arXiv subjects

Ioannis Panageas

Publications and source records attributed to Ioannis Panageas.

63 records · Page 4Linked to original sources

arXiv2017

Rock-Paper-Scissors, Differential Games and Biological Diversity

We model a situation in which a collection of species derive their fitnesses via a rock-paper-scissors-type game; however, the precise payoffs are a function of the environment. The new aspect of our model lies in adding a feedback loop: the environment changes according to the relative fitnesses of the species; in particular, it gives a boost to the species having small populations. We cast our model in the setting of a differential game and we show that for a certain setting of parameters, this dynamics cycles. Our model is a natural one, since depletion of resources used by more frequent species will shift the payoff matrix towards favoring less frequent ones. Since the dynamics cycles, no species goes extinct and diversity is maintained.

math.DS↗

arXiv2017

First-order Methods Almost Always Avoid Saddle Points

We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical systems perspective in which appropriate instantiations of the Stable Manifold Theorem allow for a global stability analysis. Thus, neither access to second-order derivative information nor randomness beyond initialization is necessary to provably avoid saddle points.

stat.ML↗

arXiv2017

Opinion Dynamics in Networks: Convergence, Stability and Lack of Explosion

Inspired by the work of [Kempe, Kleinberg, Oren, Slivkins, EC13] we introduce and analyze a model on opinion formation; the update rule of our dynamics is a simplified version of that of Kempe et. al. We assume that the population is partitioned into types whose interaction pattern is specified by a graph. Interaction leads to population mass moving from types of smaller mass to those of bigger. We show that starting uniformly at random over all population vectors on the simplex, our dynamics converges point-wise with probability one to an independent set. This settles an open problem of Kempe et. al., as applicable to our dynamics. We believe that our techniques can be used to settle the open problem for the Kempe et. al. dynamics as well. Next, we extend the model of Kempe et. al. by introducing the notion of birth and death of types, with the interaction graph evolving appropriately. Birth of types is determined by a Bernoulli process and types die when their population mass is less than a parameter $ε$. We show that if the births are infrequent, then there are long periods of "stability" in which there is no population mass that moves. Finally we show that even if births are frequent and "stability" is not attained, the total number of types does not explode: it remains logarithmic in $1/ε$.

cs.SI↗

arXiv2017

Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos

The Multiplicative Weights Update (MWU) method is a ubiquitous meta-algorithm that works as follows: A distribution is maintained on a certain set, and at each step the probability assigned to element $γ$ is multiplied by $(1 -εC(γ))>0$ where $C(γ)$ is the "cost" of element $γ$ and then rescaled to ensure that the new values form a distribution. We analyze MWU in congestion games where agents use \textit{arbitrary admissible constants} as learning rates $ε$ and prove convergence to \textit{exact Nash equilibria}. Our proof leverages a novel connection between MWU and the Baum-Welch algorithm, the standard instantiation of the Expectation-Maximization (EM) algorithm for hidden Markov models (HMM). Interestingly, this convergence result does not carry over to the nearly homologous MWU variant where at each step the probability assigned to element $γ$ is multiplied by $(1 -ε)^{C(γ)}$ even for the most innocuous case of two-agent, two-strategy load balancing games, where such dynamics can provably lead to limit cycles or even chaotic behavior.

cs.GT↗

arXiv2016

Average Case Performance of Replicator Dynamics in Potential Games via Computing Regions of Attraction

