Networks with Finite VC Dimension: Pro and Contra

arXiv:2502.02679 · stat.ML, cs.LG · Submitted 2025-02-04 · Read on arXiv

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

Tom: Today's paper: "Networks with Finite VC Dimension: Pro and Contra".

Jane: The gist:

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

Title and authors: Tom: Let's start with who wrote this stuff, "Networks with Finite VC Dimension: Pro and Contra." The authors are Věra Kůrková and Marcello Sanguineti from the Institute of Computer Science of the Czech Academy of Sciences.

Jane: They’re looking at this problem from a statistical learning theory angle, seeing how the structure of the input-output functions influences both approximation accuracy and learning consistency.

Lu: It’s interesting that they are comparing constraints based on norms, like Sobolev norms or variational norms, against constraints based on these probability distributions we talked about earlier.

Meng: So, if you restrict the network's function space by a norm constraint, you get certain guarantees about how well it learns from samples.

Tom: Right, but they show that this restriction doesn't automatically translate into good approximation when the target functions are sampled from a specific probability model for a task.

Jane: They prove that if the VC dimension is finite, the growth function is bounded by a polynomial, which means we get concentration around the mean for both approximation errors and empirical errors.

The paper's summary: Tom: So what’s the main takeaway from this paper? It lays out that finite VC dimension has a definite benefit for uniform convergence of empirical errors, but it can be a disadvantage when you are trying to approximate functions drawn from a probability distribution that models task likelihood.

Jane: They’re showing that when the growth function H(m) does not outweigh e- m lambda two/eight most functions just behave almost deterministically, meaning their errors concentrate around their mean value.

Lu: That implies we can have pretty good results for learning consistency if that condition holds, but it doesn't guarantee good accuracy for the typical function in our application space.

Meng: It’s important because it tells us that just having a small VC dimension isn't the whole story; we also need to consider the distribution of functions we are actually trying to model.

Tom: And they explore how this depends on mu H(X), which is related to the minimum mean distance from all input-output functions in the set.

The paper's improvements: Jane: The authors suggest that the key thing to watch is that minimum mean mu H(X). If this value is large, almost every function drawn according to your probability P will be hard to compute accurately with a finite VC dimension network.

Tom: That’s a big deal for practical systems. It means if the set of functions we are interested in is too diverse, even if the VC dimension is finite, we can’t reliably approximate those functions.

Lu: Conversely, if mu H(X) is small, they find that there exists at least one specific function within the class H(X) where the expected error E(eta h*) is quite small.

Meng: So, even with a finite VC dimension, if mu H(X) is low enough, we can find one specific function that actually does a good job approximating almost all functions drawn from that distribution.

Tom: It moves the conversation away from just the size of the network and toward the geometric properties of the space of functions itself.

Conclusion: Jane: To wrap up, this paper "Networks with Finite VC Dimension: Pro and Contra" shows that while finite VC dimension is a good thing for learning consistency, it can hurt approximation accuracy for functions sampled from a probability distribution.

Tom: The main point is that the success hinges on whether the growth function H(m) stays smaller than e- m lambda two/eight when dealing with the specific task distribution.

Lu: From a creative perspective, this suggests we need to design network constraints not just based on complexity bounds, but on how well those bounds align with the actual functional space of our application.

Meng: Practically, it means if your target functions are very spread out according to P, you might need more capacity than what a finite VC dimension would suggest for good approximation.

Lalam: I see this paper suggesting that our ability to model complex behaviors relies less on just fitting a bound and more on understanding the inherent geometry of the function space we are interested in.

Tom: That’s it for this one, "Networks with Finite VC Dimension: Pro and Contra." We've seen how the size of the function set interacts with approximation accuracy versus learning consistency.

Jane: It’s a complex topic, but understanding that trade-off is key to designing better learning systems. Next up, we're looking at how we detect when clients in federated learning are acting like free-riders.

Institute of Computer Science of the Czech Academy of Sciences · DIBRIS University of Genova

stat.ML, cs.LG

Submitted: 2025-02-04

Updated: 2026-10-08

License: http://creativecommons.org/publicdomain/zero/1.0/

Importance score: 83/100

The gist: The gist: Finite VC dimension is desirable for uniform convergence of empirical errors but may not be desirable for approximation of functions drawn from a probability distribution modeling the

Key concepts

VC Dimension: The VC dimension measures the capacity or complexity of a set of functions. A finite VC dimension means the class is not overly complex, which is generally good for learning algorithms because it guarantees that empirical errors will converge uniformly to their true mean values.
Uniform Convergence: This refers to a property where the error across all functions in a set decreases simultaneously as the amount of training data increases. Finite VC dimension is desirable because it helps guarantee this uniform convergence, meaning the learning process is reliable for any function in that class.
Function Approximation Error Concentration: This concept describes how approximation errors behave when dealing with functions drawn from a specific probability distribution. The paper shows that if the complexity measure (growth function) doesn't dominate another term, most approximation errors cluster tightly around their average value, making accurate approximation difficult for almost all functions.
Growth Function ($\Pi H(m)$): The growth function is a measure used to bound the probability that approximation errors are close to their mean. Its relationship with other complexity terms determines whether the concentration of errors is strong enough to guarantee good approximation performance.

