Search

The Nullstellensatz, proved by Hilbert in 1893, is a classical result that holds when the base field is algebraically closed. When the base field is finite, a version of Hilbert's Nullstellensatz is given by Terjanian in 1966. Laksov in 1987 generalized Hilbert's Nullstellensatz to a $K$-Nullstellensatz when the base field $K$ is not necessarily algebraically closed. However, unlike Tarjanian's Nullstellensatz, Laksov's Nullstellensatz is not very explicit. Later, Laksov and Westin in 1990 proposed a strengthening to Laksov's Nullstellensatz in the form of four conjectures. A projective analogue of Nullstellensatz of the classical Nullstellensatz of Hilbert is well-known for projective varieties over algebraically closed fields. For finite fields, the projective analogue of the Nullstellensatz can be derived as an application of Hilbert's Nullstellensatz, though it is not as efficient as Terjanian's Nullstellensatz. Gimenez, Ruano and San-Jos\'e in 2023 strengthened this result by identifying a computationally efficient set. Here, we introduce an even more efficient set that establishes the projective analogue of the Nullstellensatz for finite fields. Additionally, we provide counterexamples to three of the conjectures of Laksov and Westin., Comment: 15 pages

We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$ of finite degree. To obtain our results we will utilize well-known connections between these weights and properties of the Stanley-Reisner rings of a series of matroids associated to each code $C_q$. In the process, we are able to explicitly determine all the graded Betti numbers of matroids associated to $C_q$ and its elongations., Comment: 44 pages, 3 appendices, 25 tables

One of the main bottlenecks of blockchains is smart contract execution. To increase throughput, modern blockchains try to execute transactions in parallel. Unfortunately, however, common blockchain use cases introduce read-write conflicts between transactions, forcing sequentiality. We propose RapidLane, an extension for parallel execution engines that allows the engine to capture computations in conflicting parts of transactions and defer their execution until a later time, sometimes optimistically predicting execution results. This technique, coupled with support for a new construct for smart contract languages, allows one to turn certain sequential workloads into parallelizable ones. We integrated RapidLane into Block-STM, a state-of-the-art parallel execution engine used by several blockchains in production, and deployed it on the Aptos blockchain. Our evaluation shows that on commonly contended workloads, such as peer-to-peer transfers with a single fee payer and NFT minting, RapidLane yields up to $12\times$ more throughput.

Reward functions are central in specifying the task we want a reinforcement learning agent to perform. Given a task and desired optimal behavior, we study the problem of designing informative reward functions so that the designed rewards speed up the agent's convergence. In particular, we consider expert-driven reward design settings where an expert or teacher seeks to provide informative and interpretable rewards to a learning agent. Existing works have considered several different reward design formulations; however, the key challenge is formulating a reward informativeness criterion that adapts w.r.t. the agent's current policy and can be optimized under specified structural constraints to obtain interpretable rewards. In this paper, we propose a novel reward informativeness criterion, a quantitative measure that captures how the agent's current policy will improve if it receives rewards from a specific reward function. We theoretically showcase the utility of the proposed informativeness criterion for adaptively designing rewards for an agent. Experimental results on two navigation tasks demonstrate the effectiveness of our adaptive reward informativeness criterion., Comment: Longer version of the AAMAS'24 paper

mRNA translation is a crucial process that leads to protein synthesis in living cells. Therefore, it is a process that needs to work optimally for a cell to stay healthy and alive. With advancements in microscopy and novel experimental techniques, a lot of the intricate details about the translation mechanism are now known. However, the why and how of this mechanism are still ill understood, and therefore, is an active area of research. Theoretical studies of mRNA translation typically view it in terms of the Totally Asymmetric Simple Exclusion Process or TASEP. Various works have used the TASEP model in order to study a wide range of phenomena and factors affecting translation, such as ribosome traffic on an mRNA under noisy (codon-dependent or otherwise) conditions, ribosome stalling, premature termination, ribosome reinitiation and dropoff, codon-dependent elongation and competition among mRNA for ribosomes, among others. In this review, we relay the history and physics of the translation process in terms of the TASEP framework. In particular, we discuss the viability and evolution of this model and its limitations while also formulating the reasons behind its success. Finally, we also identify gaps in the existing literature and suggest possible extensions and applications that will lead to a better understanding of ribosome traffic on the mRNA., Comment: 39 pages, 5 figures, review article

