Machine Learning (Theory)

9/29/2012

Vowpal Wabbit, version 7.0

A new version of VW is out. The primary changes are:

  1. Learning Reductions: I’ve wanted to get learning reductions working and we’ve finally done it. Not everything is implemented yet, but VW now supports direct:
    1. Multiclass Classification –oaa or –ect.
    2. Cost Sensitive Multiclass Classification –csoaa or –wap.
    3. Contextual Bandit Classification –cb.
    4. Sequential Structured Prediction –searn or –dagger

    In addition, it is now easy to build your own custom learning reductions for various plausible uses: feature diddling, custom structured prediction problems, or alternate learning reductions. This effort is far from done, but it is now in a generally useful state. Note that all learning reductions inherit the ability to do cluster parallel learning.

  2. Library interface: VW now has a basic library interface. The library provides most of the functionality of VW, with the limitation that it is monolithic and nonreentrant. These will be improved over time.
  3. Windows port: The priority of a windows port jumped way up once we moved to Microsoft. The only feature which we know doesn’t work at present is automatic backgrounding when in daemon mode.
  4. New update rule: Stephane visited us this summer, and we fixed the default online update rule so that it is unit invariant.

There are also many other small updates including some contributed utilities that aid the process of applying and using VW.

Plans for the near future involve improving the quality of various items above, and of course better documentation: several of the reductions are not yet well documented.

3/15/2010

The Efficient Robust Conditional Probability Estimation Problem

I’m offering a reward of $1000 for a solution to this problem. This joins the cross validation problem which I’m offering a $500 reward for. I believe both of these problems are hard but plausibly solvable, and plausibly with a solution of substantial practical value. While it’s unlikely these rewards are worth your time on an hourly wage basis, the recognition for solving them definitely should be :-)

The Problem

The problem is finding a general, robust, and efficient mechanism for estimating a conditional probability P(y|x) where robustness and efficiency are measured using techniques from learning reductions.

In particular, suppose we have access to a binary regression oracle B which has two interfaces—one for specifying training information and one for testing. Training information is specified as B(x’,y’) where x’ is a feature vector and y’ is a scalar in [0,1] with no value returned. Testing is done according to B(x’) with a value in [0,1] returned.

A learning reduction consists of two algorithms R and R-1 which transform examples from the original input problem into examples for the oracle and then transform the oracle’s predictions into a prediction for the original problem.

The algorithm R takes as input a single example (x,y) where x is an feature vector and y is a discrete variable taking values in {1,…,k}. R then specifies a training example (x’,y’) for the oracle B. R can then create another training example for B based on all available information. This process repeats some finite number of times before halting without returning information.

A basic observation is that for any oracle algorithm, a distribution D(x,y) over multiclass examples and a reduction R induces a distribution over a sequence (x’,y’)* of oracle examples. We collapse this into a distribution D'(x’,y’) over oracle examples by drawing uniformly from the sequence.

The algorithm R-1 takes as input a single example (x,y) and returns a value in [0,1] after using (only) the testing interface of B zero or more times.

We measure the power of an oracle and a reduction according to squared-loss regret. In particular we have:


reg(D,R-1)=E(x,y)~ D[(R-1(x,y)-D(y|x))2]

and similarly letting mx’=E(x’,y’)~ D’[y'].

reg(D’,B)=E(x’,y’)~ D’(B(x’) – mx’)2

The open problem is to specify R and R-1 satisfying the following theorem:

For all multiclass distributions D(x,y), for all binary oracles B: The computational complexity of R and R-1 are O(log k)
and


reg(D,R-1) < = C reg(D’,B)

where C is a universal constant.

Alternatively, this open problem is satisfied by proving there exists no deterministic algorithms R,R-1 satisfying the above theorem statement.

Motivation

The problem of conditional probability estimation is endemic to machine learning applications. In fact, in some branches of machine learning, this is simply considered “the problem”. Typically conditional probability estimation is done in situations where the conditional probability of only one bit is required, however there are a growing number of applications where a well-estimated conditional probability over a more complex object is required. For example, all known methods for solving general contextual bandit problems require knowledge of or good estimation of P(a | x) where a is an action.

