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1. Efficient Multiple Objective Optimization for Fair Misinformation Detection

   https://doi.org/10.1109/BigData55660.2022.10020909  

   Enouen, Eric ; Mathesius, Katja ; Wang, Sean ; Carr, Arielle ; Xie, Sihong ( December 2022 , 2022 IEEE International Conference on Big Data (Big Data))

   Multiple-objective optimization (MOO) aims to simultaneously optimize multiple conflicting objectives and has found important applications in machine learning, such as simultaneously minimizing classification and fairness losses. At an optimum, further optimizing one objective will necessarily increase at least another objective, and decision-makers need to comprehensively explore multiple optima to pin-point one final solution. We address the efficiency of exploring the Pareto front that contains all optima. First, stochastic multi-gradient descent (SMGD) takes time to converge to the Pareto front with large neural networks and datasets. Instead, we explore the Pareto front as a manifold from a few initial optima, based on a predictor-corrector method. Second, for each exploration step, the predictor iteratively solves a large-scale linear system that scales quadratically in the number of model parameters, and requires one backpropagation to evaluate a second-order Hessian-vector product per iteration of the solver. We propose a Gauss-Newton approximation that scales linearly, and that requires only first-order inner-product per iteration. T hird, we explore different linear system solvers, including the MINRES and conjugate gradient methods for approximately solving the linear systems. The innovations make predictor-corrector efficient for large networks and datasets. Experiments on a fair misinformation detection task show that 1) the predictor-corrector method can find Pareto fronts better than or similar to SMGD with less time, and 2) the proposed first-order method does not harm the quality of the Pareto front identified by the second-order method, while further reducing running time. 

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2. Global Convergence and Stability of Stochastic Gradient Descent

   Patel, Vivak ; Zhang, Shushu ; Tian, Bowen ( December 2022 , Conference on Neural Information Processing Systems)

   In machine learning, stochastic gradient descent (SGD) is widely deployed to train models using highly non-convex objectives with equally complex noise models. Unfortunately, SGD theory often makes restrictive assumptions that fail to capture the non-convexity of real problems, and almost entirely ignore the complex noise models that exist in practice. In this work, we demonstrate the restrictiveness of these assumptions using three canonical models in machine learning. Then, we develop novel theory to address this shortcoming in two ways. First, we establish that SGD’s iterates will either globally converge to a stationary point or diverge under nearly arbitrary nonconvexity and noise models. Under a slightly more restrictive assumption on the joint behavior of the non-convexity and noise model that generalizes current assumptions in the literature, we show that the objective function cannot diverge, even if the iterates diverge. As a consequence of our results, SGD can be applied to a greater range of stochastic optimization problems with confidence about its global convergence behavior and stability. 

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3. Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stability.

   Damian, Alex ; Nichani, Eshaan ; Lee, Jason D. ( May 2023 , ICLR 2023)

   Traditional analyses of gradient descent show that when the largest eigenvalue of the Hessian, also known as the sharpness S(θ), is bounded by 2/η, training is "stable" and the training loss decreases monotonically. Recent works, however, have observed that this assumption does not hold when training modern neural networks with full batch or large batch gradient descent. Most recently, Cohen et al. (2021) observed two important phenomena. The first, dubbed progressive sharpening, is that the sharpness steadily increases throughout training until it reaches the instability cutoff 2/η. The second, dubbed edge of stability, is that the sharpness hovers at 2/η for the remainder of training while the loss continues decreasing, albeit non-monotonically. We demonstrate that, far from being chaotic, the dynamics of gradient descent at the edge of stability can be captured by a cubic Taylor expansion: as the iterates diverge in direction of the top eigenvector of the Hessian due to instability, the cubic term in the local Taylor expansion of the loss function causes the curvature to decrease until stability is restored. This property, which we call self-stabilization, is a general property of gradient descent and explains its behavior at the edge of stability. A key consequence of self-stabilization is that gradient descent at the edge of stability implicitly follows projected gradient descent (PGD) under the constraint S(θ)≤2/η. Our analysis provides precise predictions for the loss, sharpness, and deviation from the PGD trajectory throughout training, which we verify both empirically in a number of standard settings and theoretically under mild conditions. Our analysis uncovers the mechanism for gradient descent's implicit bias towards stability. 

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4. Improved Convergence in High Probability of Clipped Gradient Methods with Heavy Tailed Noise

   Nguyen, Ta Duy ; Nguyen, Thien Hang ; Ene, Alina ; Nguyen, Huy ( September 2023 , Advances in Neural Information Processing Systems)

   In this work, we study the convergence \emph{in high probability} of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1 more » « less
5. Improved Convergence in High Probability of Clipped Gradient Methods with Heavy Tailed Noise

   Nguyen, Ta Duy ; Nguyen, Thien Hang ; Ene, Alina ; Nguyen, Huy L ( December 2023 , Advances in neural information processing systems)

   In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1< p \leq 2$. Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of $T$, where $T$ is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on $T$ in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using existing techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters. 

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