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This paper is a collection of results on combinatorial properties of codes for the Z-channel. A Z-channel with error fraction $\tau$ takes as input a length-$n$ binary codeword and injects in an adversarial manner up to $n\tau$ asymmetric errors, i.e., errors that only zero out bits but do not flip $0$'s to $1$'s. It is known that the largest $(L-1)$-list-decodable code for the Z-channel with error fraction $\tau$ has exponential size (in $n$) if $\tau$ is less than a critical value that we call the $(L-1)$-list-decoding Plotkin point and has constant size if $\tau$ is larger than the threshold. The $(L-1)$-list-decoding Plotkin point is known to be $ L^{-\frac{1}{L-1}} - L^{-\frac{L}{L-1}} $, which equals $1/4$ for unique-decoding with $ L-1=1 $. In this paper, we derive various results for the size of the largest codes above and below the list-decoding Plotkin point. In particular, we show that the largest $(L-1)$-list-decodable code $\epsilon$-above the Plotkin point, {for any given sufficiently small positive constant $ \epsilon>0 $,} has size $\Theta_L(\epsilon^{-3/2})$ for any $L-1\ge1$. We also devise upper and lower bounds on the exponential size of codes below the list-decoding Plotkin point.

Federated learning (FL) is an emerging paradigm to train model with distributed data from numerous Internet of Things (IoT) devices. It inherently assumes a uniform capacity among participants. However, due to different conditions such as differing energy budgets or executing parallel unrelated tasks, participants have diverse computational resources in practice. Participants with insufficient computation budgets must plan for the use of restricted computational resources appropriately, otherwise they would be unable to complete the entire training procedure, resulting in model performance decline. To address this issue, we propose a strategy for estimating local models without computationally intensive iterations. Based on it, we propose Computationally Customized Federated Averaging (CC-FedAvg), which allows participants to determine whether to perform traditional local training or model estimation in each round based on their current computational budgets. Both theoretical analysis and exhaustive experiments indicate that CC-FedAvg has the same convergence rate and comparable performance as FedAvg without resource constraints. Furthermore, CC-FedAvg can be viewed as a computation-efficient version of FedAvg that retains model performance while considerably lowering computation overhead.

We derive general bounds on the probability that the empirical first-passage time $\overline{\tau}_n\equiv \sum_{i=1}^n\tau_i/n$ of a reversible ergodic Markov process inferred from a sample of $n$ independent realizations deviates from the true mean first-passage time by more than any given amount in either direction. We construct non-asymptotic confidence intervals that hold in the elusive small-sample regime and thus fill the gap between asymptotic methods and the Bayesian approach that is known to be sensitive to prior belief and tends to underestimate uncertainty in the small-sample setting. We prove sharp bounds on extreme first-passage times that control uncertainty even in cases where the mean alone does not sufficiently characterize the statistics. Our concentration-of-measure-based results allow for model-free error control and reliable error estimation in kinetic inference, and are thus important for the analysis of experimental and simulation data in the presence of limited sampling.

Ongoing advances in force field and computer hardware development enable the use of molecular dynamics (MD) to simulate increasingly complex systems with the ultimate goal of reaching cellular complexity. At the same time, rational design by high-throughput (HT) simulations is another forefront of MD. In these areas, the Martini coarse-grained force field, especially the latest version (i.e. v3), is being actively explored because it offers enhanced spatial-temporal resolution. However, the automation tools for preparing simulations with the Martini force field, accompanying the previous version, were not designed for HT simulations or studies of complex cellular systems. Therefore, they become a major limiting factor. To address these shortcomings, we present the open-source Vermouth python library. Vermouth is designed to become the unified framework for developing programs, which prepare, run, and analyze Martini simulations of complex systems. To demonstrate the power of the Vermouth library, the Martinize2 program is showcased as a generalization of the martinize script, originally aimed to set up simulations of proteins. In contrast to the previous version, Martinize2 automatically handles protonation states in proteins and post-translation modifications, offers more options to fine-tune structural biases such as the elastic network, and can convert non-protein molecules such as ligands. Finally, Martinize2 is used in two high-complexity benchmarks. The entire I-TASSER protein template database as well as a subset of 200,000 structures from the AlphaFold Protein Structure Database are converted to CG resolution and we illustrate how the checks on input structure quality can safeguard high-throughput applications.

