The Inference Report

We study the problem of lossless compression of source code, motivated by the storage demands of large-scale software archives, such as Software Heritage (https://www.softwareheritage.org/). General-purpose compressors (e.g., zstd, bzip2) offer a good trade-off between compression ratio and speed, but fail to exploit all special regularities inherent in source code. Recent approaches leverage Large Language Models (LLMs) within Shannon's symbol-ranking framework, relying on a scheme in which the predicted rank can grow arbitrarily. While effective at reducing space, this setting incurs significant throughput degradation, and leaves open the question whether it is necessary to explicitly encode all ranks. In this work, we introduce LLM-based compressors deploying two novel symbol-ranking variants that bound predictions to the top-$T$ ranks ($T=1$ or $63$), with out-of-threshold symbols stored as exceptions and compressed jointly with the rank stream via general-purpose compressors. We conduct the first large-scale evaluation of LLM-based source code compression across 30 LLMs, including general-domain, code-specialized, and quantized models. Our $T$-bounded approach outperforms prior LLM-based compressors both in compression ratio (up to 37% relative improvement) and compression throughput (40% faster). Compared to general-purpose compressors (e.g., zstd, bzip2), we obtain up to 82% relative compression gain but at a lower speed, thus offering a new trade-off point in the compression-speed spectrum. We also show that these gains are stronger on source code than on natural language, suggesting an interesting indication, namely that source code exposes regularities captured by LLMs but missed by general-purpose exact-match-based compressors. We conclude by commenting on open problems that offer theoretical and practical avenues of research.

In large-scale machine learning, distributed training commonly involves multiple workers evaluating the gradients of the model on different dataset partitions. A common challenge is the presence of straggling workers, which may significantly slow down training. Traditional gradient coding (GC) addresses this by duplicating dataset partitions across workers, allowing for the replacement of missing gradients from stragglers. However, GC requires workers to evaluate gradients on multiple dataset partitions in each step, potentially increasing overall training time. In this paper, we propose to pipeline GC, such that gradient evaluation is segmented across multiple steps and each worker evaluates gradients on just a single dataset partition per step. We develop the pipelined version for fractional repetition (FR) and cyclic repetition (CR), two representative dataset placement schemes in GC, and prove convergence guarantees for both. Through extensive simulations and experiments on cloud infrastructure, our schemes not only significantly reduce training time but also accelerate convergence compared to GC and other baselines.

The Shannon capacity $Θ(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $α(G^d)^{1/d}$ for any $d$, where $α(G^d)$ is the independence number of the $d$-th strong power of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, and $62530$ in $C_{13}^{6}$, improving the best known lower bounds for the Shannon capacity of these graphs to $Θ(C_7)\geq 134753^{1/10}>3.258020$, $Θ(C_{11})\geq 21909^{1/6}>5.289773$, and $Θ(C_{13})\geq 62530^{1/6}>6.300109$. We also improve the best known lower bounds on the independence numbers of several individual strong powers of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

In this paper, we study an unmanned aerial vehicle (UAV)-assisted integrated sensing and communication (ISAC) system, where a UAV enhances the sensing capability of a base station (BS) towards a target while ensuring reliable communication towards a downlink user. This architecture is practically attractive for future wireless networks due to the UAV's controllable mobility and adaptive sensing coverage in wireless environments. The sensing performance is characterized by the average Cramér-Rao bound (CRB), which quantifies the minimum variance of the unbiased angle-of-arrival estimation. To enhance the sensing performance, the UAV trajectory and beamforming parameters are jointly optimized under power and mobility constraints, while satisfying communication requirements to the downlink user. To address the resulting non-convex problem, we employ null-space projection for beamforming design and adopt deep reinforcement learning for the trajectory optimization over a discrete-time scale. In each time slot, beamforming is optimized based on the channel state information to improve CRB performance while mitigating interference between the BS and the communication user. Simulation results demonstrate that the proposed method significantly reduces the time-averaged CRB by over 10%, compared with the ISAC system without UAV assistance, and also achieves a higher sensing accuracy than both the fixed-UAV-trajectory and the maximum-ratio-transmission-based beamforming benchmarks.

Estimating entropy from samples is fundamental in information theory and property testing. Shannon entropy measures average uncertainty and can be estimated to constant additive accuracy over a $k$-symbol alphabet using $Θ(k/\log k)$ samples. Min-entropy depends only on the most likely symbol. Both are special cases of order-$α$ R'{e}nyi entropy, $H_α$. We characterize the sample complexity of estimating min-entropy and R'{e}nyi entropy for $k$ and integer $α>1$; our lower bounds also hold for noninteger $α\ge1.001$. We prove that min-entropy estimation to constant additive accuracy has sample complexity $Θ(k\log k)$. The upper bound uses the largest empirical frequency and concentration via dyadic grouping. The matching lower bound hides a slightly heavier symbol at a uniformly random location. Thus, min-entropy requires $Θ(\log^2 k)$ more samples than Shannon entropy and corrects a previously stated $Θ(k/\log k)$ characterization. For every integer $2\leα\le c_0\log k$, we prove the matching fixed-accuracy bound $Θ_{c_0}(αk^{1-1/α})$. Previous results gave $Ω_α(k^{1-1/α})$ for fixed integer $α>1$ and $O_{c_0}(α^2k^{1-1/α})$ for all integer $α>1$. Our upper bound analyzes an unbiased falling-factorial estimator based on $α$-way collisions, while a hidden-heavy-coordinate construction gives the matching lower bound and shows that the factor $α$ is unavoidable. For every real $1.001\leα\le c_0\log k$, we prove the uniform lower bound $Ω_{c_0}(αk^{1-1/α})$. Finally, since $0\le H_α(p)-H_\infty(p)\le\log k/(α-1)$, min-entropy uniformly approximates $H_α$ when $α$ is a sufficiently large multiple of $\log k$. Combining this reduction with our min-entropy bounds gives $Θ_\varepsilon(k\log k)$ sample complexity in the high-order regime.

Classical rate-distortion (RD) theory has long established the fundamental limits of lossy compression by quantifying the minimum number of bits required to represent a source under a prescribed distortion constraint. However, widely used distortion measures such as mean-squared error often fail to capture perceptual quality or semantic validity, which are increasingly central in modern learning-driven applications. Rate-distortion-perception (RDP) theory extends the RD framework by introducing perception as a third fundamental axis, quantified via distributional similarity between the source and reconstructed signals, leading to the rate-distortion-perception function (RDPF). This tutorial provides a structured overview of the coding principles underlying perception-aware lossy compression and surveys recent achievability results under different randomness assumptions. It then presents a unifying optimization viewpoint for computing the RDPF as defined by Blau and Michaeli, for both discrete and continuous sources under broad families of perceptual constraints, including f-divergences, alpha-divergences, and Wasserstein-based metrics. Special attention is given to computational tools such as alternating minimization schemes, Newton-based methods, and convex optimization formulations, as well as to analytically tractable cases such as Gaussian sources and the perfect-realism regime. Unlike recent broad surveys that emphasize generative architectures and AI-empowered communication systems, this tutorial focuses on the coding-theoretic and computational machinery needed to characterize, compute, and interpret the RDP limits. Finally, the tutorial outlines promising research directions at the intersection of information theory, neural compression, robust source coding, and perception-aware networked control systems.