Skip to content


Switch branches/tags

Name already in use

A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Are you sure you want to create this branch?

Latest commit


Git stats


Failed to load latest commit information.
Latest commit message
Commit time

Tilted Empirical Risk Minimization

This repository contains the implementation for the following papers

Tilted Empirical Risk Minimization, ICLR 2021

On Tilted Losses in Machine Learning: Theory and Applications, JMLR 2023

Empirical risk minimization (ERM) is typically designed to perform well on the average loss, which can result in estimators that are sensitive to outliers, generalize poorly, or treat subgroups unfairly. While many methods aim to address these problems individually, in this work, we explore them through a unified framework---tilted empirical risk minimization (TERM).

This repository contains the data, code, and experiments to reproduce our empirical results. We demonstrate that TERM can be used for a multitude of applications, such as enforcing fairness between subgroups, mitigating the effect of outliers, and handling class imbalance. TERM is not only competitive with existing solutions tailored to these individual problems, but can also enable entirely new applications, such as simultaneously addressing outliers and promoting fairness.

Getting started


As we apply TERM to a diverse set of real-world applications, the dependencies for different applications can be different.

  • if we mention that the code is based on other public codebases, then one needs to follow the same setup of those codebases.
  • otherwise, need the following dependencies (the latest versions will work):
    • python3
    • sklearn
    • numpy
    • matplotlib
    • colorsys
    • seaborn
    • scipy
    • cvxpy (optional)

Properties of TERM

Motivating examples

These figures illustrate TERM as a function of t: (a) finding a point estimate from a set of 2D samples, (b) linear regression with outliers, and (c) logistic regression with imbalanced classes. While positive values of t magnify outliers, negative values suppress them. Setting t=0 recovers the original ERM objective.

(How to generate these figures: cd TERM/toy_example & jupyter notebook , and directly run the three notebooks.)

A toy problem to visualize the solutions to TERM

TERM objectives for a squared loss problem with N=3. As t moves from - to +, t-tilted losses recover min-loss (t-->+), avg-loss (t=0), and max-loss (t-->+), and approximate median-loss (for some t). TERM is smooth for all finite t and convex for positive t.

(How to generate this figure: cd TERM/properties & jupyter notebook , and directly run the notebook.)

How to run the code for different applications

1. Robust regression

cd TERM/robust_regression
python --obj $OBJ --corrupt 1 --noise $NOISE

where $OBJ is the objective and $NOISE is the noise level (see code for options).

2. Robust classification

cd TERM/robust_classification

3. Mitigating noisy annotators

cd TERM/noisy_annotator/pytorch_resnet_cifar10
python --t -2  # TERM

4. Fair PCA

cd TERM/fair_pca
jupyter notebook

and directly run the notebook fair_pca_credit.ipynb.

  • built upon the public fair pca codebase
  • we directly extract the pre-processed Credit data dumped from the original matlab code, which are called data.csv, A.csv, and B.csv saved under TERM/fair_pca/multi-criteria-dimensionality-reduction-master/data/credit/.
  • dependencies: same as the fair pca code

5. Handling class imbalance

cd TERM/class_imbalance
python3 -m mnist.mnist_train_tilting --exp tilting  # TERM, common class=99.5%

6. Variance reduction for generalization

python --obj $OBJ $OTHER_PARAS  

where $OBJ is the objective, and $OTHER_PARAS$ are the hyperparameters associated with the objective (see code for options). We report how we select the hyperparameters along with all hyperparameter values in Appendix E of the paper. For instance, for TERM with t=50, run the following:

python --obj tilting --t 50  

7. Fair federated learning

cd TERM/fair_flearn
bash tilting 0 0 term_t0.1_seed0 > term_t0.1_seed0 2>&1 &

8. Hierarchical multi-objective tilting

cd TERM/hierarchical
python --imbalance 1 --corrupt 1 --obj tilting --t_in -2 --t_out 10  # TERM_sc
python --imbalance 1 --corrupt 1 --obj tilting --t_in 50 --t_out -2 # TERM_ca
  • TERM_{sc}: (sample level, class level)
  • TERM_{ca}: (class level, annotator level)


Please see the paper (1, 2) for more details of TERM as well as a complete list of related work.