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The majority of fault-tolerant distributed algorithms are designed assuming a nominal corruption model, in which at most a fraction $f_n$ of parties can be corrupted by the adversary. However, due to the infamous Sybil attack, nominal models are not sufficient to express the trust assumptions in open (i.e., permissionless) settings. Instead, permissionless systems typically operate in a weighted model, where each participant is associated with a weight and the adversary can corrupt a set of parties holding at most a fraction $f_w$ of total weight. In this paper, we suggest a simple way to transform a large class of protocols designed for the nominal model into the weighted model. To this end, we formalize and solve three novel optimization problems, which we collectively call the weight reduction problems, that allow us to map large real weights into small integer weights while preserving the properties necessary for the correctness of the protocols. In all cases, we manage to keep the sum of the integer weights to be at most linear in the number of parties, resulting in extremely efficient protocols for the weighted model. Moreover, we demonstrate that, on weight distributions that emerge in practice, the sum of the integer weights tends to be far from the theoretical worst-case and, often even smaller than the number of participants. While, for some protocols, our transformation requires an arbitrarily small reduction in resilience (i.e., $f_w = f_n - \epsilon$), surprisingly, for many important problems we manage to obtain weighted solutions with the same resilience ($f_w = f_n$) as nominal ones. Notable examples include asynchronous consensus, verifiable secret sharing, erasure-coded distributed storage and broadcast protocols.

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

Motivated by Carbon Emissions Trading Schemes, Treasury Auctions, and Procurement Auctions, which all involve the auctioning of homogeneous multiple units, we consider the problem of learning how to bid in repeated multi-unit pay-as-bid auctions. In each of these auctions, a large number of (identical) items are to be allocated to the largest submitted bids, where the price of each of the winning bids is equal to the bid itself. The problem of learning how to bid in pay-as-bid auctions is challenging due to the combinatorial nature of the action space. We overcome this challenge by focusing on the offline setting, where the bidder optimizes their vector of bids while only having access to the past submitted bids by other bidders. We show that the optimal solution to the offline problem can be obtained using a polynomial time dynamic programming (DP) scheme. We leverage the structure of the DP scheme to design online learning algorithms with polynomial time and space complexity under full information and bandit feedback settings. We achieve an upper bound on regret of $O(M\sqrt{T\log |\mathcal{B}|})$ and $O(M\sqrt{|\mathcal{B}|T\log |\mathcal{B}|})$ respectively, where $M$ is the number of units demanded by the bidder, $T$ is the total number of auctions, and $|\mathcal{B}|$ is the size of the discretized bid space. We accompany these results with a regret lower bound, which match the linear dependency in $M$. Our numerical results suggest that when all agents behave according to our proposed no regret learning algorithms, the resulting market dynamics mainly converge to a welfare maximizing equilibrium where bidders submit uniform bids. Lastly, our experiments demonstrate that the pay-as-bid auction consistently generates significantly higher revenue compared to its popular alternative, the uniform price auction.

We use a combination of unsupervised clustering and sparsity-promoting inference algorithms to learn locally dominant force balances that explain macroscopic pattern formation in self-organized active particle systems. The self-organized emergence of macroscopic patterns from microscopic interactions between self-propelled particles can be widely observed nature. Although hydrodynamic theories help us better understand the physical basis of this phenomenon, identifying a sufficient set of local interactions that shape, regulate, and sustain self-organized structures in active particle systems remains challenging. We investigate a classic hydrodynamic model of self-propelled particles that produces a wide variety of patterns, like asters and moving density bands. Our data-driven analysis shows that propagating bands are formed by local alignment interactions driven by density gradients, while steady-state asters are shaped by a mechanism of splay-induced negative compressibility arising from strong particle interactions. Our method also reveals analogous physical principles of pattern formation in a system where the speed of the particle is influenced by local density. This demonstrates the ability of our method to reveal physical commonalities across models. The physical mechanisms inferred from the data are in excellent agreement with analytical scaling arguments and experimental observations.

In this work, we explore a framework for contextual decision-making to study how the relevance and quantity of past data affects the performance of a data-driven policy. We analyze a contextual Newsvendor problem in which a decision-maker needs to trade-off between an underage and an overage cost in the face of uncertain demand. We consider a setting in which past demands observed under ``close by'' contexts come from close by distributions and analyze the performance of data-driven algorithms through a notion of context-dependent worst-case expected regret. We analyze the broad class of Weighted Empirical Risk Minimization (WERM) policies which weigh past data according to their similarity in the contextual space. This class includes classical policies such as ERM, k-Nearest Neighbors and kernel-based policies. Our main methodological contribution is to characterize exactly the worst-case regret of any WERM policy on any given configuration of contexts. To the best of our knowledge, this provides the first understanding of tight performance guarantees in any contextual decision-making problem, with past literature focusing on upper bounds via concentration inequalities. We instead take an optimization approach, and isolate a structure in the Newsvendor loss function that allows to reduce the infinite-dimensional optimization problem over worst-case distributions to a simple line search. This in turn allows us to unveil fundamental insights that were obfuscated by previous general-purpose bounds. We characterize actual guaranteed performance as a function of the contexts, as well as granular insights on the learning curve of algorithms.

The CONGEST and CONGEST-CLIQUE models have been carefully studied to represent situations where the communication bandwidth between processors in a network is severely limited. Messages of only $O(log(n))$ bits of information each may be sent between processors in each round. The quantum versions of these models allow the processors instead to communicate and compute with quantum bits under the same bandwidth limitations. This leads to the following natural research question: What problems can be solved more efficiently in these quantum models than in the classical ones? Building on existing work, we contribute to this question in two ways. Firstly, we present two algorithms in the Quantum CONGEST-CLIQUE model of distributed computation that succeed with high probability; one for producing an approximately optimal Steiner Tree, and one for producing an exact directed minimum spanning tree, each of which uses $\tilde{O}(n^{1/4})$ rounds of communication and $\tilde{O}(n^{9/4})$ messages, where $n$ is the number of nodes in the network. The algorithms thus achieve a lower asymptotic round and message complexity than any known algorithms in the classical CONGEST-CLIQUE model. At a high level, we achieve these results by combining classical algorithmic frameworks with quantum subroutines. An existing framework for using distributed version of Grover's search algorithm to accelerate triangle finding lies at the core of the asymptotic speedup. Secondly, we carefully characterize the constants and logarithmic factors involved in our algorithms as well as related algorithms, otherwise commonly obscured by $\tilde{O}$ notation. The analysis shows that some improvements are needed to render both our and existing related quantum and classical algorithms practical, as their asymptotic speedups only help for very large values of $n$.

We study the decay of correlation between locally constrained independent random variables in the local lemma regimes. the distribution defined by constraint satisfaction problems (CSPs) in the local lemma regime. For atomically constrained independent random variables of sufficiently large domains, we show that a decay of correlation property holds up to the local lemma condition $pD^{2+o(1)}\lesssim 1$, asymptotically matching the sampling threshold for constraint satisfaction solutions [BGG+19,GGW22]. This provides evidence for the conjectured $pD^2\lesssim 1$ threshold for the "sampling Lov\'{a}sz local lemma". We use a recursively-constructed coupling to bound the correlation decay. Our approach completely dispenses with the "freezing" paradigm originated from Beck [Bec91], which was commonly used to deal with the non-self-reducibility of the local lemma regimes, and hence can bypass the current technical barriers due to the use of $\{2,3\}$-trees.

To date, the only way to argue polynomial lower bounds for dynamic algorithms is via fine-grained complexity arguments. These arguments rely on strong assumptions about specific problems such as the Strong Exponential Time Hypothesis (SETH) and the Online Matrix-Vector Multiplication Conjecture (OMv). While they have led to many exciting discoveries, dynamic algorithms still miss out some benefits and lessons from the traditional ``coarse-grained'' approach that relates together classes of problems such as P and NP. In this paper we initiate the study of coarse-grained complexity theory for dynamic algorithms. Below are among questions that this theory can answer. What if dynamic Orthogonal Vector (OV) is easy in the cell-probe model? A research program for proving polynomial unconditional lower bounds for dynamic OV in the cell-probe model is motivated by the fact that many conditional lower bounds can be shown via reductions from the dynamic OV problem. Since the cell-probe model is more powerful than word RAM and has historically allowed smaller upper bounds, it might turn out that dynamic OV is easy in the cell-probe model, making this research direction infeasible. Our theory implies that if this is the case, there will be very interesting algorithmic consequences: If dynamic OV can be maintained in polylogarithmic worst-case update time in the cell-probe model, then so are several important dynamic problems such as $k$-edge connectivity, $(1+\epsilon)$-approximate mincut, $(1+\epsilon)$-approximate matching, planar nearest neighbors, Chan's subset union and 3-vs-4 diameter. The same conclusion can be made when we replace dynamic OV by, e.g., subgraph connectivity, single source reachability, Chan's subset union, and 3-vs-4 diameter. Lower bounds for $k$-edge connectivity via dynamic OV? (see the full abstract in the pdf file).

The growing demand for accurate control in varying and unknown environments has sparked a corresponding increase in the requirements for power supply components, including permanent magnet synchronous motors (PMSMs). To infer the unknown part of the system, machine learning techniques are widely employed, especially Gaussian process regression (GPR) due to its flexibility of continuous system modeling and its guaranteed performance. For practical implementation, distributed GPR is adopted to alleviate the high computational complexity. However, the study of distributed GPR from a control perspective remains an open problem. In this paper, a control-aware optimal aggregation strategy of distributed GPR for PMSMs is proposed based on the Lyapunov stability theory. This strategy exclusively leverages the posterior mean, thereby obviating the need for computationally intensive calculations associated with posterior variance in alternative approaches. Moreover, the straightforward calculation process of our proposed strategy lends itself to seamless implementation in high-frequency PMSM control. The effectiveness of the proposed strategy is demonstrated in the simulations.

Machine learning models often need to be robust to noisy input data. The effect of real-world noise (which is often random) on model predictions is captured by a model's local robustness, i.e., the consistency of model predictions in a local region around an input. However, the na\"ive approach to computing local robustness based on Monte-Carlo sampling is statistically inefficient, leading to prohibitive computational costs for large-scale applications. In this work, we develop the first analytical estimators to efficiently compute local robustness of multi-class discriminative models using local linear function approximation and the multivariate Normal CDF. Through the derivation of these estimators, we show how local robustness is connected to concepts such as randomized smoothing and softmax probability. We also confirm empirically that these estimators accurately and efficiently compute the local robustness of standard deep learning models. In addition, we demonstrate these estimators' usefulness for various tasks involving local robustness, such as measuring robustness bias and identifying examples that are vulnerable to noise perturbation in a dataset. By developing these analytical estimators, this work not only advances conceptual understanding of local robustness, but also makes its computation practical, enabling the use of local robustness in critical downstream applications.