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This paper refines risk estimation for learning algorithms based on preferential compression by removing the logarithmic factor from existing bounds, achieving a $O(1/\sqrt{N})$ rate. The research demonstrates that this complexity-based risk estimate matches the law-of-large-numbers rate, providing a tighter and more accurate assessment of learning algorithm performance. Furthermore, it establishes the necessity of the leading $\boldsymbol{k}/N$ term and analyzes its relationship with empirical risk across different convergence regimes.
Preferential compression risk bounds can achieve the optimal $O(1/\sqrt{N})$ convergence rate, shedding the previously unavoidable logarithmic factor.
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 $$ \frac{\boldsymbol{k}}{N} + O\left(\frac{1}{\sqrt{N}}\right), $$ 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.