Adam-like Algorithm with Smooth Clipping Attains Global Minima: Analysis Based on Ergodicity of Functional SDEs (2312.02182v1)

Abstract: In this paper, we prove that an Adam-type algorithm with smooth clipping approaches the global minimizer of the regularized non-convex loss function. Adding smooth clipping and taking the state space as the set of all trajectories, we can apply the ergodic theory of Markov semigroups for this algorithm and investigate its asymptotic behavior. The ergodic theory we establish in this paper reduces the problem of evaluating the convergence, generalization error and discretization error of this algorithm to the problem of evaluating the difference between two functional stochastic differential equations (SDEs) with different drift coefficients. As a result of our analysis, we have shown that this algorithm minimizes the the regularized non-convex loss function with errors of the form $n^{-1/2}$, $\eta^{1/4}$, $\beta^{-1} \log (\beta + 1)$ and $e^{- c t}$. Here, $c$ is a constant and $n$, $\eta$, $\beta$ and $t$ denote the size of the training dataset, learning rate, inverse temperature and time, respectively.

• The paper proves that an Adam-like algorithm with smooth clipping attains global minima through rigorous ergodic analysis of functional SDEs.
• It bridges gaps in non-convex optimization by evaluating convergence, generalization, and discretization errors in gradient-based methods.
• The study extends prior ergodicity results to cover two functional SDEs with different drift coefficients, enhancing our understanding of optimization dynamics.

In the field of machine learning, training models on complex, non-convex loss functions is a significant challenge. One algorithm often employed for this purpose is Stochastic Gradient Langevin Dynamics (SGLD), which has proven successful in attaining global minima even within non-convex settings. Building upon this, there is a family of gradient-based optimization algorithms, known as Adam-type algorithms, which include widely used methods such as Adam, AdaGrad, and RMSProp.

This paper adds to the rich tapestry of work around Adam-type algorithms, asserting a new theoretical evaluation of their performance. It explores the nuances of Adam, a variant known for leveraging past gradient information to improve convergence, and addresses a gap in existing literature: while current studies are promising, they fall short in guaranteeing convergence to global minima for non-convex objective functions.

The crux of the paper lies in the intra-play of ergodic theory and functional Stochastic Differential Equations (SDEs). By viewing state space as the entirety of trajectory sets, the paper cleverly circumvents the non-Markovian nature of Adam, applying ergodic theory to investigate asymptotic behaviors. This allows a comprehensive analysis of Adam-type algorithms and, importantly, extends prior ergodicity results to cover two functional SDEs with differing drift coefficients.

Through their rigorous analysis, the authors reveal that an Adam-like algorithm with smooth clipping shows global convergence to the global minimizer of a regularized, non-convex objective function. This refines the understanding not only of the specific algorithm at hand but could also influence the application of such techniques to other learning algorithms. Moreover, the authors provide assessments of convergence, generalization error, and discretization error in relation to Adam-type algorithms, further solidifying their theoretical standing.

In conclusion, the research showcases the power of marrying ergodic theory with the paper of Adam-like optimization algorithms. It is a worthwhile read for those intrigued by the optimization challenges stemming from complex, non-convex objective functions in machine learning. The paper's insights could pave the way for even more advanced and reliable algorithmic strategies in training deep learning models.