Machine Learning (Theory)


Exponentiated Gradient

Tags: Machine Learning,Online jl@ 4:33 pm

The Exponentiated Gradient algorithm by Manfred Warmuth and Jyrki Kivinen came out just as I was starting graduate school, so I missed it both at a conference and in class. It’s a fine algorithm which has a remarkable theoretical statement accompanying it.

The essential statement holds in the “online learning with an adversary” setting. Initially, there are of set of n weights, which might have values (1/n,…,1/n), (or any other values from a probability distribution). Everything happens in a round-by-round fashion. On each round, the following happens:

  1. The world reveals a set of features x in {0,1}n. In the online learning with an adversary literature, the features are called “experts” and thought of as subpredictors, but this interpretation isn’t necessary—you can just use feature values as experts (or maybe the feature value and the negation of the feature value as two experts).
  2. EG makes a prediction according to y’ = w . x (dot product).
  3. The world reveals the truth y in [0,1].
  4. EG updates the weights according to wi <- wie-2 c (y’ – y)xi. Here c is a small constant learning rate.
  5. The weights are renormalized to sum to 1.

The 4th line is what gives EG it’s name—exponent of the negative gradient (of squared loss in this case).

The accompanying theoretical statement (in english), is that for all sequences of observations, this algorithm does nearly as well in squared loss as the best convex combination of the features with a regret that is only logarithmic in the number of features. The mathematical theorem statement is: For all T for all T-length sequences of observations,

SumTt=1 (y’t – yt)2 <= minprobability distributions q (2/(2-c)) SumTt=1 (q.xt – y)2 + KL(q||p) / c

Here KL(q||p) = Sumi qi ln (qi / pi) is the KL-divergence between the distribution q that you compare with and the distribution p that you start with. The KL-divergence is at most log n if p is uniform.

The learning rate c plays a critical role in the theorem, and the best constant setting of c depends on how many total rounds T there are. Tong Zhang likes to think of this algorithm as the stochastic gradient descent with entropy regularization, which makes it clear that when used as an online optimization algorithm, c should be gradually decreased in value.

There are many things right here.

  1. Exponentiated Gradient competes with the best convex predictor with no caveats about how the world produces the data. This is pretty remarkable—it’s much stronger than competing with the best single feature, as many other online adversary algorithms do. The lack of assumptions about the world means this is a pretty universal algorithm.
  2. The notion of competition is logarithmic in the number of features so the algorithm can give good performance even when the number of features is extraordinarily large compared to the number of examples. In contrast gradient descent style algorithms might need a number of examples similar to the number of features. (The same paper also analyzes regret for gradient descent.)
  3. We know from a learning reductions point of view that the ability to optimize squared loss well implies that ability to optimize many other losses fairly well.

A few things aren’t right about EG. A convex combination is not as powerful a representation as a linear combination. There are natural relaxations covered in the EG paper which deal with more general combinations, as well as in some other papers. For more general combinations, the story with respect to gradient descent becomes more mixed. When the best predictor is a linear combination of many features, gradient descent style learning may be superior to EG.

EG-style algorithms are slowly coming out. For example, this paper shows that EG style updates can converge very quickly compared to other methods.


Watchword: Online Learning

It turns out that many different people use the term “Online Learning”, and often they don’t have the same definition in mind. Here’s a list of the possibilities I know of.

  1. Online Information Setting Online learning refers to a problem in which unlabeled data comes, a prediction is made, and then feedback is acquired.
  2. Online Adversarial Setting Online learning refers to algorithms in the Online Information Setting which satisfy guarantees of the form: “For all possible sequences of observations, the algorithim has regret at most log ( number of strategies) with respect to the best strategy in a set.” This is sometimes called online learning with experts.
  3. Online Optimization Constraint Online learning refers to optimizing a predictor via a learning algorithm tunes parameters on a per-example basis. This may or may not be applied in the Online Information Setting, and the strategy may or may not satisfy Adversarial setting theory.
  4. Online Computational Constraint Online learning refers to an algorithmic constraint that the amount of computation per example is constant as the number of examples increases. Again, this doesn’t imply anything in particular about the Information setting in which it is applied.
  5. Lifelong Learning Online learning refers to learning in a setting where different tasks come at you over time, and you need to rapidly adapt to past mastered tasks.


