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Communication overhead is a known bottleneck in federated learning (FL). To address this, lossy compression is commonly used on the information communicated between the server and clients during training. In horizontal FL, where each client holds a subset of the samples, such communication-compressed training methods have recently seen significant progress. However, in their vertical FL counterparts, where each client holds a subset of the features, our understanding remains limited. To address this, we propose an error feedback compressed vertical federated learning (EFVFL) method to train split neural networks. In contrast with previous communication-compressed methods for vertical FL, EFVFL does not require a vanishing compression error for the gradient norm to converge to zero for smooth nonconvex problems. By leveraging error feedback, our method can achieve a $\mathcal{O}(1/T)$ convergence rate in the full-batch case, improving over the state-of-the-art $\mathcal{O}(1/\sqrt{T})$ rate under $\mathcal{O}(1/\sqrt{T})$ compression error, and matching the rate of uncompressed methods. Further, when the objective function satisfies the Polyak-{\L}ojasiewicz inequality, our method converges linearly. In addition to improving convergence rates, our method also supports the use of private labels. Numerical experiments show that EFVFL significantly improves over the prior art, confirming our theoretical results.

We consider the problem of robustly fitting a model to data that includes outliers by formulating a percentile optimization problem. This problem is non-smooth and non-convex, hence hard to solve. We derive properties that the minimizers of such problems must satisfy. These properties lead to methods that solve the percentile formulation both for general residuals and for convex residuals. The methods fit the model to subsets of the data, and then extract the solution of the percentile formulation from these partial fits. As illustrative simulations show, such methods endure higher outlier percentages, when compared with standard robust estimates. Additionally, the derived properties provide a broader and alternative theoretical validation for existing robust methods, whose validity was previously limited to specific forms of the residuals., Comment: 5 pages, 1 figure. Submitted to IEEE Signal Processing Letters on 21-Apr-2024

Fecal Microbiota Transplant (FMT) is an FDA approved treatment for recurrent Clostridium difficile infections, and is being explored for other clinical applications, from alleviating digestive and neurological disorders, to priming the microbiome for cancer treatment, and restoring microbiomes impacted by cancer treatment. Quantifying the extent of engraftment following an FMT is important in determining if a recipient didn't respond because the engrafted microbiome didn't produce the desired outcomes (a successful FMT, but negative treatment outcome), or the microbiome didn't engraft (an unsuccessful FMT and negative treatment outcome). The lack of a consistent methodology for quantifying FMT engraftment extent hinders the assessment of FMT success and its relation to clinical outcomes, and presents challenges for comparing FMT results and protocols across studies. Here we review 46 studies of FMT in humans and model organisms and group their approaches for assessing the extent to which an FMT engrafts into three criteria: 1) Chimeric Asymmetric Community Coalescence investigates microbiome shifts following FMT engraftment. 2) Donated Microbiome Indicator Features tracks donated microbiome features as a signal of engraftment with methods such as differential abundance testing based on the current sample collection, or tracking changes in feature abundances that have been previously identified. 3) Temporal Stability examines how resistant post-FMT recipient's microbiomes are to reverting back to their baseline microbiome. Investigated together, these criteria provide a clear assessment of microbiome engraftment. We discuss the pros and cons of each of these criteria, providing illustrative examples of their application. We also introduce key terminology and recommendations on how FMT studies can be analyzed for rigorous engraftment extent assessment., Comment: 18 pages, 6 figures, 2 supplemental tables

Communication efficiency is a major challenge in federated learning (FL). In client-server schemes, the server constitutes a bottleneck, and while decentralized setups spread communications, they do not necessarily reduce them due to slower convergence. We propose Multi-Token Coordinate Descent (MTCD), a communication-efficient algorithm for semi-decentralized vertical federated learning, exploiting both client-server and client-client communications when each client holds a small subset of features. Our multi-token method can be seen as a parallel Markov chain (block) coordinate descent algorithm and it subsumes the client-server and decentralized setups as special cases. We obtain a convergence rate of $\mathcal{O}(1/T)$ for nonconvex objectives when tokens roam over disjoint subsets of clients and for convex objectives when they roam over possibly overlapping subsets. Numerical results show that MTCD improves the state-of-the-art communication efficiency and allows for a tunable amount of parallel communications.

