Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory

summary

The gist

Learning linear systems under heavy-tailed noise involves establishing non-asymptotic sample complexity bounds for least-squares estimation of vector autoregressive models using a single observed

In short

The paper analyzes estimating parameters of linear systems from a single trajectory when noise has heavy tails. It provides non-asymptotic bounds showing that if the noise has a bounded pth moment greater than two, model-based estimation remains statistically viable, though convergence slows down by a factor related to T^(1/p). This is crucial for real-world control where heavy disturbances are common.

Key concepts

Vector Autoregressive (VAR) Model: This describes a linear system where the current state depends on past states and random noise. The model is defined by z_t = Θz_t-1 + Ψξ_t, meaning the system's evolution is driven by its previous values multiplied by a parameter matrix and some random disturbance vector.
Sample Complexity: This refers to the minimum number of observations (T) required to estimate unknown system parameters accurately. The paper derives bounds for this complexity under different noise conditions, showing how quickly the estimation error decreases as more data is collected.
Bounded pth Moment Condition: This condition requires that the random noise vectors have a finite average power related to their p-th moment, where p must be greater than 2. This ensures that even though tails are heavy, they are not so extreme that the estimation problem becomes impossible or unstable.
Non-Asymptotic Analysis: Instead of relying on limits as the number of samples approaches infinity (as in classical theory), this analysis provides concrete bounds for finite sample sizes. This is important because it tells engineers exactly how accurate an estimate can be with a limited amount of data.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Retrieval-Augmented Generation (RAG) — explained in 16 episodes
Interpretability — explained in 15 episodes
Federated Learning — explained in 14 episodes
Generalization — explained in 13 episodes
Quantum Fisher Information (QFI) — explained in 13 episodes

This episode discusses

The paper

Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from A Single Trajectory · Read on arXiv

Xiaomian Yang, Sungho Shin

Massachusetts Institute of Technology

We establish non-asymptotic sample complexity bounds for the least-squares estimation of vector autoregressive models for exponentially stable systems with heavy-tailed noise based on a single observed trajectory. By assuming i.i.d. noise, bounded noise covariance, and persistent excitation, we show that the estimation error is (r 1/2T-1/2+1/p) under bounded p th moment for p > 2, where T is the number of samples, r is the noise dimension, and (times) hides logarithmic terms. We also introduce a unifying approach to sample complexity analysis applicable to broad classes of noise distributions and showcase this by deriving error bounds for sub-exponential and sub-Gaussian noise distributions. Finally, we specialize our analysis to autoregressive models with exogenous inputs and show that the dimension factor of the error bound is independent of the model order.

Transcript

Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Learning Linear Systems under Heavy-Tailed Noise".

Tom: Learning linear systems under heavy-tailed noise involves establishing non-asymptotic sample complexity bounds for least-squares estimation of vector autoregressive models using a single observed trajectory,

Jane: First, who's behind it and why it matters.

Paper summary: Tom: So, to recap where we are, this paper sets out to establish non-asymptotic sample complexity bounds for estimating vector autoregressive models when they encounter heavy-tailed noise using only a single observed trajectory. The core claim is that under specific conditions—i.i.d., zero-mean noise, bounded noise covariance, and persistent excitation—the estimation error follows the bound Oe(r(one/2T - one/two + one/p)) when the pth moment of the noise is bounded for p greater than two.

Jane: That bound shows that as the sample size T grows, we can control our estimation error by increasing T, but there’s a specific slowing factor introduced by the exponent involving p. This is important because it directly addresses situations where sub-Gaussian assumptions fail due to heavier tails in the noise distribution.

Lu: What I find especially noteworthy about this paper is their unification of the analysis framework, which they call a unifying approach to sample complexity analysis applicable to broad classes of noise distributions, and how they showcase this by deriving bounds for both sub-exponential and sub-Gaussian noise distributions. This flexibility is what makes the methodology powerful.

Meng: I'm thinking about that flexibility; if the method can handle different tail behaviors, it means we don't have to re-derive everything every time we switch from a Gaussian assumption to something heavier, which saves development cycles. But how does this affect the complexity of implementing the underlying estimation algorithm?

Lalam: I see this as an advancement in making our AI systems more resilient; when our models interact with unpredictable real-world data, having a framework that accounts for these heavy tails means the resulting control policies will be much safer and more predictable in those challenging scenarios.

Conclusion: Tom: Thinking about the full scope of this work, "Learning Linear Systems under Heavy-Tailed Noise: A Non-Asymptotic Analysis from a Single Trajectory," the authors are really showing us that model-based control isn't just theoretical fluff when we consider real-world noise profiles. They’ve rigorously shown how to quantify exactly how much data we need before our parameter estimates become unreliable in these heavy-tailed settings.

Jane: And their main implication is practical: it gives system identification practitioners a concrete, non-asymptotic guarantee on the error scale, which is much more useful than just saying "it converges as T goes to infinity." It specifically addresses the finite sample performance that matters for deployment right now.

Lu: The authors also made a significant contribution by specializing this general analysis to ARX models and showing that the dimension factor of the sample complexity only depends on the state and input dimensions, which is a nice simplification compared to older bounds that scaled with the autoregressive order. This makes it much more tractable for complex systems.

Meng: So, if we apply this knowledge practically, it means we can design controllers knowing precisely how much data they need to collect before those controllers start making bad decisions under heavy noise conditions, which is critical for setting operational limits.

Lalam: For the culture of our development teams, having these kinds of sharp theoretical tools allows us to build systems with a higher degree of confidence in their performance even when the environment is unpredictable. It reinforces a culture where we prioritize provable performance over just empirical success.

More episodes

← Home