Work on \emph{optimal} protocols for \emph{Eventual Byzantine Agreement} (EBA) -- protocols that, in a precise sense, decide as soon as possible in every run and guarantee that all nonfaulty agents decide on the same value -- has focused on emph{full-information protocols} (FIPs), where agents repeatedly send messages that completely describe their past observations to every other agent. While it can be shown that, without loss of generality, we can take an optimal protocol to be an FIP, full information exchange is impractical to implement for many applications due to the required message size. We separate protocols into two parts, the \emph{information-exchange protocol} and the \emph{action protocol}, so as to be able to examine the effects of more limited information exchange. We then define a notion of optimality with respect to an information-exchange protocol. Roughly speaking, an action protocol $P$ is optimal with respect to an information-exchange protocol $\mathcal{E}$ if, with $P$, agents decide as soon as possible among action protocols that exchange information according to $\mathcal{E}$. We present a knowledge-based EBA program for omission failures all of whose implementations are guaranteed to be correct and are optimal if the information exchange satisfies a certain safety condition. We then construct concrete programs that implement this knowledge-based program in two settings of interest that are shown to satisfy the safety condition. Finally, we show that a small modification of our program results in an FIP that is both optimal and efficiently implementable, settling an open problem posed by Halpern, Moses, and Waarts (SIAM J. Comput., 2001).
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Importance sampling is a central idea underlying off-policy prediction in reinforcement learning. It provides a strategy for re-weighting samples from a distribution to obtain unbiased estimates under another distribution. However, importance sampling weights tend to exhibit extreme variance, often leading to stability issues in practice. In this work, we consider a broader class of importance weights to correct samples in off-policy learning. We propose the use of $\textit{value-aware importance weights}$ which take into account the sample space to provide lower variance, but still unbiased, estimates under a target distribution. We derive how such weights can be computed, and detail key properties of the resulting importance weights. We then extend several reinforcement learning prediction algorithms to the off-policy setting with these weights, and evaluate them empirically.
Byzantine fault-tolerant (BFT) systems are able to maintain the availability and integrity of IoT systems, in presence of failure of individual components, random data corruption or malicious attacks. Fault-tolerant systems in general are essential in assuring continuity of service for mission critical applications. However, their implementation may be challenging and expensive. In this study, IoT Systems with Byzantine Fault-Tolerance are considered. Analytical models and solutions are presented as well as a detailed analysis for the evaluation of the availability. Byzantine Fault Tolerance is particularly important for blockchain mechanisms, and in turn for IoT, since it can provide a secure, reliable and decentralized infrastructure for IoT devices to communicate and transact with each other. The proposed model is based on continuous-time Markov chains, and it analyses the availability of Byzantine Fault-Tolerant systems. While the availability model is based on a continuous-time Markov chain where the breakdown and repair times follow exponential distributions, the number of the Byzantine nodes in the network studied follows various distributions. The numerical results presented report availability as a function of the number of participants and the relative number of honest actors in the system. It can be concluded from the model that there is a non-linear relationship between the number of servers and network availability; i.e. the availability is inversely proportional to the number of nodes in the system. This relationship is further strengthened as the ratio of break-down rate over repair rate increases.
Design activity -- constructing an artifact description satisfying given goals and constraints -- distinguishes humanity from other animals and traditional machines, and endowing machines with design abilities at the human level or beyond has been a long-term pursuit. Though machines have already demonstrated their abilities in designing new materials, proteins, and computer programs with advanced artificial intelligence (AI) techniques, the search space for designing such objects is relatively small, and thus, "Can machines design like humans?" remains an open question. To explore the boundary of machine design, here we present a new AI approach to automatically design a central processing unit (CPU), the brain of a computer, and one of the world's most intricate devices humanity have ever designed. This approach generates the circuit logic, which is represented by a graph structure called Binary Speculation Diagram (BSD), of the CPU design from only external input-output observations instead of formal program code. During the generation of BSD, Monte Carlo-based expansion and the distance of Boolean functions are used to guarantee accuracy and efficiency, respectively. By efficiently exploring a search space of unprecedented size 10^{10^{540}}, which is the largest one of all machine-designed objects to our best knowledge, and thus pushing the limits of machine design, our approach generates an industrial-scale RISC-V CPU within only 5 hours. The taped-out CPU successfully runs the Linux operating system and performs comparably against the human-designed Intel 80486SX CPU. In addition to learning the world's first CPU only from input-output observations, which may reform the semiconductor industry by significantly reducing the design cycle, our approach even autonomously discovers human knowledge of the von Neumann architecture.
