Attention-based transformers have achieved tremendous success across a variety of disciplines including natural languages. To deepen our understanding of their sequential modeling capabilities, there is a growing interest in using Markov input processes to study them. A key finding is that when trained on first-order Markov chains, transformers with two or more layers consistently develop an induction head mechanism to estimate the in-context bigram conditional distribution. In contrast, single-layer transformers, unable to form an induction head, directly learn the Markov kernel but often face a surprising challenge: they become trapped in local minima representing the unigram distribution, whereas deeper models reliably converge to the ground-truth bigram. While single-layer transformers can theoretically model first-order Markov chains, their empirical failure to learn this simple kernel in practice remains a curious phenomenon. To explain this contrasting behavior of single-layer models, in this paper we introduce a new framework for a principled analysis of transformers via Markov chains. Leveraging our framework, we theoretically characterize the loss landscape of single-layer transformers and show the existence of global minima (bigram) and bad local minima (unigram) contingent on data properties and model architecture. We precisely delineate the regimes under which these local optima occur. Backed by experiments, we demonstrate that our theoretical findings are in congruence with the empirical results. Finally, we outline several open problems in this arena.

In-context learning (ICL) is a hallmark capability of transformers, through which trained models learn to adapt to new tasks by leveraging information from the input context. Prior work has shown that ICL emerges in transformers due to the presence of special circuits called induction heads. Given the equivalence between induction heads and conditional $k$-grams, a recent line of work modeling sequential inputs as Markov processes has revealed the fundamental impact of model depth on its ICL capabilities: while a two-layer transformer can efficiently represent a conditional $1$-gram model, its single-layer counterpart cannot solve the task unless it is exponentially large. However, for higher order Markov sources, the best known constructions require at least three layers (each with a single attention head) - leaving open the question: *can a two-layer single-head transformer represent any $k^{\text{th}}$-order Markov process?* In this paper, we precisely address this and theoretically show that a two-layer transformer with one head per layer can indeed represent any conditional $k$-gram. Thus, our result provides the tightest known characterization of the interplay between transformer depth and Markov order for ICL. Building on this, we further analyze the learning dynamics of our two-layer construction, focusing on a simplified variant for first-order Markov chains, illustrating how effective in-context representations emerge during training. Together, these results deepen our current understanding of transformer-based ICL and illustrate how even shallow architectures can surprisingly exhibit strong ICL capabilities on structured sequence modeling tasks.

Data-parallel SGD is the de facto algorithm for distributed optimization, especially for large scale machine learning. Despite its merits, communication bottleneck is one of its persistent issues. Most compression schemes to alleviate this either assume noiseless communication links, or fail to achieve good performance on practical tasks. In this paper, we close this gap and introduce **LASER**: **L**ine**A**r Compre**S**sion in Wir**E**less Dist**R**ibuted Optimization. LASER capitalizes on the inherent low-rank structure of gradients and transmits them efficiently over the noisy channels. Whilst enjoying theoretical guarantees similar to those of the classical SGD, LASER shows consistent gains over baselines on a variety of practical benchmarks. In particular, it outperforms the state-of-the-art compression schemes on challenging computer vision and GPT language modeling tasks. On the latter, we obtain 50-64% improvement in perplexity over our baselines for noisy channels.

In recent years, transformer-based models have revolutionized deep learning, particularly in sequence modeling. To better understand this phenomenon, there is a growing interest in using Markov input processes to study transformers. However, our current understanding in this regard remains limited with many fundamental questions about how transformers learn Markov chains still unanswered. In this paper, we address this by focusing on first-order Markov chains and single-layer transformers, providing a comprehensive characterization of the learning dynamics in this context. Specifically, we prove that transformer parameters trained on next-token prediction loss can either converge to global or local minima, contingent on the initialization and the Markovian data properties, and we characterize the precise conditions under which this occurs. To the best of our knowledge, this is the first result of its kind highlighting the role of initialization. We further demonstrate that our theoretical findings are corroborated by empirical evidence. Based on these insights, we provide guidelines for the initialization of single-layer transformers and demonstrate their effectiveness. Finally, we outline several open problems in this arena. Code is available at: \url{https://github.com/Bond1995/Markov}.

