The naive approach: Take the derivative of the loss function which is an average of the losses calculated on every example in the dataset, a full update is powerful but it has some drawbacks… Drawbacks: . Can be extremely slow as we need to pass over the entire dataset to make a single update. . If there is a lot of redundancy in the training data, the benefit of a full update is very low The extreme approach Consider only a single example at a time and update steps based on one observation at a time, does that remind you of something?? Yes, it's the stochastic gradient descent algorithm or SGD. It can be effective even in large datasets but it also has some drawbacks… Drawbacks: . It can take longer to process one sample at a time compared to a full batch .

The use of visual sensors is flourishing, driven among others by the several applications in detection and prevention of crimes or dangerous events. While the problem of optimal camera placement for total coverage has been solved for a decade or so, that of the arrangement of cameras maximizing the recognition of objects "in-transit" is still open. The objective of this paper is to attack this problem by providing an adversarial method of proven optimality based on the resolution of Hamilton-Jacobi equations. The problem is attacked by first assuming the perspective of an adversary, i.e. computing explicitly the path minimizing the probability of detection and the quality of reconstruction. Building on this result, we introduce an optimality measure for camera configurations and perform a simulated annealing algorithm to find the optimal camera placement.

In large-scale machine learning, recent works have studied the effects of compressing gradients in stochastic optimization in order to alleviate the communication bottleneck. These works have collectively revealed that stochastic gradient descent (SGD) is robust to structured perturbations such as quantization, sparsification, and delays. Perhaps surprisingly, despite the surge of interest in large-scale, multi-agent reinforcement learning, almost nothing is known about the analogous question: Are common reinforcement learning (RL) algorithms also robust to similar perturbations? In this paper, we investigate this question by studying a variant of the classical temporal difference (TD) learning algorithm with a perturbed update direction, where a general compression operator is used to model the perturbation. Our main technical contribution is to show that compressed TD algorithms, coupled with an error-feedback mechanism used widely in optimization, exhibit the same non-asymptotic theoretical guarantees as their SGD counterparts. We then extend our results significantly to nonlinear stochastic approximation algorithms and multi-agent settings. In particular, we prove that for multi-agent TD learning, one can achieve linear convergence speedups in the number of agents while communicating just $\tilde{O}(1)$ bits per agent at each time step. Our work is the first to provide finite-time results in RL that account for general compression operators and error-feedback in tandem with linear function approximation and Markovian sampling. Our analysis hinges on studying the drift of a novel Lyapunov function that captures the dynamics of a memory variable introduced by error feedback.

When implementing the gradient descent method in low precision, the employment of stochastic rounding schemes helps to prevent stagnation of convergence caused by the vanishing gradient effect. Unbiased stochastic rounding yields zero bias by preserving small updates with probabilities proportional to their relative magnitudes. This study provides a theoretical explanation for the stagnation of the gradient descent method in low-precision computation. Additionally, we propose two new stochastic rounding schemes that trade the zero bias property with a larger probability to preserve small gradients. Our methods yield a constant rounding bias that, on average, lies in a descent direction. For convex problems, we prove that the proposed rounding methods typically have a beneficial effect on the convergence rate of gradient descent. We validate our theoretical analysis by comparing the performances of various rounding schemes when optimizing a multinomial logistic regression model and when training a simple neural network with an 8-bit floating-point format.

Federated learning enables isolated clients to train a shared model collaboratively by aggregating the locally-computed gradient updates. However, privacy information could be leaked from uploaded gradients and be exposed to malicious attackers or an honest-but-curious server. Although the additive homomorphic encryption technique guarantees the security of this process, it brings unacceptable computation and communication burdens to FL participants. To mitigate this cost of secure aggregation and maintain the learning performance, we propose a new framework called Encoded Gradient Aggregation (\emph{EGA}). In detail, EGA first encodes local gradient updates into an encoded domain with injected noises in each client before the aggregation in the server. Then, the encoded gradients aggregation results can be recovered for the global model update via a decoding function. This scheme could prevent the raw gradients of a single client from exposing on the internet and keep them unknown to the server. EGA could provide optimization and communication benefits under different noise levels and defend against gradient leakage. We further provide a theoretical analysis of the approximation error and its impacts on federated optimization. Moreover, EGA is compatible with the most federated optimization algorithms. We conduct intensive experiments to evaluate EGA in real-world federated settings, and the results have demonstrated its efficacy.

Let $V_* : \mathbb{R}^d \to \mathbb{R}$ be some (possibly non-convex) potential function, and consider the probability measure $\pi \propto e^{-V_*}$. When $\pi$ exhibits multiple modes, it is known that sampling techniques based on Wasserstein gradient flows of the Kullback-Leibler (KL) divergence (e.g. Langevin Monte Carlo) suffer poorly in the rate of convergence, where the dynamics are unable to easily traverse between modes. In stark contrast, the work of Lu et al. (2019; 2022) has shown that the gradient flow of the KL with respect to the Fisher-Rao (FR) geometry exhibits a convergence rate to $\pi$ is that \textit{independent} of the potential function. In this short note, we complement these existing results in the literature by providing an explicit expansion of $\text{KL}(\rho_t^{\text{FR}}\|\pi)$ in terms of $e^{-t}$, where $(\rho_t^{\text{FR}})_{t\geq 0}$ is the FR gradient flow of the KL divergence. In turn, we are able to provide a clean asymptotic convergence rate, where the burn-in time is guaranteed to be finite. Our proof is based on observing a similarity between FR gradient flows and simulated annealing with linear scaling, and facts about cumulant generating functions. We conclude with simple synthetic experiments that demonstrate our theoretical findings are indeed tight. Based on our numerics, we conjecture that the asymptotic rates of convergence for Wasserstein-Fisher-Rao gradient flows are possibly related to this expansion in some cases.

