In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms appearing in the ODEs penalize the two-norm of the loss gradients. We prove that the existence of similar implicit regularization in RMSProp and Adam depends on their hyperparameters and the training stage, but with a different "norm" involved: the corresponding ODE terms either penalize the (perturbed) one-norm of the loss gradients or, on the contrary, hinder its decrease (the latter case being typical). We also conduct numerical experiments and discuss how the proven facts can influence generalization.
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Stochastic gradient descent (SGD) and its variants are the main workhorses for solving large-scale optimization problems with nonconvex objective functions. Although the convergence of SGDs in the (strongly) convex case is well-understood, their convergence for nonconvex functions stands on weak mathematical foundations. Most existing studies on the nonconvex convergence of SGD show the complexity results based on either the minimum of the expected gradient norm or the functional sub-optimality gap (for functions with extra structural property) by searching the entire range of iterates. Hence the last iterations of SGDs do not necessarily maintain the same complexity guarantee. This paper shows that an $\epsilon$-stationary point exists in the final iterates of SGDs, given a large enough total iteration budget, $T$, not just anywhere in the entire range of iterates -- a much stronger result than the existing one. Additionally, our analyses allow us to measure the density of the $\epsilon$-stationary points in the final iterates of SGD, and we recover the classical $O(\frac{1}{\sqrt{T}})$ asymptotic rate under various existing assumptions on the objective function and the bounds on the stochastic gradient. As a result of our analyses, we addressed certain myths and legends related to the nonconvex convergence of SGD and posed some thought-provoking questions that could set new directions for research.
In this short communication, we describe the recent debate on whether the hazard function should be used for causal inference in time-to-event studies and consider three different potential outcomes frameworks (by Rubin, Robins, and Pearl, respectively) as well as use the single-world intervention graph to show mathematically that the hazard function has causal interpretations under all three frameworks. In addition, we argue that the hazard ratio over time can provide a useful interpretation in practical settings.
Multidimensional constellation shaping of up to 32 dimensions with different spectral efficiencies are compared through AWGN and fiber-optic simulations. The results show that no constellation is universal and the balance of required and effective SNRs should be jointly considered for the specific optical transmission scenario.
The study of the generalized Hamming weight of linear codes is a significant research topic in coding theory as it conveys the structural information of the codes and determines their performance in various applications. However, determining the generalized Hamming weights of linear codes, especially the weight hierarchy, is generally challenging. In this paper, we investigate the generalized Hamming weights of a class of linear code $\C$ over $\bF_q$, which is constructed from defining sets. These defining sets are either special simplicial complexes or their complements in $\bF_q^m$. We determine the complete weight hierarchies of these codes by analyzing the maximum or minimum intersection of certain simplicial complexes and all $r$-dimensional subspaces of $\bF_q^m$, where $1\leq r\leq {\rm dim}_{\bF_q}(\C)$.
We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability $\epsilon$, getting optimal constant factors in the leading terms in a number of different models. In the randomized model, 1) we give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error, at a small additive cost. This is an improvement over Newman's theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. 2) Using this we obtain a $(\log(n/\epsilon^2)+4)$-cost private-coin communication protocol that computes the $n$-bit Equality function, to error $\epsilon$. This improves upon the $\log(n/\epsilon^3)+O(1)$ upper bound implied by Newman's theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive $\log\log(1/\epsilon)+O(1)$. In the quantum model, 1) we exhibit a one-way protocol of cost $\log(n/\epsilon)+4$, that uses only pure states and computes the $n$-bit Equality function to error $\epsilon$. This bound was implicitly already shown by Nayak [PhD thesis'99]. 2) We show that any $\epsilon$-error one-way protocol for $n$-bit Equality that uses only pure states communicates at least $\log(n/\epsilon)-\log\log(1/\epsilon)-O(1)$ qubits. 3) We exhibit a one-way protocol of cost $\log(\sqrt{n}/\epsilon)+3$, that uses mixed states and computes the $n$-bit Equality function to error $\epsilon$. This is also tight up to an additive $\log\log(1/\epsilon)+O(1)$, which follows from Alon's result. 4) We study the number of EPR pairs required to be shared in an entanglement-assisted one-way protocol. Our upper bounds also yield upper bounds on the approximate rank and related measures of the Identity matrix.