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.)


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.


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?


Online convex optimization at COLT

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

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

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


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.


Online learning or online preservation of learning?

Tags: Machine Learning,Online jl@ 3:36 pm

In the online learning with experts setting, you observe a set of predictions, make a decision, and then observe the truth. This process repeats indefinitely. In this setting, it is possible to prove theorems of the sort:

master algorithm error count < = k* best predictor error count + c*log(number of predictors)

Is this a statement about learning or about preservation of learning? We did some experiments to analyze the new Binning algorithm which works in this setting. For several UCI datasets, we reprocessed them so that features could be used as predictors and then applied several master algorithms. The first graph confirms that Binning is indeed a better algorithm according to the tightness of the upper bound.

Here, “Best” is the performance of the best expert. “V. Bound” is the bound for Vovk‘s algorithm (the previous best). “Bound” is the bound for the Binning algorithm. “Binning” is the performance of the Binning algorithm. The Binning algorithm clearly has a tighter bound, and the performance bound is clearly a sharp constraint on the algorithm performance.

Instead of examining bounds, we can simply look at performance.

“Bin” is the performance of Binning (identical to the previous graph). BW is Binomial weighting, which is (roughly) the deterministic version of Binning. WM is Weighted Majority. Both BW and WM are deterministic algorithms which implies their performance bounds are perhaps a factor of 2 worse than Binning or Vovk’s algorithm.

In contrast, the actual performance (rather than performance bound) of the deterministic algorithms is sometimes even better than the best expert (negative regret?!). A consistent negative correlation between “online bound tightness” and “learning performance” is observed.

The question is “What’s happening here?”

  1. One reply is that we are testing in the wrong setting. These algorithms are designed to work in highly adversarial environments for which a UCI dataset does not qualify. This isn’t a convincing answer to me because many (or perhaps most) situations are not that adversarial.
  2. Another answer is “you used the wrong experts”. This is not convincing because many other learning algorithm do as well or better with the given features/experts.
  3. Another possibility is “you can start out running Binning, and when it pulls ahead of it’s bound run any learning algorithm. If the learning algorithm does badly, you can switch back to Binning and preserve the guarantee.” So, Binning is effectively a safety net.

My best current understanding is that “online learning with experts” is really “online preservation of learning”: the goal of the algorithm is to preserve whatever predictive ability the individual predictors have. This understanding fits the form of the theory statement well.

Preservation is desirable in some situations. For example, charity events sometimes work according to the following form:

  1. All participants exchange dollars for bogobucks.
  2. The participants gamble with bogobucks.
  3. The winner at the end gets some prize.

An online preservation algorithm has the property that if you acquire enough bogobucks in comparison to the number of participants, you can guarantee winning the prize. These kinds of ‘winner take all’ scenarios come up elsewhere.


A question of quantification

Tags: Definitions,Online,Reductions jl@ 7:39 am

This is about methods for phrasing and think about the scope of some theorems in learning theory. The basic claim is that there are several different ways of quantifying the scope which sound different yet are essentially the same.

  1. For all sequences of examples. This is the standard quantification in online learning analysis. Standard theorems would say something like “for all sequences of predictions by experts, the algorithm A will perform almost as well as the best expert.”
  2. For all training sets. This is the standard quantification for boosting analysis such as adaboost or multiclass boosting.
    Standard theorems have the form “for all training sets the error rate inequalities … hold”.
  3. For all distributions over examples. This is the one that we have been using for reductions analysis. Standard theorem statements have the form “For all distributions over examples, the error rate inequalities … hold”.

