We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are $n$ players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex -- this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when $r$ rounds of interaction are allowed, at least one player needs to communicate $\Omega(n^{1/20^{r+1}})$ bits. In particular, with logarithmic bandwidth, finding an MIS requires $\Omega(\log\log{n})$ rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrovi\'c, and Rubinfeld [PODC 2018] that solves MIS in $O(\log\log{n})$ rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. To prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of edge-sharing between players. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.
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In practical applications, data is used to make decisions in two steps: estimation and optimization. First, a machine learning model estimates parameters for a structural model relating decisions to outcomes. Second, a decision is chosen to optimize the structural model's predicted outcome as if its parameters were correctly estimated. Due to its flexibility and simple implementation, this ``estimate-then-optimize'' approach is often used for data-driven decision-making. Errors in the estimation step can lead estimate-then-optimize to sub-optimal decisions that result in regret, i.e., a difference in value between the decision made and the best decision available with knowledge of the structural model's parameters. We provide a novel bound on this regret for smooth and unconstrained optimization problems. Using this bound, in settings where estimated parameters are linear transformations of sub-Gaussian random vectors, we provide a general procedure for experimental design to minimize the regret resulting from estimate-then-optimize. We demonstrate our approach on simple examples and a pandemic control application.
We settle a recent conjecture on a continuous patrolling game. In this zero-sum game, an Attacker chooses a time and place to attack a network for a fixed amount of time. A Patroller patrols the network with the aim of intercepting the attack with maximum probability. The conjecture asserts that a particular patrolling strategy called the E-patrolling strategy is optimal for all tree networks. The conjecture was previously known to be true in a limited class of special cases. The E-patrolling strategy has the advantage of being straightforward to calculate and implement. We prove the conjecture by presenting $\varepsilon$-optimal strategies for the Attacker which provide upper bounds for the value of the game that come arbitrarily close to the lower bound provided by the E-patrolling strategy.
The change-point detection problem has been widely studied in time series and signal processing literature. The current methods can be resumed in the search for the appropiate partitions of a whole time series such that the problem can be approached as one of optimization; nevertheless, an exact optimization approach could result computationally expensive and approximate ones discard potential scenarios for change-points configurations in a non-rigorous manner. Thus, a framework it is presented to detect change-points in a univariate time series using a decision criterion based on the Minimum Description Length (MDL), modified such that a Bayesian analysis is included. To search for the points of change, the times where mean value deviations occur (exceedances) are analyzed and then it is evaluated which of these could constitute a change-point through a genetic algorithm using as a fitness function the previously described MDL. The effectiveness of the method it is assessed through a simulation study and on the other hand, it is analyzed its practical validity in a real dataset for the presence of Particulate Matter of less than 2.5 microns (PM2.5) in Bogot\'a, Colombia for the 2018-2020 period under different settings to understand the algorithm convergence. It is found that this definition for the objective function tends to find better results for both the number of change-points and their location in the series for most of cases reducing the error in comparison to other available methods in the literature.
We study variants of the mixed finite element method (mixed FEM) and the first-order system least-squares finite element (FOSLS) for the Poisson problem where we replace the load by a suitable regularization which permits to use $H^{-1}$ loads. We prove that any bounded $H^{-1}$ projector onto piecewise constants can be used to define the regularization and yields quasi-optimality of the lowest-order mixed FEM resp. FOSLS in weaker norms. Examples for the construction of such projectors are given. One is based on the adjoint of a weighted Cl\'ement quasi-interpolator. We prove that this Cl\'ement operator has second-order approximation properties. For the modified mixed method we show optimal convergence rates of a postprocessed solution under minimal regularity assumptions -- a result not valid for the lowest-order mixed FEM without regularization. Numerical examples conclude this work.
The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better than $2.588$ under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open $k+o(k)$ facilities. This gives a barrier to current approaches for obtaining an approximation better than $2 \sqrt{\phi} \approx 2.544$. Altogether we reduce the approximation gap of bi-point solutions by two thirds.
In the modern paradigm of federated learning, a large number of users are involved in a global learning task, in a collaborative way. They alternate local computations and two-way communication with a distant orchestrating server. Communication, which can be slow and costly, is the main bottleneck in this setting. To reduce the communication load and therefore accelerate distributed gradient descent, two strategies are popular: 1) communicate less frequently; that is, perform several iterations of local computations between the communication rounds; and 2) communicate compressed information instead of full-dimensional vectors. In this paper, we propose the first algorithm for distributed optimization and federated learning, which harnesses these two strategies jointly and converges linearly to an exact solution, with a doubly accelerated rate: our algorithm benefits from the two acceleration mechanisms provided by local training and compression, namely a better dependency on the condition number of the functions and on the dimension of the model, respectively.