There is a second intrinsic motivation which is matching the lower bound. No method faster than O(log k) can be imagined because the label y requires log2 k bits to specify and hence read. Similarly it’s easy to prove no learning reduction can provide a regret ratio with C<1.

The motivation for using the learning reduction framework to specify this problem is a combination of generality and the empirical effectiveness in application of learning reductions. Any solution to this will be general because any oracle B can be plugged in, even ones which use many strange kinds of prior information, features, and active multitask hierachical (insert your favorite adjective here) structure.

Related Results

The state of the art is summarized here which shows it’s possible to have a learning reduction satisfying the above theorem with either:

  1. C replaced by (log2 k)2 (using a binary tree structure)
  2. or the computational time increased to O(k) (using an error correcting code structure).

Hence, answering this open problem in the negative shows that there is an inherent computation vs. robustness tradeoff.

There are two other closely related problems, where similar analysis can be done.

  1. For multiclass classification, where the goal is predicting the most likely class, a result analogous to the open problem is provable using error correcting tournaments.
  2. For multiclass classification in a partial label setting, no learning reduction can provide a constant regret guarantee.

Silly tricks that don’t work

Because Learning reductions are not familiar to everyone, It’s helpful to note certain tricks which do not work here to prevent false leads and provide some intuition.

Ignore B‘s predictions and use your favorite learning algorithm instead.

This doesn’t work, because the quantification is for all D. Any specified learning algorithm will have some D on which it has nonzero regret. On the other hand, because R calls the oracle at least once, there is a defined induced distribution D’. Since the theorem must hold for all D and B, it must hold for a D your specified learning algorithm fails on and for a B for which reg(D’,B)=0 implying the theorem is not satisfied.

Feed random examples into B and vacuously satisfy the theorem by making sure that the right hand side is larger than a constant.

This doesn’t work because the theorem is stated in terms of squared loss regret rather than squared loss. In particular, if the oracle is given examples of the form (x’,y’) where y’ is uniformly at random either 0 or 1, any oracle specifying B(x’)=0.5 has zero regret.

Feed pseudorandom examples into B and vacuously satisfy the theorem by making sure that the right hand side is larger than a constant.

This doesn’t work, because the quantification is “for all binary oracles B”, and there exists one which, knowing the pseudorandom seed, can achieve zero loss (and hence zero regret).

Just use Boosting to drive the LHS to zero.

Boosting theorems require a stronger oracle—one which provides an edge over some constant baseline for each invocation. The oracle here is not limited in this fashion since it could completely err for a small fraction of invocations.

Take an existing structure, parameterize it, randomize over the parameterization, and then average over the random elements.

Employing this approach is not straightforward, because the average in D’ is over an increased number of oracle examples. Hence, at a fixed expected (over oracle examples) regret, the number of examples allowed to have a large regret is increased.

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.

5/2/2009

Wielding a New Abstraction

This post is partly meant as an advertisement for the reductions tutorial Alina, Bianca, and I are planning to do at ICML. Please come, if you are interested.

Many research programs can be thought of as finding and building new useful abstractions. The running example I’ll use is learning reductions where I have experience. The basic abstraction here is that we can build a learning algorithm capable of solving classification problems up to a small expected regret. This is used repeatedly to solve more complex problems.

In working on a new abstraction, I think you typically run into many substantial problems of understanding, which make publishing particularly difficult.

  1. It is difficult to seriously discuss the reason behind or mechanism for abstraction in a conference paper with small page limits. People rarely see such discussions and hence have little basis on which to think about new abstractions. Another difficulty is that when building an abstraction, you often don’t know the right way to state things.

