumptions which are often false for real applications, (b) state spaces are too large, and (c) make a gurantee that is rather weak. Good performance is only guaranteed after suffering the possibly catastrophic consequences of exploration.
  • Reinforcement Learning (2). Several recent papers have been attempting to analyze reinforcement learning via reduction. To date, all results are either nonconstructive or involve the use of various hints (oracle access to an optimal policy, the distribution over states of an optimal policy etc…) which short-circuit the need to explore. Obviously, these hints are not always available for real-world problems.
  • Reinforcement Learning (3). Much of the rest of reinforcement learning has something to do with exploration, but it’s difficult to summarize succinctly.
  • Online Learning. The n-armed bandit setting can be thought of as an MDP with one state and many actions. In some variants, there is even an adversary who chooses the payoffs of the arms in a non-stochastic manner. The typical result here says that you can compete well with the best constant action after some wasted actions. The exact number of wasted actions varies with the precise setting, but it is typically linear in the number of actions. This work can be traced back to (at least) Gittins indices which (unfortunately) don’t seem to have a good description available on the internet.
  • Active Learning(1) The common current use of this term has to do with “selective sampling”=choosing unlabeled samples to label so as to minimize the number of labels required to learn a good predictor (typically a classifier). A typical result has the form: Given that your classifier comes from restricted class C and the labeled data distribution obeys some constraint, the number of adaptively labeled samples required O(log (1/e)) where e is the error rate. (It turns out that the even noisy distributions are allowed.) The constraints on distributions and hypothesis spaces required to achieve these speedups are often severe.
  • Active Learning(2) Membership query learning is another example. The distinguishing difference with respect to selective sampling is that the a labeled can be requested for any unlabeled point (not just those drawn according to some natural distribution). Several relatively strong results hold for membership query learning, but there is a significant drawback: it seems that the ability to query for a label on an arbitrary point is not very natural. For example, imagine query whether a text document is about sports or politics when the text is generated at random.
  • Active Learning(3) Experimental design (which is mostly based in statistics), is often about finding the extrema of some function rather than generalization. Often, the data generating distribution is assumed to come from some specific parametric family. Unfortunately, my knowledge is sketchy here.

The striking thing about all of these models is that they fail to apply to typical real world problems. This failure is either by design (making assumptions which are simply rarely met), by failure to prove interesting results, or both.

And yet, many of the pieces are here. Active learning deals with generalization, online learning can deal with adversarial situations, and reinforcement learning deals with the situation where your choices influence what you can later learn. At a high level, there is much room for research here by design of a new model of exploration, new theoretical statements, or both.

I’ve been told “exploration is too hard”, and that’s a good warning to bear in mind, but I’m still hopeful for progress.

4/5/2006

What is state?

In reinforcement learning (and sometimes other settings), there is a notion of “state”. Based upon the state various predictions are made such as “Which action should be taken next?” or “How much cumulative reward do I expect if I take some action from this state?” Given the importance of state, it is important to examine the meaning. There are actually several distinct options and it turns out the definition variation is very important in motivating different pieces of work.

  1. Newtonian State. State is the physical pose of the world. Under this definition, there are very many states, often too many for explicit representation. This is also the definition typically used in games.
  2. Abstracted State. State is an abstracted physical state of the world. “Is the door open or closed?” “Are you in room A or not?” The number of states is much smaller here. A basic issue here is: “How do you compute the state from observations?”
  3. Mathematical State. State is a sufficient statistic of observations for making necessary predictions.
  4. Internal State. State is the internal belief/understanding/etc… which changes an agent’s actions in different circumstances. A natural question is: “How do you learn a state?” This is like the mathematical version of state, except that portions of the statistic which can not be learned are irrelevant.
  5. There are no states. There are only observations (one of which might be a reward) and actions. This is more reasonable than it might sound because state is a derived quantity and the necessity of that derivation is unclear. PSRs are an example.

The different questions these notions of state motivate can have large practical consequences on the algorithms used. It is not clear yet what the “right” notion of state is—we just don’t know what works well. I am most interested in “internal state” at the moment.

1/13/2006

Benchmarks for RL

Tags: Machine Learning,Reinforcement jl@ 12:54 am

A couple years ago, Drew Bagnell and I started the RLBench project to setup a suite of reinforcement learning benchmark problems. We haven’t been able to touch it (due to lack of time) for a year so the project is on hold. Luckily, there are several other projects such as CLSquare and RL-Glue with a similar goal, and we strongly endorse their continued development.

