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Michael Gastpar

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14 papers

2 author rows

ICLR Conference 2025 Conference Paper

Attention with Markov: A Curious Case of Single-layer Transformers

Ashok Vardhan Makkuva
Marco Bondaschi
Adway Girish
Alliot Nagle
Martin Jaggi
Hyeji Kim
Michael Gastpar

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.

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ICML Conference 2025 Conference Paper

Leveraging Sparsity for Sample-Efficient Preference Learning: A Theoretical Perspective

Yunzhen Yao
Lie He
Michael Gastpar

This paper considers the sample-efficiency of preference learning, which models and predicts human choices based on comparative judgments. The minimax optimal estimation error rate $\Theta(d/n)$ in classical estimation theory requires that the number of samples $n$ scales linearly with the dimensionality of the feature space $d$. However, the high dimensionality of the feature space and the high cost of collecting human-annotated data challenge the efficiency of traditional estimation methods. To remedy this, we leverage sparsity in the preference model and establish sharp error rates. We show that under the sparse random utility model, where the parameter of the reward function is $k$-sparse, the minimax optimal rate can be reduced to $\Theta(k/n \log(d/k))$. Furthermore, we analyze the $\ell_{1}$-regularized estimator and show that it achieves near-optimal rate under mild assumptions on the Gram matrix. Experiments on synthetic data and LLM alignment data validate our theoretical findings, showing that sparsity-aware methods significantly reduce sample complexity and improve prediction accuracy.

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NeurIPS Conference 2025 Conference Paper

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

Chanakya Ekbote
Ashok Vardhan Makkuva
Marco Bondaschi
Nived Rajaraman
Michael Gastpar
Jason Lee
Paul Liang

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.

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NeurIPS Conference 2025 Conference Paper

Which Algorithms Have Tight Generalization Bounds?

Michael Gastpar
Ido Nachum
Jonathan Shafer
Thomas Weinberger

We study which machine learning algorithms have tight generalization bounds with respect to a given collection of population distributions. Our results build on and extend the recent work of Gastpar et al. (2023). First, we present conditions that preclude the existence of tight generalization bounds. Specifically, we show that algorithms that have certain inductive biases that cause them to be unstable do not admit tight generalization bounds. Next, we show that algorithms that are sufficiently loss-stable do have tight generalization bounds. We conclude with a simple characterization that relates the existence of tight generalization bounds to the conditional variance of the algorithm's loss.

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NeurIPS Conference 2025 Conference Paper

zip2zip: Inference-Time Adaptive Tokenization via Online Compression

Saibo Geng
Nathan Ranchin
Yunzhen Yao
Maxime Peyrard
Chris Wendler
Michael Gastpar
Robert West

Tokenization efficiency plays a critical role in the performance and cost of large language models (LLMs), yet most models rely on static tokenizers optimized on general-purpose corpora. These tokenizers’ fixed vocabularies often fail to adapt to domain- or language-specific inputs, leading to longer token sequences and higher computational costs. We introduce zip2zip, a novel method for achieving context-adaptive tokenization in LLMs at inference time. Leveraging an online data compression algorithm (Lempel–Ziv–Welch), zip2zip dynamically expands its active vocabulary at inference time by continuously replacing fragmented token sequences with more compact hypertokens, which it can immediately output during generation. In doing so, the model refines its internal tokenization scheme to match the token distribution of the current context, reducing redundancy and improving representational efficiency. zip2zip consists of three key components: (1) a tokenizer based on Lempel–Ziv–Welch compression that incrementally merges co-occurring tokens into reusable hypertokens on the fly; (2) a dynamic embedding (and unembedding) layer that computes embeddings for newly formed hypertokens at runtime; and (3) a variant of autoregressive language modeling that pretrains the model to handle hypertokenized, compressed text sequences as inputs and outputs. We show that an existing LLM can be uptrained for zip2zip in 10 GPU-hours via parameter-efficient finetuning. The resulting LLM performs test-time adaptation, learning to use hypertokens in unseen contexts and reducing input and output tokens by 15–40%. Code and models are released at https: //github. com/epfl-dlab/zip2zip.

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ICLR Conference 2024 Conference Paper

Fantastic Generalization Measures are Nowhere to be Found

Michael Gastpar
Ido Nachum
Jonathan Shafer
Thomas Weinberger

We study the notion of a generalization bound being _uniformly tight_, meaning that the difference between the bound and the population loss is small for all learning algorithms and all population distributions. Numerous generalization bounds have been proposed in the literature as potential explanations for the ability of neural networks to generalize in the overparameterized setting. However, in their paper "Fantastic Generalization Measures and Where to Find Them," Jiang et al. (2020) examine more than a dozen generalization bounds, and show empirically that none of them are uniformly tight. This raises the question of whether uniformly-tight generalization bounds are at all possible in the overparameterized setting. We consider two types of generalization bounds: (1) bounds that may depend on the training set and the learned hypothesis (e.g., margin bounds). We prove mathematically that no such bound can be uniformly tight in the overparameterized setting; (2) bounds that may in addition also depend on the learning algorithm (e.g., stability bounds). For these bounds, we show a trade-off between the algorithm's performance and the bound's tightness. Namely, if the algorithm achieves good accuracy on certain distributions, then no generalization bound can be uniformly tight for it in the overparameterized setting. We explain how these formal results can, in our view, inform research on generalization bounds for neural networks, while stressing that other interpretations of these results are also possible.

