FROM OPTIMIZATION TO EQUILIBRATION: UNDERSTANDING AN EMERGING PARADIGM IN ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING

Many existing machine learning (ML) algorithms cannot be viewed as gradient descent on some single objective. The solution trajectories taken by these algorithms naturally exhibit rotation, sometimes forming cycles, a behavior that is not expected with (full-batch) gradient descent. However, these algorithms can be viewed more generally as solving for the equilibrium of a game with possibly multiple competing objectives. Moreover, some recent ML models, specifically generative adversarial networks (GANs) and its variants, are now explicitly formulated as equilibrium problems. Equilibrium problems present challenges beyond those encountered in optimization such as limit-cycles and chaotic attractors and are able to abstract away some of the difficulties encountered when training models like GANs. In this thesis, I aim to advance our understanding of equilibrium problems so as to improve state-of-the-art in GANs and related domains. In the following chapters, I will present work on designing a no-regret framework for solving monotone equilibrium problems in online or streaming settings (with applications to Reinforcement Learning), ensuring convergence when training a GAN to fit a normal distribution to data by Crossing-the-Curl, improving state-of-the-art image generation with techniques derived from theory, and borrowing tools from dynamical systems theory for analyzing the complex dynamics of GAN training

Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization

We propose the first, to our knowledge, loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches

Quantitative Analysis of Synaptic Release at the Photoreceptor Synapse

AbstractExocytosis from the rod photoreceptor is stimulated by submicromolar Ca2+ and exhibits an unusually shallow dependence on presynaptic Ca2+. To provide a quantitative description of the photoreceptor Ca2+ sensor for exocytosis, we tested a family of conventional and allosteric computational models describing the final Ca2+-binding steps leading to exocytosis. Simulations were fit to two measures of release, evoked by flash-photolysis of caged Ca2+: exocytotic capacitance changes from individual rods and postsynaptic currents of second-order neurons. The best simulations supported the occupancy of only two Ca2+ binding sites on the rod Ca2+ sensor rather than the typical four or five. For most models, the on-rates for Ca2+ binding and maximal fusion rate were comparable to those of other neurons. However, the off-rates for Ca2+ unbinding were unexpectedly slow. In addition to contributing to the high-affinity of the photoreceptor Ca2+ sensor, slow Ca2+ unbinding may support the fusion of vesicles located at a distance from Ca2+ channels. In addition, partial sensor occupancy due to slow unbinding may contribute to the linearization of the first synapse in vision

Feature Likelihood Score: Evaluating Generalization of Generative Models Using Samples

The past few years have seen impressive progress in the development of deep generative models capable of producing high-dimensional, complex, and photo-realistic data. However, current methods for evaluating such models remain incomplete: standard likelihood-based metrics do not always apply and rarely correlate with perceptual fidelity, while sample-based metrics, such as FID, are insensitive to overfitting, i.e., inability to generalize beyond the training set. To address these limitations, we propose a new metric called the Feature Likelihood Score (FLS), a parametric sample-based score that uses density estimation to provide a comprehensive trichotomic evaluation accounting for novelty (i.e., different from the training samples), fidelity, and diversity of generated samples. We empirically demonstrate the ability of FLS to identify specific overfitting problem cases, where previously proposed metrics fail. We also extensively evaluate FLS on various image datasets and model classes, demonstrating its ability to match intuitions of previous metrics like FID while offering a more comprehensive evaluation of generative models