The entropic barrier: a simple and optimal universal self-concordant barrier

Ronen Eldan and I have just uploaded our new paper on the arxiv (it should appear tomorrow, for the moment you can see it here). The abstract reads as follows:

We prove that the Fenchel dual of the log-Laplace transform of the uniform measure on a convex body in \mathbb{R}^n is a (1+o(1)) n-self-concordant barrier. This gives the first construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.

Let me briefly put this in context. As you know if you are a reader of this blog, Interior Point Methods are based on self-concordant barriers, and a basic result in this theory is Nesterov and Nemirovski’s construction of a universal self-concordant barrier whose self-concordance parameter is always O(n), see this blog post for instance. As I said at the time of writing the latter post, “this is a difficult theorem, which we state without a proof”. Similarly, in James Renegar’s book on IPM (which I highly recommend) he qualifies this as “a most impressive and theoretically important result”, but he also says “I have yet to find a relatively transparent proof of this result, and hence a proof is not contained in this book”. We hope our paper remedies this issue. In fact, our construction, which is quite different from the Nesterov-Nemirovski’s barrier, has a few advantages:

- The new barrier has a very simple interpretation as the differential entropy of a certain element in a canonical exponential family on the convex body of interest, hence the name entropic barrier.

- Its self-concordance parameter is optimal (up to the second order term). Indeed Nesterov and Nemirovski proved that on a simplex, the self-concordance parameter is at least n, and we obtain a parameter (1+o(1))n for any convex body.

- The proof is reasonably easy. For instance the self-concordance follows immediately from the fact that higher order moments of a log-concave distribution are controlled by the second order moment. (In fairness getting the tight constant 1 requires a little bit of work, but proving  a bound of O(n) is simpler.)

- Mirror Descent with the entropic barrier as a mirror map is equivalent to Cover’s continuous exponential weights, which is kind of interesting.

I would like to conclude with a comment on a result by Roland Hildebrand: he proves that for convex cones, there exists a self-concordant barrier with parameter smaller than n (yes I mean 1 \times n). This is quite amazing, and better than what we obtain since we get (1+o(1))n. However his result has two caveats: (i) as far as I understand his barrier is defined only implicitely, as the unique convex solution to a certain PDE, and (ii) while convex cones are “universal” for convex optimization, there are actually other applications where it is important to have barriers on convex bodies, see for instance some of the papers by Sasha Rakhlin. [Edit: thanks to an email by Francois Glineur we realized that a side result of our analysis is an alternative proof of Hildebrand’s bound for homogeneous cones by using a deep result of Guler. This is very mysterious and we don’t quite understand the full extent of the implication of this yet.]

Posted in Optimization | 1 Comment

What’s the (hi)story of my network?

First an announcement: we have researcher and postdoc positions available in Theory Group at MSR, and I may have a position for an intern in the summer. If you are a talented young mathematician with interests in the kind of things I blog about please get in touch with me by email for more details about these positions.

The topic of this post is the reason why the blog has been silent for the past 3 months: I have just uploaded three new network papers, written with amazing co-authors (Luc Devroye, Jian Ding, Ronen Eldan, Gabor Lugosi, Elchanan Mossel, and Miklos Racz), and which can be found here, here, and here. The overall goal of these works is to realize (a version of) the vision that Nati Linial presented in a public lecture at the Simons Institute in Berkeley about a year ago. As Nati puts it, there is a need for a statistical theory on graphs, and specifically he writes that we should “Develop a battery of generative models of graphs and methods to recover the appropriate model from the input graph”. Obviously this type of question has already been looked at by statisticians, but somehow I feel like they have been shy in the models and questions that they consider. I think that there is a huge opportunity to look at more exotic (yet practically relevant) estimation problems with network data. To simplify, and get at the heart of the matter, I will essentially discuss testing problems rather than estimation.


Is there geometry in my network?

In the space of testing problems on random graphs, the king of the hill is testing for community structure. A precise hypothesis testing problem considered by Ery Arias-Castro and Nicolas Verzelen (see here and here) is the following: Under the null hypothesis, the observed graph G on n vertices has been generated by the standard Erdos-Renyi random graph G(n,p), where each edge appears independently with probability p. Under the alternative, G comes from the model G(n,p,k,q) which is similar to G(n,p) except that there is a subset of k vertices (the community) chosen uniformly at random and such that the connection probability between two vertices from that subset is q instead of p. The case p=1/2 and q=1 corresponds to the now famous Planted Clique (PC) problem. In PC it is easy to see that the information theoretic limit for testability is when k is of order \log(n) (use for instance this), but surprisingly we do not know any polynomial-time test when k= o(\sqrt{n}). Bridging this gap is a major open problem.