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NeurIPS Conference 2024 Conference Paper

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

Alliot Nagle
Adway Girish
Marco Bondaschi
Michael Gastpar
Ashok Vardhan Makkuva
Hyeji Kim

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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ICML Conference 2024 Conference Paper

LASER: Linear Compression in Wireless Distributed Optimization

Ashok Vardhan Makkuva
Marco Bondaschi
Thijs Vogels
Martin Jaggi
Hyeji Kim
Michael Gastpar

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.

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NeurIPS Conference 2024 Conference Paper

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

Ashok V. Makkuva
Marco Bondaschi
Chanakya Ekbote
Adway Girish
Alliot Nagle
Hyeji Kim
Michael Gastpar

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}.

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JMLR Journal 2024 Journal Article

Lower Bounds on the Bayesian Risk via Information Measures

Amedeo Roberto Esposito
Adrien Vandenbroucque
Michael Gastpar

This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually any information measure, including Rényi's $\alpha$, $\varphi$-divergences, and Sibson's $\alpha$-Mutual Information. The approach considers divergences as functionals of measures and exploits the duality between spaces of measures and spaces of functions. In particular, we show that one can lower bound the risk with any information measure by upper bounding its dual via Markov's inequality. We are thus able to provide estimator-independent impossibility results thanks to the Data-Processing Inequalities that divergences satisfy. The results are then applied to settings of interest involving both discrete and continuous parameters, including the “Hide-and-Seek” problem, and compared to the state-of-the-art techniques. An important observation is that the behaviour of the lower bound in the number of samples is influenced by the choice of the information measure. We leverage this by introducing a new divergence inspired by the “Hockey-Stick” divergence, which is demonstrated empirically to provide the largest lower bound across all considered settings. If the observations are subject to privatisation, stronger impossibility results can be obtained via Strong Data-Processing Inequalities. The paper also discusses some generalisations and alternative directions. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

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ICML Conference 2024 Conference Paper

The Fundamental Limits of Least-Privilege Learning

Theresa Stadler
Bogdan Kulynych
Michael Gastpar
Nicolas Papernot
Carmela Troncoso

The promise of least-privilege learning – to find feature representations that are useful for a learning task but prevent inference of any sensitive information unrelated to this task – is highly appealing. However, so far this concept has only been stated informally. It thus remains an open question whether and how we can achieve this goal. In this work, we provide the first formalisation of the least-privilege principle for machine learning and characterise its feasibility. We prove that there is a fundamental trade-off between a representation’s utility for a given task and its leakage beyond the intended task: it is not possible to learn representations that have high utility for the intended task but, at the same time, prevent inference of any attribute other than the task label itself. This trade-off holds regardless of the technique used to learn the feature mappings that produce these representations. We empirically validate this result for a wide range of learning techniques, model architectures, and datasets.

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NeurIPS Conference 2024 Conference Paper

Transformers on Markov data: Constant depth suffices

Nived Rajaraman
Marco Bondaschi
Kannan Ramchandran
Michael Gastpar
Ashok V. Makkuva

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.

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ICLR Conference 2022 Conference Paper

A Johnson-Lindenstrauss Framework for Randomly Initialized CNNs

Ido Nachum
Jan Hazla
Michael Gastpar
Anatoly Khina

How does the geometric representation of a dataset change after the application of each randomly initialized layer of a neural network? The celebrated Johnson-Lindenstrauss lemma answers this question for linear fully-connected neural networks (FNNs), stating that the geometry is essentially preserved. For FNNs with the ReLU activation, the angle between two input contracts according to a known mapping. The question for non-linear convolutional neural networks (CNNs) becomes much more intricate. To answer this question, we introduce a geometric framework. For linear CNNs, we show that the Johnson--Lindenstrauss lemma continues to hold, namely, that the angle between two inputs is preserved. For CNNs with ReLU activation, on the other hand, the behavior is richer: The angle between the outputs contracts, where the level of contraction depends on the nature of the inputs. In particular, after one layer, the geometry of natural images is essentially preserved, whereas for Gaussian correlated inputs, CNNs exhibit the same contracting behavior as FNNs with ReLU activation.

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JMLR Journal 2021 Journal Article

Locally Differentially-Private Randomized Response for Discrete Distribution Learning

Adriano Pastore
Michael Gastpar

We consider a setup in which confidential i.i.d. samples $X_1,\dotsc,X_n$ from an unknown finite-support distribution $\boldsymbol{p}$ are passed through $n$ copies of a discrete privatization channel (a.k.a. mechanism) producing outputs $Y_1,\dotsc,Y_n$. The channel law guarantees a local differential privacy of $\epsilon$. Subject to a prescribed privacy level $\epsilon$, the optimal channel should be designed such that an estimate of the source distribution based on the channel outputs $Y_1,\dotsc,Y_n$ converges as fast as possible to the exact value $\boldsymbol{p}$. For this purpose we study the convergence to zero of three distribution distance metrics: $f$-divergence, mean-squared error and total variation. We derive the respective normalized first-order terms of convergence (as $n \to \infty$), which for a given target privacy $\epsilon$ represent a rule-of-thumb factor by which the sample size must be augmented so as to achieve the same estimation accuracy as that of a non-randomizing channel. We formulate the privacy-fidelity trade-off problem as being that of minimizing said first-order term under a privacy constraint $\epsilon$. We further identify a scalar quantity that captures the essence of this trade-off, and prove bounds and data-processing inequalities on this quantity. For some specific instances of the privacy-fidelity trade-off problem, we derive inner and outer bounds on the optimal trade-off curve. [abs] [ pdf ][ bib ] &copy JMLR 2021. ( edit, beta )

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