How did they do it? From a very high level point of view they simply combined the previous state of the art (Monte Carlo Tree Search) with the new deep learning techniques. Recall that MCTS is a technique inspired from multi-armed bandits to efficiently explore the tree of possible action sequences in a game, for more details see this very nice survey by my PhD advisor Remi Munos: From Bandits to Monte-Carlo Tree Search: The Optimistic Principle Applied to Optimization and Planning. Now in MCTS there are two key elements beyond the bandit part (i.e., how to deal with exploration v.s. exploitation): one needs a way to combine all the information collected to produce a value function for each state (this value is the key term in the upper confidence bound used by the bandit algorithm); and one needs a reasonable random policy to carry out the random rollouts once the bandit strategy has gone deep enough in the tree. In AlphaGo the initial random rollout strategy is learned via supervised deep learning on a dataset of human expert games, and the value function is learned (online, with MCTS guiding the search) via convolutional neural networks (in some sense this corresponds to a very natural inductive bias for this game).

Of course there is much more to AlphaGo than what I described above and you are invited to take a look at the paper (see this reddit thread to find the paper)!

]]>In this post I want to talk about some of the new emerging directions that the NIPS community is taking. Of course my view is completely biased as I am more representative of COLT than NIPS (though obviously the two communities have a large overlap). Also I only looked in details at about 25% of the papers so perhaps I missed the most juicy breakthrough. In any case below you will find a short summary of each of these new directions with pointers to some of the relevant papers. Before going into the fun math I wanted to first share some thoughts about the big announcement of yesterday.

**Thoughts about OpenAI**

Obvious disclaimer: the opinions expressed here represent my own and not those of my employer (or previous employer hosting this blog). Now, for those of you who missed it, yesterday Elon Musk and friends made a huge announcement: they are giving $1 billion to create a non-profit organization whose goal is the advancement of AI (see here for the official statement, and here for the New York Times covering). This is just absolutely wonderful news, and I really feel like we are watching history in the making. There are very very few places in the world solely dedicated to basic research and with that kind of money. Examples are useful to get some perspective: the Perimeter Institute for Theoretical Physics was funded with $100 million (I believe it has a major impact in the field), the Institute for Advanced Studies was funded with a similar size gift (a simple statistic give an idea of the impact: 41 out of 57 Fields medalists have been affiliated with IAS), more recently and perhaps closer to us the Simons Institute for the Theory of Computing was created with $60 million and its influence on the field keep growing (it was certainly a very influential place in my own career). Looking at what those places are doing with 1/10 of OpenAI’s budget sets the bar extremely high for OpenAI, and I am very excited to see what direction they take and what their long term plans are!

Now let’s move on to what worries me a little: the 10 founding members of OpenAI are all working on deep learning. Before explaining further why this is worrisome let me emphasize that I strongly believe that disentangling the mysteries behind the impressive practical successes of deep nets is a key challenge for the future of AI (in fact I am spending a good amount of time thinking about this issue, just like many other groups in theoretical machine learning these days). I also believe that pushing the engineering aspect of deep nets will lead to wonderful technological breakthroughs, which is why it makes sense for companies such as Facebook, Google, Baidu, Microsoft, Amazon to invest heavily in this endeavor. However it seems insane to think that the current understanding of deep nets will be sufficient to achieve even very weak forms of AI. AI is still far from being an engineering problem, and there are some fundamental theoretical questions that have to be resolved before we can brute force our way through this problem. In fact the mission statement of OpenAI mention one such fundamental question about which we know very little: currently we build systems that solve one task (e.g., image segmentation) but how do we combine these systems so that they take advantage of each other and help improving the learning of future tasks? While one can cook up heuristics to attack this problem (such as using the learned weights for one task as the initialization for another one) it seems clear to me that we are lacking the mathematical framework and tools to think properly about this question. I don’t think that deep learners are the best positioned to make conceptual progress on this question (and similar ones), though I definitely admit that they are probably the best positioned *right now* to make some practical progress. Again this is why all big companies are investing in this, but for an institution that wants to look into the more distant future it seems critical to diversify the portfolio (in fact this is exactly what Microsoft Research does) and not just follow companies who often have much shorter term objectives. I really hope that this is part of their plans.

I wish the best of luck to OpenAI and their members. The game-changing potential of this organization puts a lot of responsibility on them and I sincerely hope that they will try to seriously explore different paths to AI rather than to chase local-in-time advertisement (please don’t just solve Go with deep nets!!!).

Now time for some of the cool stuff that happened at NIPS.