This paper consider considers the problem of locating a two dimensional target from range-measurements containing outliers. Assuming that the number of outlier is known, we formulate the problem of minimizing inlier losses while ignoring outliers. This leads to a combinatorial, non-convex, non-smooth problem involving the percentile function. Using the framework of risk analysis from Rockafellar et al., we start by interpreting this formulation as a Value-at-risk (VaR) problem from portfolio optimization. To the best of our knowledge, this is the first time that a localization problem was formulated using risk analysis theory. To study the VaR formulation, we start by designing a majorizer set that contains any solution of a general percentile problem. This set is useful because, when applied to a localization scenario in 2D, it allows to majorize the solution set in terms of singletons, circumferences, ellipses and hyperbolas. Using know parametrization of these curves, we propose a grid method for the original non-convex problem. So we reduce the task of optimizing the VaR objective to that of efficiently sampling the proposed majorizer set. We compare our algorithm with four benchmarks in target localization. Numerical simulations show that our method is fast while, on average, improving the accuracy of the best benchmarks by at least 100m in a 1 Km$^2$ area., Comment: 12 pages, 4 figures. Originally submitted in -IEEE Transactions on Signal Processing- on 02-Jul-2023. The 2nd upload solves a bug in figure 3

This paper considers a distributed detection setup where agents in a network want to detect a time-varying signal embedded in temporally correlated noise. The signal of interest is the impulse response of an ARMA (auto-regressive moving average) filter, and the noise is the output of yet another ARMA filter which is fed white Gaussian noise. For this extended problem setup, which can prompt novel behaviour, we propose a comprehensive solution. First, we extend the well-known running consensus detector (RCD) to this correlated setup; then, we design an efficient implementation of the RCD by exploiting the underlying ARMA structures; and, finally, we derive the theoretical asymptotic performance of the RCD in this ARMA setup. It turns out that the error probability at each agent exhibits one of two regimes: either (a) the error probability decays exponentially fast to zero or (b) it converges to a strictly positive error floor. While regime (a) spans staple results in large deviation theory, regime (b) is new in distributed detection and is elicited by the ARMA setup. We fully characterize these two scenarios: we give necessary and sufficient conditions, phrased in terms of the zero and poles of the underlying ARMA models, for the emergence of each regime, and provide closed-form expressions for both the decay rates of regime (a) and the positive error floors of regime (b). Our analysis also shows that the ARMA setup leads to two novel features: (1) the threshold level used in RCD can influence the asymptotics of the error probabilities and (2) some agents might be weakly informative, in the sense that their observations do not improve the asymptotic performance of RCD and, as such, can be safely muted to save sensing resources. Numerical simulations illustrate and confirm the theoretical findings., Comment: 13 pages, 3 figures. Originally submitted in -IEEE Transactions on Signal Processing- on 13-Apr-2023

Expectation Maximization (EM) is the standard method to learn Gaussian mixtures. Yet its classic, centralized form is often infeasible, due to privacy concerns and computational and communication bottlenecks. Prior work dealt with data distributed by examples, horizontal partitioning, but we lack a counterpart for data scattered by features, an increasingly common scheme (e.g. user profiling with data from multiple entities). To fill this gap, we provide an EM-based algorithm to fit Gaussian mixtures to Vertically Partitioned data (VP-EM). In federated learning setups, our algorithm matches the centralized EM fitting of Gaussian mixtures constrained to a subspace. In arbitrary communication graphs, consensus averaging allows VP-EM to run on large peer-to-peer networks as an EM approximation. This mismatch comes from consensus error only, which vanishes exponentially fast with the number of consensus rounds. We demonstrate VP-EM on various topologies for both synthetic and real data, evaluating its approximation of centralized EM and seeing that it outperforms the available benchmark.

Locating a target is key in many applications, namely in high-stakes real-world scenarios, like detecting humans or obstacles in vehicular networks. In scenarios where precise statistics of the measurement noise are unavailable, applications require localization methods that assume minimal knowledge on the noise distribution. We present a scalable algorithm delimiting a tight superset of all possible target locations, assuming range measurements to known landmarks, contaminated with bounded noise and unknown distributions. This superset is of primary interest in robust statistics since it is a tight majorizer of the set of Maximum-Likelihood (ML) estimates parametrized by noise densities respecting two main assumptions: (1) the noise distribution is supported on a ellipsoidal uncertainty region and (2) the measurements are non-negative with probability one. We create the superset through convex relaxations that use Linear Fractional Representations (LFRs), a well-known technique in robust control. For low noise regimes the supersets created by our method double the accuracy of a standard semidefinite relaxation. For moderate to high noise regimes our method still improves the benchmark but the benefit tends to be less significant, as both supersets tend to have the same size (area)., Comment: 15 pages, 7 figures. Originally submitted in -IEEE Transactions on Signal Processing- on 01-Oct-2021. Accepted in -IEEE Transactions on Signal Processing- on 06-Jun-2022