We formalize the problem of prompt compression for large language models (LLMs) and present a framework to unify token-level prompt compression methods which create hard prompts for black-box models. We derive the distortion-rate function for this setup as a linear program, and provide an efficient algorithm to compute this fundamental limit via the dual of the linear program. Using the distortion-rate function as the baseline, we study the performance of existing compression schemes on a synthetic dataset consisting of prompts generated from a Markov chain, natural language queries, and their respective answers. Our empirical analysis demonstrates the criticality of query-aware prompt compression, where the compressor has knowledge of the downstream task/query for the black-box LLM. We show that there is a large gap between the performance of current prompt compression methods and the optimal strategy, and propose Adaptive QuerySelect, a query-aware, variable-rate adaptation of a prior work to close the gap. We extend our experiments to a small natural language dataset to further confirm our findings on our synthetic dataset.

Attention-based transformers have been remarkably successful at modeling generative processes across various domains and modalities. In this paper, we study the behavior of transformers on data drawn from $k^{\text{th}}$-order Markov processes, where the conditional distribution of the next symbol in a sequence depends on the previous $k$ symbols observed. We observe a surprising phenomenon empirically which contradicts previous findings: when trained for sufficiently long, a transformer with a fixed depth and $1$ head per layer is able to achieve low test loss on sequences drawn from $k^{\text{th}}$-order Markov sources, even as $k$ grows. Furthermore, this low test loss is achieved by the transformer’s ability to represent and learn the in-context conditional empirical distribution. On the theoretical side, we prove that a transformer with $O(\log_2(k))$ layers can represent the in-context conditional empirical distribution by composing induction heads to track the previous $k$ symbols in the sequence. Surprisingly, with the addition of layer normalization, we show that a transformer with a constant number of layers can represent the in-context conditional empirical distribution, concurring with our empirical observations. This result provides more insight into the benefit of soft-attention and non-linearities in the transformer architecture.

Polar codes are widely used state-of-the-art codes for reliable communication that have recently been included in the $5^{\text{th}}$ generation wireless standards ($5$G). However, there still remains room for design of polar decoders that are both efficient and reliable in the short blocklength regime. Motivated by recent successes of data-driven channel decoders, we introduce a novel $\textbf{ C}$ur${\textbf{RI}}$culum based $\textbf{S}$equential neural decoder for $\textbf{P}$olar codes (CRISP). We design a principled curriculum, guided by information-theoretic insights, to train CRISP and show that it outperforms the successive-cancellation (SC) decoder and attains near-optimal reliability performance on the $\text{Polar}(32,16)$ and $\text{Polar}(64,22)$ codes. The choice of the proposed curriculum is critical in achieving the accuracy gains of CRISP, as we show by comparing against other curricula. More notably, CRISP can be readily extended to Polarization-Adjusted-Convolutional (PAC) codes, where existing SC decoders are significantly less reliable. To the best of our knowledge, CRISP constructs the first data-driven decoder for PAC codes and attains near-optimal performance on the $\text{PAC}(32,16)$ code.

In this paper, we introduce a neural-augmented decoder for Turbo codes called TINYTURBO . TINYTURBO has complexity comparable to the classical max-log-MAP algorithm but has much better reliability than the max-log-MAP baseline and performs close to the MAP algorithm. We show that TINYTURBO exhibits strong robustness on a variety of practical channels of interest, such as EPA and EVA channels, which are included in the LTE standards. We also show that TINYTURBO strongly generalizes across different rate, blocklengths, and trellises. We verify the reliability and efficiency of TINYTURBO via over-the-air experiments.