**Scaling up sampling
**

Variational inference is a very successful paradigm in Bayesian learning where instead of trying to compute exactly the posterior distribution one searches through a parametric family for the closest (in relative entropy) distribution to the true posterior. The key observation is that one can perform stochastic gradient descent for this problem without having to compute the normalization constant in the posterior distribution (which is often an intractable problem). The only catch is that one needs to be able to sample from an element (conditioned on the observed data) of the parametric family under consideration, and this might itself be a difficult problem in large-scale applications. A basic MCMC method for this type of problems is the Langevin Monte Carlo (LMC) algorithm for which a very nice theoretical analysis was recently provided by Dalalyan in the case of convex negative log-likelihood. The issue for large-scale applications is that each step of LMC requires going through the entire data set. This is where SGLD (Stochastic Gradient Langevin Dynamics) comes in, a very nice idea of Welling and Whye Teh, where the gradient step on the convex negative log-likelihood is replaced by a stochastic gradient step. The issue is that this introduces a bias in the stationary distribution, and fixing this bias can be done in several ways such as adding an accept-reject step, or modifying appropriately the covariance matrix of the noise in the Langevin Dynamics. The jury is still out on what is the most appropriate fix, and three papers made contributions to this question at NIPS: “On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators“, “Covariance-Controlled Adaptive Langevin Thermostat for Large-Scale Bayesian Sampling“, and “A Complete Recipe for Stochastic Gradient MCMC“. Another key related question on which progress was made is to decide when to stop the chain, see “Measuring Sample Quality with Stein’s Method” (my favorite paper at this NIPS) and “Mixing Time Estimation in Reversible Markov Chains from a Single Sample Path“. My own paper “Finite-Time Analysis of Projected Langevin Monte Carlo” was also in that space: it adds nothing to the large scale picture but it shows how Langevin dynamics can cope with compactly supported distributions. Finally another related paper that I found interesting is “Sampling from Probabilistic Submodular Models“.

**When I’m a grown-up I want to do non-convex optimization!**

With deep nets in mind all the rage is about non-convex optimization. One direction in that space is to develop more efficient algorithms for specific problems where we already know polynomial-time methods under reasonable assumptions, such as low rank estimation (see “A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements“) and phase retrieval (see “Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems“). The nice thing about those new results is that they essentially show that gradient descent with a spectral initialization will work (previous evidence was already shown for alternating minimization, see also “A Nonconvex Optimization Framework for Low Rank Matrix Estimation“). Another direction in non-convex optimization is to slowly extend the class of functions that one can solve efficiently, see “Beyond Convexity: Stochastic Quasi-Convex Optimization“. Finally a thought-provoking paper which is worth mentioning is “Matrix Manifold Optimization for Gaussian Mixtures” (it comes without provable guarantees but maybe something can be done there…).

**Convex optimization strikes back**

As I said non-convex optimization is all the rage, yet there are still many things about convex optimization that we don’t understand (an interesting example is given in this paper “Information-theoretic lower bounds for convex optimization with erroneous oracles“). I blogged recently about a new understanding of Nesterov’s acceleration, but this said nothing about the Nesterov’s accelerated gradient descent. The paper “Accelerated Mirror Descent in Continuous and Discrete Time” builds on (and refines) recent advances on understanding the relation of AGD and Mirror Descent, as well as the differential equations underlying them. Talking about Mirror Descent, I was happy to see it applied to deep nets optimization in “End-to-end Learning of LDA by Mirror-Descent Back Propagation over a Deep Architecture“. Another interesting trend is the revival of second-order methods (e.g., Newton’s method) by using various low-rank approximations to the Hessian, see “Convergence rates of sub-sampled Newton methods“, “Newton-Stein Method: A Second Order Method for GLMs via Stein’s Lemma“, and “Natural Neural Networks“.

**Other topics**

There are a few other topics that caught my attention but I am running out of stamina. These include many papers on the analysis of cascades in networks (I am particularly curious about the COEVOLVE model), papers that further our understanding of random features, adaptive data analysis (see this), and a very healthy list of bandit papers (or Bayesian optimization as some like to call it).

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**Boosting**

Say that given a distribution supported on points one can find (efficiently) a classifier such that (here we are in the context of classification with the zero-one loss). Can we “boost” this weak learning algorithm into a strong learning algorithm with arbitrarily small for large enough? It turns out that this is possible, and even simple. The idea is to build a linear combination of hypotheses in with a greedy procedure. That is at time step our hypothesis is (the sign of) , and we are now looking to add with an approriate weight . A natural guess is to optimize over to minimize the training error of on our sample . This might be a difficult computational problem (how do you optimize over ?), and furthermore we would like to make use of our efficient weak learning algorithm. The key trick is that . More precisely:

where . From this we see that we would like to be a good predictor for the distribution . Thus we can pass to the weak learning algorithm, which in turns gives us with . Thus we now have:

Optimizing the above expression one finds that leads to (using )

The procedure we just described is called AdaBoost (introduce by Schapire and Freund) and we proved that it satisfies

(1)

In particular we see that our weak learner assumption implies that is realizable (and in fact realizable with margin , see next section for the definition of margin) with the hypothesis class:

This class can be thought of as a neural network with one (infinite size) hidden layer. To realize how expressive is compared to it’s a useful exercise to think about the very basic case of *decision stumps* (for which the empirical risk minimization can be implemented very efficiently):

To derive a bound on the true risk of AdaBoost it remains to calculate the VC dimension of the class where the size of the hidden layer is . This follows from more general results on the VC dimension of neural networks, and up to logarithmic factors one obtains that is of order . Putting this together with \eqref{eq:empada} we see that when is a constant, one should run AdaBoost for rounds, and then one gets .