It is not quite true that each of these is equivalent. For example, in the online learning setting, quantifying “for all sequences of examples” implies “for all distributions over examples”, but not vice-versa.

However, in the context of either boosting or reductions these are equivalent because the algorithms operate in an element-wise fashion. To see the equivalence, note that:

  1. “For any training set” is equivalent to “For any sequence of examples” because a training set is a sequence and vice versa.
  2. “For any sequence of examples” is equivalent to “For any distribution over examples” when the theorems are about unconditional example transformations because:
    1. The uniform distribution over a sufficiently long sequence of examples can approximate any distribution we care about arbitrarily well.
    2. If the theorem holds “for all distributions”, it holds for the uniform distribution over the elements in any sequence of examples.

The natural debate here is “how should the theorems be quantified?” It is difficult to answer this debate based upon mathematical grounds because we just showed an equivalence. It is nevertheless important because it strongly influences how we think about algorithms and how easy it is to integrate the knowledge across different theories. Here are the arguments I know.

  1. For all sequences of examples.
    1. Learning theory people (at least) are used to thinking about “For all sequences of examples”.
    2. (Applied) Machine learning people are not so familiar with this form of quantification.
    3. When the algorithm is example-conditional such as in online learning, the quantification is more general than “for all distributions”.
  2. For all training sets.
    1. This is very simple.
    2. It is misleadingly simple. For example, a version of the adaboost theorem also applies to test sets using the test error rates of the base classifiers. It is fairly common for this to be misunderstood.
  3. For all distributions over examples.
    1. Distributions over examples is simply how most people think about learning problems.
    2. “For all distributions over examples” is easily and often confused with “For all distributions over examples accessed by IID draws”. It seems most common to encounter this confusion amongst learning theory folks.

What quantification should be used and why?
(My thanks to Yishay Mansour for clarifying the debate.)


Fallback Analysis is a Secret to Useful Algorithms

The ideal of theoretical algorithm analysis is to construct an algorithm with accompanying optimality theorems proving that it is a useful algorithm. This ideal often fails, particularly for learning algorithms and theory. The general form of a theorem is:

If preconditions Then postconditions
When we design learning algorithms it is very common to come up with precondition assumptions such as “the data is IID”, “the learning problem is drawn from a known distribution over learning problems”, or “there is a perfect classifier”. All of these example preconditions can be false for real-world problems in ways that are not easily detectable. This means that algorithms derived and justified by these very common forms of analysis may be prone to catastrophic failure in routine (mis)application.

We can hope for better. Several different kinds of learning algorithm analysis have been developed some of which have fewer preconditions. Simply demanding that these forms of analysis be used may be too strong—there is an unresolved criticism that these algorithm may be “too worst case”. Nevertheless, it is possible to have a learning algorithm that simultaneously provides strong postconditions given strong preconditions, reasonable postconditions given reasonable preconditions, and weak postconditions given weak preconditions. Some examples of this I’ve encountered include:

  1. Sham, Matthias and Dean showing that some Bayesian regression is robust in a minimax online learning analysis.
  2. The cover tree which creates an O(n) datastructure for nearest neighbor queries while simultaneously guaranteeing O(log(n)) query time when the metric obeys a dimensionality constraint.

The basic claim is that algorithms with a good fallback analysis are significantly more likely to achieve the theoretical algorithm analysis ideal. Both of the above algorithms have been tested in practice and found capable.

Several significant difficulties occur for anyone working on fallback analysis.

  1. It’s harder. This is probably the most valid reason—people have limited time to do things. Nevertheless, it is reasonable to hope that the core techniques used by many people have had this effort put into them.
  2. It is psychologically difficult to both assume and not assume a precondition, for a researcher. A critical valuable resource here is observing multiple forms of analysis.
  3. It is psychologically difficult for a reviewer to appreciate the value of both assuming and not assuming some precondition. This is a matter of education.
  4. It is neither “sexy” nore straightforward. In particular, theoretically inclined people 1) get great joy from showing that something new is possible and 1) routinely work on papers of the form “here is a better algorithm to do X given the same assumptions”. A fallback analysis requires a change in assumption invalidating (2) and the new thing that it shows for (1) is subtle: that two existing guarantees can hold for the same algorithm. My hope here is that this subtlety becomes better appreciated in time—making useful algorithms has a fundamental sexiness of it’s own.


