In interactive coding, Alice and Bob wish to compute some function $f$ of their individual private inputs $x$ and $y$. They do this by engaging in a non-adaptive (fixed order, fixed length) protocol to jointly compute $f(x,y)$. The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions, both parties still learn $f(x,y)$. In this work, we study the optimal error resilience of such a protocol in the face of adversarial bit flip or erasures. While the optimal error resilience of such a protocol over a large alphabet is well understood, the situation over the binary alphabet has remained open. In this work, we resolve this problem of determining the optimal error resilience over binary channels. In particular, we construct protocols achieving $\frac16$ error resilience over the binary bit flip channel and $\frac12$ error resilience over the binary erasure channel, for both of which matching upper bounds are known. We remark that the communication complexity of our binary bit flip protocol is polynomial in the size of the inputs, and the communication complexity of our binary erasure protocol is linear in the size of the minimal noiseless protocol computing $f$.
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We study joint unicast and multigroup multicast transmission in single-cell massive multiple-input-multiple-output (MIMO) systems, under maximum ratio transmission. For the unicast transmission, the objective is to maximize the weighted sum spectral efficiency (SE) of the unicast user terminals (UTs) and for the multicast transmission the objective is to maximize the minimum SE of the multicast UTs. These two problems are coupled to each other in a conflicting manner, due to their shared power resource and interference. To address this, we formulate a multiobjective optimization problem (MOOP). We derive the Pareto boundary of the MOOP analytically and determine the values of the system parameters to achieve any desired Pareto optimal point. Moreover, we prove that the Pareto region is convex, hence the system should serve the unicast and multicast UTs at the same time-frequency resource.
Achieving high channel estimation accuracy and reducing hardware cost as well as power dissipation constitute substantial challenges in the design of massive multiple-input multiple-output (MIMO) systems. To resolve these difficulties, sophisticated pilot designs have been conceived for the family of energy-efficient hybrid analog-digital (HAD) beamforming architecture relying on adaptive-resolution analog-to-digital converters (RADCs). In this paper, we jointly optimize the pilot sequences, the number of RADC quantization bits and the hybrid receiver combiner in the uplink of multiuser massive MIMO systems. We solve the associated mean square error (MSE) minimization problem of channel estimation in the context of correlated Rayleigh fading channels subject to practical constraints. The associated mixed-integer problem is quite challenging due to the nonconvex nature of the objective function and of the constraints. By relying on advanced fractional programming (FP) techniques, we first recast the original problem into a more tractable yet equivalent form, which allows the decoupling of the fractional objective function. We then conceive a pair of novel algorithms for solving the resultant problems for codebook-based and codebook-free pilot schemes, respectively. To reduce the design complexity, we also propose a simplified algorithm for the codebook-based pilot scheme. Our simulation results confirm the superiority of the proposed algorithms over the relevant state-of-the-art benchmark schemes.
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aim to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By identifying the intrinsic Lyapunov function of this algorithm, we prove that it converges to a critical point of the nonconvex minimax optimization problem and achieves a computation complexity $\mathcal{O}(\kappa^{1.5}\epsilon^{-2})$, where $\epsilon$ is the desired level of accuracy and $\kappa$ is the problem's condition number. Such a computation complexity improves the state-of-the-art complexities of single-loop GDA and AltGDA algorithms (see the summary of comparison in Table 1). We demonstrate the effectiveness of our algorithm via an experiment on adversarial deep learning.
We study the problem of computing the Hamming weight of an $n$-bit string modulo $m$, for any positive integer $m \leq n$ whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is $\left\lceil n(1 - 1/m) \right\rceil$. The upper bound is via an iterative query algorithm whose core components are the well-known 1-query quantum algorithm (essentially due to Deutsch) to compute the Hamming weight a 2-bit string mod 2 (i.e., the parity of the input bits), and a new 2-query quantum algorithm to compute the Hamming weight of a 3-bit string mod 3. We show a matching lower bound (in fact for arbitrary moduli $m$) via a variant of the polynomial method [de Wolf, SIAM J. Comput., 32(3), 2003]. This bound is for the weaker task of deciding whether or not a given $n$-bit input has Hamming weight 0 modulo $m$, and it holds even in the stronger non-deterministic quantum query model where an algorithm must have positive acceptance probability iff its input evaluates to 1. For $m>2$ our lower bound exceeds $n/2$, beating the best lower bound provable using the general polynomial method [Theorem 4.3, Beals et al., J. ACM 48(4), 2001].