We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is then used to determine the number of minimum weight codewords of a projective Reed-Muller code. Various formulas for the dimension of a projective Reed-Muller code, and their equivalences are also discussed., Comment: 24 pages; to appear in Adv. Math. Commun.; some typos corrected and a reference added in this version

The Narwhal system is a state-of-the-art Byzantine fault-tolerant scalable architecture that involves constructing a directed acyclic graph (DAG) of messages among a set of validators in a Blockchain network. Bullshark is a zero-overhead consensus protocol on top of the Narwhal's DAG that can order over 100k transactions per second. Unfortunately, the high throughput of Bullshark comes with a latency price due to the DAG construction, increasing the latency compared to the state-of-the-art leader-based BFT consensus protocols. We introduce Shoal, a protocol-agnostic framework for enhancing Narwhal-based consensus. By incorporating leader reputation and pipelining support for the first time, Shoal significantly reduces latency. Moreover, the combination of properties of the DAG construction and the leader reputation mechanism enables the elimination of timeouts in all but extremely uncommon scenarios in practice, a property we name Prevalent Responsiveness" (it strictly subsumes the established and often desired Optimistic Responsiveness property for BFT protocols). We integrated Shoal instantiated with Bullshark, the fastest existing Narwhal-based consensus protocol, in an open-source Blockchain project and provide experimental evaluations demonstrating up to 40% latency reduction in the failure-free executions, and up-to 80% reduction in executions with failures against the vanilla Bullshark implementation.

Text-based games are a popular testbed for language-based reinforcement learning (RL). In previous work, deep Q-learning is commonly used as the learning agent. Q-learning algorithms are challenging to apply to complex real-world domains due to, for example, their instability in training. Therefore, in this paper, we adapt the soft-actor-critic (SAC) algorithm to the text-based environment. To deal with sparse extrinsic rewards from the environment, we combine it with a potential-based reward shaping technique to provide more informative (dense) reward signals to the RL agent. We apply our method to play difficult text-based games. The SAC method achieves higher scores than the Q-learning methods on many games with only half the number of training steps. This shows that it is well-suited for text-based games. Moreover, we show that the reward shaping technique helps the agent to learn the policy faster and achieve higher scores. In particular, we consider a dynamically learned value function as a potential function for shaping the learner's original sparse reward signals.

Block-STM is a parallel execution engine for smart contracts, built around the principles of Software Transactional Memory. Transactions are grouped in blocks, and every execution of the block must yield the same deterministic outcome. Block-STM further enforces that the outcome is consistent with executing transactions according to a preset order, leveraging this order to dynamically detect dependencies and avoid conflicts during speculative transaction execution. At the core of Block-STM is a novel, low-overhead collaborative scheduler of execution and validation tasks. Block-STM is implemented on the main branch of the Diem Blockchain code-base and runs in production at Aptos. Our evaluation demonstrates that Block-STM is adaptive to workloads with different conflict rates and utilizes the inherent parallelism therein. Block-STM achieves up to $110k$ tps in the Diem benchmarks and up to $170k$ tps in the Aptos Benchmarks, which is a $20$x and $17$x improvement over the sequential baseline with $32$ threads, respectively. The throughput on a contended workload is up to $50k$ tps and $80k$ tps in Diem and Aptos benchmarks, respectively.

Advances in blockchains have influenced the State-Machine-Replication (SMR) world and many state-of-the-art blockchain-SMR solutions are based on two pillars: Chaining and Leader-rotation. A predetermined round-robin mechanism used for Leader-rotation, however, has an undesirable behavior: crashed parties become designated leaders infinitely often, slowing down overall system performance. In this paper, we provide a new Leader-Aware SMR framework that, among other desirable properties, formalizes a Leader-utilization requirement that bounds the number of rounds whose leaders are faulty in crash-only executions. We introduce Carousel, a novel, reputation-based Leader-rotation solution to achieve Leader-Aware SMR. The challenge in adaptive Leader-rotation is that it cannot rely on consensus to determine a leader, since consensus itself needs a leader. Carousel uses the available on-chain information to determine a leader locally and achieves Liveness despite this difficulty. A HotStuff implementation fitted with Carousel demonstrates drastic performance improvements: it increases throughput over 2x in faultless settings and provided a 20x throughput increase and 5x latency reduction in the presence of faults.

