Notes on least-squares, part I

These are mainly notes for myself, but I figured that they might be of interest to some of the blog readers too. Comments on what is written below are most welcome!

Let (X, Y) \in \mathbb{R}^d \times \mathbb{R} be a pair of random variables, and let f: \mathbb{R}^d \rightarrow \mathbb{R} be the convex quadratic function defined by

    \[f(w) = \frac12 \mathbb{E} (X^{\top} w - Y)^2 .\]

We are interested in finding a minimizer w^* of f (we will also be satisfied with w such that f(w) - f(w^*) \leq \epsilon). Denoting \Sigma = \mathbb{E} X X^{\top} for the covariance matrix of the {\em features} (we refer to X as a feature vector) and \mu = \mathbb{E} Y X, one has w^* = \Sigma^{-1} \mu (if \Sigma is not invertible then the inverse should be understood as the Moore-Penrose pseudo-inverse, in which case w^* is the smallest-norm element of the affine subspace of minimizers of f). The model of computation we are interested in corresponds to large-scale problems where one has access to an unlimited stream of i.i.d. copies of (X, Y). Thus the only constraint to find w^* is our computational ability to process samples. Furthermore we want to be able to deal with large dimensional problem where \Sigma can be extremely ill-conditioned (for instance the smallest eigenvalue can be 0, or very close to 0), and thus bounds that depend on the condition number of \Sigma (ratio of largest eigenvalue to smallest eigenvalue) are irrelevant for us.

Let us introduce a few more notation before diving into the comparison of different algorithms. With the well-specified model in mind (i.e., Y = X^{\top} w^* + \epsilon where \epsilon \sim \mathcal{N}(0, \sigma^2) is independent of X) we define the noise level \sigma of the problem by assuming that \mathbb{E} \ (X^{\top} w^* - Y)^2 X X^{\top} \preceq \sigma^2 \Sigma. For simplicity we will assume (whenever necessary) that almost surely one has |X| \leq R and |Y| \leq B. All our iterative algorithms will start at w_0 = 0. We will also not worry too much about numerical constants and write \lesssim for an inequality true up to a numerical constant. Finally for the computational complexity we will count the number of elementary d-dimensional vector operations that one has to do (e.g. adding/scaling such vectors, or requesting a new i.i.d. copy of (X,Y)).

ERM

The most natural approach (also the most terrible as it will turn out) is to take a large sample (x_i, y_i)_{i \in [n]} and then minimize the corresponding empirical version \hat{f} of f, or in other words compute \hat{w} = \left(\mathbb{X} \mathbb{X}^{\top} \right)^{-1} \mathbb{X} \mathbb{Y} where \mathbb{Y} \in \mathbb{R}^n is the column vector (y_i) and \mathbb{X} \in \mathbb{R}^{d \times n} is the matrix whose i^{th} column is x_i. Since \hat{f} is a quadratic function, we can use the conjugate gradient algorithm (see for example Section 2.4 here) to get \hat{w} in d iterations. However each iteration (basically computing a gradient of \hat{f}) cost O(n) elementary vector operations, thus leading overall to O(n d) elementary vector operations (which, perhaps surprisingly, is also the cost of just forming the empirical covariance matrix \hat{\Sigma} = \mathbb{X} \mathbb{X}^{\top}). What number n of samples should one take? The answer (at least for the well-specified model) comes from a standard statistical analysis (e.g. with the Rademacher complexity described here, see also these nice lecture notes by Philippe Rigollet) which yield:

    \[\mathbb{E} \hat{f}(\hat{w}) - f(w^*) \lesssim \frac{\sigma^2 \mathrm{rank}(\Sigma)}{n} .\]

(Note that in the above bound we evaluate \hat{w} on \hat{f} instead of f, see the comments section for more details on this issue. Since this is not the main point of the post I overlook this issue in what follows.) Wrapping up we see that this “standard” approach solves our objective by taking n= \sigma^2 d / \epsilon samples and performing O(\sigma^2 d^2 / \epsilon) elementary vector operations. This d^2 number of elementary vector operations completely dooms this approach when d is very large. We note however that the statistical bound above can be shown to be minimax optimal, that is any algorithm has to use at least \sigma^2 d / \epsilon samples (in the worst case over all \Sigma) in order to satisfy our objective. Thus the only hope to do better than the above approach is to somehow process the data more efficiently.

A potential idea to trade a bit of the nice statistical property of \hat{w} for some computational easiness is to add a regularization term to \hat{f} in order to make the problem better conditioned (recall that conjugate gradient reaches an \epsilon-optimal point in \sqrt{\kappa} \log(1/\epsilon) iterations, where \kappa is the condition number).

In other words we would like to minimize, for some values of n and \lambda to be specified,

    \[\frac{1}{2n} \sum_{i=1}^n (x_i^{\top} w - y_i)^2 + \frac{\lambda}{2} |w|^2 .\]

One can write the minimizer as \hat{w}(\lambda) = \left(\mathbb{X} \mathbb{X}^{\top} + n \lambda \mathrm{I}_d \right)^{-1} \mathbb{X} \mathbb{Y}, and this point is now obtained at a cost of \tilde{O} \left(n \sqrt{\frac{\|\Sigma\|}{\lambda}}\right) elementary vector operations (\|\Sigma\| denotes the spectral norm of \Sigma and we used the fact that with high probability \|\hat{\Sigma}\| \lesssim \|\Sigma\|). Here one can be even smarter and use accelerated SVRG instead of conjugate gradient (for SVRG see Section 6.3 here, and here or here for how to accelerate it), which will reduce the number of elementary vector operations to \tilde{O}\left(n +\sqrt{n \frac{\|\Sigma\|}{\lambda}} \right). Also observe that again an easy argument based on Rademacher complexity leads to

\mathbb{E} f(\hat{w}(\lambda)) - f(w^*) \lesssim \lambda |w^*|^2 + \frac{\sigma^2 d}{n}. Thus we want to take \lambda of order \frac{\sigma^2 d}{n |w^*|^2} which in terms of \epsilon is of order \epsilon/ |w^*|^2 leading to an overall number of vector operations of

    \[\frac{\sigma^2 d}{\epsilon} + \frac{|w^*| \sqrt{\|\Sigma\| \sigma^2 d}}{\epsilon} .\]

Note that, perhaps surprisingly, anything less smart than accelerated SVRG would lead to a worse dependency on \epsilon than the one obtained above (which is also the same as the one obtained by the “standard” approach). Finally we observe that at a high level the two terms in the above bound can be viewed as “variance” and “bias” terms. The variance term comes from the inherent statistical limitation while the bias comes from the approximate optimization. In particular it makes sense that |w^*| appears in this bias term as it is the initial distance to the optimum for the optimization procedure.