Machine learning models built on datasets containing discriminative instances attributed to various underlying factors result in biased and unfair outcomes. It's a well founded and intuitive fact that existing bias mitigation strategies often sacrifice accuracy in order to ensure fairness. But when AI engine's prediction is used for decision making which reflects on revenue or operational efficiency such as credit risk modelling, it would be desirable by the business if accuracy can be somehow reasonably preserved. This conflicting requirement of maintaining accuracy and fairness in AI motivates our research. In this paper, we propose a fresh approach for simultaneous improvement of fairness and accuracy of ML models within a realistic paradigm. The essence of our work is a data preprocessing technique that can detect instances ascribing a specific kind of bias that should be removed from the dataset before training and we further show that such instance removal will have no adverse impact on model accuracy. In particular, we claim that in the problem settings where instances exist with similar feature but different labels caused by variation in protected attributes , an inherent bias gets induced in the dataset, which can be identified and mitigated through our novel scheme. Our experimental evaluation on two open-source datasets demonstrates how the proposed method can mitigate bias along with improving rather than degrading accuracy, while offering certain set of control for end user.
We consider a high-dimensional random constrained optimization problem in which a set of binary variables is subjected to a linear system of equations. The cost function is a simple linear cost, measuring the Hamming distance with respect to a reference configuration. Despite its apparent simplicity, this problem exhibits a rich phenomenology. We show that different situations arise depending on the random ensemble of linear systems. When each variable is involved in at most two linear constraints, we show that the problem can be partially solved analytically, in particular we show that upon convergence, the zero-temperature limit of the cavity equations returns the optimal solution. We then study the geometrical properties of more general random ensembles. In particular we observe a range in the density of constraints at which the systems enters a glassy phase where the cost function has many minima. Interestingly, the algorithmic performances are only sensitive to another phase transition affecting the structure of configurations allowed by the linear constraints. We also extend our results to variables belonging to $\text{GF}(q)$, the Galois Field of order $q$. We show that increasing the value of $q$ allows to achieve a better optimum, which is confirmed by the Replica Symmetric cavity method predictions.
We give a method for proactively identifying small, plausible shifts in distribution which lead to large differences in model performance. These shifts are defined via parametric changes in the causal mechanisms of observed variables, where constraints on parameters yield a "robustness set" of plausible distributions and a corresponding worst-case loss over the set. While the loss under an individual parametric shift can be estimated via reweighting techniques such as importance sampling, the resulting worst-case optimization problem is non-convex, and the estimate may suffer from large variance. For small shifts, however, we can construct a local second-order approximation to the loss under shift and cast the problem of finding a worst-case shift as a particular non-convex quadratic optimization problem, for which efficient algorithms are available. We demonstrate that this second-order approximation can be estimated directly for shifts in conditional exponential family models, and we bound the approximation error. We apply our approach to a computer vision task (classifying gender from images), revealing sensitivity to shifts in non-causal attributes.
The independent set polynomial is important in many areas. For every integer $\Delta\geq 2$, the Shearer threshold is the value $\lambda^*(\Delta)=(\Delta-1)^{\Delta-1}/\Delta^{\Delta}$ . It is known that for $\lambda < - \lambda^*(\Delta)$, there are graphs~$G$ with maximum degree~$\Delta$ whose independent set polynomial, evaluated at~$\lambda$, is at most~$0$. Also, there are no such graphs for any $\lambda > -\lambda^*(\Delta)$. This paper is motivated by the computational problem of approximating the independent set polynomial when $\lambda < - \lambda^*(\Delta)$. The key issue in complexity bounds for this problem is "implementation". Informally, an implementation of a real number $\lambda'$ is a graph whose hard-core partition function, evaluated at~$\lambda$, simulates a vertex-weight of~$\lambda'$ in the sense that $\lambda'$ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any $\lambda < - \lambda^*(\Delta)$, it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any $\lambda> \lambda^*(\Delta)$. Our result has already been used in a paper with \bezakova{} (STOC 2018) to show that it is \#P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most~$\Delta$ at any value $\lambda<-\lambda^*(\Delta)$. In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness).
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