I would like to explain why, especially in the context of criticism of other learning benchmarks. For example, sometimes the UCI Machine Learning Repository is criticized. There are two criticisms I know of:

  1. Learning algorithms have overfit to the problems in the repository. It is easy to imagine a mechanism for this happening unintentionally. Strong evidence of this would be provided by learning algorithms which perform great on the UCI machine learning repository but very badly (relative to other learning algorithms) on non-UCI learning problems. I have seen little evidence of this but it remains a point of concern. There is a natural order to how much we trust performance results:
    1. Competitions. A well run prediction competition provides some of the best evidence available about which learning algorithms work well. There are still some mechanisms for being misled (too many entries, not enough testing data, unlucky choice of learning problem that your algorithm just has the wrong bias for), but they are more controlled than anywhere else.
    2. Benchmarks. Benchmarks provide a “reasonable sanity check” that the algorithm will do something well in practice. Compared to competitions, there are additional ways to be misled and (unfortunately) to mislead. Overfitting becomes a serious issue.
    3. Simulated data Simulated data can be very misleading. Typically, there is little reason to believe that simulated data reflects the structure of real-world problems.

    This ordering should always be kept in mind when considering empirical results.

  2. The repository enforced a certain interface which many natural learning problems did not fit into. Some interface must be enforced in order for a benchmark/repository to be useful. This implies that problems fitting the interface are preferred both in submission of new problems and in choice of algorithms to solve. This is a reasonable complaint, but it’s basically a complaint that the UCI machine learning repository was too succesful. The good news here is that the internet has made it so anyone can setup a repository. Just put up a webserver and announce the dataset.

It is important to consider the benefits of a benchmark suite as well. The standard in reinforcement learning for experimental studies in reinforcement learning algorithms has been showing that some new algorithm does something reasonable on one or two reinforcement learning problems. Naturally, what this implies in practice is that the algorithms were hand tuned to work on these one-or-two problems, and there was relatively little emphasis on building a general purpose reinforcement learning algorithm.

This is strongly at odds with the end goal of the reinforcement learning community! The way to avoid this is by setting up a benchmark suite (the more problems, the better). With a large set of problems that people can easily test on, the natural emphasis in empirical algorithm development shifts towards general purpose algorithms.

Another drawback of the “one or two problems” approach is incomparability of results. When people reimplement simulators for reinforcement learning problems, it turns out these reimplementations typically differ in the details, and these details can radically alter the difficulty of the problem. For example, there are many ways to implement the “car on the hill” problem, and some of them are much more difficult to solve than others. Again, a large set of problems people can easily test on solves this problem because the precise definition of the problem is shared.

One sometimes reasonable view of the world is that if you define a problem publicly, it is as good as solved since there are plenty of smart people out there eager to prove themselves. According to this view of the world, setting up a benchmark is a vital task.

8/4/2005

Why Reinforcement Learning is Important

Tags: Reinforcement jl@ 3:04 pm

One prescription for solving a problem well is:

  1. State the problem, in the simplest way possible. In particular, this statement should involve no contamination with or anticipation of the solution.
  2. Think about solutions to the stated problem.

Stating a problem in a succinct and crisp manner tends to invite a simple elegant solution. When a problem can not be stated succinctly, we wonder if the problem is even understood. (And when a problem is not understood, we wonder if a solution can be meaningful.)

Reinforcement learning does step (1) well. It provides a clean simple language to state general AI problems. In reinforcement learning there is a set of actions A, a set of observations O, and a reward r. The reinforcement learning problem, in general, is defined by a conditional measure D( o, r | (o,r,a)*) which produces an observation o and a reward r given a history (o,r,a)*. The goal in reinforcement learning is to find a policy pi:(o,r,a)* -> a mapping histories to actions so as to maximize (or approximately maximize) the expected sum of observed rewards.

This formulation is capable of capturing almost any (all?) AI problems. (Are there any other formulations capable of capturing a similar generality?) I don’t believe we yet have good RL solutions from step (2), but that is unsurprising given the generality of the problem.

Note that solving RL in this generality is impossible (for example, it can encode classification). The two approaches that can be taken are:

  1. Simplify the problem. It is very common to consider the restricted problem where the history is summarized by the previous observation. (aka a “Markov Decision Process”). In many cases, other restrictions are added.
  2. Think about relativized solutions (such as reductions).

Both methods are options are under active investigation.

6/17/2005

Reopening RL->Classification

In research, it’s often the case that solving a problem helps you realize that it wasn’t the right problem to solve. This is the case for the “reduce RL to classification” problem with the solution hinted at here and turned into a paper here.

The essential difficulty is that the method of stating and analyzing reductions ends up being nonalgorithmic (unlike previous reductions) unless you work with learning from teleoperated robots as Greg Grudic does. The difficulty here is due to the reduction being dependent on the optimal policy (which a human teleoperator might simulate, but which is otherwise unavailable).

So, this problem is “open” again with the caveat that this time we want a more algorithmic solution.

Whether or not this is feasible at all is still unclear and evidence in either direction would greatly interest me. A positive answer might have many practical implications in the long run.

2/25/2005

Solution: Reinforcement Learning with Classification

I realized that the tools needed to solve the problem just posted were just created. I tried to sketch out the solution here (also in .lyx and .tex). It is still quite sketchy (and probably only the few people who understand reductions well can follow).

One of the reasons why I started this weblog was to experiment with “research in the open”, and this is an opportunity to do so. Over the next few days, I’ll be filling in details and trying to get things to make sense. If you have additions or ideas, please propose them.

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