Research map / Learning theory and algorithms / Sample complexity
A Universal Growth Rate for Learning with Smooth Surrogate Losses
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Shows that H-consistency bounds (and excess error bounds) for smooth margin-based binary and smooth comp-sum/constrained multi-class surrogate losses have an exact square-root growth rate near zero, and uses minimizability gaps to guide surrogate choice.
Problem
Characterizing how quickly the H-consistency bound transformation (growth rate near zero) increases for surrogate losses used in binary and multi-class classification, and what determines the choice among surrogates.
Approach
Proves Theta(t^2) growth of the H-consistency transformation T(t) for smooth margin-based binary losses and smooth comp-sum and constrained multi-class losses, via an implicit function theorem argument, giving matching upper and lower bounds; then identifies minimizability gaps as the key factor differentiating surrogates near zero.
Key results
- For smooth margin-based binary losses with Phi'(0)>0 and Phi''(0)>0, the transformation satisfies T(t)=Theta(t^2), i.e., square-root growth rate of H-consistency bounds.
- For smooth comp-sum losses (Phi'(u)<0, Phi''(u)>0 on (0,1/2]) and smooth constrained losses, T(t)=Theta(t^2) as well, giving a universal square-root rate across binary and multi-class.
- The comp-sum bound with tau in [1,2) has the form sqrt(2 n^(tau-1) t), so the number of classes scales the bound; logistic loss (tau=1) is favored when accounting for n and minimizability gaps.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a smooth margin-based surrogate satisfying the stated convexity and derivative conditions | Binary classification with a complete hypothesis set · Binary classification | H-consistency bound growth rate: T(t) = Θ (t^2) | Helped |
| Use a smooth comp-sum surrogate satisfying the stated convexity and derivative conditions | Multi-class classification with a symmetric and complete hypothesis set · Multi-class classification | H-consistency bound growth rate: T(t) = Θ(t^2) | Helped |
| Use a smooth constrained surrogate satisfying the stated convexity and derivative conditions | Multi-class classification with a symmetric and complete hypothesis set · Multi-class classification | H-consistency bound growth rate: Θ(t^2) | Helped |
| Use a polyhedral loss instead of a smooth surrogate | Binary classification · Binary classification | H-consistency bound growth rate: linear | Mixed |
| Set τ = 1 rather than another τ in the comp-sum family | Comp-sum losses · Multi-class classification | class-count scaling factor: n then vanishes | Helped |
Surprising findings
- Polyhedral losses such as hinge have linear growth rate, in contrast to the square-root rate for smooth losses.
Limitations
- Results assume complete and symmetric hypothesis sets.
- Constrained-loss result relies on an additional technical assumption that the infimum over tau is attained in a finite interval.
- Minimizability gaps are non-zero for restricted hypothesis sets, so the zero-gap bound does not apply in general.
Experimental setup
Baselines: polyhedral losses (hinge) smoothed hinge losses
Builds on: H-consistency bounds comp-sum losses constrained losses excess error bounds
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