Standard SGD

Another approach altogether, which is probably the most natural approach for anyone trained in machine learning, is to run the stochastic gradient algorithm. Writing g_i(w) = x_i (x_i^{\top} w - y_i) (this is a stochastic gradient, that is it satisfies \mathbb{E}[g_i(w) | w] = \nabla f(w)) we recall that SGD can be written as:

    \[w_{i+1} = w_i - \eta g_i = (\mathrm{I}_d - \eta x_i x_i^{\top}) w_i + \eta y_i x_i .\]

The most basic guarantee for SGD (see Theorem 6.1 here) gives that with step size \eta of order 1/\sqrt{n}, and assuming to simplify (see below for how to get around this) that |w^*| is known up to a numerical constant so that if w_{i+1} steps outside the ball of that radius then we project it back on it, one has (we also note that \mathbb{E}( |g_i(w)|^2 | w) \leq (B + |w| R)^2 \mathrm{Tr}(\Sigma), and we denote \bar{\sigma}^2 = (B + |w^*| R)^2):

    \[\mathbb{E} f\left(\frac1{n} \sum_{i=1}^n w_i\right) - f(w^*) \lesssim \sqrt{\frac{|w^*|^2 \bar{\sigma}^2 \mathrm{Tr}(\Sigma)}{n}} .\]

This means that roughly with SGD the computational cost for our main task corresponds O(|w^*|^2 \bar{\sigma}^2 \mathrm{Tr}(\Sigma) / \epsilon^2) vector operations. This is a fairly weak result compared to accelerated SVRG mainly because of: (i) dependency in \epsilon^{-2} instead of \epsilon^{-1}, and (ii) \bar{\sigma}^2 instead of \sigma^2. We will fix both issues, but let us also observe that on the positive side the term d \|\Sigma\| in accelerated SVRG is replaced by the less conservative quantity \mathrm{Tr}(\Sigma) (that is one replaces the largest eigenvalue of \Sigma by the average eigenvalue of \Sigma).

Let us see quickly check whether one can improve the performance of SGD by regularization. That is we consider applying SGD on f(w) + \frac{\lambda}{2} |w|^2 with stochastic gradient g_i(w) = (\lambda \mathrm{I}_d + x_i x_i^{\top}) w - y_i x_i which leads to (see Theorem 6.2 here):

    \[\mathbb{E} f\left(\frac1{n} \sum_{i=1}^n w_i\right) - f(w^*) \lesssim \lambda |w^*|^2 + \frac{\bar{\sigma}^2 \mathrm{Tr}(\Sigma)}{\lambda n} .\]

Optimizing over \lambda we see that we get the same rate as without regularization. Thus, unsurprisingly, standard SGD fares poorly compared to the more sophisticated approach based on accelerated SVRG described above. However we will see next that there is a very simple fix to make SGD competitive with accelerated SVRG, namely just tune up the learning rate!

Constant step size SGD (after Bach and Moulines 2013)

SGD with a constant step size \eta of order of a constant times 1/R^2 satisfies

    \[\mathbb{E} f\left(\frac1{n} \sum_{i=1}^n w_i\right) - f(w^*) \lesssim \frac{\sigma^2 d + R^2 |w^*|^2}{n}.\]

We see that this result gives a similar bias/variance decomposition as the one we saw for accelerated SVRG. In particular the variance term \sigma^2 d / n is minimax optimal, while the bias term R^2 |w^*|^2 / n matches what one would expect for a basic first order method (this term is not quite optimal as one would expect that a decrease in 1/n^2 is possible since this is the optimal rate for first-order optimization of a smooth quadratic function -without strong convexity-, and indeed a recent paper of Dieuleveut, Flammarion and Bach tackles this issue).

In part II I hope to give the intuition/proof of the above result, and in part III I will discuss other aspects of the least squares problem (dual methods, random projections, sparsity).

Posted in Optimization | 3 Comments

AlphaGo is born

Google DeepMind is making the front page of Nature (again) with a new AI for Go, named AlphaGo (see also this Nature youtube video). Computer Go is a notoriously difficult problem, and up to now AI were faring very badly compared to good human players. In their paper the DeepMind team reports that AlphaGo won 5-0 against the best European player Fan Hui!!! This is truly a jump in performance: the previous best AI, Crazy Stone, needed several handicap stones to compete with pro players. Congratulations to the DeepMind team for this breakthrough!

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)!

Posted in Uncategorized | 2 Comments

On the spirit of NIPS 2015 and OpenAI

I just came back from NIPS 2015 which was a clear success in terms of numbers (note that this growth is not all because of deep learning, only about 10% of the papers were on this topic, which is about double of those on convex optimization for example):

NIPS15growth

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).

 

Posted in Conference/workshop | 14 Comments

Convex Optimization: Algorithms and Complexity

I am thrilled to announce that my short introduction to convex optimization has just came out in the Foundations and Trends in Machine Learning series (free version on arxiv). This project started on this blog in 2013 with the lecture notes “The complexities of optimization”, it then morphed a year later into a first draft titled “Theory of Convex Optimization for Machine Learning”, and finally after one more iteration it has found the more appropriate name: “Convex Optimization: Algorithms and Complexity”. Notable additions since the last version include: a short treatment of conjugate gradient, an almost self-contained analysis of Vaidya’s 1989 cutting plane method (which attains the best of both center of gravity and ellipsoid method in terms of oracle complexity and computational complexity), and finally an algorithm with a simple geometric intuition which attains the rate of convergence of Nesterov’s accelerated gradient descent.

Posted in Optimization | Leave a comment

Crash course on learning theory, part 2

It might be useful to refresh your memory on the concepts we saw in part 1 (particularly the notions of VC dimension and Rademacher complexity). In this second and last part we will discuss two of the most successful algorithm paradigms in learning: boosting and SVM. Note that just like last time each topic we cover have its own books dedicated to them (see for example the boosting book by Schapire and Freund, and the SVM book by Scholkopf and Smola). Finally we conclude our short tour of the learning’s basics with a simple observation: stable algorithms generalize well.

 

Boosting

Say that given a distribution D supported on m points (x_i, y_i)_{i \in [m]} one can find (efficiently) a classifier h \in \mathcal{B} such that L_{D}(h) \leq 1/2 - \gamma (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 L_{\mu}(A) arbitrarily small for m large enough? It turns out that this is possible, and even simple. The idea is to build a linear combination of hypotheses in \mathcal{B} with a greedy procedure. That is at time step t-1 our hypothesis is (the sign of) f_{t-1} = \sum_{s=1}^{t-1} w_s h_s, and we are now looking to add h_{t} \in \mathcal{B} with an approriate weight w_{t} \in \mathbb{R}. A natural guess is to optimize over (w_t, h_t) to minimize the training error of f_t on our sample (x_i, y_i)_{i \in [m]}. This might be a difficult computational problem (how do you optimize over \mathcal{B}?), and furthermore we would like to make use of our efficient weak learning algorithm. The key trick is that \mathds1\{x < 0\} \leq \exp(-x). More precisely:

    \begin{eqnarray*} L_{\hat{\mu}_m}(f_t) = \frac{1}{m} \sum_{i=1}^m \mathds1\{\mathrm{sign}(f_t(x_i)) \neq y_i\} & \leq & \frac{1}{m} \sum_{i=1}^m \exp( - y_i f_t(x_i) ) \; (=: Z_t / m) \\ & = & \frac{Z_{t-1}}{m} \sum_{i=1}^m D_t(i) \exp( - y_i w_t h_t(x_i) ) , \end{eqnarray*}

where D_t(i) = \frac{1}{Z_{t-1}} \exp(- y_i f_{t-1}(i)). From this we see that we would like h_t to be a good predictor for the distribution D_t. Thus we can pass D_t to the weak learning algorithm, which in turns gives us h_t with L_{D_t}(h_t) = \epsilon_t \leq 1/2 - \gamma. Thus we now have:

    \[Z_ t = Z_{t-1} (\epsilon_t \exp(w_t) + (1-\epsilon_t) \exp(-w_t) ).\]

Optimizing the above expression one finds that w_t = \frac{1}{2} \log\left(\frac{1-\epsilon_t}{\epsilon_t}\right) leads to (using \epsilon_t \leq 1/2-\gamma)

    \[Z_t \leq Z_{t-1} \exp(-2 \gamma^2) \leq m \exp(-2 \gamma^2 t) .\]