First-order methods for stochastic optimization have undeniable relevance, in part due to their pivotal role in machine learning. Variance reduction for these algorithms has become an important research topic. In contrast to common approaches, which rarely leverage global models of the objective function, we exploit convexity and L-smoothness to improve the noisy estimates outputted by the stochastic gradient oracle. Our method, named COCO denoiser, is the joint maximum likelihood estimator of multiple function gradients from their noisy observations, subject to co-coercivity constraints between them. The resulting estimate is the solution of a convex Quadratically Constrained Quadratic Problem. Although this problem is expensive to solve by interior point methods, we exploit its structure to apply an accelerated first-order algorithm, the Fast Dual Proximal Gradient method. Besides analytically characterizing the proposed estimator, we show empirically that increasing the number and proximity of the queried points leads to better gradient estimates. We also apply COCO in stochastic settings by plugging it in existing algorithms, such as SGD, Adam or STRSAGA, outperforming their vanilla versions, even in scenarios where our modelling assumptions are mismatched., Comment: 25 pages, 14 figures

In recent work, we proposed a distributed Banach-Picard iteration (DBPI) that allows a set of agents, linked by a communication network, to find a fixed point of a locally contractive (LC) map that is the average of individual maps held by said agents. In this work, we build upon the DBPI and its local linear convergence (LLC) guarantees to make several contributions. We show that Sanger's algorithm for principal component analysis (PCA) corresponds to the iteration of an LC map that can be written as the average of local maps, each map known to each agent holding a subset of the data. Similarly, we show that a variant of the expectation-maximization (EM) algorithm for parameter estimation from noisy and faulty measurements in a sensor network can be written as the iteration of an LC map that is the average of local maps, each available at just one node. Consequently, via the DBPI, we derive two distributed algorithms - distributed EM and distributed PCA - whose LLC guarantees follow from those that we proved for the DBPI. The verification of the LC condition for EM is challenging, as the underlying operator depends on random samples, thus the LC condition is of probabilistic nature., Comment: 13 pages, 2 figures

The Banach-Picard iteration is widely used to find fixed points of locally contractive (LC) maps. This paper extends the Banach-Picard iteration to distributed settings; specifically, we assume the map of which the fixed point is sought to be the average of individual (not necessarily LC) maps held by a set of agents linked by a communication network. An additional difficulty is that the LC map is not assumed to come from an underlying optimization problem, which prevents exploiting strong global properties such as convexity or Lipschitzianity. Yet, we propose a distributed algorithm and prove its convergence, in fact showing that it maintains the linear rate of the standard Banach-Picard iteration for the average LC map. As another contribution, our proof imports tools from perturbation theory of linear operators, which, to the best of our knowledge, had not been used before in the theory of distributed computation., Comment: 7 pages

We present the Stochastic alternate Linearization Method (StochaLM), a token-based method for distributed optimization. This algorithm finds the solution of a consensus optimization problem by solving a sequence of subproblems where some components of the cost are linearized around specific anchor points. StochaLM can be interpreted as a dual block coordinate ascent method whose block components are selected using the state of an ergodic Markov chain. This sampling process is neither essentially cyclic nor independent over time, preventing us from using proofs of convergence of dual block coordinate ascent methods done in previous works. The proof of convergence of our method is, therefore, also novel. We show that, if the cost is strongly convex and the network is fully connected, then, with probability one, the primal sequence generated by StochaLM converges to the solution. Our method is application-friendly, as it has no global hyperparameters to tune; any hyperparameters can be tuned locally by the agents, using information regarding their private cost and neighbourhood. Our method is, therefore, decentralized in the truest sense. Numerical experiments evidence that our method converges to the solution faster than other token-based methods, even when these methods' hyperparameters are tuned for optimal performance., Comment: Updated with correct proof of convergence

Social interaction between microbes can be described at many levels of details, ranging from the biochemistry of cell-cell interactions to the ecological dynamics of populations. Choosing the best level to model microbial communities without losing generality remains a challenge. Here we propose to model cross-feeding interactions at an intermediate level between genome-scale metabolic models of individual species and consumer-resource models of ecosystems, which is suitable to empirical data. We applied our method to three published examples of multi-strain Escherichia coli communities with increasing complexity consisting of uni-, bi-, and multi-directional cross-feeding of either substitutable metabolic byproducts or essential nutrients. The intermediate-scale model accurately described empirical data and could quantify exchange rates elusive by other means, such as the byproduct secretions, even for a complex community of 14 amino acid auxotrophs. We used the three models to study each community's limits of robustness to perturbations such as variations in resource supply, antibiotic treatments and invasion by other "cheaters" species. Our analysis provides a foundation to quantify cross-feeding interactions from experimental data, and highlights the importance of metabolic exchanges in the dynamics and stability of microbial communities., Comment: 6 figures