The common cause principle for two random variables $A$ and $B$ is examined in the case of causal insufficiency, when their common cause $C$ is known to exist, but only the joint probability of $A$ and $B$ is observed. As a result, $C$ cannot be uniquely identified (the latent confounder problem). We show that the generalized maximum likelihood method can be applied to this situation and allows identification of $C$ that is consistent with the common cause principle. It closely relates to the maximum entropy principle. Investigation of the two binary symmetric variables reveals a non-analytic behavior of conditional probabilities reminiscent of a second-order phase transition. This occurs during the transition from correlation to anti-correlation in the observed probability distribution. The relation between the generalized likelihood approach and alternative methods, such as predictive likelihood and the minimum common cause entropy, is discussed. The consideration of the common cause for three observed variables (and one hidden cause) uncovers causal structures that defy representation through directed acyclic graphs with the Markov condition.

Given the high incidence of cardio and cerebrovascular diseases (CVD), and its association with morbidity and mortality, its prevention is a major public health issue. A high level of blood pressure is a well-known risk factor for these events and an increasing number of studies suggest that blood pressure variability may also be an independent risk factor. However, these studies suffer from significant methodological weaknesses. In this work we propose a new location-scale joint model for the repeated measures of a marker and competing events. This joint model combines a mixed model including a subject-specific and time-dependent residual variance modeled through random effects, and cause-specific proportional intensity models for the competing events. The risk of events may depend simultaneously on the current value of the variance, as well as, the current value and the current slope of the marker trajectory. The model is estimated by maximizing the likelihood function using the Marquardt-Levenberg algorithm. The estimation procedure is implemented in a R-package and is validated through a simulation study. This model is applied to study the association between blood pressure variability and the risk of CVD and death from other causes. Using data from a large clinical trial on the secondary prevention of stroke, we find that the current individual variability of blood pressure is associated with the risk of CVD and death. Moreover, the comparison with a model without heterogeneous variance shows the importance of taking into account this variability in the goodness-of-fit and for dynamic predictions.

In this work, drawing inspiration from the type of noise present in real hardware, we study the output distribution of random quantum circuits under practical non-unital noise sources with constant noise rates. We show that even in the presence of unital sources like the depolarizing channel, the distribution, under the combined noise channel, never resembles a maximally entropic distribution at any depth. To show this, we prove that the output distribution of such circuits never anticoncentrates $\unicode{x2014}$ meaning it is never too "flat" $\unicode{x2014}$ regardless of the depth of the circuit. This is in stark contrast to the behavior of noiseless random quantum circuits or those with only unital noise, both of which anticoncentrate at sufficiently large depths. As consequences, our results have interesting algorithmic implications on both the hardness and easiness of noisy random circuit sampling, since anticoncentration is a critical property exploited by both state-of-the-art classical hardness and easiness results.

We consider a decision maker allocating one unit of renewable and divisible resource in each period on a number of arms. The arms have unknown and random rewards whose means are proportional to the allocated resource and whose variances are proportional to an order $b$ of the allocated resource. In particular, if the decision maker allocates resource $A_i$ to arm $i$ in a period, then the reward $Y_i$ is$Y_i(A_i)=A_i \mu_i+A_i^b \xi_{i}$, where $\mu_i$ is the unknown mean and the noise $\xi_{i}$ is independent and sub-Gaussian. When the order $b$ ranges from 0 to 1, the framework smoothly bridges the standard stochastic multi-armed bandit and online learning with full feedback. We design two algorithms that attain the optimal gap-dependent and gap-independent regret bounds for $b\in [0,1]$, and demonstrate a phase transition at $b=1/2$. The theoretical results hinge on a novel concentration inequality we have developed that bounds a linear combination of sub-Gaussian random variables whose weights are fractional, adapted to the filtration, and monotonic.

We study causal inference and efficient estimation for the expected number of recurrent events in the presence of a terminal event. We define our estimand as the vector comprising both the expected number of recurrent events and the failure survival function evaluated along a sequence of landmark times. We identify the estimand in the presence of right-censoring and causal selection as an observed data functional under coarsening at random, derive the nonparametric efficiency bound, and propose a multiply-robust estimator that achieves the bound and permits nonparametric estimation of nuisance parameters. Throughout, no absolute continuity assumption is made on the underlying probability distributions of failure, censoring, or the observed data. Additionally, we derive the class of influence functions when the coarsening distribution is known and review how published estimators may belong to the class. Along the way, we highlight some interesting inconsistencies in the causal lifetime analysis literature.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.