Testing for community structure is of course very important in practice (and it’s also a beautiful mathematical question), but it seems a bit silly to restrict all the attention to this single question. In my joint work with Jian, Ronen and Miki we propose to test for geometric structure. This is also quite important in practice, as learning representations for the vertices of a graph is a fundamental primitive to many network analysis techniques. The abstract of our paper reads as follows, and I refer you to the paper itself for more details:

We study the problem of detecting the presence of an underlying high-dimensional geometric structure in a random graph. Under the null hypothesis, the observed graph is a realization of an Erdos-Renyi random graph G(n,p). Under the alternative, the graph is generated from the G(n,p,d) model, where each vertex corresponds to a latent independent random vector uniformly distributed on the sphere \mathbb{S}^{d-1}, and two vertices are connected if the corresponding latent vectors are close enough. In the dense regime (i.e., p is a constant), we propose a near-optimal and computationally efficient testing procedure based on a new quantity which we call signed triangles. The proof of the detection lower bound is based on a new bound on the total variation distance between a Wishart matrix and an appropriately normalized GOE matrix. In the sparse regime, we make a conjecture for the optimal detection boundary. We conclude the paper with some preliminary steps on the problem of estimating the dimension in G(n,p,d).


Where did my network come from?

The two other papers (1, 2) are follow-ups on the project that I described a few months ago in this blog post. In a nutshell the goal was to understand if the Preferential Attachment model depends on its initial condition. See for example these beautiful pictures by Igor Kortchemski of the Preferential Attachment tree on 5000 vertices with different seeds:



(click here to see the seeds for the above pictures, and see here and here for pictures with other seeds). With Elchanan and Miki we conjectured that any initial condition leads to a different model, and we were able to partially prove this conjecture. The full conjecture was proved in a beautiful follow-up paper by a group of fellow French, Nicolas Curien, Thomas Duquesne, Igor Kortchemski, and Ioan Manolescu. Using their new proof technique (but with a different statistic) we (Ronen, Elchanan, Miki, and I) were then able to extend the theorem to the Uniform Attachment model, in which case the result is perhaps even more surprising (in fact I was originally convinced that the seed could not possibly have an influence in the Uniform Attachment model!). The paper can be found here.

The Seed Theorem emerging from these works opens up the problem of actually finding the seed. This is the problem we looked at with Luc and Gabor (in the special case where the seed is a single vertex), and the abstract of the paper reads as follows:

We investigate algorithms to find the first vertex in large trees generated by either the uniform attachment or preferential attachment model. We require the algorithm to output a set of K vertices, such that, with probability at least 1-\epsilon, the first vertex is in this set. We show that for any \epsilon, there exist such algorithms with K independent of the size of the input tree. Moreover, we provide almost tight bounds for the best value of K as a function of \epsilon. In the uniform attachment case we show that the optimal K is subpolynomial in 1/\epsilon, and that it has to be at least superpolylogarithmic. On the other hand, the preferential attachment case is exponentially harder, as we prove that the best K is polynomial in 1/\epsilon. We conclude the paper with several open problems.


Open problems

The four papers (1, 2, 3, 4) contain about a dozen open problems, and we could have easily extended the list to twenty open problems (for example I have added a few open problems in my talk at Oberwolfach two weeks ago, see here for Louigi Addario-Berry‘s notes). Can you solve one of them? Or even better, can you propose new interesting graph hypothesis testing problems?

Posted in Random graphs | 1 Comment

Komlos conjecture, Gaussian correlation conjecture, and a bit of machine learning

Today I would like to talk (somewhat indirectly) about a beautiful COLT 2014 paper by Nick Harvey and Samira Samadi. The problem studied in this paper goes as follows: imagine that you have a bunch of data points in \mathbb{R}^n with a certain mean value \mu. Your goal is to generate a sequence out these data points, such that the partial averages of this new sequence are as close as possible to \mu. The “herding” procedure of Max Welling has the property that the average of the t^{th} first terms in the new sequence is at distance at most O(1/t) from \mu (this was shown by Francis Bach, Simon Lacoste-Julien and Guillaume Obozinski by viewing the herding procedure as a simple Frank-Wolfe). In their paper Nick and Samira improves the dependency on the dimension n in the O notation by using a new algorithm inspired by the theory of discrepancy. In the rest of this post I would like to discuss the basic notions of this theory. What follows is a transcription of what Ronen Eldan (a postdoc in the theory group at MSR) explained to me (of course all errors are mine). I will use c and C to denote absolute numerical constants that take whatever value I need them to take.

Discrepancy theory

Consider a large finite set of vertices V, and a size-n set \mathcal{P} of subsets of vertices (in other words (V, \mathcal{P}) is an hypergraph). For sake of simplicity we will assume here that elements of \mathcal{P} are also of size n. We are now interested in the following combinatorial optimization problem:

    \begin{align*} & \qquad \;\; \mathrm{min.} \qquad \;\;\max_{A \in \mathcal{P}} \left| \sum_{v \in A} f(v) \right| \\ & f : V \rightarrow \{-1,1\} . \end{align*}

In words, one is looking for a partitioning f of the vertices, so as to have the hyperedges A \in \mathcal{P} as balanced as possible.

Alright, so let’s try something simple, just randomly partition the vertices: clearly a simple union bounds shows that the obtained objective value in this case is of order \sqrt{n \log n}. Denote D (for Discrepancy) the smallest objective value (in particular we just saw that D \leq C \sqrt{n \log n}). A beautiful result of Joel Spencer shows that one can always do much better than random partitioning:

Theorem (Spencer’s six deviations, 1985): D \leq 6 \sqrt{n}.