The augmented Lagrangian method (ALM) is a classical optimization tool that solves a given "difficult" (constrained) problem via finding solutions of a sequence of "easier"(often unconstrained) sub-problems with respect to the original (primal) variable, wherein constraints satisfaction is controlled via the so-called dual variables. ALM is highly flexible with respect to how primal sub-problems can be solved, giving rise to a plethora of different primal-dual methods. The powerful ALM mechanism has recently proved to be very successful in various large scale and distributed applications. In addition, several significant advances have appeared, primarily on precise complexity results with respect to computational and communication costs in the presence of inexact updates and design and analysis of novel optimal methods for distributed consensus optimization. We provide a tutorial-style introduction to ALM and its variants for solving convex optimization problems in large scale and distributed settings. We describe control-theoretic tools for the algorithms' analysis and design, survey recent results, and provide novel insights in the context of two emerging applications: federated learning and distributed energy trading.

Gut microbial composition has been linked to multiple health outcomes. Yet, temporal analysis of this composition had been limited to deterministic models. In this paper, we introduce a probabilistic model for the dynamics of intestinal microbiomes that takes into account interaction among bacteria as well as external effects such as antibiotics. The model successfully deals with pragmatic issues such as random measurement error and varying time intervals between measurements through latent space modeling. We demonstrate utility of the model by using latent state features to predict the clinical events of intestinal domination and bacteremia, improving accuracy over existing methods. We further leverage this framework to validate known links between antibiotics and clinical outcomes, while discovering new ones., Comment: Experiment code available at https://github.com/ZhongRuiqi1997/NIPS2017MLCB, software at https://github.com/ZhongRuiqi1997/Kalman-Filter-Intestinal-Microbiota

Consider a set of agents that wish to estimate a vector of parameters of their mutual interest. For this estimation goal, agents can sense and communicate. When sensing, an agent measures (in additive gaussian noise) linear combinations of the unknown vector of parameters. When communicating, an agent can broadcast information to a few other agents, by using the channels that happen to be randomly at its disposal at the time. To coordinate the agents towards their estimation goal, we propose a novel algorithm called FADE (Fast and Asymptotically efficient Distributed Estimator), in which agents collaborate at discrete time-steps; at each time-step, agents sense and communicate just once, while also updating their own estimate of the unknown vector of parameters. FADE enjoys five attractive features: first, it is an intuitive estimator, simple to derive; second, it withstands dynamic networks, that is, networks whose communication channels change randomly over time; third, it is strongly consistent in that, as time-steps play out, each agent's local estimate converges (almost surely) to the true vector of parameters; fourth, it is both asymptotically unbiased and efficient, which means that, across time, each agent's estimate becomes unbiased and the mean-square error (MSE) of each agent's estimate vanishes to zero at the same rate of the MSE of the optimal estimator at an almighty central node; fifth, and most importantly, when compared with a state-of-art consensus+innovation (CI) algorithm, it yields estimates with outstandingly lower mean-square errors, for the same number of communications -- for example, in a sparsely connected network model with 50 agents, we find through numerical simulations that the reduction can be dramatic, reaching several orders of magnitude.

We present an algorithm for the problem of linear distributed estimation of a parameter in a network where a set of agents are successively taking measurements. The approach considers a roaming token in a network that carries the estimate, and jumps from one agent to another in its vicinity according to the probabilities of a Markov chain. When the token is at an agent it records the agent's local information. We analyze the proposed algorithm and show that it is consistent and asymptotically optimal, in the sense that its mean-square-error (MSE) rate of decay approaches the centralized one as the number of iterations increases. We show these results for a scenario where the network changes over time, and we consider two different set of assumptions on the network instantiations: they are i.i.d. and connected on the average, or they are deterministic and strongly connected for every finite time window of a fixed size. Simulations show our algorithm is competitive with consensus+innovations type algorithms, achieving a smaller MSE at each iteration in all considered scenarios.

