Online nonconvex optimization has been an active area of research recently. Previous studies either considered the global regret with full information about the objective functions, or studied the local regret with window-smoothed objective functions, which required access to unlimited number of gradient oracles per time step. In this paper, we focus on the more challenging and practical setting, where access to only a single oracle is allowed per time step, and take the local regret of the original (i.e., unsmoothed) objective functions as the performance metric. Specifically, for both settings respectively with a single exact and stochastic gradient oracle feedback, we derive lower bounds on the local regret and show that the classical online (stochastic) gradient descent algorithms are optimal. Moreover, for the more challenging setting with a single function value oracle feedback, we develop an online algorithm based on a one-point running difference gradient estimator, and show that such an algorithm achieves a local regret that a generic stochastic gradient oracle can best achieve.
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Optimization
With the widespread adoption of machine learning systems, the need to curtail their behavior has become increasingly apparent. This is evidenced by recent advancements towards developing models that satisfy robustness, safety, and fairness requirements. These requirements can be imposed (with generalization guarantees) by formulating constrained learning problems that can then be tackled by dual ascent algorithms. Yet, though these algorithms converge in objective value, even in non-convex settings, they cannot guarantee that their outcome is feasible. Doing so requires randomizing over all iterates, which is impractical in virtually any modern applications. Still, final iterates have been observed to perform well in practice. In this work, we address this gap between theory and practice by characterizing the constraint violation of Lagrangian minimizers associated with optimal dual variables, despite lack of convexity. To do this, we leverage the fact that non-convex, finite-dimensional constrained learning problems can be seen as parametrizations of convex, functional problems. Our results show that rich parametrizations effectively mitigate the issue of feasibility in dual methods, shedding light on prior empirical successes of dual learning. We illustrate our findings in fair learning tasks.
Optimization · Convex
It has been demonstrated that the training problem for a variety of (non) linear two-layer neural networks (such as two-layer perceptrons, convolutional networks, and self-attention) can be posed as equivalent convex optimization problems, with an induced regularizer which encourages low rank. However, this regularizer becomes prohibitively expensive to compute at moderate scales, impeding training convex neural networks. To this end, we propose applying the Burer-Monteiro factorization to convex neural networks, which for the first time enables a Burer-Monteiro perspective on neural networks with non-linearities. This factorization leads to an equivalent yet computationally tractable non-convex alternative with no spurious local minima. We develop a novel relative optimality bound of stationary points of the Burer-Monteiro factorization, providing verifiable conditions under which any stationary point is a global optimum. Further, for the first time, we show that linear self-attention with sufficiently many heads has no spurious local minima. Our experiments validate the novel relative optimality bound and the utility of the Burer-Monteiro factorization for scaling convex neural networks.
Miscellaneous Aspects of Machine Learning · Everything Else
Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the $\ell_1$ regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far reaching generalization of the Lasso’s celebrated ``Irrepresentability Condition’’. To provide additional insight into this condition (consequently on the types of recoverable costs) we work out in detail the Gaussian case. Surprisingly, varying the entropic regularizer provides evidence that the Gaussian iOT interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation, an important problem in ML.
Many machine learning methods operate by inverting a neural network at inference time, which has become a popular technique for solving inverse problems in computer vision, robotics, and graphics. However, these methods often involve gradient descent through a highly non-convex loss landscape, causing the optimization process to be unstable and slow. We introduce a method that learns a loss landscape where gradient descent is efficient, bringing massive improvement and acceleration to the inversion process. We demonstrate this advantage on a number of methods for both generative and discriminative tasks, including GAN inversion, adversarial defense, and 3D human pose reconstruction.
We introduce a novel method to estimate the essential matrix for two-view Structure from Motion (SfM). We show that every 3 by 3 essential matrix can be embedded in a 4 by 4 rotation, having its bottom right entry fixed to zero; we call the latter the quintessential matrix. This embedding leads to rich relations with the space of 4-D rotations, quaternions, and the classical twisted-pair ambiguity in two-view SfM. We use this structure to derive a succession of semidefinite relaxations that require fewer parameters than the existing non-minimal solvers and yield faster convergence with certifiable optimality. We then exploit the low-rank geometry of these relaxations to reduce them to an equivalent optimization on a Riemannian manifold and solve them via the Riemannian Staircase method. The experimental evaluation confirms that our algorithm always finds the globally optimal solution and outperforms the existing non-minimal methods. We make our implementations open source.
We describe a method to parse a complex, cluttered indoor scene into primitives which offer a parsimonious abstraction of scene structure. Our primitives are simple convexes. Our method uses a learned regression procedure to parse a scene into a fixed number of convexes from RGBD input, and can optionally accept segmentations to improve the decomposition. The result is then polished with a descent method which adjusts the convexes to produce a very good fit, and greedily removes superfluous primitives. Because the entire scene is parsed, we can evaluate using traditional depth, normal, and segmentation error metrics. Our evaluation procedure demonstrates that the error from our primitive representation is comparable to that of predicting depth from a single image.
An oft-cited open problem of federated learning is the existence of data heterogeneity among clients. One path- way to understanding the drastic accuracy drop in feder- ated learning is by scrutinizing the behavior of the clients' deep models on data with different levels of "difficulty", which has been left unaddressed. In this paper, we investi- gate a different and rarely studied dimension of FL: ordered learning. Specifically, we aim to investigate how ordered learning principles can contribute to alleviating the hetero- geneity effects in FL. We present theoretical analysis and conduct extensive empirical studies on the efficacy of or- derings spanning three kinds of learning: curriculum, anti- curriculum, and random curriculum. We find that curricu- lum learning largely alleviates non-IIDness. Interestingly, the more disparate the data distributions across clients the more they benefit from ordered learning. We provide analysis explaining this phenomenon, specifically indicating how curriculum training appears to make the objective land- scape progressively less convex, suggesting fast converging iterations at the beginning of the training procedure. We derive quantitative results of convergence for both convex and nonconvex objectives by modeling the curriculum train- ing on federated devices as local SGD with locally biased stochastic gradients. Also, inspired by ordered learning, we propose a novel client selection technique that benefits from the real-world disparity in the clients. Our proposed approach to client selection has a synergic effect when applied together with ordered learning in FL.
Point-spread-function (PSF) engineering is a powerful computational imaging technique wherein a custom phase mask is integrated into an optical system to encode additional information into captured images. Used in combination with deep learning, such systems now offer state-of-the-art performance at monocular depth estimation, extended depth-of-field imaging, lensless imaging, and other tasks. Inspired by recent advances in spatial light modulator (SLM) technology, this paper answers a natural question: Can one encode additional information and achieve superior performance by changing a phase mask dynamically over time? We first prove that the set of PSFs described by static phase masks is non-convex and that, as a result, time-averaged PSFs generated by dynamic phase masks are fundamentally more expressive. We then demonstrate, in simulation, that time-averaged dynamic (TiDy) phase masks can leverage this increased expressiveness to offer substantially improved monocular depth estimation and extended depth-of-field imaging performance.