We will now see a splendid (partial) proof of this result. Let \gamma be the Gaussian measure in \mathbb{R}^d (in the following we will take d=|V|). First let me remind you of a basic fact about this measure, and a related conjecture. A set of the form S_{v,t} = \{x : |v^{\top} x| \leq t\}, for some v \in \mathbb{R}^n, t >0, is called a slab.

Theorem (Sidak, 1967): Let A be a symmetric convex set and B be a slab, then

    \[\gamma(A \cap B) \geq \gamma(A) \gamma(B) .\]

Obviously there is equality if A and B are two orthogonal slabs, and the theorem says (in particular) that an angle between the slabs would imply positive correlation.

Conjecture (Gaussian correlation): Let A and B be two symmetric convex sets, then

    \[\gamma(A \cap B) \geq \gamma(A) \gamma(B) .\]

Ok, now back to the proof of Spencer’s theorem. First, let’s rewrite things suggestively in terms of the slabs S_A := S_{\frac{\mathds{1}_A}{\sqrt{n}}, C}, A \in \mathcal{P}. We want to show that there exists f \in \{-1,1\}^d such that f \in \cap_{A \in \mathcal{P}} S_A. Using Sidak’s theorem one clearly has \gamma(\cap_{A \in \mathcal{P}} S_A) \geq c^d (with c being roughly like 1-2 \exp(-C^2/2)). From there it is now a fairly short route to show that, if c \geq 0.99 (say), then the intersection of \{-1,1\}^d with \cap_{A \in \mathcal{P}} S_A must be non-empty. Ronen showed me a non-trivial (and algorithmically non-efficient) 5 lines argument for this but I think that it’s yet unpublished work so I won’t reproduce it here. Another proof of this, which yields an efficient algorithm for finding a point in the intersection, is given in this very nice paper of Lovett and Meka.


In what we discussed so far the underlying hypergraph was completely arbitrary. For some specific graphs one can clearly improve the bound on D. Indeed if all hyperedges are disjoint, then D=0. More interestingly, Beck and Fiala showed in 1981 that, if each vertex appears in at most t hyperedges, then D \leq 2 t. They also conjectured that the correct bound should be O(\sqrt{t}). In 1998 Banaszczyk showed a bound of O(\sqrt{t \log(n)}) using the following theorem (which is a variant of the result mentioned above):

Theorem (Banaszczyk, 1998): Let A be a convex set such that \gamma(A) \geq 1/2, and u_1, \hdots u_m \in \mathbb{R}^d such that \|u_i\|_2 \leq C. Then there exists f \in \{-1,1\}^m such that \sum_{i=1}^m f_i u_i \in A.

The longstanding Komlos’ conjecture says that in fact this holds for A=[-1,1]^n (despite the fact that the Gaussian measure of this set is c^n).

Finally let me mention another open problem, which is the matrix version of Spencer’s six deviations, see this blog post by Raghu Meka for details.


Posted in Optimization, Probability theory, Theoretical Computer Science | 5 Comments

A zest of number theory

I just encountered an amazing number theoretic result. It is probably very well known, but for those who never saw it it’s quite something, so I thought I would share it.

Let n be a positive integer. A partition of n is a multiset of positive integers such that the sum of its elements is equal to n (for example \{1,1,2\} is a partition of 4). The partition function p(n) is defined as the number of partitions of n. The amazing result that I want to tell you about is the formula for the asymptotic equivalent of p(n). Before giving away the formula let’s try to get some intuition about the order of magnitude one can expect. For this let me first look at the simpler quantity q(n), defined as the number of multiset of [n] with n elements (for example q(2)=3). With the stars and bars argument (thanks to Luc Devroye for reminding me about this elementary combinatorial technique!) one can easily see that q(n) \geq 2^n-1. Moreover it is also easy to see that there is an injection from multisets of [\sqrt{n}] with \sqrt{n} elements to partitions of n (observe that the sum of the elements of such a multiset is necessarily smaller than n). Thus

    \[p(n) \geq q(\sqrt{n}) \geq 2^{\sqrt{n}}-1.\]

A more involved argument, based on Young Diagram representation of partitions, shows that p(n) \leq 2^{100 \sqrt{n}} (don’t quote me on the 100, I think it’s correct but I have only rough calculations here).

Now here we go for the asymptotic formula, discovered by Hardy and Ramanujan in 1918:

    \[p(n) \sim \frac{1}{4 \sqrt{3} n} \exp\left(\pi \sqrt{2 n /3}\right) .\]

In my opinion this is a beautiful connection between \pi, e, and a basic property of integers. The proof is based on the circle method, an elementary technique in complex analysis (see here for the details).

Posted in Theoretical Computer Science | 1 Comment

Probability in high dimension

The Barcelona events have just ended, and I’m happy to report that everything went very smoothly. In my opinion the quality of the works presented at COLT and at the Foundations of Learning Theory workshop were truly outstanding. I hope to blog about the results that I found the most exciting next week.

The point of this post is to advertise the lecture notes of my friend (and colleague) Ramon van Handel. These notes cover all the basics of modern high dimensional probability. Acquaintance with this material is extremely useful in all the fields that I’m interested in (machine learning, statistics, optimization, and theoretical computer science). It is the first time that I see this material presented in such a cohesive manner, and it is an excellent companion to the book of Boucheron, Lugosi and Massart. Last but not least Ramon is an amazing mathematical writer (for instance if you want to learn about filtering I highly recommend his lecture notes on Hidden Markov Models).