Processing data collected by a network of agents often boils down to solving an optimization problem. The distributed nature of these problems calls for methods that are, themselves, distributed. While most collaborative learning problems require agents to reach a common (or consensus) model, there are situations in which the consensus solution may not be optimal. For instance, agents may want to reach a compromise between agreeing with their neighbors and minimizing a personal loss function. We present DJAM, a Jacobi-like distributed algorithm for learning personalized models. This method is implementation-friendly: it has no hyperparameters that need tuning, it is asynchronous, and its updates only require single-neighbor interactions. We prove that DJAM converges with probability one to the solution, provided that the personal loss functions are strongly convex and have Lipschitz gradient. We then give evidence that DJAM is on par with state-of-the-art methods: our method reaches a solution with error similar to the error of a carefully tuned ADMM in about the same number of single-neighbor interactions., Comment: 5 pages, 1 figure

Predicting evolution of expanding populations is critical to control biological threats such as invasive species and cancer metastasis. Expansion is primarily driven by reproduction and dispersal, but nature abounds with examples of evolution where organisms pay a reproductive cost to disperse faster. When does selection favor this 'survival of the fastest?' We searched for a simple rule, motivated by evolution experiments where swarming bacteria evolved into an hyperswarmer mutant which disperses $ \sim 100\%$ faster but pays a growth cost of $\sim 10 \%$ to make many copies of its flagellum. We analyzed a two-species model based on the Fisher equation to explain this observation: the population expansion rate ($v$) results from an interplay of growth ($r$) and dispersal ($D$) and is independent of the carrying capacity: $v=2\sqrt{rD}$. A mutant can take over the edge only if its expansion rate ($v_2$) exceeds the expansion rate of the established species ($v_1$); this simple condition ($v_2 > v_1$) determines the maximum cost in slower growth that a faster mutant can pay and still be able to take over. Numerical simulations and time-course experiments where we tracked evolution by imaging bacteria suggest that our findings are general: less favorable conditions delay but do not entirely prevent the success of the fastest. Thus, the expansion rate defines a traveling wave fitness, which could be combined with trade-offs to predict evolution of expanding populations.

We construct a novel preorder on the set of nodes of a simple undirected graph. We prove that the preorder (induced by the topology of the graph) is preserved, e.g., by the logistic dynamical system (both in discrete and continuous time). Moreover, the underlying equivalence relation of the preorder corresponds to the coarsest equitable partition (CEP). This will further imply that the logistic dynamical system on a graph preserves its coarsest equitable partition. The results provide a nontrivial invariant set for the logistic and the like dynamical systems, as we show. We note that our construct provides a functional characterization for the CEP as an alternative to the pure set theoretical iterated degree sequences characterization. The construct and results presented might have independent interest for analysis on graphs or qualitative analysis of dynamical systems over networks., Comment: 19 pages, 2 figures

We establish a functional weak law of large numbers for observable macroscopic state variables of interacting particle systems (e.g., voter and contact processes) over fast time-varying sparse random networks of interactions. We show that, as the number of agents $N$ grows large, the proportion of agents $\left(\overline{Y}_{k}^{N}(t)\right)$ at a certain state $k$ converges in distribution -- or, more precisely, weakly with respect to the uniform topology on the space of \emph{c\`adl\`ag} sample paths -- to the solution of an ordinary differential equation over any compact interval $\left[0,T\right]$. Although the limiting process is Markov, the prelimit processes, i.e., the normalized macrostate vector processes $\left(\mathbf{\overline{Y}}^{N}(t)\right)=\left(\overline{Y}_{1}^{N}(t),\ldots,\overline{Y}_{K}^{N}(t)\right)$, are non-Markov as they are tied to the \emph{high-dimensional} microscopic state of the system, which precludes the direct application of standard arguments for establishing weak convergence. The techniques developed in the paper for establishing weak convergence might be of independent interest., Comment: 26 pages, 5 figures

Fictitious play (FP) is a canonical game-theoretic learning algorithm which has been deployed extensively in decentralized control scenarios. However standard treatments of FP, and of many other game-theoretic models, assume rather idealistic conditions which rarely hold in realistic control scenarios. This paper considers a broad class of best response learning algorithms, that we refer to as FP-type algorithms. In such an algorithm, given some (possibly limited) information about the history of actions, each individual forecasts the future play and chooses a (myopic) best action given their forecast. We provide a unifed analysis of the behavior of FP-type algorithms under an important class of perturbations, thus demonstrating robustness to deviations from the idealistic operating conditions that have been previously assumed. This robustness result is then used to derive convergence results for two control-relevant relaxations of standard game-theoretic applications: distributed (network-based) implementation without full observability and asynchronous deployment (including in continuous time). In each case the results follow as a direct consequence of the main robustness result.