    Here’s my current attempt: The process of abstraction for learning reductions can start with sample complexity bounds (or online learning against an adversary analysis). A very simple sample complexity bound is that for all sets of hypotheses H, for all distributions D on examples (x,y), and for all confidence parameters d

    Pr(x,y)m~Dm(for all h in H: |e(h,D)-e(h,(x,y)m)| < (ln( |H|/ d )/m)0.5 ) > 1 – d

    Here (x,y)m is a sequence of m IID samples, e(h,D) is the error rate of h on D and e(h,(x,y)m) is the empirical error rate of h on the set of IID samples.

    The previous bound is a very simple example, and yet remarkably complex both to state and to interpret—many people have been lost by the meaning of d. The impact of this complexity is that it is difficult to effectively use these bounds in practical learning algorithm design, particularly in solving more complex learning problems where much more than one bit of prediction is required. This was a central frustration that I ran into in my thesis work. Some progress has been made since then, but it is still quite difficult. The abstraction in the learning reduction setting is:

    1. You throw away d, because it only has a logarithmic dependence anyways.
    2. You eliminate H and m on the theory that intelligent choices for H and m are made in practice.
    3. You eliminate the IID assumption, because it is no longer needed to define things

    The statement then is

    e(A((x,y)m),D)-e(h*,D) < eps

    where A() is the hypothesis output by the learning algorithm, h* is the best possible predictor, and eps is used to parameterize the theorems. This abstraction is radical in some sense, but something radical was needed to yield tractable and useful analysis on the complex problems people need to solve in practice.

  2. A consequence of lack of familiarity, is that people often misread. In reading a paper, there is a temptation to not read carefully and fill in your understanding of things. Most of the time this works out well, but not here. For example, we saw many instances where people inserted IID sample assumptions or other things that simply weren’t there.
  3. Once you get past the lack of familiarity and misunderstandings, there is a feeling that the new abstraction is cheating. To some extent I understand, as I remember learning about abstractions in class, and I remember feeling that they were in some real sense cheating by dropping important details. For example:
    1. Big-O notation provides an upper bound specified up to constants. For example O(log n) computational complexity means there exists a constant c such that the number of operations requires is less than c log n. Big-O can be abused by hiding “constants” larger than the plausible values for the parameters. In machine learning, a particularly egregious case occurs in Bandit analysis where the punchline of some papers is “logarithmic regret”, hiding an arbitrarily large problem dependent constant.
    2. TCP provides a mechanism for reliable transport over an unreliable network. It is a very commonly used mechanism for sending information over the internet—you used TCP in reading this. TCP is both a programming construct and a mechanism for abstracting communicating over a network. The TCP abstraction is broken when the network is too unreliable for it to recover, such as on sketchy wireless networks where the programmer built for the TCP abstraction which wasn’t delivered.
    3. Dimensional analysis is a technique for quick analysis in physics. The basic idea is to just look at the units when estimating some quantity and combine them to get the right unit answer. For example, to compute the distance d traveled after time t with acceleration a, you simply use at2, since that formula is the only way to combine a with units of distance/time2 and t with units of time to get units of distance. This answer is off by a factor of 2 from what a more detailed analysis using integration yields, which is typical. Dimensional analysis can be misleading when the constants are very large. One example is in Gravitation where there is a table with time and distance equated since they are related by a constant—the speed of light 3*108 m/s. For example, E=mc2 becomes E=m.

    Although the above breakages are real, the usefulness of these abstractions, in terms of allowing us to quickly think about and make decisions more than offsets the drawbacks. Indeed, even the breakages stated above are thought provoking or useful enough that I can’t even say it is wrong to consider them. This property that abstractions can be abused is generically essential to the process of abstraction itself. Abstraction is about neglecting details, and when these details are not neglectable, the abstraction is abused or ineffective. Because of this, any abstraction is insufficient for analyzing and solving real problems where the neglected details matter.

    Just as for these abstractions, the learning reduction abstraction can be abused—the chosen learning algorithm can be pathetic yielding vacuous bounds, or the reduction can scramble the feature information with an encryption algorithm making it so no reasonable learning algorithm could yield other than pathetic performance. Similarly, there are situations in which I don’t know how to effectively use a learning reduction to build a learning algorithm, and it seems implausible that observation changes as more is learned in the future.

