Source-linked AI summary
The Feature Importance Ranking Measure
Alexander Zien, Nicole Kraemer, Soeren Sonnenburg, Gunnar Raetsch
TL;DR
Complex learning machines can predict accurately but are difficult to interpret, while simpler interpretable models may sacrifice predictive power. FIRM retrospectively evaluates features for arbitrary learning machines, using feature correlations to improve relevance assessment and interpretation.
Problem
Feature-importance methods often restrict the learning machine, arbitrarily select among correlated features, or become misguided by feature rescaling.
Method
FIRM generalizes substring-impact analysis to arbitrary continuous features and exploits feature-distribution and correlation information to rank features retrospectively.
Results
FIRM identifies GATTACA and edit-distance-one sequences as important, whereas raw feature weighting assigns random or erratic importance.
Takeaways & Limitations
FIRM provides a universal, objective feature-ranking measure for gaining insights from learning machines where alternative techniques struggle.
Takeaways & Limitations
FIRM depends on the input-feature distribution, which is generally unavailable, so practical approximations require assumptions that may not always hold.
Abstract
from arXiv · showhide
Most accurate predictions are typically obtained by learning machines with complex feature spaces (as e.g. induced by kernels). Unfortunately, such decision rules are hardly accessible to humans and cannot easily be used to gain insights about the application domain. Therefore, one often resorts to linear models in combination with variable selection, thereby sacrificing some predictive power for presumptive interpretability. Here, we introduce the Feature Importance Ranking Measure (FIRM), which by retrospective analysis of arbitrary learning machines allows to achieve both excellent predictive performance and superior interpretation. In contrast to standard raw feature weighting, FIRM takes the underlying correlation structure of the features into account. Thereby, it is able to discover the most relevant features, even if their appearance in the training data is entirely prevented by noise. The desirable properties of FIRM are investigated analytically and illustrated in simulations.
1 Introduction
The paper motivates FIRM as a universal, objective feature-importance measure that interprets arbitrary learning machines while accounting for feature dependencies. It addresses shortcomings of feature weighting and selection, including sensitivity to rescaling, correlated features, and restricted model classes.
- Limitations of existing approaches: Feature selection can be unstable with correlated features, may choose among equivalent subsets, and can reduce predictive accuracy through enforced sparsity.These drawbacks also arise when sparsity is imposed at the kernel-space level in multiple kernel learning.
- Limitations of existing approaches: Raw feature weights are not invariant to positive rescaling and fail to account for correlations between features, even when predictions remain unchanged.Rescaling an input dimension and inversely rescaling its weight can arbitrarily change the assigned importance without changing the learned predictor.
- Contribution: FIRM is designed to evaluate any feature for arbitrary learning machines, including nonlinear models and features absent from the model's output function.It applies to classification, regression, and ranking when predictions are represented by a real-valued output function.
- Contribution: The paper seeks an importance measure that is universal, objective under correlated features and rescaling, and intelligent about connections among related features.These criteria are presented as goals for interpreting high-performing learning machines without restricting the primary learner.
- Method: FIRM estimates a feature's total impact on a trained predictor's score, extending sequence-based POIMs to real-valued features and incorporating dependency structure.Under normality assumptions, the paper states that FIRM generalizes sensitivity analysis while accounting for correlations.
2 The Feature Importance Ranking Measure (FIRM)
FIRM ranks features by the variability of their conditional expected score, extending this idea beyond binary features to continuous and unobserved features. Its formulations account for feature dependence and are designed to be robust across learning machines and transformations.
- Definition: FIRM defines a feature’s importance as the standard deviation of its conditional expected score.The conditional expected score is q_f(t) = E[s(X) | f(X) = t].
- Generalization: FIRM extends POIMs from binary sequence features to continuous features.POIMs are recovered as a special case of the broader measure.
- Scope: FIRM applies to classification, regression, ranking, nonlinear functions, arbitrary feature spaces, and features absent from the predictor’s output function.The method can assess higher-degree polynomial features after training a lower-degree kernel machine.
- Computation: For general scores and input distributions, exact FIRM computation is usually intractable, but normality assumptions yield a first-order analytical approximation.The approximation is exact when the transformed function is linear.
- Binary cases: For independent features, FIRM reduces to the associated linear weights, while conjunction and XOR analyses expose interaction effects that ordinary linear weights may miss.The XOR feature receives zero conditional expected scores, revealing the linear model’s inability to capture that dependence.
- Properties: Unlike raw weights, FIRM incorporates predictor dependence through covariance-based weighting and avoids arbitrary changes from feature rescaling in analytically tractable cases.For normally distributed features, Q = D^-1Σw; for uncorrelated features, the result reduces to a rescaled weight vector.
3 Simulation Studies
Simulation studies show that FIRM recovers meaningful feature importance across Boolean, continuous, and sequence data, including features obscured by correlations, noise, or mutations. Compared with raw SVM weights, FIRM more closely reflects the underlying discriminative structure.
- 3.1 Binary Data: FIRM more closely recovered the Boolean formula than raw SVM weights, identifying x1 ∧¬x2 and ¬x1/2 as the most important features.The raw SVM instead assigned its largest weights to x2 and x1 ∧x2, while the degree-2 kernel underfit the function.
- 3.1 Binary Data: FIRM based on true labels found all seven 2-tuples leading to true output, with only ¬x1 ∧x2 receiving a stronger false-value score.This provides a close approximation to the underlying truth, while combinations involving x3 received slightly negative values.
- 3.2 Gaussian Data: In continuous data, dimension two showed the strongest slope and discriminative power, whereas the third, noise-only dimension was identified as uninformative.The experiment used a linear SVM and modeled conditional expected scores for each dimension.
- 3.3 Sequence Data: For sequence data, FIRM identified GATTACA as most important at positions 20–50 and assigned significant importance to strings with edit distance 1.Raw feature weights gave random importance to mutated sequences and behaved erratically even for GATTACA.
- 3.3 Sequence Data: By modeling correlations among short substrings and longer motifs, FIRM recovered the ideal feature even when the exact consensus sequence was unreliable because of mutations.This behavior arose automatically from the feature correlations and required no additional domain knowledge beyond a uniform zeroth-order Markov distribution.
4 Summary and Conclusions
The paper presents FIRM as a general feature-ranking measure that evaluates arbitrary features through a trained scoring function while accounting for feature dependencies. Its applicability is broad, but computation depends on assumptions or estimates of the input distribution, which may not hold in real-world settings.
- Summary and Conclusions: FIRM generalizes POIMs from sequence analysis to arbitrary continuous features and imposes no restriction on the learning method.The measure is described as objective, universal across feature representations, and invariant to translation with reasonable rescaling invariance.
- Summary and Conclusions: FIRM can identify relevant features absent from the training data or not directly represented by the primary learning machine by using feature dependencies.This includes longer sequence motifs and other correlated features that the learner does not explicitly contain.
- Summary and Conclusions: FIRM depends on the input-feature distribution, which is generally unavailable, so practical computation requires assumptions or approximations.Under real-world violations of those assumptions, derived formulas can still be interpreted as estimates based on first- and second-order statistics.
- Summary and Conclusions: FIRM can be used with any learning framework and even without a prior learning step, although computed importance depends on the accuracy of the trained learning machine.Using training labels as scores usually gives results similar to using a learned function.
- Summary and Conclusions: The indirect procedure may improve results by smoothing label errors, extending labeled data to the full input space, and exploiting distributional information.The paper identifies understanding and exploiting these effects as an avenue for future research.