Landmark codes underpin reliable physical layer communication, e.g., Reed-Muller, BCH, Convolution, Turbo, LDPC, and Polar codes: each is a linear code and represents a mathematical breakthrough. The impact on humanity is huge: each of these codes has been used in global wireless communication standards (satellite, WiFi, cellular). Reliability of communication over the classical additive white Gaussian noise (AWGN) channel enables benchmarking and ranking of the different codes. In this paper, we construct KO codes, a computationally efficient family of deep-learning driven (encoder, decoder) pairs that outperform the state-of-the-art reliability performance on the standardized AWGN channel. KO codes beat state-of-the-art Reed-Muller and Polar codes, under the low-complexity successive cancellation decoding, in the challenging short-to-medium block length regime on the AWGN channel. We show that the gains of KO codes are primarily due to the nonlinear mapping of information bits directly to transmit symbols (bypassing modulation) and yet possess an efficient, high-performance decoder. The key technical innovation that renders this possible is design of a novel family of neural architectures inspired by the computation tree of the {\bf K}ronecker {\bf O}peration (KO) central to Reed-Muller and Polar codes. These architectures pave way for the discovery of a much richer class of hitherto unexplored nonlinear algebraic structures.

Reed-Muller (RM) codes are conjectured to achieve the capacity of any binary-input memoryless symmetric (BMS) channel, and are observed to have a comparable performance to that of random codes in terms of scaling laws. On the negative side, RM codes lack efficient decoders with performance close to that of a maximum likelihood decoder for general parameters. Also, they only admit certain discrete sets of rates. In this paper, we focus on subcodes of RM codes with flexible rates that can take any code dimension from 1 to$n$. where$n$is the blocklength. We first extend the recursive projection-aggregation (RPA) algorithm proposed recently by Ye and Abbe for decoding RM codes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different ways for pruning the projections. We then derive the soft-decision based version of our algorithm, called soft-subRPA, that is shown to improve upon the performance of subRPA. Furthermore, it enables training a machine learning (ML) model to search for good sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly reduced number of projections. For instance, our simulation results on a (64,14) RM subcode show almost identical performance for full-projection decoding and pruned-projection decoding with 15 projections picked via training our ML model. This is equivalent to lowering the complexity by a factor of more than 4 without sacrificing the decoding performance.

Gating is a key feature in modern neural networks including LSTMs, GRUs and sparsely-gated deep neural networks. The backbone of such gated networks is a mixture-of-experts layer, where several experts make regression decisions and gating controls how to weigh the decisions in an input-dependent manner. Despite having such a prominent role in both modern and classical machine learning, very little is understood about parameter recovery of mixture-of-experts since gradient descent and EM algorithms are known to be stuck in local optima in such models.In this paper, we perform a careful analysis of the optimization landscape and show that with appropriately designed loss functions, gradient descent can indeed learn the parameters accurately. A key idea underpinning our results is the design of two {\em distinct} loss functions, one for recovering the expert parameters and another for recovering the gating parameters. We demonstrate the first sample complexity results for parameter recovery in this model for any algorithm and demonstrate significant performance gains over standard loss functions in numerical experiments.

In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel minimax optimization. Building upon recent advances in the field of input convex neural networks, we propose a new framework to estimate the optimal transport mapping as the gradient of a convex function that is trained via minimax optimization. Numerical experiments confirm the accuracy of the learned transport map. Our approach can be readily used to train a deep generative model. When trained between a simple distribution in the latent space and a target distribution, the learned optimal transport map acts as a deep generative model. Although scaling this to a large dataset is challenging, we demonstrate two important strengths over standard adversarial training: robustness and discontinuity. As we seek the optimal transport, the learned generative model provides the same mapping regardless of how we initialize the neural networks. Further, a gradient of a neural network can easily represent discontinuous mappings, unlike standard neural networks that are constrained to be continuous. This allows the learned transport map to match any target distribution with many discontinuous supports and achieve sharp boundaries.