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

(1)   \begin{equation*}  L_{\hat{\mu}_m}(f_t) \leq \exp(- 2 \gamma^2 t) . \end{equation*}

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

    \[\mathcal{H} = \left\{ \mathrm{sign} \left( \sum_{t \in \mathbb{N}} w_t h_t(\cdot) \right) : \forall t \in \mathbb{N}, w_t \in \mathbb{R}, h_t \in \mathcal{B} \right\} .\]

This class can be thought of as a neural network with one (infinite size) hidden layer. To realize how expressive \mathcal{H} is compared to \mathcal{B} 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):

    \[\mathcal{B} = \{x \mapsto \mathrm{sign}(\theta - x_i), i \in [n], \theta \in \mathbb{R}\} .\]

To derive a bound on the true risk L_{\mu}(f_t) of AdaBoost it remains to calculate the VC dimension of the class \mathcal{H}_t where the size of the hidden layer is t. This follows from more general results on the VC dimension of neural networks, and up to logarithmic factors one obtains that \mathrm{VC}(\mathcal{H}_t) is of order t \times \mathrm{VC}(\mathcal{B}). Putting this together with \eqref{eq:empada} we see that when \gamma is a constant, one should run AdaBoost for t=\Theta(\log(m)) rounds, and then one gets L_{\mu}(f_t) \lesssim \sqrt{\frac{\mathrm{VC}(\mathcal{B})}{m}}.

 

Margin

We consider \mathcal{D} to be the set of distributions \mu such that there exists w^* with \mathbb{P}_{\mu}(Y \langle w^{*} , X \rangle > 0) =1 (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 w with minimal Euclidean norm and such that Y_i \langle w , X_i \rangle \geq 1, \forall i \in [m]. 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):

    \[\mathcal{W}_S = \{w \in \mathbb{R}^n : Y_i \langle w , X_i \rangle \geq 1, \forall i \in [m] \; \text{and} \; |w| \leq |w^*|\} .\]

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 \mathds1\{\mathrm{sign}(\langle w, x\rangle) \neq y\} by the (Lipschitz!) “ramp loss” \min(1,\max(0,1-y\langle w , x \rangle)) makes no difference for the optimum w^*, and our estimated weight still has 0 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 2 |w^*| / \sqrt{m} (assuming the examples X_i are normalized to be in the Euclidean ball). Now it is easy to see that the existence of w^* with |w^*| \leq 1/\gamma (and \mathbb{P}_{\mu}(Y \langle w^{*} , X \rangle \geq 1) =1) exactly corresponds to a geometric margin of \gamma between positive and negative examples (indeed the margin is exactly \min_i y_i \langle \frac{w^*}{|w^*|} , x \rangle). To summarize we just saw that under the \gamma-margin condition the SVM algorithm has a sample complexity of order 1/ (\gamma \epsilon)^2. 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 \mathcal{H} (as we will see in the next subsection this notation will be consistent with \mathcal{H} being the hypothesis class). Denote \psi for this map, and let K(x,x') = \langle \psi(x), \psi(x') \rangle_{\mathcal{H}} 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 \psi(x_1), \hdots, \psi(x_m) (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 w=\sum_{i=1}^m \alpha_i \psi(x_i) (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 (K(x_i, x_j))_{i, j \in [m]} as we only need to consider \langle w, \psi(x_i)\rangle = \sum_{j=1}^m \alpha_j K(x_j, x_i) and |w|^2 = \sum_{i,j=1}^m \alpha_i \alpha_j K(x_i, x_j). In particular we never need to compute the points \psi(x_i) (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: \psi(x) = (\prod_{i \in J} x_i)_{J \subset [n], |J| =k} leads to K(x,x') = (\langle x, x'\rangle)^k. 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 m^2 (the kernel matrix (K(x_i,x_j))) and in our big data days this can be a disaster. We really want to focus on methods with computational complexity linear in m, 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 K rather than the embedding \psi. In particular since the only thing that matter are inner products \langle \psi(x), \psi(x') \rangle_{\mathcal{H}} we might as well assume that \psi(x) := K(x,\cdot), and \mathcal{H} is the completion of \mathrm{Span}(K(x, \cdot), x \in \mathcal{X}) where the inner product is defined by \langle K(x, \cdot), K(x', \cdot) \rangle_{\mathcal{H}} = K(x,x') (and the definition is extended to \mathcal{H} by linearity). Assuming that K is positive definite (that is Gram matrices built from K are positive definite) one obtains a well-defined Hilbert space \mathcal{H}. Furthermore this Hilbert space has a special property: for any f \in \mathcal{H}, x \in \mathcal{X}, f(x) = \langle f, K(x, \cdot) \rangle_{\mathcal{H}}. In other words \mathcal{H} 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

    \begin{align*} & \mathrm{min}. \; \sum_{i,j=1}^m \alpha_i \alpha_j K(x_i,x_j) \\ & \text{s.t.} \; \alpha \in \mathbb{R}^m \; \text{and} \; \forall j \in [m], y_j \sum_{i=1}^m \alpha_i K(x_i, x_j) \geq 1 , \end{align*}

as

    \begin{align*} & \mathrm{min}. \; \|f\|_{\mathcal{H}} \\ & \text{s.t.} \; f \in \mathcal{H} \; \text{and} \; \forall j \in [m], y_j f(x_j) \geq 1 . \end{align*}

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 \mathcal{H} with smallest norm. In other words, thinking in terms of inductive bias, one should choose a kernel for which the norm \|\cdot\|_{\mathcal{H}} 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 \ell is defined on \mathcal{Y} \times \mathbb{R} instead of \mathcal{Y} \times \mathcal{Y})

    \[\min_{w \in \mathbb{R}^n}. \; \frac1{m} \sum_{i=1}^m \ell(y_i, \langle w, x_i \rangle) + \lambda |w|^2 ,\]

one can consider the much more exciting

    \[\min_{f \in \mathcal{H}}. \; \frac1{m} \sum_{i=1}^m \ell(y_i, f(x_i)) + \lambda \|f\|_{\mathcal{H}}^2 ,\]

since this can be equivalently written as

    \[\min_{\alpha \in \mathbb{R}^m}. \; \frac1{m} \sum_{i=1}^m \ell\left(y_i, \sum_{j=1}^m \alpha_j K(x_j, x_i)\right) + \lambda \sum_{i,j=1}^m \alpha_i \alpha_j K(x_i,x_j) .\]

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 K(x,x')=k(x-x'). A beautiful theorem of Bochner characterizes the continuous maps k: \mathbb{R}^n \rightarrow \mathbb{R} (with k(0)=1) for which such a K is a positive definite kernel: it is necessary and sufficient that k is the characteristic function of a probability measure \mu, that is

    \[k(x) = \int_{\mathbb{R}^n} \exp(i \langle \omega, x \rangle ) d\mu(\omega) .\]

An important example in practice is the Gaussian kernel: K(x,x')=\exp\left( - \frac{|x-x'|^2}{2 \sigma^2}\right) (this corresponds to mapping x to the function \exp\left( - \frac{|x- \cdot |^2}{2 \sigma^2}\right)). One can check that in this case \mu is itself a Gaussian (centered and with covariance \sigma^{-2} \mathrm{I}_n).

Now let us restrict our attention to the case where \mu has a density h with respect to the Lebesgue measure, that is \frac{d\mu}{d\omega}=h. A standard calculation then shows that

    \[K(x,x')= \frac{1}{(2\pi)^{n}} \int_{\mathbb{R}^n} \frac{\widehat{K(x, \cdot)}(\omega)\overline{\widehat{K(x, \cdot)}(\omega)}}{h(\omega)} d\omega ,\]

which implies in particular

    \[\| f \|_{\mathcal{H}} = \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\mathbb{R}^n} \frac{|\widehat{f}(\omega)|^2}{h(\omega)} d\omega .\]

Note that for the Gaussian kernel one has h(\omega)^{-1} \propto \exp\left(\frac{\sigma^2 |\omega|^2}{2}\right), that is the high frequency in f are severely penalized in the RKHS norm. Also note that smaller values of \sigma correspond to less regularization, which is what one would have expected from the feature map representation (indeed the features \exp\left( - \frac{|x- \cdot |^2}{2 \sigma^2}\right) are more localized around the data point x for larger values of \sigma).

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 K be a translation invariant kernel and \mu its corresponding probability measure. Let’s rewrite K in a convenient form, using \cos(\theta-\theta') = \frac{1}{\pi} \int_{0}^{\pi} 2 \cos(\theta + \phi) \cos(\theta'+\phi) d\phi and Bochner’s theorem,

    \[K(x,x') = \mathbb{E}_{\omega \sim \mu} \cos(\langle \omega, x-x' \rangle) = 2 \mathbb{E}_{\omega \sim \mu, \phi \sim \mathrm{unif}([0,\pi])} \cos(\langle \omega, x \rangle + \phi) \cos(\langle \omega, x' \rangle + \phi).\]

A simple idea is thus to build the following random feature map: given \omega_1, \hdots, \omega_d and \phi_1, \hdots, \phi_d, i.i.d. draws from respectively \mu and \mathrm{unif}([0, \pi]), let \Psi: \mathbb{R}^n \rightarrow \mathbb{R}^d be defined by \Psi(x)=(\Psi_1(x), \hdots, \Psi_d(x)) where

    \[\Psi_i(x) = \sqrt{\frac{2}{d}} \cos(\langle \omega_i, x \rangle + \phi_i) .\]

For d \gtrsim n / \epsilon^2 it is an easy exercise to verify that with probability at least 1-\epsilon (provided \mu has a second moment at most polynomial in n) one will have for any x, x' in some compact set (with diameter at most polynomial in n),

    \[|K(x,x') - \langle \Psi(x), \Psi(x') \rangle | \leq \epsilon .\]

The SVM problem can now be approximated by:

    \begin{align*} & \mathrm{min}. \; |w| \\ & \text{s.t.} \; w \in \mathbb{R}^d \; \text{and} \; y_j \langle w, \Psi(x_j) \geq 1 , \forall j \in [m]. \end{align*}

This optimization problem is potentially much simpler than the vanilla kernel SVM when m is much bigger than n (essentially n replaces m 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 S'=(Z_1', \hdots, Z_m') be an independent copy of S, and denote S^i = (Z_1, \hdots, Z_{i-1}, Z_i', Z_{i+1}, \hdots, Z_m). Then one can write, using the slight abuse of notation \ell(h, (x,y)) :=\ell(h(x), y),

    \begin{eqnarray*} \mathbb{E}_S (L_{\mu}(A(S)) - L_{\hat{\mu}_m}(A(S))) & = & \mathbb{E}_{S, S', i \sim \mathrm{unif}([m])} (\ell(A(S), Z_i') - \ell(A(S), Z_i)) \\ & = & \mathbb{E}_{S, S', i \sim \mathrm{unif}([m])} (\ell(A(S^i), Z_i) - \ell(A(S), Z_i)) . \end{eqnarray*}

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

    \[w(S) \in \argmin_{w \in \mathbb{R}} L_{\hat{\mu}_m}(h_w) + \lambda |w|^2 ,\]

is 2 / (\lambda m)-stable for a convex and 1-Lipschitz loss. Denote f_S(w) for the above objective function, then one has

    \[f_S(w) - f_S(w') = f_{S^i}(w) - f_{S^i}(w') + \frac{\ell(h_w, Z_i) - \ell(h_{w'}, Z_i)}{m} + \frac{\ell(h_{w'}, Z_i') - \ell(h_{w}, Z_i')}{m} ,\]

and thus by Lipschitzness

    \begin{eqnarray*} f_S(w(S^i))-f_S(w(S)) & \leq & \frac{\ell(h_{w(S^i)}, Z_i) - \ell(h_{w(S)}, Z_i)}{m} + \frac{\ell(h_{w(S)}, Z_i') - \ell(h_{w(S^i)}, Z_i')}{m} \\ & \leq & \frac{2}{m} |w(S) - w(S^i)| . \end{eqnarray*}

On the other hand by strong convexity one has

    \[f_S(w) - f_S(w(S)) \geq \lambda |w-w(S)|^2 ,\]

and thus with the above we get |w(S) - w(S^i)| \leq 2 / (\lambda m) which implies (by Lipschitzness)

    \[\ell(h_{w(S^i)}, Z_i) - \ell(h_{w(S)}, Z_i) \leq \frac{2}{\lambda m} ,\]

or in other words regularized empirical risk minimization (\mathrm{R-ERM}) is stable. In particular denoting w^* for the minimizer of w \mapsto L_{\mu}(h_w) we have:

    \begin{eqnarray*} L_{\mu}(\mathrm{R-ERM}) & \leq & \mathbb{E}_S L_{\hat{\mu}_m}(\mathrm{R-ERM}) + \frac{2}{\lambda m} \\ & \leq & \mathbb{E}_S L_{\hat{\mu}_m}(w^*) + \lambda |w^*| + \frac{2}{\lambda m} \\ & = & L_{\mu}(w^*) + \lambda |w^*| + \frac{2}{\lambda m} . \end{eqnarray*}

Assuming |w^*| \leq 1 and optimizing over \lambda we recover the bound we obtained previously via Rademacher complexity.

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Crash course on learning theory, part 1

This week and next week I’m giving 90 minutes lectures at MSR on the fundamentals of learning theory. Below you will find my notes for the first course, where we covered the basic setting of statistical learning theory, Glivenko-Cantelli classes, Rademacher complexity, VC dimension, linear regression, logistic regression, soft-SVM, sample complexity of classification with halfspaces, and convex learning with SGD. All of this (and much more) can be found in the excellent book by Shai Shalev-Shwartz and Shai Ben-David (freely available online) which I heavily used to prepare these lectures. Next week we will cover boosting, margin and kernels, stability, and an introduction to neural networks [note: part 2 can be found here].

 

What statistical learning is about

The basic ingredients in statistical learning theory are as follows:

  • Two measurable spaces \mathcal{X} and \mathcal{Y} (the input and output sets).
  • A loss function \ell : \mathcal{Y} \times \mathcal{Y} \rightarrow \mathbb{R}.
  • A set \mathcal{H} \subset \mathcal{Y}^{\mathcal{X}} of maps from \mathcal{X} to \mathcal{Y} (the hypothesis class)
  • A set \mathcal{D} of probability measures on \mathcal{X} \times \mathcal{Y}.

The mathematically most elementary goal in learning can now be described as follows: given an i.i.d. sequence S := ((X_i, Y_i))_{i \in [m]} drawn from some unknown measure \mu \in \mathcal{D}, one wants to find a map from \mathcal{X} to \mathcal{Y} that is doing almost as well as the best map in \mathcal{H} in terms of predicting Y given X when (X,Y) is drawn from the unknown \mu and the prediction mistakes are evaluated according to the loss function \ell. In other words, denoting for h \in \mathcal{Y}^{\mathcal{X}}, L_{\mu}(h) = \mathbb{E}_{(X,Y) \sim \mu} \ell(h(X), Y), one is interested in finding a map A : (\mathcal{X} \times \mathcal{Y})^m \rightarrow \mathcal{Y}^{\mathcal{X}} that minimizes:

    \[R_m(A) := \sup_{\mu \in \mathcal{D}} R_{m}(A; \mu) ,\]

where

    \[R_{m}(A; \mu) := \mathbb{E}_{S \sim \mu^{\otimes m}} \; L_{\mu}(A(S)) - \inf_{h \in \mathcal{H}} L_{\mu}(h) .\]

Let us make a few comments/observations about this problem and some of its variants. In all our discussion we restrict to \mathcal{X} \subset \mathbb{R}^n and \mathcal{H} = \{h_w : w \in \mathcal{W} \subset \mathbb{R}^d\}.

  • The two canonical examples of the above framework are the problem of classification with halfspaces (\mathcal{Y} = \{-1,1\}, h_w(x) = \mathrm{sign}(\langle w, x\rangle), and \ell(y,y') = \mathds1\{y \neq y'\}), and the problem of linear regression (\mathcal{Y} = \mathbb{R}, h_w(x) = \langle w, x \rangle, and \ell(y,y') = (y-y')^2). 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 \mathcal{H} (this is called the discriminative approach) or restricting the set of possible distributions \mathcal{D} (the generative approach).
  • One says that (\mathcal{D}, \mathcal{H}, \ell) is learnable if there exists A such that \lim_{m\rightarrow+\infty}R_m(A) \leq 0. The sample complexity of the problem (at scale \epsilon >0) is the smallest m such that there exists A with R_m(A) \leq \epsilon.
  • Obviously some problems are not learnable, say in classification with the zero-one loss one cannot learn if \mathcal{D} is the set of all distributions on [0,1]^n \times \{-1,1\} and \mathcal{H} = \{-1,1\}^{[0,1]^n}. 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 \mathcal{D} to be the set of distributions such that the conditional probability \eta(x) = \mathbb{P}(Y=1 | X=x) is 1-Lipschitz, then the nearest neighbor rule (i.e., A(S)(x) outputs the label Y_i of the closest point X_i to x) shows that the problem is (essentially) learnable with sample complexity 1/\epsilon^{n+1}. 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 L_{\mu}(A(S)) to \inf_{h \in \mathcal{H}} L_{\mu}(h) by a convergence in probability (and this convergence should be uniform in \mu \in \mathcal{D}). Furthermore in basic PAC learning one assumes that \mathcal{D} is restricted to measures such that \inf_{h \in \mathcal{H}} L_{\mu}(h) = 0, also called the realizable case in the context of classification. On the other hand in agnostic PAC learning \mathcal{D} is the set of all measures on \mathcal{X} \times \mathcal{Y}.
  • A weaker notion of learning is to simply require consistency, that is for any \mu \in \mathcal{D} one has R_m(A; \mu) going to zero. One can show that k-Nearest Neighbor is “universally consistent” provided that k 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 A is restricted to map into \mathcal{H}.
  • 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 \mathcal{F} \subset \mathbb{R}^{\mathcal{Z}} is a Glivenko-Cantelli class if for any measure \mu on \mathcal{Z}, denoting \hat{\mu}_m = \frac{1}{m} \sum_{i=1}^m \delta_{z_i} for the empirical measure of z \sim \mu^{\otimes m}, one has that \sup_{f \in \mathcal{F}} |\mathbb{E}_{\hat{\mu}_m} f - \mathbb{E}_{\mu} f| goes to 0 in probability as m goes to infinity. A useful technique to get a handle on Glivenko-Cantelli is the following symmetrization argument, where \epsilon \sim \mathrm{unif}(\{-1,1\}^m),

    \begin{eqnarray*} \mathbb{E}_z \sup_{f \in \mathcal{F}} |\mathbb{E}_{\hat{\mu}_m} f - \mathbb{E}_{\mu} f| & = & \mathbb{E}_z \sup_{f \in \mathcal{F}} \left| \mathbb{E}_{z'} \frac{1}{m} \sum_{i=1}^m (f(z_i) - f(z_i')) \right| \\ & \leq & \mathbb{E}_{z, z', \epsilon} \sup_{f \in \mathcal{F}} \left| \frac{1}{m} \sum_{i=1}^m \epsilon_i (f(z_i) - f(z_i')) \right| \\ & \leq & 2 \mathbb{E}_{z, \epsilon} \sup_{f \in \mathcal{F}} \left|\frac{1}{m} \sum_{i=1}^m \epsilon_i f(z_i)\right| . \end{eqnarray*}

This last quantity is known as the Rademacher complexity of \mathcal{F} on \mu, and its supremum over \mu is simply called the Rademacher complexity of \mathcal{F}, denoted \mathrm{Rad}_m(\mathcal{F}). In particular McDiarmid’s inequality together with boundedness of f directly implies that with probability at least 1-\delta,

(1)   \begin{equation*} \sup_{f \in \mathcal{F}} |\mathbb{E}_{\hat{\mu}_m} f - \mathbb{E}_{\mu} f| \lesssim \mathrm{Rad}_m(\mathcal{F}) + \sqrt{\frac{\log(1/\delta)}{m}}. \end{equation*}

In other words if the Rademacher complexity of \mathcal{F} goes to 0, then \mathcal{F} is a Glivenko-Cantelli class.

In learning theory the class of interest is \mathcal{F} = \{(x,y) \mapsto \ell(h(x), y), h \in \mathcal{H}\} (in particular \mathcal{Z} = \mathcal{X} \times \mathcal{Y}). If \mathcal{F} is a Glivenko-Cantelli class (that is the true loss and the empirical loss are close for any h \in \mathcal{H}) then one obtains a consistent method with the \mathrm{ERM} rule (empirical risk minimization):

    \[\mathrm{ERM}(S) = \mathrm{argmin}_{h \in \mathcal{H}} \frac{1}{m} \sum_{i=1}^m \ell(h(X_i), Y_i) .\]

Furthermore if the Glivenko-Cantelli convergence is uniform in \mu \in \mathcal{D} (which is the case if say the Rademacher complexity \mathrm{Rad}_m(\mathcal{F}) goes to 0) 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 \mathcal{Y} \subset \mathbb{R} and x \mapsto \ell(h(x), y) is 1-Lipschitz for any h and y, then \mathrm{Rad}_m(\mathcal{F}) \leq \mathrm{Rad}_m(\mathcal{H}). Thus, if one wants to show that linear regression is PAC learnable (say with \mathcal{Y} = [-1,1]) it suffices to study the Rademacher complexity of linear functions. For instance if \mathcal{X} and \mathcal{W} are both the unit Euclidean ball \mathrm{B}_2^n then the Rademacher complexity of linear functions is bounded by 2/\sqrt{m} (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)   \begin{equation*} \mathrm{ERM} : (X_i, Y_i)_{i \in [m]} \mapsto \mathrm{argmin}_{w \in \mathrm{B}_2^n} \frac{1}{m} \sum_{i=1}^m (\langle w, X_i\rangle - Y_i)^2 , \end{equation*}

satisfies for any \mu, with probability at least 1- \delta,

    \[L_{\mu}(\mathrm{ERM}(S)) - \inf_{h \in \mathcal{H}} L_{\mu}(h) \lesssim \sqrt{\frac{\log(1/\delta)}{m}} .\]

Also observe that one can rewrite the optimization problem (2) by Lagrangian duality as follows, for some \lambda>0 and with obvious notations for \mathbb{X} and \mathbb{Y},

    \[\mathrm{argmin}_{w \in \mathbb{R}^n} \frac1{m} |\mathbb{X} w - \mathbb{Y}|^2 + \lambda |w|^2 ,\]

which is sometimes called the ridge regression problem. The added penalty \lambda | \cdot | 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 \lambda leads to smaller Rademacher complexity but also to a smaller comparison set \mathcal{H}. This two types of error should be thought of as the estimation and approximation errors.

