Data Sampling Affects the Complexity of Online SGD over Dependent Data

Conventional machine learning applications typically assume that data samples are independently and identically distributed (i.i.d.). However, practical scenarios often involve a data-generating process that produces highly dependent data samples, which are known to heavily bias the stochastic optimization process and slow down the convergence of learning. In this paper, we conduct a fundamental study on how different stochastic data sampling schemes affect the sample complexity of online stochastic gradient descent (SGD) over highly dependent data. Specifically, with a $\phi$-mixing model of data dependence, we show that online SGD with proper periodic data-subsampling achieves an improved sample complexity over the standard online SGD in the full spectrum of the data dependence level. Interestingly, even subsampling a subset of data samples can accelerate the convergence of online SGD over highly dependent data. Moreover, we show that online SGD with mini-batch sampling can further substantially improve the sample complexity over online SGD with periodic data-subsampling over highly dependent data. Numerical experiments validate our theoretical results.

A Fast and Convergent Proximal Algorithm for Regularized Nonconvex and Nonsmooth Bi-level Optimization

Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from a high computation complexity in nonconvex bi-level optimization. In this work, we study a proximal gradient-type algorithm that adopts the approximate implicit differentiation (AID) scheme for nonconvex bi-level optimization with possibly nonconvex and nonsmooth regularizers. In particular, the algorithm applies the Nesterov's momentum to accelerate the computation of the implicit gradient involved in AID. We provide a comprehensive analysis of the global convergence properties of this algorithm through identifying its intrinsic potential function. In particular, we formally establish the convergence of the model parameters to a critical point of the bi-level problem, and obtain an improved computation complexity $\mathcal{O}(\kappa^{3.5}\epsilon^{-2})$ over the state-of-the-art result. Moreover, we analyze the asymptotic convergence rates of this algorithm under a class of local nonconvex geometries characterized by a {\L}ojasiewicz-type gradient inequality. Experiment on hyper-parameter optimization demonstrates the effectiveness of our algorithm.

Delving into the Estimation Shift of Batch Normalization in a Network

Batch normalization (BN) is a milestone technique in deep learning. It normalizes the activation using mini-batch statistics during training but the estimated population statistics during inference. This paper focuses on investigating the estimation of population statistics. We define the estimation shift magnitude of BN to quantitatively measure the difference between its estimated population statistics and expected ones. Our primary observation is that the estimation shift can be accumulated due to the stack of BN in a network, which has detriment effects for the test performance. We further find a batch-free normalization (BFN) can block such an accumulation of estimation shift. These observations motivate our design of XBNBlock that replace one BN with BFN in the bottleneck block of residual-style networks. Experiments on the ImageNet and COCO benchmarks show that XBNBlock consistently improves the performance of different architectures, including ResNet and ResNeXt, by a significant margin and seems to be more robust to distribution shift.

Sense Embeddings are also Biased--Evaluating Social Biases in Static and Contextualised Sense Embeddings

Sense embedding learning methods learn different embeddings for the different senses of an ambiguous word. One sense of an ambiguous word might be socially biased while its other senses remain unbiased. In comparison to the numerous prior work evaluating the social biases in pretrained word embeddings, the biases in sense embeddings have been relatively understudied. We create a benchmark dataset for evaluating the social biases in sense embeddings and propose novel sense-specific bias evaluation measures. We conduct an extensive evaluation of multiple static and contextualised sense embeddings for various types of social biases using the proposed measures. Our experimental results show that even in cases where no biases are found at word-level, there still exist worrying levels of social biases at sense-level, which are often ignored by the word-level bias evaluation measures.

Listing Maximal k-Plexes in Large Real-World Graphs

Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. Clique, as the densest model, has been widely investigated. However, in practice, communities rarely form as cliques for various reasons, e.g., data noise. Therefore, $k$-plex, -- graph with each vertex adjacent to all but at most $k$ vertices, is introduced as a relaxed version of clique. Often, to better simulate cohesive communities, an emphasis is placed on connected $k$-plexes with small $k$. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O^*(\gamma^D)$ time for each constant $k$, where $\gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less than the vertex number $n$ in real-word graphs. Compared to the trivial bound of $2^n$, the improvement is significant, and our bound is better than all previously known results. In practice, we further use several techniques to accelerate listing $k$-plexes of a given size, such as structural-based prune rules, cache-efficient data structures, and parallel techniques. All these together result in a very practical algorithm. Empirical results show that our approach outperforms the state-of-the-art solutions by up to orders of magnitude.