Posted in Probability theory | Leave a comment

Theory of Convex Optimization for Machine Learning

I am extremely happy to release the first draft of my monograph based on the lecture notes published last year on this blog. (Comments on the draft are welcome!) The abstract reads as follows:

This monograph presents the main mathematical ideas in convex optimization. Starting from the fundamental theory of black-box optimization, the material progresses towards recent advances in structural optimization and stochastic optimization. Our presentation of black-box optimization, strongly influenced by the seminal book of Nesterov, includes the analysis of the Ellipsoid Method, as well as (accelerated) gradient descent schemes. We also pay special attention to non-Euclidean settings (relevant algorithms include Frank Wolfe, Mirror Descent, and Dual Averaging) and discuss their relevance in machine learning. We provide a gentle introduction to structural optimization with FISTA (to optimize a sum of a smooth and a simple non-smooth term), Saddle-Point Mirror Prox (Nemirovski’s alternative to Nesterov’s smoothing), and a concise description of Interior Point Methods. In stochastic optimization we discuss Stochastic Gradient Descent, mini-batches, Random Coordinate Descent, and sublinear algorithms. We also briefly touch upon convex relaxation of combinatorial problems and the use of randomness to round solutions, as well as random walks based methods.

Posted in Optimization | 13 Comments

COLT 2014 accepted papers

The accepted papers for COLT 2014 have just been posted! This year we had a record number of 140 submissions, out of which 52 were accepted (38 for a 20mn presentation and 14 for a 5mn presentation). In my opinion the program is very strong, I am looking forward to listen to all these talks in Barcelona! By the way now may be a very good time to register, the deadline for early registration is coming up fast!

Here is a list of the accepted papers together with links when I could find one (if you want me to add a link just send me an email).

COLT 2014 accepted papers

- Jacob Abernethy, Chansoo Lee, Abhinav Sinha and Ambuj Tewari. Learning with Perturbations via Gaussian Smoothing

- Alekh Agarwal, Animashree Anandkumar, Prateek Jain, Praneeth Netrapalli and Rashish Tandon. Learning Sparsely Used Overcomplete Dictionaries

- Alekh Agarwal, Ashwin Badanidiyuru, Miroslav Dudik, Robert Schapire and Aleksandrs Slivkins. Robust Multi-objective Learning with Mentor Feedback- Morteza Alamgir, -

- Ulrike von Luxburg and Gabor Lugosi. Density-preserving quantization with application to graph downsampling

- Joseph Anderson, Mikhail Belkin, Navin Goyal, Luis Rademacher and James Voss. The More, the Merrier: the Blessing of Dimensionality for Learning Large Gaussian Mixtures

- Sanjeev Arora, Rong Ge and Ankur Moitra. New Algorithms for Learning Incoherent and Overcomplete Dictionaries

- Ashwinkumar Badanidiyuru, John Langford and Aleksandrs Slivkins. Resourceful Contextual Bandits

- Shai Ben-David and Ruth Urner. The sample complexity of agnostic learning under deterministic labels

- Aditya Bhaskara, Moses Charikar and Aravindan Vijayaraghavan. Uniqueness of Tensor Decompositions with Applications to Polynomial Identifiability

- Evgeny Burnaev and Vladimir Vovk. Efficiency of conformalized ridge regression

- Karthekeyan Chandrasekaran and Richard M. Karp. Finding a most biased coin with fewest flips

- Yudong Chen, Xinyang Yi and Constantine Caramanis. A Convex Formulation for Mixed Regression: Minimax Optimal Rates

- Amit Daniely, Nati Linial and Shai Shalev-Shwartz. The complexity of learning halfspaces using generalized linear methods

- Amit Daniely and Shai Shalev-Shwartz. Optimal Learners for Multiclass Problems

- Constantinos Daskalakis and Gautam Kamath. Faster and Sample Near-Optimal Algorithms for Proper Learning Mixtures of Gaussians

- Ofer Dekel, Jian Ding, Tomer Koren and Yuval Peres. Online Learning with Composite Loss Functions Can Be Hard

- Tim van Erven, Wojciech Kotlowski and Manfred K. Warmuth. Follow the Leader with Dropout Perturbations

- Vitaly Feldman and Pravesh Kothari. Learning Coverage Functions and Private Release of Marginals

- Vitaly Feldman and David Xiao. Sample Complexity Bounds on Differentially Private Learning via Communication Complexity

- Pierre Gaillard, Gilles Stoltz and Tim van Erven. A Second-order Bound with Excess Losses

- Eyal Gofer. Higher-Order Regret Bounds with Switching Costs

- Sudipto Guha and Kamesh Munagala. Stochastic Regret Minimization via Thompson Sampling

- Moritz Hardt, Raghu Meka, Prasad Raghavendra and Benjamin Weitz. Computational Limits for Matrix Completion

- Moritz Hardt and Mary Wootters. Fast Matrix Completion Without the Condition Number

- Elad Hazan, Zohar Karnin and Raghu Meka. Volumetric Spanners: an Efficient Exploration Basis for Learning

