Tyler Sypherd

Gowtham R. Kurri, Monica Welfert, Tyler Sypherd et al.

We prove a two-way correspondence between the min-max optimization of general CPE loss function GANs and the minimization of associated $f$-divergences. We then focus on $α$-GAN, defined via the $α$-loss, which interpolates several GANs (Hellinger, vanilla, Total Variation) and corresponds to the minimization of the Arimoto divergence. We show that the Arimoto divergences induced by $α$-GAN equivalently converge, for all $α\in \mathbb{R}_{>0}\cup\{\infty\}$. However, under restricted learning models and finite samples, we provide estimation bounds which indicate diverse GAN behavior as a function of $α$. Finally, we present empirical results on a toy dataset that highlight the practical utility of tuning the $α$ hyperparameter.

Tyler Sypherd, Nathan Stromberg, Richard Nock et al.

There is a growing need for models that are interpretable and have reduced energy and computational cost (e.g., in health care analytics and federated learning). Examples of algorithms to train such models include logistic regression and boosting. However, one challenge facing these algorithms is that they provably suffer from label noise; this has been attributed to the joint interaction between oft-used convex loss functions and simpler hypothesis classes, resulting in too much emphasis being placed on outliers. In this work, we use the margin-based $α$-loss, which continuously tunes between canonical convex and quasi-convex losses, to robustly train simple models. We show that the $α$ hyperparameter smoothly introduces non-convexity and offers the benefit of "giving up" on noisy training examples. We also provide results on the Long-Servedio dataset for boosting and a COVID-19 survey dataset for logistic regression, highlighting the efficacy of our approach across multiple relevant domains.

Tyler Sypherd, Richard Nock, Lalitha Sankar

Properness for supervised losses stipulates that the loss function shapes the learning algorithm towards the true posterior of the data generating distribution. Unfortunately, data in modern machine learning can be corrupted or twisted in many ways. Hence, optimizing a proper loss function on twisted data could perilously lead the learning algorithm towards the twisted posterior, rather than to the desired clean posterior. Many papers cope with specific twists (e.g., label/feature/adversarial noise), but there is a growing need for a unified and actionable understanding atop properness. Our chief theoretical contribution is a generalization of the properness framework with a notion called twist-properness, which delineates loss functions with the ability to "untwist" the twisted posterior into the clean posterior. Notably, we show that a nontrivial extension of a loss function called $α$-loss, which was first introduced in information theory, is twist-proper. We study the twist-proper $α$-loss under a novel boosting algorithm, called PILBoost, and provide formal and experimental results for this algorithm. Our overarching practical conclusion is that the twist-proper $α$-loss outperforms the proper $\log$-loss on several variants of twisted data.

Gowtham R. Kurri, Tyler Sypherd, Lalitha Sankar

We introduce a tunable GAN, called $α$-GAN, parameterized by $α\in (0,\infty]$, which interpolates between various $f$-GANs and Integral Probability Metric based GANs (under constrained discriminator set). We construct $α$-GAN using a supervised loss function, namely, $α$-loss, which is a tunable loss function capturing several canonical losses. We show that $α$-GAN is intimately related to the Arimoto divergence, which was first proposed by Österriecher (1996), and later studied by Liese and Vajda (2006). We also study the convergence properties of $α$-GAN. We posit that the holistic understanding that $α$-GAN introduces will have practical benefits of addressing both the issues of vanishing gradients and mode collapse.

Tyler Sypherd, Mario Diaz, Lalitha Sankar et al.

We analyze the optimization landscape of a recently introduced tunable class of loss functions called $α$-loss, $α\in (0,\infty]$, in the logistic model. This family encapsulates the exponential loss ($α= 1/2$), the log-loss ($α= 1$), and the 0-1 loss ($α= \infty$) and contains compelling properties that enable the practitioner to discern among a host of operating conditions relevant to emerging learning methods. Specifically, we study the evolution of the optimization landscape of $α$-loss with respect to $α$ using tools drawn from the study of strictly-locally-quasi-convex functions in addition to geometric techniques. We interpret these results in terms of optimization complexity via normalized gradient descent.

Tyler Sypherd, Mario Diaz, John Kevin Cava et al.

We introduce a tunable loss function called $α$-loss, parameterized by $α\in (0,\infty]$, which interpolates between the exponential loss ($α= 1/2$), the log-loss ($α= 1$), and the 0-1 loss ($α= \infty$), for the machine learning setting of classification. Theoretically, we illustrate a fundamental connection between $α$-loss and Arimoto conditional entropy, verify the classification-calibration of $α$-loss in order to demonstrate asymptotic optimality via Rademacher complexity generalization techniques, and build-upon a notion called strictly local quasi-convexity in order to quantitatively characterize the optimization landscape of $α$-loss. Practically, we perform class imbalance, robustness, and classification experiments on benchmark image datasets using convolutional-neural-networks. Our main practical conclusion is that certain tasks may benefit from tuning $α$-loss away from log-loss ($α= 1$), and to this end we provide simple heuristics for the practitioner. In particular, navigating the $α$ hyperparameter can readily provide superior model robustness to label flips ($α> 1$) and sensitivity to imbalanced classes ($α< 1$).

Tyler Sypherd, Mario Diaz, Lalitha Sankar et al.

We present $α$-loss, $α\in [1,\infty]$, a tunable loss function for binary classification that bridges log-loss ($α=1$) and $0$-$1$ loss ($α= \infty$). We prove that $α$-loss has an equivalent margin-based form and is classification-calibrated, two desirable properties for a good surrogate loss function for the ideal yet intractable $0$-$1$ loss. For logistic regression-based classification, we provide an upper bound on the difference between the empirical and expected risk at the empirical risk minimizers for $α$-loss by exploiting its Lipschitzianity along with recent results on the landscape features of empirical risk functions. Finally, we show that $α$-loss with $α= 2$ performs better than log-loss on MNIST for logistic regression.