As machine learning becomes more widespread throughout society, aspects including data privacy and fairness must be carefully considered, and are crucial for deployment in highly regulated industries. Unfortunately, the application of privacy enhancing technologies can worsen unfair tendencies in models. In particular, one of the most widely used techniques for private model training, differentially private stochastic gradient descent (DPSGD), frequently intensifies disparate impact on groups within data. In this work we study the fine-grained causes of unfairness in DPSGD and identify gradient misalignment due to inequitable gradient clipping as the most significant source. This observation leads us to a new method for reducing unfairness by preventing gradient misalignment in DPSGD. The increasingly widespread use of machine learning throughout society has brought into focus social, ethical, and legal considerations surrounding its use. In highly regulated industries, such as healthcare and banking, regional laws and regulations require data collection and analysis to respect the privacy of individuals. Other regulations focus on the fairness of how models are developed and used. As machine learning is progressively adopted in highly regulated industries, the privacy and fairness aspects of models must be considered at all stages of the modelling lifecycle. There are many privacy enhancing technologies including differential privacy (Dwork et al., 2006), federated learning (McMahan et al., 2017), secure multiparty computation (Yao, 1986), and homomorphic encryption (Gentry, 2009) that are used separately or jointly to protect the privacy of individuals whose data is used for machine learning (Choquette-Choo et al., 2020; Adnan et al., 2022; Kalra et al., 2021).

Time-varying stochastic optimization problems frequently arise in machine learning practice (e.g. gradual domain shift, object tracking, strategic classification). Although most problems are solved in discrete time, the underlying process is often continuous in nature. We exploit this underlying continuity by developing predictor-corrector algorithms for time-varying stochastic optimizations. We provide error bounds for the iterates, both in presence of pure and noisy access to the queries from the relevant derivatives of the loss function. Furthermore, we show (theoretically and empirically in several examples) that our method outperforms non-predictor corrector methods that do not exploit the underlying continuous process.

In this paper, we propose energy-based sample adaptation at test time for domain generalization. Where previous works adapt their models to target domains, we adapt the unseen target samples to source-trained models. To this end, we design a discriminative energy-based model, which is trained on source domains to jointly model the conditional distribution for classification and data distribution for sample adaptation. The model is optimized to simultaneously learn a classifier and an energy function. To adapt target samples to source distributions, we iteratively update the samples by energy minimization with stochastic gradient Langevin dynamics. Moreover, to preserve the categorical information in the sample during adaptation, we introduce a categorical latent variable into the energy-based model. The latent variable is learned from the original sample before adaptation by variational inference and fixed as a condition to guide the sample update. Experiments on six benchmarks for classification of images and microblog threads demonstrate the effectiveness of our proposal. Deep neural networks are vulnerable to domain shifts and suffer from lack of generalization on test samples that do not resemble the ones in the training distribution (Recht et al., 2019; Zhou et al., 2021; Krueger et al., 2021; Shen et al., 2022). To deal with the domain shifts, domain generalization has been proposed (Muandet et al., 2013; Gulrajani & Lopez-Paz, 2020; Cha et al., 2021). Domain generalization strives to learn a model exclusively on source domains in order to generalize well on unseen target domains. The major challenge stems from the large domain shifts and the unavailability of any target domain data during training. To address the problem, domain invariant learning has been widely studied, e.g., (Motiian et al., 2017; Zhao et al., 2020; Nguyen et al., 2021), based on the assumption that invariant representations obtained on source domains are also valid for unseen target domains. However, since the target data is inaccessible during training, it is likely an "adaptivity gap" (Dubey et al., 2021) exists between representations from the source and target domains. Therefore, recent works try to adapt the classification model with target samples at test time by further fine-tuning model parameters (Sun et al., 2020; Wang et al., 2021) or by introducing an extra network module for adaptation (Dubey et al., 2021). Rather than adapting the model to target domains, Xiao et al. (2022) adapt the classifier for each sample at test time. Nevertheless, a single sample would not be able to adjust the whole model due to the large number of model parameters and the limited information contained in the sample.

An increasing number of data science and machine learning problems rely on computation with tensors, which better capture the multi-way relationships and interactions of data than matrices. When tapping into this critical advantage, a key challenge is to develop computationally efficient and provably correct algorithms for extracting useful information from tensor data that are simultaneously robust to corruptions and ill-conditioning. This paper tackles tensor robust principal component analysis (RPCA), which aims to recover a low-rank tensor from its observations contaminated by sparse corruptions, under the Tucker decomposition. To minimize the computation and memory footprints, we propose to directly recover the low-dimensional tensor factors -- starting from a tailored spectral initialization -- via scaled gradient descent (ScaledGD), coupled with an iteration-varying thresholding operation to adaptively remove the impact of corruptions. Theoretically, we establish that the proposed algorithm converges linearly to the true low-rank tensor at a constant rate that is independent with its condition number, as long as the level of corruptions is not too large. Empirically, we demonstrate that the proposed algorithm achieves better and more scalable performance than state-of-the-art matrix and tensor RPCA algorithms through synthetic experiments and real-world applications.