- Prateek Jain and Sewoong Oh. Learning Mixtures of Discrete Product Distributions using Spectral Decompositions

- Kevin Jamieson, Matthew Malloy, Robert Nowak and Sebastien Bubeck. lil’ UCB: An Optimal Exploration Algorithm for Multi-Armed Bandits

- Satyen Kale. Multiarmed Bandits With Limited Expert Advice

- Varun Kanade and Justin Thaler. Distribution-Independent Reliable Learning

- Ravindran Kannan, Santosh S. Vempala and David Woodruff. Principal Component Analysis and Higher Correlations for Distributed Data

- Emilie Kaufmann, Olivier Cappé and Aurélien Garivier. On the Complexity of A/B Testing

- Matthäus Kleindessner and Ulrike von Luxburg. Uniqueness of ordinal embedding

- Kfir Levy, Elad Hazan and Tomer Koren. Logistic Regression: Tight Bounds for Stochastic and Online Optimization

- Ping Li, Cun-Hui Zhang and Tong Zhang. Compressed Counting Meets Compressed Sensing

- Che-Yu Liu and Sébastien Bubeck. Most Correlated Arms Identification

- Stefan Magureanu, Richard Combes and Alexandre Proutière. Lipschitz Bandits:Regret Lower Bounds and Optimal Algorithms

- Shie Mannor, Vianney Perchet and Gilles Stoltz. Approachability in unknown games: Online learning meets multi-objective optimization

- Brendan McMahan and Francesco Orabona. Unconstrained Online Linear Learning in Hilbert Spaces: Minimax Algorithms and Normal Approximations

- Shahar Mendelson. Learning without Concentration

- Aditya Menon and Robert Williamson. Bayes-Optimal Scorers for Bipartite Ranking

- Elchanan Mossel, Joe Neeman and Allan Sly. Belief Propagation, Robust Reconstruction and Optimal Recovery of Block Models

- Andreas Maurer, Massimiliano Pontil and Bernardino Romera-Paredes. An Inequality with Applications to Structured Sparsity and Multitask Dictionary Learning

- Alexander Rakhlin and Karthik Sridharan. Online Nonparametric Regression

- Harish Ramaswamy, Balaji S.B., Shivani Agarwal and Robert Williamson. On the Consistency of Output Code Based Learning Algorithms for Multiclass Learning Problems

- Samira Samadi and Nick Harvey. Near-Optimal Herding

- Rahim Samei, Pavel Semukhin, Boting Yang and Sandra Zilles. Sample Compression for Multi-label Concept Classes

- Ingo Steinwart, Chloe Pasin and Robert Williamson. Elicitation and Identification of Properties

- Ilya Tolstikhin, Gilles Blanchard and Marius Kloft. Localized Complexities for Transductive Learning

- Robert Williamson. The Geometry of Losses

- Jiaming Xu, Marc Lelarge and Laurent Massoulie. Edge Label Inference in Generalized Stochastic Block Models: from Spectral Theory to Impossibility Results

- Se-Young Yun and Alexandre Proutiere. Community Detection via Random and Adaptive Sampling

- Yuchen Zhang, Martin Wainwright and Michael Jordan. Lower bounds on the performance of polynomial-time algorithms for sparse linear regression

Posted in Conference/workshop | 3 Comments

On the influence of the seed graph in the preferential attachment model


The preferential attachment model, introduced in 1992 by Mahmoud and popularized in 1999 by Barabási and Albert, has attracted a lot of attention in the last decade. In its simplest form it describes the evolution of a random tree. Formally we denote by \mathrm{PA}(n) the preferential attachment tree on n vertices which is defined by induction as follows. First \mathrm{PA}(2) is the unique tree on two vertices. Then, given \mathrm{PA}(n), \mathrm{PA}(n+1) is formed from \mathrm{PA}(n) by adding a new vertex u and a new edge uv where v is selected at random among vertices in \mathrm{PA}(n) according to the following probability distribution:

    \[\mathbb{P}\left(v = i \ \middle| \, \mathrm{PA}(n) \right) = \frac{d_{\mathrm{PA}(n)}(i)}{2 \left( n - 1 \right)} ,\]

where d_T(u) denotes the degree of vertex u in a tree T. In other words vertices of large degrees are more likely to attract the new nodes. This model of evolution is argued to be a good approximation for things such a network of citations, or the internet network.

One of the main reason for the success of the preferential attachment model is the following theorem, which shows that the degree distribution in \mathrm{PA}(n) follows a power-law, a feature that many real-world networks (such as the internet) exhibit but which is not reproduced by standard random graph models such as the Erdös-Rényi model.

Theorem [Bollobás, Riordan, Spencer and Tusnády (2001)Let d be fixed. Then as n \to +\infty, the proportion of vertices with degree d tends in probability to

    \[\frac{4}{(d+1)(d+2)(d+3)} \sim \frac{4}{d^3} .\]

While the above theorem is a fine and interesting mathematical result, I do not view it as the critical aspect of the preferential attachment model (note that Wikipedia disagrees). In my opinion \mathrm{PA}(n) is simply interesting merely because of its natural rule of evolution.