On-line learning of regular decision rules

Many decision problems can be represented in the form
FOR n=1,2,…:
— Reality chooses a datum xn.
— Decision Maker chooses his decision dn.
— Reality chooses an observation yn.
— Decision Maker suffers loss L(yn,dn).
The observation yn can be, for example, tomorrow’s stock price and the decision dn the number of shares Decision Maker chooses to buy. The datum xn ideally contains all information that might be relevant in making this decision. We do not want to assume anything about the way Reality generates the observations and data.

Suppose there is a good and not too complex decision rule D mapping each datum x to a decision D(x). Can we perform as well, or almost as well, as D, without knowing it? This is essentially a special case of the problem of on-line learning.

This is a simple result of this kind. Suppose the data xn are taken from [0,1] and L(y,d)=|yd|. A norm ||h|| of a function h on [0,1] is defined by

||h||2 = (Integral01 h(t) dt)2 + Integral01 (h'(t))2 dt.
Decision Maker has a strategy that guarantees, for all N and all D with finite ||D||,
Sumn=1N L(yn,dn) is at most Sumn=1N L(yn,D(xn)) + (||2D–1|| + 1) N1/2.
Therefore, Decision Maker is competitive with D provided the “mean slope” (Integral01 (D'(t))2dt)1/2 of D is significantly less than N1/2. This can be extended to general reproducing kernel Hilbert spaces of decision rules and to many other loss functions.

It is interesting that the only known way of proving this result uses a non-standard (although very old) notion of probability. The standard definition (“Kolmogorov’s axiomatic“) is that probability is a normalized measure. The alternative is to define probability in terms of games rather than measure (this was started by von Mises and greatly advanced by Ville, who in 1939 replaced von Mises’s awkward gambling strategies with what he called martingales). This game-theoretic probability allows one to state the most important laws of probability as continuous strategies for gambling against the forecaster: the gambler becomes rich if the forecasts violate the law. In 2004 Akimichi Takemura noticed that for every such strategy the forecaster can play in such a way as not to lose a penny. Takemura’s procedure is simple and constructive: for any continuous law of probability we have an explicit forecasting strategy that satisfies that law perfectly. We call this procedure defensive forecasting. Takemura’s version was about binary observations, but it has been greatly extended since.

Now let us see how we can achieve the goal Sumn=1 N L(yn,dn) < Sumn=1N L(yn,D(xn)) + (something small). This would be easy if we knew the true probabilities for the observations. At each step we would choose the decision with the smallest expected loss and the law of large numbers would allow us to prove that our goal would be achieved. Alas, we do not know the true probabilities (if they exist at all). But with defensive forecasting at our disposal we do not need them: the fake probabilities produced by defensive forecasting are ideal as far as the law of large numbers is concerned. The rest of the proof is technical details.


Online Learning as the Mathematics of Accountability

Tags: Online jl@ 9:59 am

Accountability is a social problem. When someone screws up, do you fire them? Or do you accept the error and let them continue? This is a very difficult problem and we all know of stories where the wrong decision was made.

Online learning (as meant here), is a subfield of learning theory which analyzes the online learning model.

In the online learning model, there are a set of hypotheses or “experts”. On any instantance x, each expert makes a prediction y. A master algorithm A uses these predictions to form it’s own prediction yA and then learns the correct prediction y*. This process repeats.

The goal of online learning is to find a master algorithm A which uses the advice of the experts to make good predictions. In particular, we typically want to guarantee that the master algorithm performs almost as well as the best expert. If L(e) is the loss of expert e and L(A) is the loss of the master algorithm, it is often possible to prove:

L(A) less than mine L(e) + log(number of experts)

over all sequences.