Terminology

VC Dimension — 38× in this paper · explained in 2 episodes
Accuracy — 26× in this paper · explained in 1 episode
Consistency — 17× in this paper · explained in 3 episodes
Loss Function — 4× in this paper · explained in 2 episodes
Activation Function — 3× in this paper · explained in 1 episode
Robustness — 2× in this paper · explained in 5 episodes
Approximation Bound — 1× in this paper · explained in 1 episode
Federated Learning — 1× in this paper · explained in 14 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes

Summary

The gist: Finite VC dimension is desirable for uniform convergence of empirical errors but may not be desirable for approximation of functions drawn from a probability distribution modeling the likelihood that they occur in a given type of application.

Introduction and Motivation

The investigation explores the trade-offs between constraints on classes of functions to be approximated and bounds on various measures of network complexity, focusing on constraints defined by probability distributions rather than norms 1 − ΠH(m) e − mλ 2/8 1 − ΠH(m) e− mλ 2/8." This result implies that when ΠH(m) does not outweigh e− mλ 2/8, errors in approximation of almost all functions behave almost deterministically, they concentrate around their mean value <ref:2502.03111, When ΠH(m) does not outweigh e− mλ 2/8, then most functions cannot be approximated with better accuracy than 2 − λ.

Conclusion on VC Dimension

For sets of I/O functions with finite VC dimension, the growth function is bounded by a polynomial <ref:2502.03114, So, any set with a finite VC dimension has a growth function that is bounded by a polynomial. This implies that errors in approximation concentrate around their mean values <ref:2502.03114, Theorem 4.1 implies that errors in approximation by networks with sets of I/O functions with finite VC dimension concentrate around their mean values. The paper concludes that while the finiteness of the VC dimension is desirable for learning (uniform convergence), it may be a disadvantage for function approximation <ref:2502.03115, "We have shown that while it is well-known that the finiteness of the VC dimension of a set H(X) of network I/O functions is desirable for learning (it guarantees uniform convergence of empirical errors), it may be a disadvantage for function approximation."

Discussion and Practical Implications

The concentration of approximation errors depends on the minimum µH(X) <ref:2502.03116, It depends on the minimum µH(X) of the mean values of distances from all I/O functions, around which approximation errors tightly concentrate. If µH(X) is large, almost all functions cannot be computed accurately by networks with finite VC dimension <ref:2502.03116, If µH(X) is large, almost all functions drawn according to the probability P, which characterizes the likelihood that a function occurs in a given type of task, cannot be computed accurately. Conversely, when µH(X) is small, there exists some h∗ ∈ H(X), for which E(ηh∗) is sufficiently small <ref:2502.03116, This implies that there exists some h∗ ∈ H(X), for which E(ηh∗) is sufficiently small. This suggests that one specific function can well approximate almost all functions chosen according to P <ref:2502.03116, So, almost all functions chosen randomly according to P can be well approximated by the function h∗.

How it works

The paper investigates approximation and learning of binary-valued functions on finite data sets X = ≃ R m <ref:2502.026794, We investigate approximation and learning of classifiers on finite data sets.
It models the probability distribution P on B(X) as a product distribution P(f):= Ym i=1 Pi(f(xi)) = Ym i=1 Pi(f(xi)xi), implying independence of random variables f(x1),..., f(xm) <ref:2502.03010, We consider the case when P can be expressed as the product P(f):= Ym i=1 Pi(f(xi)) = Ym i=1 Pi(f(xi)xi), which implies that the random variables f(x1),..., f(xm) are independent.
It uses the McDiarmid Bound to show that both approximation and empirical errors satisfy a coordinate-wise Lipschitz property, guaranteeing concentration of values around their mean <ref:2502.03011, We employ a concentration inequality that holds for functions of independent random variables satisfying a smoothness assumption that can be seen as a coordinate-wise Lipschitz property.
The growth function ΠH(m) is used to bound the probability that approximation errors are close to their mean value, showing this depends on whether ΠH(m) outweighs e− mλ 2/8 <ref:2502.03114, The smaller ΠH(m), the larger probability that an accuracy of approximation of a random function is close to µH(X).
For deep ReLU networks, the growth function depends polynomially on the size of the domain m, with degree equal to L and W <ref:2502.

Improvements for AI systems

Confidence in Approximation for Large Datasets: The system can leverage networks with finite VC dimension to achieve almost deterministic behavior for both approximation and empirical errors when processing large data sets, as shown by the result that both errors in approximation and empirical errors behave almost deterministically.
Tailored Approximation Accuracy: The system can determine if a desired accuracy is achievable for a function drawn from a probability distribution by checking the condition derived from Theorem 4.1, specifically assessing whether the growth function ΠH(m) does not outweigh e − mλ28.
Adaptive Learning Consistency: The system can monitor empirical errors and use the concentration bounds to ensure that empirical errors El,n,h converge to the theoretical error El(h) uniformly for all h ∈ H(X), thereby guaranteeing consistency in learning from samples of data.

Abstract

Approximation and learning of classifiers of large data sets by neural networks in terms of high-dimensional geometry and statistical learning theory are investigated. The influence of the VC dimension of sets of input-output functions of networks on approximation capabilities is compared with its influence on consistency in learning from samples of data. It is shown that, whereas finite VC dimension is desirable for uniform convergence of empirical errors, it may not be desirable for approximation of functions drawn from a probability distribution modeling the likelihood that they occur in a given type of application. Based on the concentration-of-measure properties of high dimensional geometry, it is proven that both errors in approximation and empirical errors behave almost deterministically for networks implementing sets of input-output functions with finite VC dimensions in processing large data sets. Practical limitations of the universal approximation property, the trade-offs between the accuracy of approximation and consistency in learning from data, and the influence of depth of networks with ReLU units on their accuracy and consistency are discussed.