For a good abstraction, the drawbacks are matched by the advantages. The principle advantage is that there is a new way to examine and solve problems. This has several interesting effects.

  1. A good abstraction can capture a more complete specification of the problem. As an example, the sample complexity view of learning is broken in practice, because when insufficient performance is achieved people choose a different set of hypotheses H by throwing in additional features or choosing a different learning algorithm. Capturing this process in the sample complexity view requires an additional level of complexity. In the reduction view, this is entirely natural, because any means for achieving a better generalization—more/better features, a better learning algorithm, a better prior, sticking a human in the learning process, etc… are legitimate. This is particularly powerful when architecting solutions, providing a partial answer to the “What?” question Yehuda pointed out.
  2. A higher level abstraction can let you accidentally solve problems in other areas as well. A good example of this is error correcting tournaments which are useful for tournament design to select the best player/team/paper in real tournaments. Recently, I was amused to learn that a standard betting procedure for basketball tournaments exactly mirrors the importance weights suggested for the final elimination of ECTs. The first phase of ECTs provides a sound and practical method to seed a final elimination tournament, eliminating the need for (and biases of) a committee.
  3. Perhaps the most interesting effect is that the new abstraction can aid you in finding effective solutions to new problems. For learning reductions, there are about 3 compelling instances I’ve seen so far.
    1. Given training-time access to a good policy oracle, Searn provides a method for decomposing any complex prediction problem into simple problems, such that low regret solutions to the simple problems imply a low regret solution to the original problem. While Searn competes well (computationally and prediction-wise) with existing methods for linear chain style structured prediction, it really shines on more complex problems. Hal used Searn for automatic document summarization (see section 6.2) which previously wasn’t really solved via ML. More generally, when I learn about the details of other complex prediction systems for machine translation or vision, the base algorithms are tweaked, typically in ways that Searn would suggest. This suggests that Searn formalizes and automates the intuitions of practical people.
    2. The “one step RL” reduction in Bianca‘s thesis (page 119) provided tractable and effective approaches to learning in partial feedback problems where only the loss of a chosen label is learned. An even simpler reduction exists as a matter of folklore—estimate the the value of each label and then take an argmax. However, we have found classification approaches generally work better, where applicable, and as the theory suggests.
    3. Many commonly used algorithms for prediction have a running time linear (or worse) in the number of labels with decision trees a good exception. While simply predicting faster isn’t normally solving a “new problem”, an exponential improvement in computational time seems to merit this description because it allows entirely new kinds of applications. It turns out that it is both very easy to do logarithmic time prediction wrong, and that this problem is often fixable. Furthermore, it appears logarthmic time prediction can really work in practice over very many labels.

When we started working on learning reductions, I had no idea what either the difficulties or rewards were going to be—it simply seemed like a natural and compelling direction of investigation. Given the substantial difficulties encountered, it’s not at all clear that this pursuit was personally worthwhile. It has cost much time which could have been put to good use in other ways.

On the other hand, the advantages are also substantial. I’ve learned something about architecting solutions to problems, both expanding the domain of application for the field and providing a personal edge that I can bring to many conversations about ML. It’s also progress towards the AI goal, which interests me. When I think of what I could have worked on instead to achieve these goals, I don’t have any more compelling answer yet. Learning reductions seem to have accomplished more per unit thought than any other theoretical approach I can identify over the last 5 or 6 years. Furthermore, they are composable by design, so they should stay relevant (and perhaps even become more so), when people use an online active deep semisupervised probabilistic convolutional algorithm to solve a problem, particularly for complex problems.

As I said at the beginning, please join us for the tutorial, if you are interested.