Another interesting example is the LASSO problem

    \[\mathrm{argmin}_{w \in \mathbb{R}^n} \frac1{m} |\mathbb{X} w - \mathbb{Y}|^2 + \lambda \|w\|_1 ,\]

which corresponds to \mathcal{W} being the \ell_1-unit ball (up to scaling). In this case if \mathcal{X} is the \ell_{\infty}-unit ball one can show that the Rademacher complexity of linear functions is of order \sqrt{\log(n) / m}. 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. \mathcal{X} = [-1,1]^d, in which case the Rademacher complexity (with \mathcal{W} = \mathrm{B}_2^n) scales with the Euclidean radius which is \sqrt{d}. 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 d is the number of words in the english dictionary, and our data points x_i 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 A \subset \mathrm{B}_2^m then

    \[\mathbb{E}_{\epsilon}\sup_{a \in A} \left|\frac{1}{m} \sum_{i=1}^m \epsilon_i a_i\right|\leq \frac{\sqrt{2 \log |A|}}{m} .\]

Recall that in learning with the zero-one loss one has \mathcal{F} = \{(x,y) \mapsto \mathds1\{h(x) \neq y\}, h \in \mathcal{H}\}.

Thus we need to count the set of vectors \{(\mathds1\{h(x_1) \neq y_1\}, \hdots, \mathds1\{h(x_m) \neq y_m\}), h \in \mathcal{H}\}. The Vapnik-Chervonenkis dimension is introduced exactly for this task: for an hypothesis class \mathcal{H} \subset \{-1,1\}^{\mathcal{X}}, \mathrm{VC}(\mathcal{H}) is the largest integer k such that one can find a set \{x_1, \hdots, x_k\} \subset \mathcal{X} shattered by \mathcal{H}, that is |\{(h(x_1), \hdots, h(x_k)), h \in \mathcal{H}\}| = 2^k. The Sauer-Shelah lemma then gives

    \[\left|\left\{(\mathds1\{h(x_1) \neq y_1\}, \hdots, \mathds1\{h(x_m) \neq y_m\}), h \in \mathcal{H}\right\}\right| \leq \left(\frac{e m}{\mathrm{VC}(\mathcal{H})}\right)^{\mathrm{VC}(\mathcal{H})}.\]

In particular we obtain:

    \[\mathrm{Rad}(\mathcal{F}) \leq \sqrt{\frac{2 \mathrm{VC}(\mathcal{H}) \log\left( \frac{e m}{\mathrm{VC}(\mathcal{H})} \right)}{m}} .\]

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 \mathrm{VC}(\mathcal{H}) \leq \log_2(|\mathcal{H}|), and the VC dimension of halfspaces in \mathbb{R}^d is d.

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 \ell so that we can then write L_{\mu}^{0-1} \leq L_{\mu}^{\ell}. 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):

    \[\ell(h_w, (x,y)) = \max(0, 1 - y \langle w, x\rangle) ,\]

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

    \[\ell(h_w, (x,y)) = \log_2(1+\exp(- y \langle w, x\rangle)).\]

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 (x, y) \in \mathcal{X} \times \mathcal{Y} the map w \in \mathcal{W} \mapsto \ell(h_w(x), y) is convex (in particular \mathcal{W} is a convex set).

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

Just as an example assume that for any w \in \mathcal{W}, |w| \leq R, and that for any x, y, w, g \in \partial_w \ell(h_w(x), y) (the set of subgradients for \ell) one has |g| \leq L.

Then running one-pass SGD with step-size given by \frac{R}{L} \sqrt{\frac{2}{t}} gives R_m(SGD) \leq R L \sqrt{\frac{2}{m}} (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).

Posted in Optimization, Probability theory, Theoretical Computer Science | 9 Comments

Some stuff I learned this week

This week I had the pleasure to spend 3 days at a wonderful workshop (disclaimer: I was part of the organization, with Ofer Dekel, Yuval Peres and James Lee, so I might be a bit biased). Below you will find three results that I particularly enjoyed (one from each day). Obviously many more topics were covered at the workshop, from finding correlations in subquadratic time, to the basic problem of learning halfspaces (see for instances these two COLT’15 papers, by Daniely and by Awasthi, Balcan, Haghtalab and Urner), or how stability can explain why deep nets trained with SGD are not overfitting (upcoming work by Hardt-Recht-Singer; also on a related note I learned from Yuval about the Kolmogrov-Arnold representation theorem, which seems very much related to the famous Barron’1990 result on the universal approximation properties of neural networks with 1 hidden layer). Finally two more things: (i) we also have a blog where we collected all the open problems discussed during the workshop, and (ii) many of the talks have been recorded (including the ones I mention below, as well as the nice tutorial by Sergiu Hart on dynamics in games). I will post the link to the videos as soon as we upload them.

 

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 \mu on \{0,1\}^n. We say that \mu is positively correlated if for any f,g monotone increasing one has \int f g d\mu \geq \int f d\mu \int g d\mu. On the other hand we say that \mu is negatively correlated if for any f,g monotone increasing and such that f and g depends on disjoint subset of variables, one has \int f g d\mu \leq \int f d\mu \int g d\mu. Positive correlation is particularly easy to check thanks to the famous FKG theorem which states that if \mu is log-supermodular (i.e. - \log(\mu) is a submodular function) then \mu 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 i,j and X = (X_1, \hdots, X_n) \sim \mu,

    \[\mathbb{E} X_i X_j \leq \mathbb{E} X_i \mathbb{E} X_j .\]

Writing this inequality in terms of the generating function F(x) = \mathbb{E} \prod_{k=1}^n x_i^{X_i} one has (with \mathds{1}=(1,\hdots,1)):

    \[F(\mathds{1}) \frac{\partial^2 F}{\partial x_i \partial x_j}(\mathds{1}) \leq \frac{\partial F}{\partial x_i}(\mathds{1}) \frac{\partial F}{\partial x_j}(\mathds{1}) .\]

One says that \mu is strongly Rayleigh if the above equation holds true for any x \in \mathbb{R}^n instead of just at x= \mathds{1}. 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 F is a stable polynomial (that is F does not have any roots in \mathbb{H}^n where \mathbb{H} 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 G 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 \nu_0 or \nu_1 (say the players also know that the choice I \in \{0,1\} of the measure was a coin flip). Question: assuming that at the end of each day the players reveal their conditional expectation for I (given all the information they currently have) to their neighbors in G, 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 I=0 and I=1 (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 G 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 \mathrm{SBM}(n,p,W) where n \in \mathbb{N} is the number of vertices, p \in [0,1]^k represents the proportion of each community, and W \in [0,1]^{k \times k} represents the matrix of connection probabilities. More precisely the random graph \mathrm{SBM}(n,p,W) is defined as follows: First each vertex i \in [n] is assigned a latent label \sigma_i \in [k], where the latent label vector \sigma \in [k]^n is drawn uniformly at random conditionally on the fact that the number of vertices with label \sigma \in [k] is exactly p_{\sigma} n. The edges are then drawn independently (conditionally on \sigma), and each edge \{i,j\} is present with probability W_{\sigma_i, \sigma_j}. This model has received a LOT of attention in the last thirty years. The basic objective is to recover the latent label vector \sigma from the observation of the unlabelled graph \mathrm{SBM}(n,p,W). 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 x, y \in \mathbb{R}^k:

    \[D(x,y) = \max_{t \in [0,1]} \sum_{i=1}^k (t x_i + (1-t) y_i - x_i^t y_i^{1-t}) .\]

Theorem (Abbe-Sandon 2015): Asymptotically almost sure exact recovery of the latent feature vector in \mathrm{SBM}(n,p, Q \frac{\log(n)}{n}) is possible if and only if for any i, j \in [k], i \neq j,

    \[D \big( (\mathrm{diag}(p) Q)_i, (\mathrm{diag}(p) Q)_j \big) \geq 1 .\]

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 1-\epsilon 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 \log(n)). 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.

Posted in Conference/workshop, Probability theory | Leave a comment

A solution to bandit convex optimization

Ronen Eldan and I have just uploaded to the arXiv our newest paper which finally proves that for online learning with bandit feedback, convex functions are not much harder than linear functions. The quest for this result started in 2004 with a paper by Robert Kleinberg, and one by Flaxman, Kalai, McMahan, and over the years it has become the flagship open problem in bandit theory (see for instance Hazan and Levy 2015 where they refer to the attainable regret for convex bandit as the most important open question in bandit learning).

Our result with Ronen however is far from the end of the story for this problem: currently our dependency on the dimension n is n^{11}, 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 \mathrm{poly}(n) \sqrt{T} 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 f. Now suppose that there is some unknown convex function g defined on the same domain as f and for which you would like to know if g is equal to f. I also guarantee you that if g is not equal to f then g takes a negative value somewhere on its domain (say even less than - \epsilon). You can obtain information on g through noisy queries, that is you can ask what is the value of g at x and you obtain g(x) + \xi where \xi is a standard Gaussian. Can you design a non-adaptive query strategy such that after at most \mathrm{poly}(n) / \epsilon^2 (with possibly extra logarithmic factors) you will be able to correctly decide if g equals f or not with probability at least 1- \epsilon? In the paper we give a positive answer by constructing a map f \mapsto \mu where \mu is a distribution supported on the domain of f which “explores f 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!

Posted in Optimization | 2 Comments

Revisiting Nesterov’s Acceleration

Nesterov’s accelerated gradient descent (AGD) is hard to understand. Since Nesterov’s 1983 paper people have tried to explain “why” acceleration is possible, with the hope that the answer would go beyond the mysterious (but beautiful) algebraic manipulations of the original proof. This search for a good “why” has accelerated in the recent years under the impulsion of the machine learning community, as one would like to generalize Nesterov’s acceleration in various ways to deal with machine learning applications. Obviously generalizing is difficult when even the basic phenomenon is not well understood. Recent works in the “why” direction are re-interpreting the AGD algorithm from different point of views:

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 \mathbb{R}^n a function f which is \alpha-strongly convex and \beta smooth (see here for definitions). Denote x^* the minimizer of f, \kappa = \beta / \alpha the condition number of f, \mathrm{B}(x, r^2) for a Euclidean ball centered at x and of radius squared r^2,

    \[x^+ = x - \frac{1}{\beta} \nabla f(x), \;\; x^{++} = x - \frac{1}{\alpha} \nabla f(x) ,\]

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

    \[x^* \in \mathrm{B}\left(x^{++}, \frac{|\nabla f(x)|^2}{\alpha^2} \left(1 - \frac{1}{\kappa}\right) - \frac{2}{\alpha} (f(x^+) - f(x^*)) \right) .\]

Finally we need some fact on the intersection of balls. Take a ball with radius squared R^2, 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 R^2. An easy calculation shows that if the radius squared of one ball shrinks by 1-\epsilon then the intersection is included in a ball of radius squared (1-\epsilon) R^2. Furthermore if both balls shrink (additively) by the same amount (which is an \epsilon fraction of the radius squared for one of the balls), then the intersection is included in a ball of radius squared (1-\sqrt{\epsilon}) R^2. As we will see this fact is the reason why one can accelerate.

A suboptimal algorithm

Assume that we are given x \in \mathbb{R}^n such that x^* \in \mathrm{B}(x,R^2). As we saw before one also has

    \[x^* \in \mathrm{B}\left(x - \frac{1}{\alpha} \nabla f(x), \frac{|\nabla f(x)|^2}{\alpha^2} \left(1 - \frac{1}{\kappa}\right) \right) .\]

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

    \[x^* \in \mathrm{B}\left(x', \left(1 - \frac{1}{\kappa}\right) R^2\right) .\]

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

Acceleration

Now assume that the point x from the previous section and its guarantee R^2 on the distance to the optimum came from the local information at some other point x_0, that is x = x_0^{++} and R^2 = R_0^2 - \frac{2}{\alpha} (f(x_0^+) - f(x^*)) where R_0^2 = \frac{|\nabla f(x_0)|^2}{\alpha^2} \left(1 - \frac{1}{\kappa}\right). Assume furthermore that x is at least as good as a step of gradient descent at x_0, that is f(x) \leq f(x_0^+). Then we can use the local information at x to reduce the ball around x, indeed one has

    \[f(x^*) \leq f(x^+) \leq f(x) - \frac{1}{2 \beta} \nabla f(x) \leq f(x_0^+) - \frac{1}{2 \beta} \nabla f(x) ,\]

and thus

    \[x^* \in \mathrm{B}\left(x, R_0^2 - \frac{|\nabla f(x)|^2}{\alpha^2 \kappa} \right) .\]

Furthermore, as in the last section, one has

    \[x^* \in \mathrm{B}\left(x - \frac{1}{\alpha} \nabla f(x), \frac{|\nabla f(x)|^2}{\alpha^2} \left(1 - \frac{1}{\kappa}\right) \right) .\]

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

    \[x^* \in \mathrm{B}\left(x', \left(1 - \frac{1}{\sqrt{\kappa}}\right) R_0^2 \right) .\]

The only issue at this point is that to iterate the map from x to x' one would need that x' is at least as good as a step of gradient descent on x, which is not necessarily true (just as the assumption f(x_0^{++}) \leq f(x_0^+) is not necessarily true). We circumvent this difficulty by doing a line search on the line going through x' and x^+. The details (which are straightforward given the above discussion) are in the paper, specifically in Section 3. We also report encouraging preliminary experimental results.

Posted in Optimization | 7 Comments

COLT 2015 accepted papers and some cool videos

Like last year I compiled a list of the COLT 2015 accepted papers together with links to the arxiv version whenever I could find one. These papers were selected from 180 submissions, a number that keep rising in the recent years (of course this is true for all the major learning conferences, for instance ICML had over 1000 submissions this year). This strong program, together with a pretty good location (Paris, 5th), should make COLT 2015 quite attractive! Also, following the trend of COLT 2013 and COLT 2014, we will have some “pre-COLT” activity, with an optimization workshop co-organized by Vianney Perchet (also the main organizer for COLT itself) and Patrick Louis Combettes.