This paper gives tight logarithmic lower bounds on the solo step complexity of leader election in an asynchronous shared-memory model with single-writer multi-reader (SWMR) registers, for randomized obstruction-free algorithms. The approach extends to lower bounds for randomized obstruction-free algorithms using multi-writer registers under bounded write concurrency, showing a trade-off between the solo step complexity of a leader election algorithm, and the worst-case contention incurred by a processor in an execution.

Existing committee-based Byzantine state machine replication (SMR) protocols, typically deployed in production blockchains, face a clear trade-off: (1) they either achieve linear communication cost in the happy path, but sacrifice liveness during periods of asynchrony, or (2) they are robust (progress with probability one) but pay quadratic communication cost. We believe this trade-off is unwarranted since existing linear protocols still have asymptotic quadratic cost in the worst case. We design Ditto, a Byzantine SMR protocol that enjoys the best of both worlds: optimal communication on and off the happy path (linear and quadratic, respectively) and progress guarantee under asynchrony and DDoS attacks. We achieve this by replacing the view-synchronization of partially synchronous protocols with an asynchronous fallback mechanism at no extra asymptotic cost. Specifically, we start from HotStuff, a state-of-the-art linear protocol, and gradually build Ditto. As a separate contribution and an intermediate step, we design a 2-chain version of HotStuff, Jolteon, which leverages a quadratic view-change mechanism to reduce the latency of the standard 3-chain HotStuff. We implement and experimentally evaluate all our systems. Notably, Jolteon's commit latency outperforms HotStuff by 200-300ms with varying system size. Additionally, Ditto adapts to the network and provides better performance than Jolteon under faulty conditions and better performance than VABA (a state-of-the-art asynchronous protocol) under faultless conditions. This proves our case that breaking the robustness-efficiency trade-off is in the realm of practicality., Comment: arXiv admin note: text overlap with arXiv:2103.03181

We consider the problem of teaching via demonstrations in sequential decision-making settings. In particular, we study how to design a personalized curriculum over demonstrations to speed up the learner's convergence. We provide a unified curriculum strategy for two popular learner models: Maximum Causal Entropy Inverse Reinforcement Learning (MaxEnt-IRL) and Cross-Entropy Behavioral Cloning (CrossEnt-BC). Our unified strategy induces a ranking over demonstrations based on a notion of difficulty scores computed w.r.t. the teacher's optimal policy and the learner's current policy. Compared to the state of the art, our strategy doesn't require access to the learner's internal dynamics and still enjoys similar convergence guarantees under mild technical conditions. Furthermore, we adapt our curriculum strategy to the setting where no teacher agent is present using task-specific difficulty scores. Experiments on a synthetic car driving environment and navigation-based environments demonstrate the effectiveness of our curriculum strategy., Comment: NeurIPS 2021

The popularity of permissioned blockchain systems demands BFT SMR protocols that are efficient under good network conditions (synchrony) and robust under bad network conditions (asynchrony). The state-of-the-art partially synchronous BFT SMR protocols provide optimal linear communication cost per decision under synchrony and good leaders, but lose liveness under asynchrony. On the other hand, the state-of-the-art asynchronous BFT SMR protocols are live even under asynchrony, but always pay quadratic cost even under synchrony. In this paper, we propose a BFT SMR protocol that achieves the best of both worlds -- optimal linear cost per decision under good networks and leaders, optimal quadratic cost per decision under bad networks, and remains always live.