Now think about the application of the PA model to the internet. Of course there is a few obvious objections, such as the fact that in \mathrm{PA}(n) a website can only link to one other website. While this is clearly ridiculous I think that \mathrm{PA}(n) still contains the essence of what one would like to capture to model the evolution of the internet. However there is one potentially important aspect which is overlooked: in the early days of the internet the PA model was probably very far from being a good approximation to the evolution of the network. It is perhaps reasonable to assume that after 1995 the network was evolving according to PA, but certainly from 1970 to 1995 the evolution followed fundamentally different rules. This observation suggests to study the preferential attachment model with a seed.

Thus we are now interested in \mathrm{PA}(n, T), where T is a finite seed tree. Formally \mathrm{PA}(n, T) is also defined by induction, where \mathrm{PA}(|T|, T) = T and \mathrm{PA}(n+1,T) is formed from \mathrm{PA}(n, T) as before. A very basic question which seems to have been overlooked in the literature is the following: what is the influence of the seed T on \mathrm{PA}(n, T) as n goes to infinity?

In our recent joint work with Elchanan Mossel and Miklos Racz we looked exactly at this question. More precisely we ask the following: given two seed trees T and S, do the distributions \mathrm{PA}(n, S) and \mathrm{PA}(n, T) remain separated (say in total variation distance) as n goes to infinity? In other words we are interested in the following quantity:

    \[\delta(S, T) = \lim_{n \to \infty} \mathrm{TV}(\mathrm{PA}(n, S), \mathrm{PA}(n, T)) .\]

A priori it could be that \delta(S, T) = 0 for any S and T, which would mean that the seed has no influence and that the preferential attachment “forgets” its initial conditions. We prove that this is far from true:

Theorem [Bubeck, Mossel and Racz (2014)] Let S and T be two finite trees on at least 3 vertices. If the degree distributions in S and T are different, then \delta \left( S, T \right) > 0.

If I wanted to make a bold statement I could say that this theorem implies the following: by looking at the internet network today, one can still “see” the influence of the topological structure of the internet back in the 90’s. In other words to a certain extent one can go back in time and potentially infer some properties that people may have believed to be lost (perhaps some army’s secrets hidden in the structure of the ARPANET?). Of course at this stage this is pure science fiction, but the theorem certainly leaves that possibility open. Note that we believe that the theorem can even be strengthen to the following statement:

Conjecture Let S and T be two finite trees on at least 3 vertices. If S and T are non-isomorphic, then \delta \left( S, T \right) > 0.

These statements show that even when n is large one can still “see” in \mathrm{PA}(n, T) the influence of the original seed T. However by considering the total variation distance we allow global statistics that depend on entire tree \mathrm{PA}(n, T). What about local statistics that could be computed by an agent looking only at a finite neighborhood around her? Mathematically this question can be interpreted in the framework of the Benjamini-Schramm limit. Recall that a sequence of random graphs (G_n) tends to a random infinite rooted tree (\mathcal{T}, Z) (Z is the random root) if for any r \in \mathbb{N}, the random ball of radius r around a random vertex k_n in G_n tends in distribution to the random ball of radius r around Z in \mathcal{T}. In other words when n is large enough a random agent cannot tell if she is in G_n or in \mathcal{T} by looking at a finite neighborhood around her. One has the following theorem for the weak limit of the PA model:

Theorem [Berger, Borgs, Chayes and Saberi (2014)The Benjamini-Schramm limit of \mathrm{PA}(n) is the Pólya-point graph with m=1.

We extend this result to an arbitrary seed and show that locally the seed has no influence:

Theorem [Bubeck, Mossel and Racz (2014)For any tree T the Benjamini-Schramm limit of \mathrm{PA}(n, T) is the Pólya-point graph with m=1.

Thus, while the army’s secret of the 90’s might be at risk if one looks at the overall topology of the current internet network, these secrets are safe from any local agent who would only access a (random) finite part of the network.

These new results on the PA model naturally lead to a host of new problems. We end the paper with a list of 7 open problems, I recommend to take a look at them (and try to solve them)!

Posted in Random graphs | 6 Comments

Nesterov’s Accelerated Gradient Descent for Smooth and Strongly Convex Optimization


About a year ago I described Nesterov’s Accelerated Gradient Descent in the context of smooth optimization. As I mentioned previously this has been by far the most popular post on this blog. Today I have decided to revisit this post to give a slightly more geometrical proof (though unfortunately still magical in various parts). I will focus on unconstrained optimization of a smooth and strongly convex function f (in the previous post I dealt only with the smooth case). Precisely f is \alpha-strongly convex and \beta-smooth, and we denote by Q=\beta / \alpha the condition number of f. As explained in this post, in this case the basic gradient descent algorithm requires O(Q \log(1/\epsilon)) iterations to reach \epsilon-accuracy. As we shall see below Nesterov’s Accelerated Gradient Descent attains the improved oracle complexity of O(\sqrt{Q} \log(1/\epsilon)). This is particularly relevant in Machine Learning applications since the strong convexity parameter \alpha can often be viewed as a regularization term, and 1/\alpha can be as large as the sample size. Thus reducing the number of step from “\mathrm{sample \; size}” to “\sqrt{\mathrm{sample \; size}}” can be a huge deal, especially in large scale applications.

