Learning without Concentration (1401.0304v2)

Abstract: We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used here relies on a small-ball' assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets. The resulting estimates scale correctly with thenoise level' of the problem, and when applied to the classical, bounded scenario, always improve the known bounds.

• The paper derives sharp ERM performance bounds by leveraging a small-ball condition to overcome limitations of concentration inequalities in heavy-tailed settings.
• It introduces two complexity parameters, β*_N and α*_N, quantifying the version space size and noise interaction within the convex function class.
• The findings promote robust learning by relaxing boundedness assumptions and extending empirical risk minimization to high-dimensional, heavy-tailed scenarios.

An Expert Overview of "Learning without Concentration"

The paper "Learning without Concentration" by Shahar Mendelson addresses the performance of Empirical Risk Minimization (ERM) within a convex class and under squared loss conditions, challenging the conventional reliance on concentration inequalities. This research derives sharp bounds without the boundedness assumption on the class members and target, employing a "small-ball" condition to manage instances involving heavy-tailed distributions.

Summary of Key Contributions

The paper's central contribution lies in the derivation of performance bounds for ERM when dealing with potentially heavy-tailed data. Unlike classical approaches that depend on concentration inequalities applicable to bounded or well-behaved distributions, this work leverages the small-ball condition. This condition provides lower bounds on the probability that functions exceed certain thresholds, enabling the management of variability without assuming boundedness.

Analysis and Findings

1. Theoretical Reassessment: Through rigorous analysis, the research demonstrates that classical theorems maintaining boundedness assumptions, such as Theorem 1.1 in the paper, are restrictive and yield suboptimal results in cases of Gaussian and heavy-tailed noise. The results reveal that these traditional approaches fail to capture the estimation error's scaling correctly with the problem's intrinsic parameters or noise level, highlighting the necessity of an alternative framework.
2. Introduction of Two Complexity Parameters:
   • β*_N: This parameter is concerned with the "size" of the version space. It provides an upper estimate on the L2 diameter of the set of functions within the convex class matching the observed data closely.
   • α*_N: This parameter captures the interaction between the noise and the class members. It measures the correlation between the random noise and class functions, offering insights into the noise's influence by concentrating on its primary interactions.
3. Implementation of Small-Ball Condition: A novel aspect of the framework is the small-ball condition, which facilitates lower-bound probability estimates for heavy-tailed functions. This condition is essential because many real-world functions do not exhibit strong concentration properties but can still be bounded from below using a small-ball approach.
4. Optimality Considerations: The potential optimality of the bounds is considered, showing that for a variety of function classes, the proposed parameters are likely to capture the estimation error effectively. The separation of the problem into "low noise" and "high noise" regimes underscores the complexity of these interactions.

Implications and Future Work

The implications of this work are profound, removing significant barriers for applying ERM in real-world scenarios where data distributions are not necessarily bounded or exhibit heavy tails. The findings encourage further exploration into learning frameworks where small-ball conditions replace classic concentration inequalities. Moreover, the reduction in assumptions extends the applicability of empirical processes and relates closely to robust statistics, particularly in high-dimensional settings.

Looking forward, the research sets the stage for deeper investigations into non-centered loss functions and their associated learning problems, potentially generalizing these methods to diverse statistical and machine learning tasks.

In conclusion, Mendelson’s work paves the way for open-ended exploration in machine learning, pointing towards a more generalized and less assumption-dependent paradigm in theoretical and applied research. This shift is crucial for developing algorithms capable of managing the complexities inherent in modern, high-dimensional datasets with less restrictive conditions, enhancing robustness and flexibility.