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

– Arpit Agarwal and Shivani Agarwal. On Consistent Surrogate Risk Minimization and Property Elicitation
Noga Alon, Nicolò Cesa-Bianchi, Ofer Dekel and Tomer Koren. Online Learning with Feedback Graphs: Beyond Bandits
– Sanjeev Arora, Rong Ge, Tengyu Ma and Ankur Moitra. Simple, Efficient, and Neural Algorithms for Sparse Coding
– Pranjal Awasthi, Maria Florina Balcan, Nika Haghtalab and Ruth Urner. Efficient Learning of Linear Separators under Bounded Noise
– Pranjal Awasthi, Moses Charikar, Kevin Lai and Andrej Risteski. Label optimal regret bounds for online local learning
– Maria-Florina Balcan, Avrim Blum and Santosh Vempala. Efficient Representations for Lifelong Learning and Autoencoding
– Akshay Balsubramani and Yoav Freund. Optimally Combining Classifiers Using Unlabeled Data
– Peter Bartlett, Wouter Koolen, Alan Malek, Eiji Takimoto and Manfred Warmuth. Minimax Fixed-Design Linear Regression
– Alexandre Belloni, Tengyuan Liang, Hari Narayanan and Alexander Rakhlin. Escaping the Local Minima via Simulated Annealing: Optimization of Approximately Convex Functions
Sébastien Bubeck, Ofer Dekel, Tomer Koren and Yuval Peres. Bandit Convex Optimization: sqrt{T} Regret in One Dimension
– Yang Cai, Constantinos Daskalakis and Christos Papadimitriou. Optimum Statistical Estimation with Strategic Data Sources
Nicolo Cesa-Bianchi, Yishay Mansour and Ohad Shamir. On the Complexity of Learning with Kernels
– Yuxin Chen, Hamed Hassani, Amin Karbasi and Andreas Krause. Sequential Information Maximization: When is Greedy Near-optimal?
Hubie Chen and Matt Valeriote. Learnability of Solutions to Conjunctive Queries: The Full Dichotomy
– Dehua Cheng, Yu Cheng, Yan Liu, Richard Peng and Shang-Hua Teng. Efficient Sampling for Gaussian Graphical Models via Spectral Sparsification
– Corinna Cortes, Vitaly Kuznetsov, Mehryar Mohri and Manfred Warmuth. On-Line Learning Algorithms for Path Experts with Non-Additive Losses
– Rachel Cummings, Stratis Ioannidis and Katrina Ligett. Truthful Linear Regression
– Gautam Dasarathy, Robert Nowak and Xiaojin Zhu. $S^2$: An Efficient Graph Based Active Learning Algorithm with Application to Nonparametric Classification
Miroslav Dudik, Katja Hofmann, Robert E. Schapire, Aleksandrs Slivkins and Masrour Zoghi. Contextual Dueling Bandits
– Miroslav Dudík, Robert Schapire and Matus Telgarsky. Convex Risk Minimization and Conditional Probability Estimation
– Justin Eldridge, Mikhail Belkin and Yusu Wang. Beyond Hartigan Consistency: Merge Distortion Metric for Hierarchical Clustering
– Moein Falahatgar, Ashkan Jafarpour, Alon Orlitsky, Venkatadheeraj Pichapati and Ananda Theertha Suresh. Faster Algorithms for Testing under Conditional Sampling
Uriel Feige, Yishay Mansour and Robert Schapire. Learning and inference in the presence of corrupted inputs
Nicolas Flammarion and Francis Bach. From Averaging to Acceleration, There is Only a Step-size
Dean Foster, Howard Karloff and Justin Thaler. Variable Selection is Hard
– Rafael Frongillo and Ian Kash. Vector-Valued Property Elicitation
Roy Frostig, Rong Ge, Sham Kakade and Aaron Sidford. Competing with the Empirical Risk Minimizer in a Single Pass
Pierre Gaillard and Sébastien Gerchinovitz. A Chaining Algorithm for Online Nonparametric Regression
– Nicolas Goix, Anne Sabourin and Stéphan Clémençon. Learning the dependence structure of rare events: a non-asymptotic study
– Zaid Harchaoui, Anatoli Juditsky, Arkadi Nemirovski and Dmitry Ostrovsky. Adaptive recovery of signals by convex optimization
– Sam Hopkins, Jonathan Shi and David Steurer. Tensor principal component analysis
Prateek Jain and Praneeth Netrapalli. Fast Exact Matrix Completion with Finite Samples
– Majid Janzamin, Animashree Anandkumar and Rong Ge. Learning Overcomplete Latent Variable Models through Tensor Methods
– Parameswaran Kamalaruban, Robert Williamson and Xinhua Zhang. Exp-Concavity of Proper Composite Losses
– Sudeep Kamath, Alon Orlitsky, Venkatadheeraj Pichapati and Ananda Theertha Suresh. On Learning Distributions from their Samples
Varun Kanade and Elchanan Mossel. MCMC Learning
Zohar Karnin and Edo Liberty. Online PCA with Spectral Bounds
Junpei Komiyama, Junya Honda, Hisashi Kashima and Hiroshi Nakagawa. Regret Lower Bound and Optimal Algorithm in Dueling Bandit Problem
– Samory Kpotufe, Ruth Urner and Shai Ben-David. Hierarchical label queries with data-dependent partitions
– Rasmus Kyng, Anup Rao, Sushant Sachdeva and Daniel A. Spielman. Algorithms for Lipschitz Learning on Graphs
Jan Leike and Marcus Hutter. Bad Universal Priors and Notions of Optimality
– Tengyuan Liang, Alexander Rakhlin and Karthik Sridharan. Learning with Square Loss: Localization through Offset Rademacher Complexity
– Haipeng Luo and Robert Schapire. Achieving All with No Parameters: Adaptive NormalHedge
– Mehrdad Mahdavi, Lijun Zhang and Rong Jin. Lower and Upper Bounds on the Generalization of Stochastic Exponentially Concave Optimization
– Konstantin Makarychev, Yury Makarychev and Aravindan Vijayaraghavan. Correlation Clustering with Noisy Partial Information
Issei Matsumoto, Kohei Hatano and Eiji Takimoto. Online Density Estimation of Bradley-Terry Models
– Behnam Neyshabur, Ryota Tomioka and Nathan Srebro. Norm-Based Capacity Control in Neural Networks
– Christos Papadimitriou and Santosh Vempala. Cortical Learning via Prediction
Richard Peng, He Sun and Luca Zanetti. Partitioning Well-Clustered Graphs: Spectral Clustering Works!
– Vianney Perchet, Philippe Rigollet, Sylvain Chassang and Erik Snowberg. Batched Bandit Problems
Patrick Rebeschini and Amin Karbasi. Fast Mixing for Discrete Point Processes
– Mark Reid, Rafael Frongillo, Robert Williamson and Nishant Mehta. Generalized Mixability via Entropic Duality
Hans Simon. An Almost Optimal PAC Algorithm
– Jacob Steinhardt and John Duchi. Minimax rates for memory-bounded sparse linear regression
– Christos Thrampoulidis, Samet Oymak and Babak Hassibi. Regularized Linear Regression: A Precise Analysis of the Estimation Error
– Santosh Vempala and Ying Xiao. Max vs Min: Tensor Decomposition and ICA with nearly Linear Sample Complexity
– Huizhen Yu and Richard Sutton. On Convergence of Emphatic Temporal-Difference Learning
Posted in Conference/workshop | 3 Comments