Without further due let’s get to the algorithm, which can be described quite succintly. Note that everything written below is simply a condensed version of the calculations appearing on pages 71–81 in Nesterov 2004’s book. Start at an arbitrary initial point x_1 = y_1 and then iterate the following equations for s \geq 1,

    \begin{eqnarray*} y_{s+1} & = & x_s - \frac{1}{\beta} \nabla f(x_s) , \\ x_{s+1} & = & \left(1 + \frac{\sqrt{Q}-1}{\sqrt{Q}+1} \right) y_{s+1} - \frac{\sqrt{Q}-1}{\sqrt{Q}+1} y_s . \end{eqnarray*}

Theorem Let f be \alpha-strongly convex and \beta-smooth, then Nesterov’s Accelerated Gradient Descent satisfies

    \[f(y_t) - f(x^*) \leq \frac{\alpha + \beta}{2} \|x_1 - x^*\|^2 \exp\left(- \frac{t-1}{\sqrt{Q}} \right).\]

Proof: We define \alpha-strongly convex quadratic functions \Phi_s, s \geq 1 by induction as follows:

(1)   \begin{eqnarray*} \Phi_1(x) & = & f(x_1) + \frac{\alpha}{2} \|x-x_1\|^2 , \notag \\ \Phi_{s+1}(x) & = & \left(1 - \frac{1}{\sqrt{Q}}\right) \Phi_s(x) \notag \\ & & + \frac{1}{\sqrt{Q}} \left(f(x_s) + \nabla f(x_s)^{\top} (x-x_s) + \frac{\alpha}{2} \|x-x_s\|^2 \right).  \end{eqnarray*}

Intuitively \Phi_s becomes a finer and finer approximation (from below) to f in the following sense:

(2)   \begin{equation*}  \Phi_{s+1}(x) \leq f(x) + \left(1 - \frac{1}{\sqrt{Q}}\right)^s (\Phi_1(x) - f(x)). \end{equation*}

The above inequality can be proved immediately by induction, using the fact that by \alpha-strong convexity one has

    \[f(x_s) + \nabla f(x_s)^{\top} (x-x_s) + \frac{\alpha}{2} \|x-x_s\|^2 \leq f(x) .\]

Equation (2) by itself does not say much, for it to be useful one needs to understand how “far” below f is \Phi_s. The following inequality answers this question:

(3)   \begin{equation*}  f(y_s) \leq \min_{x \in \mathbb{R}^n} \Phi_s(x) . \end{equation*}

The rest of the proof is devoted to showing that (3) holds true, but first let us see how to combine (2) and (3) to obtain the rate given by the theorem (we use that by \beta-smoothness one has f(x) - f(x^*) \leq \frac{\beta}{2} \|x-x^*\|^2):

    \begin{eqnarray*} f(y_t) - f(x^*) & \leq & \Phi_t(x^*) - f(x^*) \\ & \leq & \left(1 - \frac{1}{\sqrt{Q}}\right)^{t-1} (\Phi_1(x^*) - f(x^*)) \\ & \leq & \frac{\alpha + \beta}{2} \|x_1-x^*\|^2 \left(1 - \frac{1}{\sqrt{Q}}\right)^{t-1} . \end{eqnarray*}

We now prove (3) by induction (note that it is true at s=1 since x_1=y_1). Let \Phi_s^* = \min_{x \in \mathbb{R}^n} \Phi_s(x). Using the definition of y_{s+1} (and \beta-smoothness), convexity, and the induction hypothesis, one gets

    \begin{align*} & f(y_{s+1}) \\ & \leq f(x_s) - \frac{1}{2 \beta} \| \nabla f(x_s) \|^2 \\ & = \left(1 - \frac{1}{\sqrt{Q}}\right) f(y_s) + \left(1 - \frac{1}{\sqrt{Q}}\right)(f(x_s) - f(y_s)) + \frac{f(x_s)}{\sqrt{Q}} - \frac{1}{2 \beta} \| \nabla f(x_s) \|^2 \\ & \leq \left(1 - \frac{1}{\sqrt{Q}}\right) \Phi_s^* + \left(1 - \frac{1}{\sqrt{Q}}\right) \nabla f(x_s)^{\top} (x_s - y_s) + \frac{f(x_s)}{\sqrt{Q}} - \frac{1}{2 \beta} \| \nabla f(x_s) \|^2 . \end{align*}

Thus we now have to show that

(4)   \begin{equation*}  \Phi_{s+1}^* \geq \left(1 - \frac{1}{\sqrt{Q}}\right) \Phi_s^* + \left(1 - \frac{1}{\sqrt{Q}}\right) \nabla f(x_s)^{\top} (x_s - y_s) + \frac{f(x_s)}{\sqrt{Q}} - \frac{1}{2 \beta} \| \nabla f(x_s) \|^2 . \end{equation*}

To prove this inequality we have to understand better the functions \Phi_s. First note that \nabla^2 \Phi_s(x) = \alpha \mathrm{I}_n (immediate by induction) and thus \Phi_s has to be of the following form:

    \[\Phi_s(x) = \Phi_s^* + \frac{\alpha}{2} \|x - v_s\|^2 ,\]

for some v_s \in \mathbb{R}^n. Now observe that by differentiating (1) and using the above form of \Phi_s one obtains

    \[\nabla \Phi_{s+1}(x) = \alpha \left(1 - \frac{1}{\sqrt{Q}}\right) (x-v_s) + \frac{1}{\sqrt{Q}} \nabla f(x_s) + \frac{\alpha}{\sqrt{Q}} (x-x_s) .\]

