Sébastien Roch

ICML Conference 2025 Conference Paper

Sample Complexity of Branch-length Estimation by Maximum Likelihood

• David Clancy Jr.
• Hanbaek Lyu
• Sébastien Roch

We consider the branch-length estimation problem on a bifurcating tree: a character evolves along the edges of a binary tree according to a two-state symmetric Markov process, and we seek to recover the edge transition probabilities from repeated observations at the leaves. This problem arises in phylogenetics, and is related to latent tree graphical model inference. In general, the log-likelihood function is non-concave and may admit many critical points. Nevertheless, simple coordinate maximization has been known to perform well in practice, defying the complexity of the likelihood landscape. In this work, we provide the first theoretical guarantee as to why this might be the case. We show that deep inside the Kesten-Stigum reconstruction regime, provided with polynomially many $m$ samples (assuming the tree is balanced), there exists a universal parameter regime (independent of the size of the tree) where the log-likelihood function is strongly concave and smooth with high probability. On this high-probability likelihood landscape event, we show that the standard coordinate maximization algorithm converges exponentially fast to the maximum likelihood estimator, which is within $O(1/\sqrt{m})$ from the true parameter, provided a sufficiently close initial point.

SODA Conference 2016 Conference Paper

Species Trees from Gene Trees Despite a High Rate of Lateral Genetic Transfer: A Tight Bound (Extended Abstract)

• Constantinos Daskalakis
• Sébastien Roch

Reconstructing the tree of life from molecular sequences is a fundamental problem in computational biology. Modern data sets often contain a large number of genes which can complicate the reconstruction problem due to the fact that different genes may undergo different evolutionary histories. This is the case in particular in the presence of lateral genetic transfer (LGT), whereby a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from (but related to) the species phylogeny. In previous work, a stochastic model of LGT was introduced and it was shown that the species phylogeny can be reconstructed from gene trees despite surprisingly high rates of LGT. Both lower and upper bounds on this rate were obtained, but a large gap remained. Here we close this gap, up to a constant. Specifically, we show that the species phylogeny can be reconstructed perfectly even when each edge of the tree has a constant probability of being the location of an LGT event. Our new reconstruction algorithm builds the tree recursively from the leaves. We also provide a matching bound in the negative direction (up to a constant).

FOCS Conference 2008 Conference Paper

Sequence Length Requirement of Distance-Based Phylogeny Reconstruction: Breaking the Polynomial Barrier

• Sébastien Roch

We introduce a new distance-based phylogeny reconstruction technique which provably achieves, at sufficiently short branch lengths, a sequence length requirement growing slower than any polynomial. The technique is based on a new averaging procedure that implicitly reconstructs ancestral sequences. In the same token, we extend previous results on phase transitions in phylogeny reconstruction to general time-reversible models. More precisely, we show that in the so-called Kesten-Stigum zone---roughly, a region of the parameter space where ancestral sequences are well approximated by ``linear combinations'' of observed sequences---sequences of length e radiclog n suffice for reconstruction. Here n is the number of extant species. We improve this result to poly(log n) the ultrametric case. Surprisingly, this last result suggests that a UPGMA-type algorithm may in some sense be "optimal'' under a molecular clock. Our results challenge---to some extent---the conventional wisdom that estimates of evolutionary distances alone carry significantly less information about phylogenies than full sequence datasets.

STOC Conference 2007 Conference Paper

First to market is not everything: an analysis of preferential attachment with fitness

• Christian Borgs
• Jennifer T. Chayes
• Constantinos Daskalakis
• Sébastien Roch

The design of algorithms on complex networks, such as routing, ranking or recommendation algorithms, requires a detailed understanding of the growth characteristics of the networks of interest, such as the Internet,the web graph, social networks or online communities. To this end, preferential attachment, in which the popularity (or relevance) of a node is determined by its degree, is a well-known and appealing random graph model, whose predictions are in accordance with experiments on the web graph and several social networks. However, its central assumption, that the popularity of the nodes dependsonly on their degree, is not a realistic one, since every node has potentially some intrinsic quality which can differentiate its attractiveness from other nodes with similar degrees.

STOC Conference 2007 Conference Paper

On the submodularity of influence in social networks

• Elchanan Mossel
• Sébastien Roch

We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the spread of influence in social networks. A social network can be represented by a directed graph where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or "word-of-mouth" effects on such a graph is to consider an increasing process of "infected" (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by KKT in [7,8] where the authors also impose several natural assumptions: the threshold values are (uniformly) random to account for our lack of knowledge of the true values; and the activation functions are monotone and submodular, i.e. have "diminishing returns." The monotonicity condition indicates that a node is more likely to become active if more of its neighbors are active, while the submodularity condition, indicates that the marginal effect of each neighbor is decreasing when the set of active neighbors increases. For an initial set of active nodes s , let σ(S) denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function σ(S) is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination. In other words, our results demonstrate that "local" submodularity is preserved "globally" under diffusion processes. This is of natural computational interest, as many optimization problems have good approximation algorithms for submodular functions. In particular, our results coupled with an argument in [7] imply that a greedy algorithm gives an (1-1/e-ε)-approximation algorithm for maximizing σ(S) among all sets s of a given size. This result has important practical implications for many social network analysis problems, notably viral marketing.

STOC Conference 2006 Conference Paper

Optimal phylogenetic reconstruction

• Constantinos Daskalakis
• Elchanan Mossel
• Sébastien Roch

One of the major tasks of evolutionary biology is the reconstruction of phylogenetic trees from molecular data. The evolutionary model is given by a Markov chain on the true evolutionary tree. Given samples from this Markov chain at the leaves of the tree, the goal is to reconstruct the evolutionary tree.It is well known that in order to reconstruct a tree on n leaves, sequences of length Ω(log n) are needed. It was conjectured by M. Steel that for the CFN evolutionary model, if the mutation probability on all edges of the tree is less than p* = (√2-1)/2 3/2 , then the tree can be recovered from sequences of length O(log n). This was proven by the second author in the special case where the tree is "balanced". The second author also proved that if all edges have mutation probability larger than p* then the length needed is n Ω(1) . This "phase-transition" in the number of samples needed is closely related to the phase transition for the reconstruction problem (or extremality of free measure) studied extensively in statistical physics, probability and computer science.Here we complete the proof of Steel's conjecture and give a reconstruction algorithm using optimal (up to a multiplicative constant) sequence length. Our results further extend to obtain an optimal reconstruction algorithm for the Jukes-Cantor model with short edges. All reconstruction algorithms run in polynomial time.

FOCS Conference 2006 Conference Paper

The Kesten-Stigum Reconstruction Bound Is Tight for Roughly Symmetric Binary Channels

• Christian Borgs
• Jennifer T. Chayes
• Elchanan Mossel
• Sébastien Roch

We establish the exact threshold for the reconstruction problem for a binary asymmetric channel on the b-ary tree, provided that the asymmetry is sufficiently small. This is the first exact reconstruction threshold obtained in roughly a decade. We discuss the implications of our result for Glauber dynamics, phylogenetic reconstruction, noisy communication and the so-called "replica symmetry breaking" in spin glasses and random satisfiability problems

STOC Conference 2005 Conference Paper

Learning nonsingular phylogenies and hidden Markov models

• Elchanan Mossel
• Sébastien Roch