In particular, there is no assumption of independent samples and there is no assumption that the experts perform well (or can perform well). This is not a high probability statement: it simply always holds. These assumption-free qualities are very important for application to the accountability problem, because the experts really can be adversarial.

In any situation where we have a set of human experts giving advice on the same subject, we can hope to apply online learning algorithms to better distill collective advice into single prediction. Examples include:

  1. stock picking Many of the big stocks have ‘analysts’ who predict whether they will go up or down. For an example, look here.
  2. surveys It is common to see things like “A gain of 1.4 percent was expected, based on the median estimate in a Bloomberg survey of 53 economists” in news articles. Presumably, these economists are reused frequently implying they have a record to which an online algorithm could be applied.

This application of online learning isn’t trivial. Even for the above examples, it isn’t clear how to handle issues like:

  1. A new expert starts making predictions. There are some reasonable ad-hoc mechanisms for coping with this in the context of particular algorithms.
  2. An expert declines to make a prediction. The modified “sleeping experts” setting handles this, but the results are not quite as tight.
  3. The loss associated with individual predictions is highly variable rather than something simple like “0/1-loss” or “squared error loss”. One approach to this is to combine regret minimizing learning reductions with online learning algorithms (drawback: the reduced predictions may not be of intuitive things). Another approach is simply trying to make very flexible master algorithms (drawback: flexibility often comes with a weakening in the theoretical guarantees).
  4. In the real world, we may not have feedback about a prediction until after the next 10 predictions (or more) need to be made.
  5. In the real world, there may be uncertainty about the measured truth. Quantifying GDP growth requires a lot of work and has some fundamental uncertainty associated with it, especially when immediate feedback is required.


Structured Regret Minimization

Tags: Online,Papers jl@ 3:50 pm

Geoff Gordon made an interesting presentation at the snowbird learning workshop discussing the use of no-regret algorithms for the use of several robot-related learning problems. There seems to be a draft here. This seems interesting in two ways:

  1. Drawback Removal One of the significant problems with these online algorithms is that they can’t cope with structure very easily. This drawback is addressed for certain structures.
  2. Experiments One criticism of such algorithms is that they are too “worst case”. Several experiments suggest that protecting yourself against this worst case does not necessarily incur a great loss.


Binomial Weighting

Tags: Online,Papers,Problems jl@ 8:16 pm

Suppose we have a set of classifiers c making binary predictions from an input x and we see examples in an online fashion. In particular, we repeatedly see an unlabeled example x, make a prediction y’(possibly based on the classifiers c), and then see the correct label y.

When one of these classifiers is perfect, there is a great algorithm available: predict according to the majority vote over every classifier consistent with every previous example. This is called the Halving algorithm. It makes at most log2 |c| mistakes since on any mistake, at least half of the classifiers are eliminated.

Obviously, we can’t generally hope that the there exists a classifier which never errs. The Binomial Weighting algorithm is an elegant technique allowing a variant Halving algorithm to cope with errors by creating a set of virtual classifiers for every classifier which occasionally disagree with the original classifier. The Halving algorithm on this set of virtual classifiers satisfies a theorem of the form:

errors of binomial weighting algorithm less than minc f(number of errors of c, number of experts)

The Binomial weighting algorithm takes as a parameter the maximal minimal number of mistakes of a classifier. By introducing a “prior” over the number of mistakes, it can be made parameter free. Similarly, introducing a “prior” over the set of classifiers is easy and makes the algorithm sufficiently flexible for common use.

However, there is a problem. The minimal value of f() is 2 times the number of errors of any classifier, regardless of the number of classifiers. This is frustrating because a parameter-free learning algorithm taking an arbitrary “prior” and achieving good performance on an arbitrary (not even IID) set of examples is compelling for implementation and use, if we had a good technique for removing the factor of 2. How can we do that?