In particular \Phi_{s+1} is by definition minimized at v_{s+1} which can now be defined by induction using the above identity, precisely:

(5)   \begin{equation*}  v_{s+1} = \left(1 - \frac{1}{\sqrt{Q}}\right) v_s + \frac{1}{\sqrt{Q}} x_s - \frac{1}{\alpha \sqrt{Q}} \nabla f(x_s) . \end{equation*}

Using the form of \Phi_s and \Phi_{s+1}, as well as the original definition (1) one gets the following identity by evaluating \Phi_{s+1} at x_s:

(6)   \begin{equation*}  \Phi_{s+1}^* + \frac{\alpha}{2} \|x_s - v_{s+1}\|^2 = \left(1 - \frac{1}{\sqrt{Q}}\right) \Phi_s^* + \frac{\alpha}{2} \left(1 - \frac{1}{\sqrt{Q}}\right) \|x_s - v_s\|^2 + \frac{1}{\sqrt{Q}} f(x_s) . \end{equation*}

Note that thanks to (5) one has

    \begin{eqnarray*} \|x_s - v_{s+1}\|^2   & = & \left(1 - \frac{1}{\sqrt{Q}}\right)^2 \|x_s - v_s\|^2 + \frac{1}{\alpha^2 Q} \|\nabla f(x_s)\|^2 \\ & & - \frac{2}{\alpha \sqrt{Q}} \left(1 - \frac{1}{\sqrt{Q}}\right) \nabla f(x_s)^{\top}(v_s-x_s) , \end{eqnarray*}

which combined with (6) yields

    \begin{eqnarray*} \Phi_{s+1}^* & = & \left(1 - \frac{1}{\sqrt{Q}}\right) \Phi_s^* + \frac{1}{\sqrt{Q}} f(x_s) + \frac{\alpha}{2 \sqrt{Q}} \left(1 - \frac{1}{\sqrt{Q}}\right) \|x_s - v_s\|^2 \\ & & \qquad - \frac{1}{2 \beta} \| \nabla f(x_s) \|^2 + \frac{1}{\sqrt{Q}} \left(1 - \frac{1}{\sqrt{Q}}\right) \nabla f(x_s)^{\top}(v_s-x_s) . \end{eqnarray*}

Finally we show by induction that v_s - x_s = \sqrt{Q}(x_s - y_s), which concludes the proof of (4) and thus also concludes the proof of the theorem:

    \begin{eqnarray*} v_{s+1} - x_{s+1} & = & \left(1 - \frac{1}{\sqrt{Q}}\right) v_s + \frac{1}{\sqrt{Q}} x_s - \frac{1}{\alpha \sqrt{Q}} \nabla f(x_s) - x_{s+1} \\ & = & \sqrt{Q} x_s - (\sqrt{Q}-1) y_s - \frac{\sqrt{Q}}{\beta} \nabla f(x_s) - x_{s+1} \\ & = & \sqrt{Q} y_{s+1} - (\sqrt{Q}-1) y_s - x_{s+1} \\ & = & \sqrt{Q} (x_{s+1} - y_{s+1}) , \end{eqnarray*}

where the first equality comes from (5), the second from the induction hypothesis, the third from the definition of y_{s+1} and the last one from the definition of x_{s+1}.


Posted in Optimization | 5 Comments

COLT deadline next week

COLT deadline is approaching fast, don’t forget to send your awesome learning theory paper(s) before Friday 7th! Also recall that if you get a paper into COLT it will (i) give you an excuse to spend a few days in Barcelona at the best time of the year and (ii) you could stay a few more days to attend our awesome Foundations of Learning Theory workshop.

In other news I attended ITCS in Princeton last month and it was absolutely great. Here are a few papers that I really liked:

- Amir Shpilka, Avishay Tal and Ben Lee Volk, On the Structure of Boolean Functions with Small Spectral Norm

- Andrew Wan, John Wright and Chenggang Wu. Decision Trees, Protocols, and the Fourier Entropy-Influence Conjecture

- Cristopher Moore and Leonard Schulman, Tree Codes and a Conjecture on Exponential Sums

- Fernando Brandao, Aram Harrow, James Lee and Yuval Peres, Adversarial hypothesis testing and a quantum Stein’s Lemma for restricted measurements

- Yossi Azar, Uriel Feige, Michal Feldman and Moshe Tennenholtz, Sequential Decision Making with Vector Outcomes

- Manor Mendel and Assaf Naor, Expanders with respect to Hadamard spaces and random graphs

- David Gamarnik and Madhu Sudan, Limits of local algorithms over sparse random graphs

- Rishi Gupta, Tim Roughgarden and C. Seshadhri, Decompositions of Triangle-Dense Graphs

During my visit to the Theory Group at MSR I also learned about the following topics which my readers will probably like too: Tucker’s lemma (see also the related Borsuk-Ulam theorem), the lamplighter graph and random walks on it, sparse regularity lemma, and algorithmic applications of evolving sets.

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