While classical scaling, just like principal component analysis, is parameter-free, other methods for embedding multivariate data require the selection of one or several tuning parameters. This tuning can be difficult due to the unsupervised nature of the situation. We propose a simple, almost obvious, approach to supervise the choice of tuning parameter(s): minimize a notion of stress. We apply this approach to the selection of the patch size in a prototypical patch-stitching embedding method, both in the multidimensional scaling (aka network localization) setting and in the dimensionality reduction (aka manifold learning) setting. In our study, we uncover a new bias--variance tradeoff phenomenon.
Recently, machine learning of the branch and bound algorithm has shown promise in approximating competent solutions to NP-hard problems. In this paper, we utilize and comprehensively compare the outcomes of three neural networks--graph convolutional neural network (GCNN), GraphSAGE, and graph attention network (GAT)--to solve the capacitated vehicle routing problem. We train these neural networks to emulate the decision-making process of the computationally expensive Strong Branching strategy. The neural networks are trained on six instances with distinct topologies from the CVRPLIB and evaluated on eight additional instances. Moreover, we reduced the minimum number of vehicles required to solve a CVRP instance to a bin-packing problem, which was addressed in a similar manner. Through rigorous experimentation, we found that this approach can match or improve upon the performance of the branch and bound algorithm with the Strong Branching strategy while requiring significantly less computational time. The source code that corresponds to our research findings and methodology is readily accessible and available for reference at the following web address: //isotlaboratory.github.io/ml4vrp
As large language models (LLMs) become more prevalent, there is a growing need for new and improved quantization methods that can meet the computationalast layer demands of these modern architectures while maintaining the accuracy. In this paper, we present TEQ, a trainable equivalent transformation that preserves the FP32 precision of the model output while taking advantage of low-precision quantization, especially 3 and 4 bits weight-only quantization. The training process is lightweight, requiring only 1K steps and fewer than 0.1 percent of the original model's trainable parameters. Furthermore, the transformation does not add any computational overhead during inference. Our results are on-par with the state-of-the-art (SOTA) methods on typical LLMs. Our approach can be combined with other methods to achieve even better performance. The code is available at //github.com/intel/neural-compressor.
As artificial intelligence (AI) models continue to scale up, they are becoming more capable and integrated into various forms of decision-making systems. For models involved in moral decision-making, also known as artificial moral agents (AMA), interpretability provides a way to trust and understand the agent's internal reasoning mechanisms for effective use and error correction. In this paper, we provide an overview of this rapidly-evolving sub-field of AI interpretability, introduce the concept of the Minimum Level of Interpretability (MLI) and recommend an MLI for various types of agents, to aid their safe deployment in real-world settings.
Graph neural networks (GNNs) have been demonstrated to be a powerful algorithmic model in broad application fields for their effectiveness in learning over graphs. To scale GNN training up for large-scale and ever-growing graphs, the most promising solution is distributed training which distributes the workload of training across multiple computing nodes. However, the workflows, computational patterns, communication patterns, and optimization techniques of distributed GNN training remain preliminarily understood. In this paper, we provide a comprehensive survey of distributed GNN training by investigating various optimization techniques used in distributed GNN training. First, distributed GNN training is classified into several categories according to their workflows. In addition, their computational patterns and communication patterns, as well as the optimization techniques proposed by recent work are introduced. Second, the software frameworks and hardware platforms of distributed GNN training are also introduced for a deeper understanding. Third, distributed GNN training is compared with distributed training of deep neural networks, emphasizing the uniqueness of distributed GNN training. Finally, interesting issues and opportunities in this field are discussed.
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