We study broadcasting on multiple-access channels under adversarial packet injection. Leaky-bucket adversaries model packet injection. There is a fixed set of stations attached to a channel. Additional constrains on the model include bounds on the number of stations activated at a round, individual injection rates, and randomness in generating and injecting packets. Broadcast algorithms that we concentrate on are deterministic and distributed. We demonstrate that some broadcast algorithms designed for ad-hoc channels have bounded latency for wider ranges of injection rates when executed on channels with a fixed number of stations against adversaries that can activate at most one station per round. Individual injection rates are shown to impact latency, as compared to the model of general leaky bucket adversaries. Outcomes of experiments are given that compare the performance of broadcast algorithms against randomized adversaries. The experiments include randomized backoff algorithms.
We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.
The group testing problem consists of determining a small set of defective items from a larger set of items based on a number of possibly-noisy tests, and is relevant in applications such as medical testing, communication protocols, pattern matching, and more. We study the noisy version of this problem, where the outcome of each standard noiseless group test is subject to independent noise, corresponding to passing the noiseless result through a binary channel. We introduce a class of algorithms that we refer to as Near-Definite Defectives (NDD), and study bounds on the required number of tests for asymptotically vanishing error probability under Bernoulli random test designs. In addition, we study algorithm-independent converse results, giving lower bounds on the required number of tests under Bernoulli test designs. Under reverse Z-channel noise, the achievable rates and converse results match in a broad range of sparsity regimes, and under Z-channel noise, the two match in a narrower range of dense/low-noise regimes. We observe that although these two channels have the same Shannon capacity when viewed as a communication channel, they can behave quite differently when it comes to group testing. Finally, we extend our analysis of these noise models to a general binary noise model (including symmetric noise), and show improvements over known existing bounds in broad scaling regimes.
The one-fifth success rule is one of the best-known and most widely accepted techniques to control the parameters of evolutionary algorithms. While it is often applied in the literal sense, a common interpretation sees the one-fifth success rule as a family of success-based updated rules that are determined by an update strength $F$ and a success rate. We analyze in this work how the performance of the (1+1) Evolutionary Algorithm on LeadingOnes depends on these two hyper-parameters. Our main result shows that the best performance is obtained for small update strengths $F=1+o(1)$ and success rate $1/e$. We also prove that the running time obtained by this parameter setting is, apart from lower order terms, the same that is achieved with the best fitness-dependent mutation rate. We show similar results for the resampling variant of the (1+1) Evolutionary Algorithm, which enforces to flip at least one bit per iteration.
We present a randomized $O(m \log^2 n)$ work, $O(\text{polylog } n)$ depth parallel algorithm for minimum cut. This algorithm matches the work bounds of a recent sequential algorithm by Gawrychowski, Mozes, and Weimann [ICALP'20], and improves on the previously best parallel algorithm by Geissmann and Gianinazzi [SPAA'18], which performs $O(m \log^4 n)$ work in $O(\text{polylog } n)$ depth. Our algorithm makes use of three components that might be of independent interest. Firstly, we design a parallel data structure that efficiently supports batched mixed queries and updates on trees. It generalizes and improves the work bounds of a previous data structure of Geissmann and Gianinazzi and is work efficient with respect to the best sequential algorithm. Secondly, we design a parallel algorithm for approximate minimum cut that improves on previous results by Karger and Motwani. We use this algorithm to give a work-efficient procedure to produce a tree packing, as in Karger's sequential algorithm for minimum cuts. Lastly, we design an efficient parallel algorithm for solving the minimum $2$-respecting cut problem.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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