**Margin**

We consider to be the set of distributions such that there exists with (again we are in the context of classification with the zero-one loss, and this assumption means that the data is almost surely realizable). The SVM idea is to search for with minimal Euclidean norm and such that . Effectively this is doing empirical risk minimization over the following data dependent hypothesis class (which we write in terms of the set of admissible weight vectors):

The key point is that we can now use the contraction lemma to bound the Rademacher complexity of this class. Indeed replacing the zero-one loss by the (Lipschitz!) “ramp loss” makes no difference for the optimum , and our estimated weight still has training error while its true loss is only surestimated. Using the argument from previous sections we see that the Rademacher complexity of our hypothesis class (with respect to the ramp loss) is bounded by (assuming the examples are normalized to be in the Euclidean ball). Now it is easy to see that the existence of with (and ) exactly corresponds to a geometric margin of between positive and negative examples (indeed the margin is exactly ). To summarize we just saw that under the -margin condition the SVM algorithm has a sample complexity of order . This suggests that from an estimation perspective one should map the points into a high-dimensional space so that one could hope to have the separability condition (with margin). However this raises computational challenges, as the QP given by the SVM can suddenly look daunting. This is where kernels come into the picture.

**Kernels**

So let’s go overboard and map the points to an infinite dimensional Hilbert space (as we will see in the next subsection this notation will be consistent with being the hypothesis class). Denote for this map, and let be the kernel associated with it. The key point is that we are not using all the dimensions of our Hilbert space in the SVM optimization problem, but rather we are effectively working in the subspace spanned by (this is because we are only working with inner products with those vectors, and we are trying to minimize the norm of the resulting vector). Thus we can restrict our search to (this fact is called Mercer representer theorem and the previous sentence is the proof…). The beauty is that now we only need to compute the Gram matrix as we only need to consider and . In particular we never need to compute the points (which anyway could be infinite dimensional, so we couldn’t really write them down…). Note that the same trick would work with soft SVM (i.e., regularized hinge loss). To drive the point home let’s see an example: leads to . I guess it doesn’t get much better than this :). Despite all this beauty, one should note that we now have to manipulate an object of size (the kernel matrix ) and in our big data days this can be a disaster. We really want to focus on methods with computational complexity linear in , and thus one is led to the problem of kernel approximation, which we will explore below. But first let us explore a bit further what kernel SVM is really doing.

**RKHS and the inductive bias of kernels**

As we just saw in kernel methods the true central element is the kernel rather than the embedding . In particular since the only thing that matter are inner products we might as well assume that , and is the completion of where the inner product is defined by (and the definition is extended to by linearity). Assuming that is positive definite (that is Gram matrices built from are positive definite) one obtains a well-defined Hilbert space . Furthermore this Hilbert space has a special property: for any , . In other words is a *reproducing kernel Hilbert space* (RKHS), and in fact any RKHS can be obtained with the above construction (this is a simple consequence of Riesz representation theorem). Now observe that we can rewrite the kernel SVM problem

as

While the first formulation is computationally more effective, the second sheds light on what we are really doing: simply searching the consistent (with margin) hypothesis in with smallest norm. In other words, thinking in terms of inductive bias, one should choose a kernel for which the norm represents the kind of *smoothness* one expects in the mapping from input to output (more on that next).

It should also be clear now that one can “kernelize” any regularized empirical risk minimization, that is instead of the boring (note that here the loss is defined on instead of )

one can consider the much more exciting

since this can be equivalently written as

This gives the kernel ridge regression, kernel logistic regression, etc…

**Translation invariant kernels and low-pass filtering**

We will now investigate a bit further the RKHS that one obtains with translation invariant kernels, that is . A beautiful theorem of Bochner characterizes the continuous maps (with ) for which such a is a positive definite kernel: it is necessary and sufficient that is the characteristic function of a probability measure , that is

An important example in practice is the Gaussian kernel: (this corresponds to mapping to the function ). One can check that in this case is itself a Gaussian (centered and with covariance ).

Now let us restrict our attention to the case where has a density with respect to the Lebesgue measure, that is . A standard calculation then shows that

which implies in particular

Note that for the Gaussian kernel one has , that is the high frequency in are severely penalized in the RKHS norm. Also note that smaller values of correspond to less regularization, which is what one would have expected from the feature map representation (indeed the features are more localized around the data point for larger values of ).

To summarize, SVM with translation invariant kernels correspond to some kind of soft low-pass filtering, where the exact form of the penalization for higher frequency depends on the specific kernel being used (smoother kernels lead to more penalization).

**Random features**

Let us now come back to computational issues. As we pointed out before, the vanilla kernel method has at least a quadratic cost in the number of data points. A common approach to reduce this cost is to use a low rank approximation of the Gram matrix (indeed thanks to the i.i.d. assumption there is presumably a lot of redundancy in the Gram matrix), or to resort to online algorithms (see for example the forgetron of Dekel, Shalev-Shwartz and Singer). Another idea is to approximate the feature map itself (a priori this doesn’t sound like a good idea, since as we explained above the beauty of kernels is that we avoid computing this feature map). We now describe an elegant and powerful approximation of the feature map (for translation invariant kernels) proposed by Rahimi and Recht which is based on *random features*.

Let be a translation invariant kernel and its corresponding probability measure. Let’s rewrite in a convenient form, using and Bochner’s theorem,

A simple idea is thus to build the following random feature map: given and , i.i.d. draws from respectively and , let be defined by where

For it is an easy exercise to verify that with probability at least (provided has a second moment at most polynomial in ) one will have for any in some compact set (with diameter at most polynomial in ),

The SVM problem can now be approximated by:

This optimization problem is potentially much simpler than the vanilla kernel SVM when is much bigger than (essentially replaces for most computational aspects of the problem, including space/time complexity of prediction after training).