3/8/2009

Prediction Science

One view of machine learning is that it’s about how to program computers to predict well. This suggests a broader research program centered around the more pervasive goal of simply predicting well.
There are many distinct strands of this broader research program which are only partially unified. Here are the ones that I know of:

  1. Learning Theory. Learning theory focuses on several topics related to the dynamics and process of prediction. Convergence bounds like the VC bound give an intellectual foundation to many learning algorithms. Online learning algorithms like Weighted Majority provide an alternate purely game theoretic foundation for learning. Boosting algorithms yield algorithms for purifying prediction abiliity. Reduction algorithms provide means for changing esoteric problems into well known ones.
  2. Machine Learning. A great deal of experience has accumulated in practical algorithm design from a mixture of paradigms, including bayesian, biological, optimization, and theoretical.
  3. Mechanism Design. The core focus in game theory is on equilibria, mostly typically Nash equilibria, but also many other kinds of equilibria. The point of equilibria, to a large extent, is predicting how agents will behave. When this is employed well, principally in mechanism design for auctions, it can be a very powerful concept.
  4. Prediction Markets. The basic idea in a prediction market is that commodities can be designed so that their buy/sell price reflects a form of wealth-weighted consensus estimate of the probability of some event. This is not simply mechanism design, because (a) the thin market problem must be dealt with and (b) the structure of plausible guarantees is limited.
  5. Predictive Statistics. Part of statistics focuses on prediction, essentially becoming indistinguishable from machine learning. The canonical example of this is tree building algorithms such as CART, random forests, and some varieties of boosting. Similarly the notion of probability, counting, and estimation are all handy.
  6. Robust Search. I have yet to find an example of robust search which isn’t useful—and there are several varieties. This includes active learning, robust min finding, and (more generally) compressed sensing and error correcting codes.

The lack of unification is fertile territory for new research, so perhaps it’s worthwhile to think about how these different research programs might benefit from each other.

  1. Learning Theory. The concept of mechanism design is mostly missing from learning theory, but it is sure to be essential when interactive agents are learning. We’ve found several applications for robust search as well as new settings for robust search such as active learning, and error correcting tournaments, but there are surely others.
  2. Machine Learning and Predictive Statistics. Machine learning has been applied to auction design. There is a strong relationship between incentive compatibility and choice of loss functions, both for choosing proxy losses and approximating the real loss function imposed by the world. It’s easy to imagine designer loss functions from the study of incentive compatibility mechanisms giving learning algorithm an edge. I found this paper thought provoking that way. Since machine learning and information markets share a design goal, are there hybrid approaches which can outperform either?
  3. Mechanism Design. There are some notable similarities between papers in ML and mechanism design. For example there are papers about learning on permutations and pricing in combinatorial markets. I haven’t yet taken the time to study these carefully, but I could imagine that one suggests advances for the other, and perhaps vice versa. In general, the idea of using mechanism design with context information (as is done in machine learning), could also be extremely powerful.
  4. Prediction Markets. Prediction markets are partly an empirical field and partly a mechanism design field. There seems to be relatively little understanding about how well and how exactly information from multiple agents is supposed to interact to derive a good probability estimate. For example, the current global recession reminds us that excess leverage is a very bad idea. The same problem comes up in machine learning and is solved by the weighted majority algorithm (and even more thoroughly by the hedge algorithm). Can an information market be designed with the guarantee that an imperfect but best player decides the vote after not-too-many rounds? How would this scale as a function of the ratio of a participants initial wealth to the total wealth?
  5. Robust Search. Investigations into robust search are extremely diverse, essentially only unified in a mathematically based analysis. For people interested in robust search, machine learning and information markets provide a fertile ground for empirical application and new settings. Can all mechanisms for robust search be done with context information, as is common in learning? Do these approaches work empirically in machine learning or information markets?

There are almost surely many other interesting research topics and borrowable techniques here, and probably even other communities oriented around prediction. While the synthesis of these fields is almost sure to eventually happen, I’d like to encourage it sooner rather than later. For someone working on one of these branches, attending a conference on one of the other branches might be a good start. At a lesser time investment, Oddhead is a good start.

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