Biological processes at the cellular level are stochastic in nature, and the immune response system is no different. Therefore, models that attempt to explain this system need to also incorporate noise or fluctuations that can account for the observed variability. In this work, a stochastic model of the immune response system is presented in terms of the dynamics of the T cells and the virus particles. Making use of the Green's function and the Wilemski-Fixman approximation, this model is then solved to obtain the analytical expression for the joint probability density function of these variables in the early and late stages of infection. This is then also used to calculate the average level of virus particles in the system. Upon comparing the theoretically predicted average virus levels to those of COVID-19 patients, it is hypothesized that the long lived dynamics that are characteristic of such viral infections are due to the long range correlations in the temporal fluctuations of the virions. This model therefore provides an insight into the effects of noise on viral dynamics.

Let $G$ be a graph on $n$ nodes. In the stochastic population protocol model, a collection of $n$ indistinguishable, resource-limited nodes collectively solve tasks via pairwise interactions. In each interaction, two randomly chosen neighbors first read each other's states, and then update their local states. A rich line of research has established tight upper and lower bounds on the complexity of fundamental tasks, such as majority and leader election, in this model, when $G$ is a clique. Specifically, in the clique, these tasks can be solved fast, i.e., in $n \operatorname{polylog} n$ pairwise interactions, with high probability, using at most $\operatorname{polylog} n$ states per node. In this work, we consider the more general setting where $G$ is an arbitrary graph, and present a technique for simulating protocols designed for fully-connected networks in any connected regular graph. Our main result is a simulation that is efficient on many interesting graph families: roughly, the simulation overhead is polylogarithmic in the number of nodes, and quadratic in the conductance of the graph. As a sample application, we show that, in any regular graph with conductance $\phi$, both leader election and exact majority can be solved in $\phi^{-2} \cdot n \operatorname{polylog} n$ pairwise interactions, with high probability, using at most $\phi^{-2} \cdot \operatorname{polylog} n$ states per node. This shows that there are fast and space-efficient population protocols for leader election and exact majority on graphs with good expansion properties. We believe our results will prove generally useful, as they allow efficient technology transfer between the well-mixed (clique) case, and the under-explored spatial setting., Comment: 42 pages, 5 figures

Following Johnsen and Verdure (2013), we can associate to any linear code $C$ an abstract simplicial complex and in turn, a Stanley-Reisner ring $R_C$. The ring $R_C$ is a standard graded algebra over a field and its projective dimension is precisely the dimension of $C$. Thus $R_C$ admits a graded minimal free resolution and the resulting graded Betti numbers are known to determine the generalized Hamming weights of $C$. The question of purity of the minimal free resolution of $R_C$ was considered by Ghorpade and Singh (2020) when $C$ is the generalized Reed-Muller code. They showed that the resolution is pure in some cases and it is not pure in many other cases. Here we give a complete characterization of the purity of graded minimal free resolutions of Stanley-Reisner rings associated to generalized Reed-Muller codes of an arbitrary order., Comment: 8 pages

We study a security threat to reinforcement learning where an attacker poisons the learning environment to force the agent into executing a target policy chosen by the attacker. As a victim, we consider RL agents whose objective is to find a policy that maximizes reward in infinite-horizon problem settings. The attacker can manipulate the rewards and the transition dynamics in the learning environment at training-time, and is interested in doing so in a stealthy manner. We propose an optimization framework for finding an optimal stealthy attack for different measures of attack cost. We provide lower/upper bounds on the attack cost, and instantiate our attacks in two settings: (i) an offline setting where the agent is doing planning in the poisoned environment, and (ii) an online setting where the agent is learning a policy with poisoned feedback. Our results show that the attacker can easily succeed in teaching any target policy to the victim under mild conditions and highlight a significant security threat to reinforcement learning agents in practice., Comment: Journal version of ICML'20 paper. New theoretical results for jointly poisoning rewards and transitions

The advent of Deep Learning models like VGG-16 and Resnet-50 has considerably revolutionized the field of image classification, and by using these Convolutional Neural Networks (CNN) architectures, one can get a high classification accuracy on a wide variety of image datasets. However, these Deep Learning models have a very high computational complexity and so incur a high computational cost of running these algorithms as well as make it hard to interpret the results. In this paper, we present a shallow CNN architecture which gives the same classification accuracy as the VGG-16 and Resnet-50 models for thin blood smear RBC slide images for detection of malaria, while decreasing the computational run time by an order of magnitude. This can offer a significant advantage for commercial deployment of these algorithms, especially in poorer countries in Africa and some parts of the Indian subcontinent, where the menace of malaria is quite severe., Comment: 8 pages, 4 figures, 1 table

One of the central challenges faced by a reinforcement learning (RL) agent is to effectively learn a (near-)optimal policy in environments with large state spaces having sparse and noisy feedback signals. In real-world applications, an expert with additional domain knowledge can help in speeding up the learning process via \emph{shaping the environment}, i.e., making the environment more learner-friendly. A popular paradigm in literature is \emph{potential-based reward shaping}, where the environment's reward function is augmented with additional local rewards using a potential function. However, the applicability of potential-based reward shaping is limited in settings where (i) the state space is very large, and it is challenging to compute an appropriate potential function, (ii) the feedback signals are noisy, and even with shaped rewards the agent could be trapped in local optima, and (iii) changing the rewards alone is not sufficient, and effective shaping requires changing the dynamics. We address these limitations of potential-based shaping methods and propose a novel framework of \emph{environment shaping using state abstraction}. Our key idea is to compress the environment's large state space with noisy signals to an abstracted space, and to use this abstraction in creating smoother and more effective feedback signals for the agent. We study the theoretical underpinnings of our abstraction-based environment shaping, and show that the agent's policy learnt in the shaped environment preserves near-optimal behavior in the original environment.

We study a security threat to reinforcement learning where an attacker poisons the learning environment to force the agent into executing a target policy chosen by the attacker. As a victim, we consider RL agents whose objective is to find a policy that maximizes average reward in undiscounted infinite-horizon problem settings. The attacker can manipulate the rewards or the transition dynamics in the learning environment at training-time and is interested in doing so in a stealthy manner. We propose an optimization framework for finding an \emph{optimal stealthy attack} for different measures of attack cost. We provide sufficient technical conditions under which the attack is feasible and provide lower/upper bounds on the attack cost. We instantiate our attacks in two settings: (i) an \emph{offline} setting where the agent is doing planning in the poisoned environment, and (ii) an \emph{online} setting where the agent is learning a policy using a regret-minimization framework with poisoned feedback. Our results show that the attacker can easily succeed in teaching any target policy to the victim under mild conditions and highlight a significant security threat to reinforcement learning agents in practice., Comment: ICML 2020

Machine teaching studies the interaction between a teacher and a student/learner where the teacher selects training examples for the learner to learn a specific task. The typical assumption is that the teacher has perfect knowledge of the task---this knowledge comprises knowing the desired learning target, having the exact task representation used by the learner, and knowing the parameters capturing the learning dynamics of the learner. Inspired by real-world applications of machine teaching in education, we consider the setting where teacher's knowledge is limited and noisy, and the key research question we study is the following: When does a teacher succeed or fail in effectively teaching a learner using its imperfect knowledge? We answer this question by showing connections to how imperfect knowledge affects the teacher's solution of the corresponding machine teaching problem when constructing optimal teaching sets. Our results have important implications for designing robust teaching algorithms for real-world applications.

Fast convolutions via transforms, either Winograd or FFT, had emerged as a preferred way of performing the computation of convolutional layers, as it greatly reduces the number of required operations. Recent work shows that, for many layer structures, a well--designed implementation of fast convolutions can greatly utilize modern CPUs, significantly reducing the compute time. However, the generous amount of shared L3 cache present on modern CPUs is often neglected, and the algorithms are optimized solely for the private L2 cache. In this paper we propose an efficient `L3 Fusion` algorithm that is specifically designed for CPUs with significant amount of shared L3 cache. Using the hierarchical roofline model, we show that in many cases, especially for layers with fewer channels, the `L3 fused` approach can greatly outperform standard 3 stage one provided by big vendors such as Intel. We validate our theoretical findings, by benchmarking our `L3 fused` implementation against publicly available state of the art.