**Stability**

We conclude our tour of the basic topics in statistical learning with a different point of view on generalization that was put forward by Bousquet and Elisseeff.

Let’s start with a simple observation. Let be an independent copy of , and denote . Then one can write, using the slight abuse of notation ,

This last quantity can be interpreted as a stability notion, and we see that controlling it would in turn control how different is the true risk compared to the empirical risk. Thus stable methods generalize.

We will now show that regularization can be interpreted as a stabilizer. Precisely we show that

is -stable for a convex and -Lipschitz loss. Denote for the above objective function, then one has

and thus by Lipschitzness

On the other hand by strong convexity one has

and thus with the above we get which implies (by Lipschitzness)

or in other words regularized empirical risk minimization () is stable. In particular denoting for the minimizer of we have:

Assuming and optimizing over we recover the bound we obtained previously via Rademacher complexity.

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**What statistical learning is about**

The basic ingredients in statistical learning theory are as follows:

- Two measurable spaces and (the
*input*and*output*sets). - A loss function .
- A set of maps from to (the
*hypothesis class*) - A set of probability measures on .

The mathematically most elementary goal in learning can now be described as follows: given an i.i.d. sequence drawn from some unknown measure , one wants to find a map from to that is doing almost as well as the best map in in terms of predicting given when is drawn from the unknown and the prediction mistakes are evaluated according to the loss function . In other words, denoting for , one is interested in finding a map that minimizes:

where

Let us make a few comments/observations about this problem and some of its variants. In all our discussion we restrict to and .

- The two canonical examples of the above framework are the problem of classification with halfspaces (, , and ), and the problem of linear regression (, , and ). The loss functions just defined are respectively called the zero-one loss and the least-squares loss.
- The prior knowledge about the relation between inputs and outputs (also called the inductive bias) is expressed by either restricting the comparison set (this is called the
*discriminative approach*) or restricting the set of possible distributions (the*generative approach*). - One says that is learnable if there exists such that . The sample complexity of the problem (at scale ) is the smallest such that there exists with .
- Obviously some problems are not learnable, say in classification with the zero-one loss one cannot learn if is the set of all distributions on and . This is sometimes called the no-free lunch theorem (note however that this observation really doesn’t deserve the name “theorem”). Interestingly it turns out that it doesn’t take much (in some sense) to make the problem learnable: if one restricts to be the set of distributions such that the conditional probability is -Lipschitz, then the nearest neighbor rule (i.e., outputs the label of the closest point to ) shows that the problem is (essentially) learnable with sample complexity . However this is not so good news, as this sample complexity is exponential in the dimension (this is the so-called “curse of dimensionality”). To get anything practical we will have to strengthen our inductive bias.
- The stronger notion of PAC learnability replaces the convergence in expectation of to by a convergence in probability (and this convergence should be uniform in ). Furthermore in basic PAC learning one assumes that is restricted to measures such that , also called the
*realizable case*in the context of classification. On the other hand in*agnostic*PAC learning is the set of all measures on . - A weaker notion of learning is to simply require
*consistency*, that is for any one has going to zero. One can show that -Nearest Neighbor is “universally consistent” provided that grows to infinity at a sublinear rate. This fact is not too surprising if you recall Lusin’s theorem. - Proper learning refers to the situation where is restricted to map into .
- Of course we are interested in the computational complexity of learning too, and the “real” objective of machine learning is to find large and expressive hypothesis classes that are learnable in polynomial sample complexity and polynomial time.

**Glivenko-Cantelli and Rademacher complexity**

Recall that a set of functions is a *Glivenko-Cantelli* class if for any measure on , denoting for the empirical measure of , one has that goes to in probability as goes to infinity. A useful technique to get a handle on Glivenko-Cantelli is the following symmetrization argument, where ,

This last quantity is known as the *Rademacher complexity* of on , and its supremum over is simply called the Rademacher complexity of , denoted . In particular McDiarmid’s inequality together with boundedness of directly implies that with probability at least ,

(1)

In other words if the Rademacher complexity of goes to , then is a Glivenko-Cantelli class.

In learning theory the class of interest is (in particular ). If is a Glivenko-Cantelli class (that is the true loss and the empirical loss are close for any ) then one obtains a consistent method with the rule (empirical risk minimization):

Furthermore if the Glivenko-Cantelli convergence is uniform in (which is the case if say the Rademacher complexity goes to ) then the problem is agnostic PAC learnable (thanks to (1))

Bounding the Rademacher complexity (and variants such as when the Rademacher random variables are replaced by Gaussians) is one of the main objective in the rich theory of empirical processes. Many tools have been developed for this task and we rapidly cover some of them below.

**Lipschitz loss**

The contraction lemma says that if and is 1-Lipschitz for any and , then . Thus, if one wants to show that linear regression is PAC learnable (say with ) it suffices to study the Rademacher complexity of linear functions. For instance if and are both the unit Euclidean ball then the Rademacher complexity of linear functions is bounded by (in particular it is independent of the dimension, more on that below). Thus summarizing this discussion we see that in this case the map

(2)

satisfies for any , with probability at least ,

Also observe that one can rewrite the optimization problem (2) by Lagrangian duality as follows, for some and with obvious notations for and ,

which is sometimes called the *ridge regression* problem. The added penalty is thought of as a *regularizer*, and from the perspective of this section it enforces a small Rademacher complexity. The tradeoff is that larger values of leads to smaller Rademacher complexity but also to a smaller comparison set . This two types of error should be thought of as the *estimation and approximation* errors.

