Benign overfitting in leaky ReLU networks with moderate input dimension

The problem of benign overfitting asks whether it is possible for a model to perfectly fit noisy training data and still generalize well. We study benign overfitting in two-layer leaky ReLU networks trained with the hinge loss on a binary classification task. We consider input data which can be decomposed into the sum of a common signal and a random noise component, which lie on subspaces orthogonal to one another. We characterize conditions on the signal to noise ratio (SNR) of the model parameters giving rise to benign versus non-benign, or harmful, overfitting: in particular, if the SNR is high then benign overfitting occurs, conversely if the SNR is low then harmful overfitting occurs. We attribute both benign and non-benign overfitting to an approximate margin maximization property and show that leaky ReLU networks trained on hinge loss with Gradient Descent (GD) satisfy this property. In contrast to prior work we do not require near orthogonality conditions on the training data: notably, for input dimension $d$ and training sample size $n$, while prior work shows asymptotically optimal error when $d = \Omega(n^2 \log n)$, here we require only $d = \Omega\left(n \log \frac{1}{\epsilon}\right)$ to obtain error within $\epsilon$ of optimal.

Training shallow ReLU networks on noisy data using hinge loss: when do we overfit and is it benign?

We study benign overfitting in two-layer ReLU networks trained using gradient descent and hinge loss on noisy data for binary classification. In particular, we consider linearly separable data for which a relatively small proportion of labels are corrupted or flipped. We identify conditions on the margin of the clean data that give rise to three distinct training outcomes: benign overfitting, in which zero loss is achieved and with high probability test data is classified correctly; overfitting, in which zero loss is achieved but test data is misclassified with probability lower bounded by a constant; and non-overfitting, in which clean points, but not corrupt points, achieve zero loss and again with high probability test data is classified correctly. Our analysis provides a fine-grained description of the dynamics of neurons throughout training and reveals two distinct phases: in the first phase clean points achieve close to zero loss, in the second phase clean points oscillate on the boundary of zero loss while corrupt points either converge towards zero loss or are eventually zeroed by the network. We prove these results using a combinatorial approach that involves bounding the number of clean versus corrupt updates across these phases of training.

Detecting and Mitigating Indirect Stereotypes in Word Embeddings

Societal biases in the usage of words, including harmful stereotypes, are frequently learned by common word embedding methods. These biases manifest not only between a word and an explicit marker of its stereotype, but also between words that share related stereotypes. This latter phenomenon, sometimes called "indirect bias,'' has resisted prior attempts at debiasing. In this paper, we propose a novel method called Biased Indirect Relationship Modification (BIRM) to mitigate indirect bias in distributional word embeddings by modifying biased relationships between words before embeddings are learned. This is done by considering how the co-occurrence probability of a given pair of words changes in the presence of words marking an attribute of bias, and using this to average out the effect of a bias attribute. To evaluate this method, we perform a series of common tests and demonstrate that measures of bias in the word embeddings are reduced in exchange for minor reduction in the semantic quality of the embeddings. In addition, we conduct novel tests for measuring indirect stereotypes by extending the Word Embedding Association Test (WEAT) with new test sets for indirect binary gender stereotypes. With these tests, we demonstrate the presence of more subtle stereotypes not addressed by previous work. The proposed method is able to reduce the presence of some of these new stereotypes, serving as a crucial next step towards non-stereotyped word embeddings.