AI Insight
This paper proves that risk bounds for machine learning algorithms based on compression schemes can achieve an optimal convergence rate of O(1/√N), removing an unnecessary logarithmic factor present in previous results. The authors show that for compression-based learning, the risk can be bounded by k/N + O(1/√N) with high probability, where k is a sample-dependent complexity measure and N is the sample size, matching the rate at which empirical probability estimates converge. They further demonstrate conditions under which this complexity-based estimator maintains its convergence rate even when traditional empirical risk estimators fail or converge more slowly.
Why it matters
This result provides tighter theoretical guarantees for compression-based learning algorithms, which have applications in model selection, sample compression, and developing computationally efficient learning methods. The improved bounds could lead to better understanding of when and why certain learning algorithms generalize well, particularly in settings where traditional empirical risk minimization may be unreliable.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Compression schemes provide a general framework for deriving data-dependent risk bounds for learning algorithms. For learning rules based on preferent compression, existing results upper-bound the risk in terms of a sample-dependent complexity $boldsymbol{k}$, defined as the cardinality of the compression augmented by the training examples on which the reconstructed hypothesis is inappropriate. These results, however, yield a margin of order $O(sqrt{ln N/N})$ above $boldsymbol{k}/N$. In this paper, we establish that the logarithmic factor is not intrinsic. We prove that, for every fixed confidence level $1-delta$, the risk is bounded with probability at least $1-delta$ by $$ frac{boldsymbol{k}}{N} + Oleft(frac{1}{sqrt{N}}right), $$ uniformly over all possible values of $boldsymbol{k}$. Thus, risk estimates for learning algorithms based on preferent compression attain the same law-of-large-numbers rate as the empirical estimate of the probability of a fixed event. We also show that the leading term $boldsymbol{k}/N$ is necessary and investigate its relation to the empirical risk. In particular, we identify regimes in which the empirical risk retains the $O(1/sqrt{N})$ convergence rate, as well as regimes in which it converges more slowly or ceases to be consistent, while the complexity-based estimator preserves the $O(1/sqrt{N})$ rate.
Source: Learning by Compression: Risk Estimates Attain the Law-of-Large-Numbers Rate