A.2.5 Loss Function and Optimizer For all the experiments except N-MNIST in this work, we use cross entropy and stochastic gradient

Many real-world data are sequentially collected over time and often exhibit skewed class distributions, resulting in imbalanced data streams. While existing approaches have explored several strategies, such as resampling and reweighting, for imbalanced data stream learning, our work distinguishes itself by addressing the imbalance problem through training modification, particularly focusing on gradient descent techniques. We introduce the harmonized gradient descent (HGD) algorithm, which aims to equalize the norms of gradients across different classes. By ensuring the gradient norm balance, HGD mitigates under-fitting for minor classes and achieves balanced online learning. Notably, HGD operates in a streamlined implementation process, requiring no data-buffer, extra parameters, or prior knowledge, making it applicable to any learning models utilizing gradient descent for optimization. Theoretical analysis, based on a few common and mild assumptions, shows that HGD achieves a satisfied sub-linear regret bound. The proposed algorithm are compared with the commonly used online imbalance learning methods under several imbalanced data stream scenarios. Extensive experimental evaluations demonstrate the efficiency and effectiveness of HGD in learning imbalanced data streams.

Dynamic game theory offers a toolbox for formalizing and solving for both cooperative and non-cooperative strategies in multi-agent scenarios. However, the optimal configuration of such games remains largely unexplored. While there is existing literature on the parametrization of dynamic games, little research examines this parametrization from a strategic perspective where each agent's configuration choice is influenced by the decisions of others. In this work, we introduce the concept of a game of configuration, providing a framework for the strategic fine-tuning of differential games. We define a game of configuration as a two-stage game within the setting of finite-horizon, affine-quadratic, AQ, differential games. In the first stage, each player chooses their corresponding configuration parameter, which will impact their dynamics and costs in the second stage. We provide the subgame perfect solution concept and a method for computing first stage cost gradients over the configuration space. This then allows us to formulate a gradient-based method for searching for local solutions to the configuration game, as well as provide necessary conditions for equilibrium configurations over their downstream (second stage) trajectories. We conclude by demonstrating the effectiveness of our approach in example AQ systems, both zero-sum and general-sum.

Different from these gradient-based methods, we focus on the gradient-free methods in this paper. We are also aware of many recent works on the algorithmic design in the structured nonsmooth nonconvex optimization. Then, we proceed to prove the second statement. In this section, we present some technical lemmas for analyzing the convergence property of gradient-free method and its two-phase version. We also give the proofs of Theorem 3.2 and 3.4.

The vital part of every learning-based algorithm is an optimization procedure, e.g., Stochastic Gradient Descent.

Optimization is a key component for training machine learning models and has a strong impact on their generalization. In this paper, we consider a particular optimization method--the stochastic gradient Langevin dynamics (SGLD) algorithm--and investigate the generalization of models trained by SGLD.