We find large deviations rates for consensus-based distributed inference for directed networks. When the topology is deterministic, we establish the large deviations principle and find exactly the corresponding rate function, equal at all nodes. We show that the dependence of the rate function on the stochastic weight matrix associated with the network is fully captured by its left eigenvector corresponding to the unit eigenvalue. Further, when the sensors' observations are Gaussian, the rate function admits a closed form expression. Motivated by these observations, we formulate the optimal network design problem of finding the left eigenvector which achieves the highest value of the rate function, for a given target accuracy. This eigenvector therefore minimizes the time that the inference algorithm needs to reach the desired accuracy. For Gaussian observations, we show that the network design problem can be formulated as a semidefinite (convex) program, and hence can be solved efficiently. When observations are identically distributed across agents, the system exhibits an interesting property: the graph of the rate function always lies between the graphs of the rate function of an isolated node and the rate function of a fusion center that has access to all observations. We prove that this fundamental property holds even when the topology and the associated system matrices change randomly over time, with arbitrary distribution. Due to generality of its assumptions, the latter result requires more subtle techniques than the standard large deviations tools, contributing to the general theory of large deviations., Comment: 35 pages, submitted to Transactions on Signal Processing

With the help of an in-band full-duplex relay station, it is possible to simultaneously transmit and receive signals from multiple users. The performance of such system can be greatly increased when the relay station is equipped with a large number of antennas on both transmitter and receiver sides. In this paper, we exploit the use of massive arrays to effectively suppress the loopback interference (LI) of a decode-and-forward relay (DF) and evaluate the performance of the end-to-end (e2e) transmission. This paper assumes imperfect channel state information is available at the relay and designs a minimum mean-square error (MMSE) filter to mitigate the interference. Subsequently, we adopt zero-forcing (ZF) filters for both detection and beamforming. The performance of such system is evaluated in terms of bit error rate (BER) at both relay and destinations, and an optimal choice for the transmission power at the relay is shown. We then propose a complexity efficient optimal power allocation (OPA) algorithm that, using the channel statistics, computes the minimum power that satisfies the rate constraints of each pair. The results obtained via simulation show that when both MMSE filtering and OPA method are used, better values for the energy efficiency are attained., Comment: Accepted to the 16th IEEE International Workshop on Signal Processing Advances in Wireless Communications - SPAWC, Stockholm, Sweden 2015

The paper studies the highly prototypical Fictitious Play (FP) algorithm, as well as a broad class of learning processes based on best-response dynamics, that we refer to as FP-type algorithms. A well-known shortcoming of FP is that, while players may learn an equilibrium strategy in some abstract sense, there are no guarantees that the period-by-period strategies generated by the algorithm actually converge to equilibrium themselves. This issue is fundamentally related to the discontinuous nature of the best response correspondence and is inherited by many FP-type algorithms. Not only does it cause problems in the interpretation of such algorithms as a mechanism for economic and social learning, but it also greatly diminishes the practical value of these algorithms for use in distributed control. We refer to forms of learning in which players learn equilibria in some abstract sense only (to be defined more precisely in the paper) as weak learning, and we refer to forms of learning where players' period-by-period strategies converge to equilibrium as strong learning. An approach is presented for modifying an FP-type algorithm that achieves weak learning in order to construct a variant that achieves strong learning. Theoretical convergence results are proved., Comment: 22 pages

Empirical Centroid Fictitious Play (ECFP) is a generalization of the well-known Fictitious Play (FP) algorithm designed for implementation in large-scale games. In ECFP, the set of players is subdivided into equivalence classes with players in the same class possessing similar properties. Players choose a next-stage action by tracking and responding to aggregate statistics related to each equivalence class. This setup alleviates the difficult task of tracking and responding to the statistical behavior of every individual player, as is the case in traditional FP. Aside from ECFP, many useful modifications have been proposed to classical FP, e.g., rules allowing for network-based implementation, increased computational efficiency, and stronger forms of learning. Such modifications tend to be of great practical value; however, their effectiveness relies heavily on two fundamental properties of FP: robustness to alterations in the empirical distribution step size process, and robustness to best-response perturbations. The main contribution of the paper is to show that similar robustness properties also hold for the ECFP algorithm. This result serves as a first step in enabling practical modifications to ECFP, similar to those already developed for FP., Comment: Submitted for publication. Initial Submission: Mar. 2015