Another interesting example is the LASSO problem

which corresponds to being the -unit ball (up to scaling). In this case if is the -unit ball one can show that the Rademacher complexity of linear functions is of order . This corresponds in some sense to the inductive bias that the optimal weight is sparse.

Finally let us make an important remark about scaling issues. In practice features are often normalized individually, i.e. , in which case the Rademacher complexity (with ) scales with the Euclidean radius which is . However if the examples are all sparse then in fact the dimension is replaced by the sparsity, which can make a huge difference in practice (think of the case where is the number of words in the english dictionary, and our data points are articles, which are typically much shorter than the dictionary). We see that ridge regression and LASSO are after different kind of sparsity, the former is about sparsity in the features while the latter is about sparsity of the features that matter for prediction.

**Zero-one loss**

For the zero-one loss we cannot use the contraction lemma as this loss is not Lipschitz. Instead we use Massart’s lemma which states that if then

Recall that in learning with the zero-one loss one has .

Thus we need to count the set of vectors . The Vapnik-Chervonenkis dimension is introduced exactly for this task: for an hypothesis class , is the largest integer such that one can find a set *shattered* by , that is . The Sauer-Shelah lemma then gives

In particular we obtain:

As a rule of thumb the VC dimension is of order of the number of parameters defining the hypothesis class (obviously this is not always true…). For instance one always has , and the VC dimension of halfspaces in is .

Unfortunately the ERM rule for the zero-one loss on halfspaces is computationally hard to implement (on the other hand in the realizable case it can be written as an LP, we will come back to this a bit later). In practice what we do is to relax the zero-one loss by some convex surrogate so that we can then write . Thus if somehow we manage to have a small error with the surrogate loss then we also have a small error with the zero-one loss. Two important examples of surrogate losses for classification with halfspaces are the hinge loss (which gives the soft-SVM algorithm, more on that later):

and the logistic loss (which gives regularized logistic regression):

Using the contraction lemma from the previous section one can easily control the Rademacher complexity with those losses.

**Convex learning**

We say that the loss is convex if for any the map is convex (in particular is a convex set).

The following situations are learnable: convex loss with either [bounded gradients, bounded ], or [bounded second derivative, bounded ], or [strongly convex, bounded gradients].Theorem:

Just as an example assume that for any , and that for any (the set of subgradients for ) one has .

Then running *one-pass SGD *with step-size given by gives (again this is dimension independent!). Observe that if one runs multiple passes of SGD then this bound does not apply, and one is in fact implementing the ERM rule from the previous section (for which bounds can then be derived via Rademacher complexity as we showed before).

**Negative association**

Robin Pemantle gave an excellent tutorial on negative association, see the slides here. The punchline can be summarized pretty succinctly. Consider a measure on . We say that is positively correlated if for any monotone increasing one has . On the other hand we say that is negatively correlated if for any monotone increasing and such that and depends on disjoint subset of variables, one has . Positive correlation is particularly easy to check thanks to the famous FKG theorem which states that if is log-supermodular (i.e. is a submodular function) then is positively correlated (this is nice because submodularity is a *local* condition). One may hope that log-submodular distributions are negatively correlated, but unfortunately this is not true in general. The route to an easily checkable condition for negative correlation is a bit more tortuous and it goes as follows. First observe that negative association clearly implies pairwise negative correlation, that is for any and ,

Writing this inequality in terms of the generating function one has (with ):

One says that is *strongly Rayleigh* if the above equation holds true for any instead of just at . It turns out that this condition is more restrictive than negative correlation, but it is also much more tractable, in particular because it is equivalent to the fact that is a *stable polynomial* (that is does not have any roots in where is the complex upper half plane). Once you know that latter equivalence it becomes trivial to check that determinantal measures are negatively correlated since they are strongly Rayleigh (just use the fact that the characteristic polynomial of a symmetric matrix only has roots on the real line). A particularly useful property of strongly Rayleigh measures is that they have the usual Gaussian tails for any Lipschitz functional, just like product measures (this was conjectured by Elchanan Mossel, and recently proved by Robin Pemantle and Yuval Peres). Recently this set of measures has also been useful in theoretical computer science, see the recent work of Shayan Oveis Gharan and his co-authors (where they use negative correlation properties of random spanning trees). Finally I briefly mention that the sampling problem for negatively associated measures is a quite rich question, see the slides by Robin and also this COLT’15 paper by Rebeschini and Karbasi.

**Learning in networks**

Elchanan Mossel talked about a beautiful set of open questions. The basic problem goes as follows: let be a graph, and assume that each vertex is a player that receives an independent random variable distributed according to some measure. The measure used to produce the random variables is the same for all vertices, and it is either or (say the players also know that the choice of the measure was a coin flip). Question: assuming that at the end of each day the players reveal their conditional expectation for (given all the information they currently have) to their neighbors in , how many days do we need to reach an agreement? If convergence happens, is the conditional expectation that is reached the optimal one (that is the conditional expectation given all the random variables)? What if instead of revealing your belief you only reveal what is the most likely to you between and (revealed action game)? Mossel-Sly-Tamuz show that both in the revealed belief and the revealed action game there will eventually be an agreement. Furthermore in the revealed beliefs this will be close to the optimal posterior provided that is large enough. Both of these theorems currently lack a convergence rate.