What does it mean to fully understand the behavior of a network of adaptive agents? The golden standard typically is the behavior of learning dynamics in potential games, where many evolutionary dynamics, e.g., replicator, are known to converge to sets of equilibria. Even in such classic settings many critical questions remain unanswered. We examine issues such as: Point-wise convergence: Does the system actually equilibrate even in the presence of continuums of equilibria? Computing regions of attraction: Given point-wise convergence can we compute the region of asymptotic stability of each equilibrium (e.g., estimate its volume, geometry)? System invariants: Invariant functions remain constant along every system trajectory. This notion is orthogonal to the game theoretic concept of a potential function, which always strictly increases/decreases along system trajectories. Do dynamics in potential games exhibit invariant functions? If so, how many? How do these functions look like? Based on these geometric characterizations, we propose a novel quantitative framework for analyzing the efficiency of potential games with many equilibria. The predictions of different equilibria are weighted by their probability to arise under evolutionary dynamics given uniformly random initial conditions. This average case analysis is shown to offer novel insights in classic game theoretic challenges, including quantifying the risk dominance in stag-hunt games and allowing for more nuanced performance analysis in networked coordination and congestion games with large gaps between price of stability and price of anarchy.

cs.GT↗

arXiv2016

Gradient Descent Only Converges to Minimizers: Non-Isolated Critical Points and Invariant Regions

Given a non-convex twice differentiable cost function f, we prove that the set of initial conditions so that gradient descent converges to saddle points where \nabla^2 f has at least one strictly negative eigenvalue has (Lebesgue) measure zero, even for cost functions f with non-isolated critical points, answering an open question in [Lee, Simchowitz, Jordan, Recht, COLT2016]. Moreover, this result extends to forward-invariant convex subspaces, allowing for weak (non-globally Lipschitz) smoothness assumptions. Finally, we produce an upper bound on the allowable step-size.

math.DS↗

arXiv2016

Mutation, Sexual Reproduction and Survival in Dynamic Environments

A new approach to understanding evolution [Val09], namely viewing it through the lens of computation, has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [CLPV14]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [WVA05], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability. On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game. Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation.

q-bio.PE↗

arXiv2015

The Complexity of Genetic Diversity

A key question in biological systems is whether genetic diversity persists in the long run under evolutionary competition or whether a single dominant genotype emerges. Classic work by Kalmus in 1945 has established that even in simple diploid species (species with two chromosomes) diversity can be guaranteed as long as the heterozygote individuals enjoy a selective advantage. Despite the classic nature of the problem, as we move towards increasingly polymorphic traits (e.g. human blood types) predicting diversity and understanding its implications is still not fully understood. Our key contribution is to establish complexity theoretic hardness results implying that even in the textbook case of single locus diploid models predicting whether diversity survives or not given its fitness landscape is algorithmically intractable. We complement our results by establishing that under randomly chosen fitness landscapes diversity survives with significant probability. Our results are structurally robust along several dimensions (e.g., choice of parameter distribution, different definitions of stability/persistence, restriction to typical subclasses of fitness landscapes). Technically, our results exploit connections between game theory, nonlinear dynamical systems, complexity theory and biology and establish hardness results for predicting the evolution of a deterministic variant of the well known multiplicative weights update algorithm in symmetric coordination games which could be of independent interest.

q-bio.PE↗

arXiv2014

Natural Selection as an Inhibitor of Genetic Diversity: Multiplicative Weights Updates Algorithm and a Conjecture of Haploid Genetics

In a recent series of papers a surprisingly strong connection was discovered between standard models of evolution in mathematical biology and Multiplicative Weights Updates Algorithm, a ubiquitous model of online learning and optimization. These papers establish that mathematical models of biological evolution are tantamount to applying discrete Multiplicative Weights Updates Algorithm, a close variant of MWUA, on coordination games. This connection allows for introducing insights from the study of game theoretic dynamics into the field of mathematical biology. Using these results as a stepping stone, we show that mathematical models of haploid evolution imply the extinction of genetic diversity in the long term limit, a widely believed conjecture in genetics. In game theoretic terms we show that in the case of coordination games, under minimal genericity assumptions, discrete MWUA converges to pure Nash equilibria for all but a zero measure of initial conditions. This result holds despite the fact that mixed Nash equilibria can be exponentially (or even uncountably) many, completely dominating in number the set of pure Nash equilibria. Thus, in haploid organisms the long term preservation of genetic diversity needs to be safeguarded by other evolutionary mechanisms such as mutations and speciation.

math.DS↗