some notes on uncertainty in machine learning, particularly deep learning

1.11. uncertainty

1.11.1. basics

  • calibration - predicted probabilities should match real probabilities

    • platt scaling - given trained classifier and new calibration dataset, basically just fit a logistic regression from the classifier predictions -> labels

    • isotonic regression - nonparametric, requires more data than platt scaling

      • piecewise-constant non-decreasing function instead of logistic regression

  • confidence - predicted probability = confidence, max margin, entropy of predicted probabilities across classes

  • ensemble uncertainty - ensemble predictions yield uncertainty (e.g. variance within ensemble)

  • quantile regression - use quantile loss to penalize models differently + get confidence intervals

1.11.2. complementarity rejection learning

  • rejection learning - allow models to reject (not make a prediction) when they are not confidently accurate (chow 1957, cortes et al. 2016)

  • To Trust Or Not To Trust A Classifier (jiang, kim et al 2018) - find a trusted region of points based on nearest neighbor density (in some embedding space)

    • trust score uses density over some set of nearest neighbors

    • do clustering for each class - trust score = distance to once class’s cluster vs the other classes complementarity

  • complementarity - ML should focus on points hard for humans + seek human input on points hard for ML

    • note: goal of perception isn’t to learn categories but learn things that are associated with actions

  • Predict Responsibly: Improving Fairness and Accuracy by Learning to Defer (madras et al. 2018) - adaptive rejection learning - build on rejection learning considering the strengths/weaknesses of humans

  • Learning to Complement Humans (wilder et al. 2020) - 2 approaches for how to incorporate human input:

    • discriminative approach - jointly train predictive model and policy for deferring to human (witha cost for deferring)

    • decision-theroetic approach - train predictive model + policy jointly based on value of information (VOI)

    • do real-world experiments w/ humans to validate: scientific discovery (a galaxy classification task) and medical diagnosis (detection of breast cancer metastasis)

  • Gaining Free or Low-Cost Transparency with Interpretable Partial Substitute (wang, 2019) - given a black-box model, find a subset of the data for which predictions can be made using a simple rule-list (tong wang has a few papers like this)

1.11.3. outlier-detection

  • overview from sklearn

  • elliptic envelope - assume data is Gaussian and fit elliptic envelop (maybe robustly) to tell when data is an outlier

  • local outlier factor (breunig et al. 2000) - score based on nearest neighbor density

  • idea: gradients should be larger if you are on the image manifold

  • isolation forest (liu et al. 2008) - lower average number of random splits required to isolate a sample means more outlier

  • one-class svm - estimates the support of a high-dimensional distribution using a kernel (2 approaches:)

    • separate the data from the origin (with max margin between origin and points) (scholkopf et al. 2000)

    • find a sphere boundary around a dataset with the volume of the sphere minimized (tax & duin 2004)

  • detachment index (kuenzel 2019) - based on random forest

    • for covariate \(j\), detachment index \(d^j(x) = \sum_i^n w (x, X_i) \vert X_i^j - x^j \vert\)

      • \(w(x, X_i) = \underbrace{1 / T\sum_{t=1}^{T}}_{\text{average over T trees}} \frac{\overbrace{1\{ X_i \in L_t(x) \}}^{\text{is } X_i \text{ in the same leaf?}}}{\underbrace{\vert L_t(x) \vert}_{\text{num points in leaf}}}\) is \(X_i\) relevant to the point \(x\)?

1.11.4. bayesian approaches

  • epistemic uncertainty - uncertainty in the DNN model parameters

    • without good estimates of this, often get aleatoric uncertainty wrong (since \(p(y\vert x) = \int p(y \vert x, \theta) p(\theta \vert data) d\theta\)

  • aleatoric uncertainty - inherent and irreducible data noise (e.g. features contradict each other)

    • this can usually be gotten by predicting a distr. \(p(y \vert x)\) instead of a point estimate

    • ex. logistic reg. already does this

    • ex. regression - just predict mean and variance of Gaussian

  • gaussian processes

1.11.5. neural networks directly predict uncertainty nearest-neighbor methods ensemble approaches bayesian neural networks