Single-shot Hyper-parameter Optimization for Federated Learning: A General Algorithm & Analysis

We address the relatively unexplored problem of hyper-parameter optimization (HPO) for federated learning (FL-HPO). We introduce Federated Loss SuRface Aggregation (FLoRA), a general FL-HPO solution framework that can address use cases of tabular data and any Machine Learning (ML) model including gradient boosting training algorithms and therefore further expands the scope of FL-HPO. FLoRA enables single-shot FL-HPO: identifying a single set of good hyper-parameters that are subsequently used in a single FL training. Thus, it enables FL-HPO solutions with minimal additional communication overhead compared to FL training without HPO. We theoretically characterize the optimality gap of FL-HPO, which explicitly accounts for the heterogeneous non-IID nature of the parties' local data distributions, a dominant characteristic of FL systems. Our empirical evaluation of FLoRA for multiple ML algorithms on seven OpenML datasets demonstrates significant model accuracy improvements over the considered baseline, and robustness to increasing number of parties involved in FL-HPO training.

On the Convergence of Gradient Extrapolation Methods for Unbalanced Optimal Transport

We study the Unbalanced Optimal Transport (UOT) between two measures of possibly different masses with at most $n$ components, where marginal constraints of the standard Optimal Transport (OT) are relaxed via Kullback-Leibler divergence with regularization factor $\tau$. We propose a novel algorithm based on Gradient Extrapolation Method (GEM-UOT) to find an $\varepsilon$-approximate solution to the UOT problem in $O\big( \kappa n^2 \log\big(\frac{\tau n}{\varepsilon}\big) \big)$, where $\kappa$ is the condition number depending on only the two input measures. Compared to the only known complexity ${O}\big(\tfrac{\tau n^2 \log(n)}{\varepsilon} \log\big(\tfrac{\log(n)}{{\varepsilon}}\big)\big)$ for solving the UOT problem via the Sinkhorn algorithm, ours is better in $\varepsilon$ and lifts Sinkhorn's linear dependence on $\tau$, which hindered its practicality to approximate the standard OT via UOT. Our proof technique is based on a novel dual formulation of the squared $\ell_2$-norm regularized UOT objective, which is of independent interest and also leads to a new characterization of approximation error between UOT and OT in terms of both the transportation plan and transport distance. To this end, we further present an algorithm, based on GEM-UOT with fine tuned $\tau$ and a post-process projection step, to find an $\varepsilon$-approximate solution to the standard OT problem in $O\big( \kappa n^2 \log\big(\frac{ n}{\varepsilon}\big) \big)$, which is a new complexity in the literature of OT. Extensive experiments on synthetic and real datasets validate our theories and demonstrate the favorable performance of our methods in practice.

Accelerated Proximal Alternating Gradient-Descent-Ascent for Nonconvex Minimax Machine Learning

Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aim to solve a nonconvex minimax optimization problem. However, the existing studies show that it suffers from a high computation complexity in nonconvex minimax optimization. In this paper, we develop a single-loop and fast AltGDA-type algorithm that leverages proximal gradient updates and momentum acceleration to solve regularized nonconvex minimax optimization problems. By identifying the intrinsic Lyapunov function of this algorithm, we prove that it converges to a critical point of the nonconvex minimax optimization problem and achieves a computation complexity $\mathcal{O}(\kappa^{1.5}\epsilon^{-2})$, where $\epsilon$ is the desired level of accuracy and $\kappa$ is the problem's condition number. Such a computation complexity improves the state-of-the-art complexities of single-loop GDA and AltGDA algorithms (see the summary of comparison in Table 1). We demonstrate the effectiveness of our algorithm via an experiment on adversarial deep learning.