arXiv subjects

Ashok Vardhan Makkuva

Publications and source records attributed to Ashok Vardhan Makkuva.

17 recordsLinked to original sources

arXiv2026

From Markov to Laplace: How Mamba In-Context Learns Markov Chains

While transformer-based language models have driven the AI revolution thus far, their computational complexity has spurred growing interest in viable alternatives, such as structured state space sequence models (SSMs) and Selective SSMs. Among these, Mamba (S6) and its variant Mamba-2 have shown remarkable inference speed-ups over transformers while achieving comparable or superior performance on complex language modeling tasks. However, despite these architectural innovations and empirical successes, the fundamental learning capabilities of Mamba remain poorly understood. In this paper, we address this gap by studying in-context learning (ICL) on Markov chains and uncovering an interesting phenomenon: even a single-layer Mamba efficiently learns the in-context Laplacian smoothing estimator, which is both Bayes and minimax optimal. To explain this, we theoretically characterize the representation capacity of Mamba and reveal the fundamental role of convolution in enabling it to represent the optimal Laplacian smoothing. These theoretical insights align strongly with empirical results and, to the best of our knowledge, represent the first formal connection between Mamba and optimal statistical estimators. Finally, we outline promising research directions inspired by these findings.

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arXiv2026

TERMINATOR: Learning Optimal Exit Points for Early Stopping in Chain-of-Thought Reasoning

Large Reasoning Models (LRMs) achieve impressive performance on complex reasoning tasks via Chain-of-Thought (CoT) reasoning, which enables them to generate intermediate thinking tokens before arriving at the final answer. However, LRMs often suffer from significant overthinking, spending excessive compute time even after the answer is generated early on. Prior work has identified the existence of an optimal reasoning length such that truncating reasoning at this point significantly shortens CoT outputs with virtually no change in performance. However, determining optimal CoT lengths for practical datasets is highly non-trivial as they are fully task and model-dependent. In this paper, we precisely address this and design Terminator, an early-exit strategy for LRMs at inference to mitigate overthinking. The central idea underpinning Terminator is that the first arrival of an LRM's final answer is often predictable, and we leverage these first answer positions to create a novel dataset of optimal reasoning lengths to train Terminator. Powered by this approach, Terminator achieves significant reductions in CoT lengths of 14%-55% on average across four challenging practical datasets: MATH-500, AIME 2025, HumanEval, and GPQA, while outperforming current state-of-the-art methods and reducing inference latency by more than 2x compared to the original LRM.

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arXiv2025

What One Cannot, Two Can: Two-Layer Transformers Provably Represent Induction Heads on Any-Order Markov Chains

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 kth-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.

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arXiv2025

Attention with Markov: A Framework for Principled Analysis of Transformers via Markov Chains

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. Code is available at https://github.com/Bond1995/Markov .

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arXiv2024

Fundamental Limits of Prompt Compression: A Rate-Distortion Framework for Black-Box Language Models

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.

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arXiv2024

Transformers on Markov Data: Constant Depth Suffices

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 \kth 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 \kth 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, our main result is that a transformer with a single head and three layers can represent the in-context conditional empirical distribution for \kth Markov sources, concurring with our empirical observations. Along the way, we prove that \textit{attention-only} transformers 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. These results provide more insight into our current understanding of the mechanisms by which transformers learn to capture context, by understanding their behavior on Markov sources.

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arXiv2024

Local to Global: Learning Dynamics and Effect of Initialization for Transformers

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 transformer parameters and demonstrate their effectiveness. Finally, we outline several open problems in this arena. Code is available at: https://github.com/Bond1995/Markov.

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arXiv2024

LASER: Linear Compression in Wireless Distributed Optimization

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: LineAr CompreSsion in WirEless DistRibuted 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.

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arXiv2023

Machine Learning-Aided Efficient Decoding of Reed-Muller Subcodes

Reed-Muller (RM) codes achieve the capacity of general binary-input memoryless symmetric channels and are conjectured to have a comparable performance to that of random codes in terms of scaling laws. However, such results are established assuming maximum-likelihood decoders for general code parameters. Also, RM codes only admit limited sets of rates. Efficient decoders such as successive cancellation list (SCL) decoder and recently-introduced recursive projection-aggregation (RPA) decoders are available for RM codes at finite lengths. In this paper, we focus on subcodes of RM codes with flexible rates. We first extend the RPA decoding algorithm to RM subcodes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different approaches to prune the projections. Next, we derive the soft-decision based version of our algorithm, called soft-subRPA, that not only improves upon the performance of subRPA but also enables a differentiable decoding algorithm. Building upon the soft-subRPA algorithm, we then provide a framework for training a machine learning (ML) model to search for \textit{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 smaller number of projections. We also show that the choice of the projections in decoding RM subcodes matters significantly, and our ML-aided projection pruning scheme is able to find a \textit{good} selection, i.e., with negligible performance degradation compared to the full-projection case, given a reasonable number of projections.

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arXiv2023

CRISP: Curriculum based Sequential Neural Decoders for Polar Code Family

Polar codes are widely used state-of-the-art codes for reliable communication that have recently been included in the 5th generation wireless standards (5G). However, there remains room for the 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 Polar(32,16) and 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 PAC(32,16) code.

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arXiv2021

KO codes: Inventing Nonlinear Encoding and Decoding for Reliable Wireless Communication via Deep-learning

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 computationaly 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 real 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. The code is available at \href{https://github.com/deepcomm/KOcodes}{https://github.com/deepcomm/KOcodes}

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arXiv2021

Reed-Muller Subcodes: Machine Learning-Aided Design of Efficient Soft Recursive Decoding

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 in this paper, 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 \textit{good} sets of projections in the sense of minimizing 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.

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arXiv2020

Learning in Gated Neural Networks

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.

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arXiv2020

Optimal transport mapping via input convex neural networks

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 where the gradient of one convex function represents the optimal transport mapping. Numerical experiments confirm that we learn the optimal transport mapping. This approach ensures that the transport mapping we find is optimal independent of how we initialize the neural networks. Further, target distributions from a discontinuous support can be easily captured, as gradient of a convex function naturally models a {\em discontinuous} transport mapping.

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arXiv2019

Breaking the gridlock in Mixture-of-Experts: Consistent and Efficient Algorithms

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 paramters 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.

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arXiv2019

Learning One-hidden-layer Neural Networks under General Input Distributions

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 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.

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arXiv2016

Equivalence of additive-combinatorial linear inequalities for Shannon entropy and differential entropy

This paper addresses the correspondence between linear inequalities of 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 of 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 \cite{KM14} can be deduced from their discrete counterparts due to Tao \cite{Tao10} in a unified manner. Generalizations to certain abelian groups are also obtained. Our proof of extending inequalities of Shannon entropy to differential entropy relies on a result of Rényi \cite{Renyi59} 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; the converse uses the asymptotics of the differential entropy of convolutions with weak additive noise.

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