**The complexity of stochastic block models**

The stochastic block model is a canonical model of a random graph with communities. The general version of this model is denoted where is the number of vertices, represents the proportion of each community, and represents the matrix of connection probabilities. More precisely the random graph is defined as follows: First each vertex is assigned a latent label , where the latent label vector is drawn uniformly at random conditionally on the fact that the number of vertices with label is exactly . The edges are then drawn independently (conditionally on ), and each edge is present with probability . This model has received a LOT of attention in the last thirty years. The basic objective is to recover the latent label vector from the observation of the unlabelled graph . The following very recent result by Emmanuel Abbe and Colin Sandon answers the most basic question one can ask: when is this objective achievable? First we introduce the Abbe-Sandon divergence for vectors :

Theorem(Abbe-Sandon 2015): Asymptotically almost sure exact recovery of the latent feature vector in is possible if and only if for any , ,

In their paper Abbe and Sandon also give an efficient algorithm which achieves the information theoretic limit given above. It remains an outstanding open problem to obtain such a precise picture for the problem of *weak recovery* where one asks to recover correctly only a fraction of the latent label vector. This question is particularly interesting in the *sparse* regime where the average degree is constant (note that in the above theorem the average degree diverges at the rate ). Recently a lot of progress has been made in this mathematically challenging regime too, see in particular the papers by Mossel-Neeman-Sly, Massoulie and Bordenave-Lelarge-Massoulie.

Our result with Ronen however is far from the end of the story for this problem: currently our dependency on the dimension is , which can certainly be improved, and even more importantly we do not yet know an *algorithm *which attains our regret bound! Indeed our proof follows the trail of my COLT’15 paper with Ofer Dekel, Tomer Koren and Yuval Peres where we suggest to attack the *dual Bayesian problem* (in the latter paper we solved this problem in dimension one). Thus finding an algorithm with for bandit convex optimization remains as open as before, but now at least we know that such an algorithm exist!!

Along the way of our new proof we solve an interesting property testing problem on convex functions, which we feel could have applications beyond convex bandits. The problem can be described as follows. Assume that you are given a convex and non-negative convex function . Now suppose that there is some unknown convex function defined on the same domain as and for which you would like to know if is equal to . I also guarantee you that if is not equal to then takes a *negative* value somewhere on its domain (say even less than ). You can obtain information on through noisy queries, that is you can ask what is the value of at and you obtain where is a standard Gaussian. Can you design a non-adaptive query strategy such that after at most (with possibly extra logarithmic factors) you will be able to correctly decide if equals or not with probability at least ? In the paper we give a positive answer by constructing a map where is a distribution supported on the domain of which “explores at every scale simultaneously”. At this point this map is still somewhat mysterious, and there are many open questions around it (can you compute the map efficiently? can this be useful beyond Bayesian convex bandits? can you find a better construction than ours? what is the optimal number of queries?). See the paper for all the details!

– Allen-Zhu and Orrechia view AGD as a linear coupling of Gradient Descent and Mirror Descent. This gives a nice new proof where the acceleration results from a primal-dual point of view on the progress of the algorithm.

– Su, Boyd and Candes (building on earlier work of O’Donoghue and Candes) view AGD as the discretization of a certain second-order ODE. Other papers in a somewhat similar direction include the work of Lessard, Recht and Packard; as well as Flammarion and Bach. Note that these papers do not give alternative proofs of acceleration for general smooth functions.

– Other relevant references include a recent paper of Arjevani, Shalev-Shwartz and Shamir, and a blog post by Hardt.

In joint work with Yin-Tat Lee and Mohit Singh we discovered a simple geometric reason for the possiblity of acceleration, which can be turned into a new algorithm with the same theoretical guarantees as AGD, and improved performances in practice. The proof of the convergence rate is 3 lines, assuming a completely elementary geometric fact (in dimension 2) which itself takes less than half a page to prove. The general principle is loosely inspired from the ellipsoid method (except that we work with balls, which makes the algorithm much more efficient). Let me guide you through this.

**Preliminaries**

Ok so we are trying to minimize on a function which is -strongly convex and smooth (see here for definitions). Denote the minimizer of , the condition number of , for a Euclidean ball centered at and of radius squared ,

which are respectively a standard and long step of gradient descent. The crucial observation is that the local information at and the assumptions on allow to enclose the minimizer in some ball, precisely:

Finally we need some fact on the intersection of balls. Take a ball with radius squared , and intersect it with some other ball which has the center of the previous ball on its boundary. This intersection is obviously contained in a ball of radius squared . An easy calculation shows that if the radius squared of one ball shrinks by then the intersection is included in a ball of radius squared . Furthermore if both balls shrink (additively) by the same amount (which is an fraction of the radius squared for one of the balls), then the intersection is included in a ball of radius squared . As we will see this fact is the reason why one can accelerate.

**A suboptimal algorithm**

Assume that we are given such that . As we saw before one also has

Taking the intersection of these two balls and using the fact from the previous section we see that there exists such that

Thus by iterating the map from to (defined implicitely above) we obtain a new algorithm with the same rate of convergence as gradient descent.

**Acceleration**

Now assume that the point from the previous section and its guarantee on the distance to the optimum came from the local information at some other point , that is and where . Assume furthermore that is at least as good as a step of gradient descent at , that is . Then we can use the local information at to reduce the ball around , indeed one has

and thus

Furthermore, as in the last section, one has

Taking the intersection of the above balls and enclosing it in another ball one obtains (see preliminaries) that there exists such that

The only issue at this point is that to iterate the map from to one would need that is at least as good as a step of gradient descent on , which is not necessarily true (just as the assumption is not necessarily true). We circumvent this difficulty by doing a line search on the line going through and . The details (which are straightforward given the above discussion) are in the paper, specifically in Section 3. We also report encouraging preliminary experimental results.

]]>On a completely different topic, I wanted to share some videos which many readers of this blog will enjoy. These are the videos of the 2015 Breakthrough Prize in Mathematics Symposium, with speakers (and prize winners) Jacob Lurie, Terence Tao, Maxim Kontsevich, Richard Taylor, and Simon Donaldson. They were asked to give talks to a general audience, and they succeeded at different levels. Both Taylor and Lurie took this request very seriously, perhaps a bit too much even, and their talks (here and here) are very elementary (yet still entertaining!). Tao talks about the Polymath projects, and the video can be safely skipped unless you have never heard of Polymath. I understood nothing of Kontsevich’s talk (it’s pretty funny to think that his talk was prepared with the guideline of aiming at a general audience). My favorite talk by far was the one by Donaldson. Thanks to him I finally understand what the extra 7 unobserved dimensions of our universe could look like! There is also a panel discussion led by Yuri Milner with the 5 mathematicians. Unfortunately the questions are a bit dull, so there is not much that the panelists can do to make this interesting. Yet there are a few gems in the answers, such as Tao claiming that *universality* (such as in the Central Limit Theorem) is behind the unreasonable effectiveness of mathematics in physics, and Kontsevich who replies to Tao that this is a valid point at the macroscopic level, but the fact that mathematics work so well at a microscopic level (e.g., quantum mechanics) makes him question whether we live in a simulation. Kontsevich also says that there is no fundamental obstacle to building an A.I., and he even claims that he gave some thoughts to this problem, though I could not find any paper written by him on this matter.

**COLT 2015 accepted papers**

On Consistent Surrogate Risk Minimization and Property Elicitation

Minimax Fixed-Design Linear Regression

Sequential Information Maximization: When is Greedy Near-optimal?

– Learnability of Solutions to Conjunctive Queries: The Full Dichotomy

Efficient Sampling for Gaussian Graphical Models via Spectral Sparsification

On-Line Learning Algorithms for Path Experts with Non-Additive Losses

Truthful Linear Regression

$S^2$: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification

Convex Risk Minimization and Conditional Probability Estimation

Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering

– Learning and inference in the presence of corrupted inputs

Vector-Valued Property Elicitation

Learning the dependence structure of rare events: a non-asymptotic study

Adaptive recovery of signals by convex optimization

Tensor principal component analysis

Learning Overcomplete Latent Variable Models through Tensor Methods

Exp-Concavity of Proper Composite Losses

On Learning Distributions from their Samples

– Online PCA with Spectral Bounds

– Regret Lower Bound and Optimal Algorithm in Dueling Bandit Problem

Hierarchical label queries with data-dependent partitions

– Bad Universal Priors and Notions of Optimality

Lower and Upper Bounds on the Generalization of Stochastic Exponentially Concave Optimization

Correlation Clustering with Noisy Partial Information

– Online Density Estimation of Bradley-Terry Models

Cortical Learning via Prediction

– Partitioning Well-Clustered Graphs: Spectral Clustering Works!

Batched Bandit Problems

– Fast Mixing for Discrete Point Processes

– An Almost Optimal PAC Algorithm

Minimax rates for memory-bounded sparse linear regression

Regularized Linear Regression: A Precise Analysis of the Estimation Error

Max vs Min: Tensor Decomposition and ICA with nearly Linear Sample Complexity

On Convergence of Emphatic Temporal-Difference Learning

]]>So what is deep learning? Well, from my current limited understanding, it seems to be a pretty simple supervised learning technique which can be explained in a few lines. Of course, simple problems can be very deep (ever heard of the Collatz conjecture?), and it looks like this may be the case with deep learning. After explaining what is deep learning (or more accurately what is my understanding of deep learning), I will point out some recent theoretical progress towards understanding the basic phenomenona at play. Let me be clear right away: it seems to me that no one has a real clue about what’s going on.

**Deep learning: it’s all about the features**

Ok, so, you have a data set with and . Typically is an image, and is say if there is a pinguin in this image. A very efficient algorithm to learn a mapping from to is to use some kind of linear classification, such as multi-class logistic regression (see also this post for some background on linear classification). However applying a linear classification directly on typically does not work very well, and this why we usually first map the point to , where is some feature map. Designing has been part of the art of machine learning, and the idea behind deep learning is to incorporate the problem of finding a good map as part of the learning process (again in the previous post on linear classification I also discussed this possibility in the context of SVM, though we ended up with a semi-definite program which does not scale well with large data sets).

Ok so what kind of do we look at in deep learning? Well they are of the following form, for some sequence of matrices , with :

The (non-linear) mappings are fixed and part of the design choice (i.e., they are not learned). The parameters are also specified beforehand. Furthermore one usually impose extra constraints on the matrices , such as having all columns bounded in -norm (say by ).

The game now is simply to incorporate the matrices as free variables when one performs the optimization of the multiclass logistic loss. This optimization is always performed with Stochastic Gradient Descent (or a simple variant of it, such as mini-batch SGD, see this post and Section 6.2 in my survey): take an example in your data set ; compute the gradient of the logistic loss at this example (with respect to the matrices and the logistic regression weights), and do a small step in the opposite direction of the gradient.

**Design choices**

Part of the success of deep learning is in the choices left open above, such as the choice of the non-linear maps, and the set of constraints on the matrices. Let me partly describe some of these choices that Krizhevsky, Sutskever and Hinton used for their computer vision breakthrough where they basically halfed the error rate of the competition on the ImageNet dataset (the details can be found here).

First they take . Then and are the so-called ReLU function (Rectified Linear Unit), which simply take the positive part coordinate-wise. On the other hand are composition of ReLU with *max-pooling*. The max-pooling is a dimension reduction operation, which simply takes adjacent coordinates (in the case of an image, adjacent pixels) and returns the maximal element. Regarding the constraints on the matrices, and are left unconstrained (besides perhaps the energy control on the columns), while are highly structure *convolutional networks*, or ConvNets as they are usually called. Essentially this means that in these matrices, the columns up to are translations of some very sparse vector. Precisely, for , they take and each sparse vector corresponds to a image. In other words the vector corresponds to a collection of images, each one of them being the convolution of with a small patch (whose weights are being learned).

While this post is not about the history of deep learning (this has been discussed at length pretty much everywhere on the internet), I still would like to point out that the apparently arbitrary choice behind max-pooling and the ConvNets are due to Yann LeCun, who has experimented with these for many many years.

**Some mysterious steps**

So far I would say that the method described above kind of make sense (apart from the max-pooling operation which I really do not understand). But things are now going to turn out a bit more wacky than you might have expected. First note that there is one key element missing from my description of the learning algorithm: what does “small” mean in the sentence “take a small step in the opposite direction of the gradient”. If you know about convex optimization and machine learning, then you know that a lot of work usually goes into understanding precisely what kind of learning rate one should take, and in particular at which rate one should decrease it over time. In deep learning all of this theory goes out of the window since the objective function is not convex (note that there is *some *convexity in the sense that the objective is convex in – thanks to Nicolas Le Roux for fixing a mistake in the first version of the previous sentence). Here it seems that the current state of the art is to start with some guess, say , then look at how the training error decays as more examples are being processed, and if the training error plateau then just divide the learning rate by .

Another point which seems key to the success of such an architecture is to control overfitting. Again for a convex problem we know that one-pass through the data with SGD and the appropriate learning rate would work just fine in terms of generalization error. In deep learning the algorithm makes multiple passes on the data, and furthermore the problem is non-convex. This strongly suggests that overfitting will be a real problem, especially given the number of free parameters. Thankfully Hinton and his colleagues invented a quite smart technique, called “dropout”. The idea is simply to take the gradient with respect to only half of the columns of the matrices , where the half is chosen at random at each iteration. In the network view of the architecture this amounts to “dropping” each unit with probability . Intuitively this forces the network to be more robust, and to “repeat” itself, thus avoiding overfitting.

**Hacking through life**

Of course, if you want to beat everyone else in the world at something, you better have every parts of your system completely optimized. In that respect the deep learning community took the implementation of the above ideas to the next level by using GPUs, thus making the computation of the gradients much faster (by the way, the ReLU non-linearity also helps here, compared to more traditional non-linearities such as sigmoids). Various other tricks are being used, such as artificially expanding the data set by altering the (e.g., by translating the images) or by running the network backward.

**Towards an understanding of deep learning?**

I think it’s fair to say that we do not understand why deep nets work at all, and we understand even less why they are doing so much better than anything else. The three main questions are: (i) why is SGD meaningful for this non-convex problem, (ii) why is dropout enough to avoid overfitting, (iii) can we explain why ConvNets are a sensible way to learn representations?

Very limited progress has been made on these questions, though I will cite some papers that at least try to attack these problems. I try to summarize each work with one sentence, see the papers for more details (and in particular the details will reveal that these papers do not make direct progress on the above questions).

– Sanjeev Arora and his co-authors proved that a “random and sparse” deep net can be learned in polynomial time.

– Roi Livni, Shai Shalev-Shwartz and Ohad Shamir obtain some results on the approximation power of a neural network with a non-linearity of the form .

– Gerard Ben Arous, Yann LeCun and their students draw an interesting connection between fully connected neural networks (such as the last 3 layers in the above architecture) and spherical spin glasses.

– Stephane Mallat gives some insights on (iii) via the scattering transform. Arnab Paul and Suresh Venkat borrow ideas from group theory to shed some light on (iii).

– Francis Bach proves some generalization bounds for neural networks with one hidden layer.

I apologize for misrepresenting deep learning so badly in this post. I know I missed a lot, and you’re welcome to correct me in the comments!

**Edit. Some further references added thanks to various comments:**

– Approaches for (ii): David McAllester with PAC-Bayesian bounds, and Percy Liang and students with a link to AdaGrad.

– Approach to (iii) by a group at Facebook on the relevance of ConvNets for certain natural distributions.

– I said in the post that all optimization theory goes out of the window when the problem is non-convex. This is of course not true, and for instance a 1998 paper by Leon Bottou has results for SGD in more general scenario. However in the non-convex case we only have very weak guarantees which basically say nothing about the global optimum.

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