Significant advances have been made recently on training neural networks, where the main challenge is in solving an optimization problem with abundant critical points. However, existing approaches to address this issue crucially rely on a restrictive assumption: the training data is drawn from a Gaussian distribution. In this paper, we provide a novel unified framework to design loss functions with desirable landscape properties for a wide range of general input distributions. On these loss functions, remarkably, stochastic gradient descent theoretically recovers the true parameters with \emph{global} initializations and empirically outperforms the existing approaches. Our loss function design bridges the notion of score functions with the topic of neural network optimization. Central to our approach is the task of estimating the score function from samples, which is of basic and independent interest to theoretical statistics. Traditional estimation methods (example: kernel based) fail right at the outset; we bring statistical methods of local likelihood to design a novel estimator of score functions, that provably adapts to the local geometry of the unknown density.

Mixture-of-Experts (MoE) is a widely popular model for ensemble learning and is a basic building block of highly successful modern neural networks as well as a component in Gated Recurrent Units (GRU) and Attention networks. However, present algorithms for learning MoE, including the EM algorithm and gradient descent, are known to get stuck in local optima. From a theoretical viewpoint, finding an efficient and provably consistent algorithm to learn the parameters remains a long standing open problem for more than two decades. In this paper, we introduce the first algorithm that learns the true parameters of a MoE model for a wide class of non-linearities with global consistency guarantees. While existing algorithms jointly or iteratively estimate the expert parameters and the gating parameters in the MoE, we propose a novel algorithm that breaks the deadlock and can directly estimate the expert parameters by sensing its echo in a carefully designed cross-moment tensor between the inputs and the output. Once the experts are known, the recovery of gating parameters still requires an EM algorithm; however, we show that the EM algorithm for this simplified problem, unlike the joint EM algorithm, converges to the true parameters. We empirically validate our algorithm on both the synthetic and real data sets in a variety of settings, and show superior performance to standard baselines.

A blockchain is a database of sequential events that is maintained by a distributed group of nodes. A key consensus problem in blockchains is that of determining the next block (data element) in the sequence. Many blockchains address this by electing a new node to propose each new block. The new block is (typically) appended to the tip of the proposer's local blockchain, and subsequently broadcast to the rest of the network. Without network delay (or adversarial behavior), this procedure would give a perfect chain, since each proposer would have the same view of the blockchain. A major challenge in practice is forking. Due to network delays, a proposer may not yet have the most recent block, and may therefore create a side chain that branches from the middle of the main chain. Forking reduces throughput, since only one a single main chain can survive, and all other blocks are discarded. We propose a new P2P protocol for blockchains called Barracuda, in which each proposer, prior to proposing a block, polls ℓ other nodes for their local blocktree information. Under a stochastic network model, we prove that this lightweight primitive improves throughput as if the entire network were a factor of ℓ faster. We provide guidelines on how to implement Barracuda in practice, guaranteeing robustness against several real-world factors.

This paper addresses the correspondence between linear inequalities for Shannon entropy and differential entropy for sums of independent group-valued random variables. We show that any balanced (with the sum of coefficients being zero) linear inequality for Shannon entropy holds if and only if its differential entropy counterpart also holds; moreover, any linear inequality for differential entropy must be balanced. In particular, our result shows that recently proved differential entropy inequalities by Kontoyiannis and Madiman can be deduced from their discrete counterparts due to Tao in a unified manner. Generalizations to certain abelian groups are also obtained. Our proof of extending inequalities for Shannon entropy to differential entropy relies on a result of Rényi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing that the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum; the converse uses the asymptotics of the differential entropy of convolutions with weak additive noise.

To be considered for the 2016 IEEE Jack Keil Wolf ISIT Student Paper Award. This paper addresses the question of to what extent do discrete entropy inequalities for weighted sums of independent group-valued random variables continue to hold for differential entropies. We show that all balanced affine inequalities (with the sum of coefficients being zero) of Shannon entropy extend to differential entropy; conversely, any affine inequality for differential entropy must be balanced. In particular, this result recovers recently proved differential entropy inequalities by Kontoyiannis and Madiman [1] from their discrete counterparts due to Tao [2] in a unified manner. Our proof relies on a result of Rényi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum.