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Empirical Bernstein Bounds and Sample Variance Penalization
Andreas Maurer, Massimiliano Pontil
TL;DR
The paper addresses the need for confidence bounds that adapt to hypothesis-specific variance rather than treating a hypothesis class uniformly. It develops empirical Bernstein bounds, extends them to function classes with polynomial growth, and introduces sample variance penalization; in supported settings, SVP can have excess risk of order 1/n while ERM can be of order 1/√n. The paper also reports preliminary experiments and discusses sample-compression applications.
Problem
Hoeffding-style bounds provide a uniformly blurred view because their confidence intervals do not depend on the hypothesis variance.
Method
The paper develops observable empirical Bernstein bounds, extends them using growth-function control, and minimizes the resulting variance-sensitive bounds through sample variance penalization.
Results
When an optimal hypothesis has zero variance, SVP's excess risk decreases as 1/n, while an example shows ERM cannot decrease faster than O(1/√n).
Takeaways & Limitations
Variance-sensitive, data-dependent bounds can support a learning method with substantially better excess-risk guarantees than ERM in simple but potentially practical scenarios.
Takeaways & Limitations
The paper leaves efficient implementation and real-life dataset comparisons open, partly because SVP is non-convex where ERM may be convex.
Abstract
from arXiv · showhide
We give improved constants for data dependent and variance sensitive confidence bounds, called empirical Bernstein bounds, and extend these inequalities to hold uniformly over classes of functionswhose growth function is polynomial in the sample size n. The bounds lead us to consider sample variance penalization, a novel learning method which takes into account the empirical variance of the loss function. We give conditions under which sample variance penalization is effective. In particular, we present a bound on the excess risk incurred by the method. Using this, we argue that there are situations in which the excess risk of our method is of order 1/n, while the excess risk of empirical risk minimization is of order 1/sqrt/{n}. We show some experimental results, which confirm the theory. Finally, we discuss the potential application of our results to sample compression schemes.
1 Introduction
The paper develops observable, variance-sensitive confidence bounds and uses them to motivate sample variance penalization as an alternative to ERM. It extends these bounds uniformly to function classes with polynomial growth and identifies settings where SVP can achieve faster excess-risk decay.
- Evaluation and extensions: The paper reports preliminary experiments comparing SVP and ERM and discusses applying empirical Bernstein bounds to sample-compression schemes.The compression application is motivated by the tightness of individual performance estimates.
- Confidence bounds: For zero-variance hypotheses, the confidence-interval diameter decays as O(1/n).For nonzero variances, the interval can still be substantially smaller than the variance-independent Hoeffding interval.
- Confidence bounds: Empirical Bernstein bounds make confidence intervals data-dependent and potentially tighter for hypotheses with small sample variance.The resulting view is less precise for high-variance hypotheses and more focused for low-variance hypotheses.
- Confidence bounds: The empirical Bernstein bound is observable because it replaces the unobservable variance in Bennett-style bounds with the sample variance.This yields a data-dependent estimate whose interval diameter can be determined from the observed sample.
- Uniform extensions: Uniform versions extend the bounds from finite hypothesis spaces to classes whose growth function can be controlled, including classes with polynomial growth in n.The growth-function formulation uses covering numbers such as N∞(1/n, F, n).
- SVP algorithm: Sample variance penalization selects hypotheses by minimizing variance-sensitive uniform upper bounds rather than relying only on empirical risk.Its excess risk can be O(1/n) when an optimal hypothesis has zero variance, whereas ERM can be no faster than O(1/√n) in an example.
2 Empirical Bernstein bounds and variance estimation
The paper develops concentration results for sample variance and uses them to derive empirical Bernstein bounds for independent bounded variables. These observable bounds are sharper for low-variance hypotheses and extend Bennett-style guarantees beyond identically distributed samples.
- Variance estimation: The analysis proves concentration inequalities for the variance of bounded random variables, including independent non-identically distributed samples.The results are developed for independent variables, not necessarily identically distributed.
- Proof strategy: The proof uses auxiliary results on self-bounding random variables and conditional expectations to control deviations of the sample variance.The derivation includes a technical lemma for i.i.d. variables in an interval and a self-boundedness theorem for the variance analysis.
- Variance estimation: Theorem 10 establishes concentration bounds for the sample variance of independent variables taking values in [0,1].The theorem explicitly treats vectors of independent [0,1]-valued random variables.
- Empirical Bernstein bounds: The empirical Bernstein bound combines observable sample-variance information with concentration control and reduces to Theorem 4 for identically distributed variables.The resulting bound is presented for independent [0,1]-valued variables.
3 Empirical Bernstein bounds for function classes of polynomial growth
The paper extends empirical Bernstein bounds uniformly over function classes by adapting a doublesample argument to the nonlinear empirical standard-deviation functional. Covering numbers replace finite-class cardinality, yielding bounds applicable to classes with controlled growth.
- Proof construction: The proof adapts the classical doublesample method to handle the nonlinearity of the empirical standard-deviation functional.It introduces functionals Φ and Ψ and uses randomized sample exchanges.
- Symmetrization: A symmetrization lemma bounds the probability that Φ on one randomized sample exceeds Ψ on the exchanged sample by 5e^-t.The bound is established for fixed functions and fixed doubled samples.
- Concentration control: The argument combines Bennett’s inequality with sample-variance concentration to control the two deviation probabilities underlying the symmetrization bound.The two components contribute bounds of e^-t and 3e^-t before the final estimate.
- Covering-number reduction: Uniformization replaces the potentially infinite class by a finite subset whose size is controlled by the covering number N(ε, F, 2n).The proof selects an ε-net and transfers the comparison from F to its finite representative set.
- Final bound: The resulting uniform bound scales with N(ε, F, 2n), with ε set to 1/n and the confidence parameter determined by equating the failure probability to δ.The final estimate contains the factor 10N(ε, F, 2n)e^-t and uses n ≥16.
- Further consequence: The paper also notes that a simplified version yields uniform bounds for the standard deviation and variance over the function class.This extension uses the sample-variance concentration results directly.
4 Sample variance penalization versus empirical risk minimization
The paper turns empirical Bernstein bounds into sample variance penalization, selecting hypotheses by an empirical-risk objective augmented with a variance term. Its analysis identifies low-variance optimal hypotheses as the setting where SVP can outperform ERM, with theory and experiments supporting this comparison.
- Method: Sample variance penalization minimizes empirical Bernstein bounds by incorporating the empirical variance of each hypothesis’s loss.Setting λ = 0 reduces SVP to empirical risk minimization.
- Theory: The excess-risk bound for SVP scales in part with the standard deviation of an optimal hypothesis.For an optimal hypothesis with zero variance, the stated rate becomes (ln M(n))/n, and equals 1/n for finite classes.
- Example: In the constructed two-hypothesis example, the optimal hypothesis has zero variance while the inferior hypothesis has excess risk ε and variance 1/4 − ε^2.The example is designed to contrast SVP’s variance sensitivity with ERM’s empirical-risk selection.
- Comparison with ERM: ERM’s excess risk cannot have a faster rate than O(1/√n) in the example, whereas SVP decays as 1/n for a suitable λ.The ERM lower bound holds with probability at least δ for a suitable δ depending on n and ε.
- Scope and interpretation: The authors characterize the example as artificial and state that SVP’s practical advantage is small relative to the optimal risks, around 1/4.They suggest SVP may be useful when optimal hypotheses have substantial risk but very small variance.
- Experiment: The experiment uses K = 500 coordinate hypotheses, B = 1/4, sample sizes from 10 to 500, and compares λ = 0 with λ = 2.5.True risks were averaged over 10000 randomly generated distributions; Figure 1 reports the comparison.
5 Application to sample compression
The paper applies sample variance penalization and empirical Bernstein bounds to sample compression schemes, where hypotheses are trained on subsamples and evaluated on held-out data. The resulting uniform guarantee is especially favorable when variance is small and the compression sets are sparse.
- Compression setup: Sample compression trains on subsamples and uses the remaining examples to evaluate the resulting hypotheses.The selected subsample encodes problem-relevant information when its corresponding hypothesis has small risk.
- Compression setup: SVP combines sample compression with variance-sensitive performance estimates based on empirical Bernstein bounds.The method evaluates fixed-size compression sets using empirical risk and sample variance on the complementary data.
- Guarantee: Theorem 17 provides a uniform high-probability guarantee over all compression sets and the corresponding SVP choice.Its proof uses a union bound and Theorem 4, with additional uniform bounds because the optimal compression set is sample-dependent.
- Interpretation: The excess-risk interpretation is more subtle because the optimal hypothesis is sample-dependent.Under a near-optimal generative model with many small independent noise contributions, small variance yields tight excess-risk control.
- Guarantee: The bound can substantially improve on Hoeffding’s inequality when the variance is small and d/n is small.The sparsity condition is supported by ln |C| ≤ d ln(ne/d).
6 Conclusion
The paper presents sample variance penalization as an alternative to empirical risk minimization, analyzed through empirical Bernstein bounds and concentration of empirical standard deviation. It reports a potential advantage in simple practical settings while identifying implementation and evaluation questions that remain open.
- Conclusion: SVP is analyzed using empirical Bernstein bounds and concentration properties of the empirical standard deviation.
- Open questions: Efficient implementation, non-convexity relative to convex ERM, real-life dataset comparisons, and compression applications remain open issues.