We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch from which the effective resistance between any pair of nodes can be estimated, to $(1 \pm \epsilon)$-multiplicative accuracy, in $\widetilde{O}(1)$-time. Consequently, we obtain an $\widetilde{O}(m\epsilon^{-1})$-time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of $\widetilde{O}(m\epsilon^{-3/2})$ [Chu et. al. 2018] and $\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018] for general graphs and $\widetilde{O}(m + n\epsilon^{-2})$ for expanders [Li, Sachdeva 2022]. We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require $\widetilde{\Omega}(n^2 \epsilon^{-1/2})$-time, improving upon the previous best such lower bound of $\widetilde{\Omega}(n^2 \epsilon^{-1/13})$ [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.
In this work we investigate whether it is plausible to use the performance of a reinforcement learning (RL) agent to estimate the difficulty measured as the player completion rate of different levels in the mobile puzzle game Lily's Garden.For this purpose we train an RL agent and measure the number of moves required to complete a level. This is then compared to the level completion rate of a large sample of real players.We find that the strongest predictor of player completion rate for a level is the number of moves taken to complete a level of the ~5% best runs of the agent on a given level. A very interesting observation is that, while in absolute terms, the agent is unable to reach human-level performance across all levels, the differences in terms of behaviour between levels are highly correlated to the differences in human behaviour. Thus, despite performing sub-par, it is still possible to use the performance of the agent to estimate, and perhaps further model, player metrics.
We consider deterministic distributed algorithms for reaching agreement in synchronous networks of arbitrary topologies. Links are bi-directional and prone to failures while nodes stay non-faulty at all times. A faulty link may omit messages. Agreement among nodes is understood as holding in each connected component of a network obtained by removing faulty links. We call ``disconnected agreement'' the algorithmic problem of reaching such agreement. We introduce the concept of stretch, which is the number of connected components of a network, obtained by removing faulty links, minus~$1$ plus the sum of diameters of connected components. We define the concepts of ``fast'' and ``early-stopping'' algorithms for disconnected agreement by referring to stretch. A network has $n$ nodes and $m$ links. Nodes are normally assumed to know their own names and ability to associate communication with local ports. If we additionally assume that a bound~$\Lambda$ on stretch is known to all nodes, then there is an algorithm for disconnected agreement working in time $O(\Lambda)$ using messages of $O(\log n)$ bits. We give a general disconnected agreement algorithm operating in~$n+1$ rounds that uses messages of $O(\log n)$ bits. Let~$\lambda$ be an unknown stretch occurring in an execution; we give an algorithm working in time~$(\lambda+2)^3$ and using messages of $O(n\log n)$ bits. We show that disconnected agreement can be solved in the optimal $O(\lambda)$ time, but at the cost of increasing message size to~$O(m\log n)$. We also design an algorithm that uses only~$O(n)$ non-faulty links and works in time~$O(n m)$, while nodes start with their ports mapped to neighbors and messages carry $O(m\log n)$ bits. We prove lower bounds on the performance of disconnected-agreement solutions that refer to the parameters of evolving network topologies and the knowledge available to nodes.
Offering a viable alternative architecture to centrally-controlled global digital platforms for social networking is an open challenge. Here we present a grassroots architecture for serverless, permissionless, peer-to-peer social networks termed grassroots social networking. The architecture is geared for roaming (address-changing) agents communicating over an unreliable network, e.g., smartphones communicating via UDP. The architecture incorporates (i) a decentralized social graph, where each member controls, maintains and stores only their local neighbourhood in the graph; (ii) member-created feeds, with authors and followers; and (iii) a novel grassroots dissemination protocol, in which communication occurs only along the edges of the social graph. The architecture realizes these components using the blocklace data structure -- a distributed partially-ordered counterpart of the replicated totally-ordered blockchain. We provide two example grassroots social networking protocols -- Twitter/LinkedIn-like and WhatsApp-like -- and address their safety, liveness, privacy, and spam/deep-fake resistance, demonstrating how centrally-controlled social networks could be supplanted by a grassroots architecture.
For a given function $F$ from $\mathbb F_{p^n}$ to itself, determining whether there exists a function which is CCZ-equivalent but EA-inequivalent to $F$ is a very important and interesting problem. For example, K\"olsch \cite{KOL21} showed that there is no function which is CCZ-equivalent but EA-inequivalent to the inverse function. On the other hand, for the cases of Gold function $F(x)=x^{2^i+1}$ and $F(x)=x^3+{\rm Tr}(x^9)$ over $\mathbb F_{2^n}$, Budaghyan, Carlet and Pott (respectively, Budaghyan, Carlet and Leander) \cite{BCP06, BCL09FFTA} found functions which are CCZ-equivalent but EA-inequivalent to $F$. In this paper, when a given function $F$ has a component function which has a linear structure, we present functions which are CCZ-equivalent to $F$, and if suitable conditions are satisfied, the constructed functions are shown to be EA-inequivalent to $F$. As a consequence, for every quadratic function $F$ on $\mathbb F_{2^n}$ ($n\geq 4$) with nonlinearity $>0$ and differential uniformity $\leq 2^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to $F$. Also for every non-planar quadratic function on $\mathbb F_{p^n}$ $(p>2, n\geq 4)$ with $|\mathcal W_F|\leq p^{n-1}$ and differential uniformity $\leq p^{n-3}$, we explicitly construct functions which are CCZ-equivalent but EA-inequivalent to $F$.
This paper is concerned with the optimal allocation of detection resources (sensors) to mitigate multi-stage attacks, in the presence of the defender's uncertainty in the attacker's intention. We model the attack planning problem using a Markov decision process and characterize the uncertainty in the attacker's intention using a finite set of reward functions -- each reward represents a type of the attacker. Based on this modeling framework, we employ the paradigm of the worst-case absolute regret minimization from robust game theory and develop mixed-integer linear program (MILP) formulations for solving the worst-case regret minimizing sensor allocation strategies for two classes of attack-defend interactions: one where the defender and attacker engage in a zero-sum game, and another where they engage in a non-zero-sum game. We demonstrate the effectiveness of our framework using a stochastic gridworld example.
A core capability of intelligent systems is the ability to quickly learn new tasks by drawing on prior experience. Gradient (or optimization) based meta-learning has recently emerged as an effective approach for few-shot learning. In this formulation, meta-parameters are learned in the outer loop, while task-specific models are learned in the inner-loop, by using only a small amount of data from the current task. A key challenge in scaling these approaches is the need to differentiate through the inner loop learning process, which can impose considerable computational and memory burdens. By drawing upon implicit differentiation, we develop the implicit MAML algorithm, which depends only on the solution to the inner level optimization and not the path taken by the inner loop optimizer. This effectively decouples the meta-gradient computation from the choice of inner loop optimizer. As a result, our approach is agnostic to the choice of inner loop optimizer and can gracefully handle many gradient steps without vanishing gradients or memory constraints. Theoretically, we prove that implicit MAML can compute accurate meta-gradients with a memory footprint that is, up to small constant factors, no more than that which is required to compute a single inner loop gradient and at no overall increase in the total computational cost. Experimentally, we show that these benefits of implicit MAML translate into empirical gains on few-shot image recognition benchmarks.
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