See the weighted majority algorithm for an example of a similar algorithm which can remove a factor of 2 using randomization and at the expense of introducing a parameter. There are known techniques for eliminating this parameter, but they appear not as tight (and therefore practically useful) as introducing a “prior” over the number of errors.


Prior, “Prior” and Bias

Many different ways of reasoning about learning exist, and many of these suggest that some method of saying “I prefer this predictor to that predictor” is useful and necessary. Examples include Bayesian reasoning, prediction bounds, and online learning. One difficulty which arises is that the manner and meaning of saying “I prefer this predictor to that predictor” differs.

  1. Prior (Bayesian) A prior is a probability distribution over a set of distributions which expresses a belief in the probability that some distribution is the distribution generating the data.
  2. “Prior” (Prediction bounds & online learning) The “prior” is a measure over a set of classifiers which expresses the degree to which you hope the classifier will predict well.
  3. Bias (Regularization, Early termination of neural network training, etc…) The bias is some (often implicitly specified by an algorithm) way of preferring one predictor to another.

This only scratches the surface—there are yet more subtleties. For example the (as mentioned in meaning of probability) shifts from one viewpoint to another.


Problem: Online Learning

Tags: Online,Problems jl@ 8:27 am

Despite my best intentions, this is not a fully specified problem, but rather a research direction.

Competitive online learning is one of the more compelling pieces of learning theory because typical statements of the form “this algorithm will perform almost as well as a large set of other algorithms” rely only on fully-observable quantities, and are therefore applicable in many situations. Examples include Winnow, Weighted Majority, and Binomial Weighting. Algorithms with this property haven’t taken over the world yet. Here might be some reasons:

  1. Lack of caring. Many people working on learning theory don’t care about particular applications much. This means constants in the algorithm are not optimized, usable code is often not produced, and empirical studies aren’t done.
  2. Inefficiency. Viewed from the perspective of other learning algorithms, online learning is terribly inefficient. It requires that every hypothesis (called an expert in the online learning setting) be enumerated and tested on every example. (This is similar to the inefficiency of using Bayes law as an algorithm directly, and there are strong similarities in the algorithms.)

For an example of (1), there is a mysterious factor of 2 in the Binomial Weighting algorithm which has not been fully resolved. Some succesful applications also exist such as those based on SNoW.

The way to combat problem (2) is to introduce more structure into the hypothesis/experts. Some efforts have already been made in this direction. For example, it’s generally feasible to introduce an arbitrary bias or “prior” over the experts in the form of some probability distribution, and perform well with respect to that bias. Another nice piece of work by Adam Kalai and Santosh Vempala discusses how to efficiently handle several forms of structured experts. At an intuitive level, further development discovering how to efficiently work with new forms of structure might payoff well.

The basic research direction here is to address the gap between theory and practice, possibly by solving the above problems and possibly by discovering and addressing other problems.


NIPS: Online Bayes

Tags: Bayesian,Online DrewBagnell@ 4:21 pm

One nice use for this blog is to consider and discuss papers that that have appeared at recent conferences. I really enjoyed Andrew Ng and Sham Kakade’s paper Online Bounds for Bayesian Algorithms. From the paper:

The philosophy taken in the Bayesian methodology is often at odds with
that in the online learning community…. the online learning setting
makes rather minimal assumptions on the conditions under which the
data are being presented to the learner —usually, Nature could provide
examples in an adversarial manner. We study the performance of
Bayesian algorithms in a more adversarial setting… We provide
competitive bounds when the cost function is the log loss, and we
compare our performance to the best model in our model class (as in
the experts setting).

It’s a very nice analysis of some of my favorite algorithms that all hinges around a beautiful theorem:

Let Q be any distribution over parameters theta. Then for all sequences S:

L_{Bayes}(S) leq L_Q(S) + KL(Q|P)

where P is our prior and the losses L are: first, log-loss for the Bayes algorithm (run online) and second, expected log-loss with respect to an arbitrary distribution Q.

Any thoughts? Any other papers you thought we have to read?

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