A common architectural choice for deep metric learning is a convolutional neural network followed by global average pooling (GAP). Albeit simple, GAP is a highly effective way to aggregate information. One possible explanation for the effectiveness of GAP is considering each feature vector as representing a different semantic entity and GAP as a convex combination of them. Following this perspective, we generalize GAP and propose a learnable generalized sum pooling method (GSP). GSP improves GAP with two distinct abilities: i) the ability to choose a subset of semantic entities, effectively learning to ignore nuisance information, and ii) learning the weights corresponding to the importance of each entity. Formally, we propose an entropy-smoothed optimal transport problem and show that it is a strict generalization of GAP, i.e., a specific realization of the problem gives back GAP. We show that this optimization problem enjoys analytical gradients enabling us to use it as a direct learnable replacement for GAP. We further propose a zero-shot loss to ease the learning of GSP. We show the effectiveness of our method with extensive evaluations on 4 popular metric learning benchmarks. Code is available at: GSP-DML Framework
Improving Lens Flare Removal with General-Purpose Pipeline and Multiple Light Sources Recovery
PDF ↗When taking images against strong light sources, the resulting images often contain heterogeneous flare artifacts. These artifacts can importantly affect image visual quality and downstream computer vision tasks. While collecting real data pairs of flare-corrupted/flare-free images for training flare removal models is challenging, current methods utilize the direct-add approach to synthesize data. However, these methods do not consider automatic exposure and tone mapping in image signal processing pipeline (ISP), leading to the limited generalization capability of deep models training using such data. Besides, existing methods struggle to handle multiple light sources due to the different sizes, shapes and illuminance of various light sources. In this paper, we propose a solution to improve the performance of lens flare removal by revisiting the ISP and remodeling the principle of automatic exposure in the synthesis pipeline and design a more reliable light sources recovery strategy. The new pipeline approaches realistic imaging by discriminating the local and global illumination through convex combination, avoiding global illumination shifting and local over-saturation. Our strategy for recovering multiple light sources convexly averages the input and output of the neural network based on illuminance levels, thereby avoiding the need for a hard threshold in identifying light sources. We also contribute a new flare removal testing dataset containing the flare-corrupted images captured by ten types of consumer electronics. The dataset facilitates the verification of the generalization capability of flare removal methods. Extensive experiments show that our solution can effectively improve the performance of lens flare removal and push the frontier toward more general situations.
Shape primitives decomposition has been an important and long-standing task in 3D shape analysis. Prior arts heavily rely on 3D point clouds or voxel data for shape primitives extraction, which are less practical in real-world scenarios. This paper proposes to learn shape primitives from multi-view images by introducing implicit surface rendering. It is challenging since implicit shapes have a high degree of freedom, which violates the simplicity property of shape primitives. In this work, a novel regularization term named Implicit Convexity Regularization (ICR) imposed on implicit primitive learning is proposed to tackle this problem. We start with the convexity definition of general 3D shapes, and then derive the equivalent expression for implicit shapes represented by signed distance functions (SDFs). Further, instead of directly constraining the output SDF values which cause unstable optimization, we alternatively impose constraint on second order directional derivatives on line segments inside the shapes, which proves to be a tighter condition for 3D convexity. Implicit primitives constrained by the proposed ICR are combined into a whole object via softmax-weighted-sum operation over all primitive SDFs. Experiments on synthetic and real-world datasets show that our method is able to decompose objects into simple and reasonable shape primitives without the need of segmentation labels or 3D data. Code and data is publicly available in https://github.com/seanywang0408/ICR.
Federated learning (FL) is an emerging machine learning paradigm that allows multiple parties to train a shared model collaboratively in a privacy-preserving manner. Existing horizontal FL methods generally assume that the FL server and clients hold the same model structure. However, due to system heterogeneity and the need for personalization, enabling clients to hold models with diverse structures has become an important direction. Existing model-heterogeneous FL approaches often require publicly available datasets and incur high communication and/or computational costs, which limit their performances. To address these limitations, we propose a simple but effective Federated Global prediction Header (FedGH) approach. It is a communication and computation-efficient model-heterogeneous FL framework which trains a shared generalized global prediction header with representations extracted by heterogeneous extractors for clients' models at the FL server. The trained generalized global prediction header learns from different clients. The acquired global knowledge is then transferred to clients to substitute each client's local prediction header. We derive the non-convex convergence rate of FedGH. Extensive experiments on two real-world datasets demonstrate that FedGH achieves significantly more advantageous performance in both model-homogeneous and -heterogeneous FL scenarios compared to seven state-of-the-art personalized FL models, beating the best-performing baseline by up to 8.87% (for model-homogeneous FL) and 1.83% (for model-heterogeneous FL) in terms of average test accuracy, while saving up to 85.53% of communication overhead.
When Measures are Unreliable: Imperceptible Adversarial Perturbations toward Top-k Multi-Label Learning
PDF ↗With the great success of deep neural networks, adversarial learning has received widespread attention in various studies, ranging from multi-class learning to multi-label learning. However, existing adversarial attacks toward multi-label learning only pursue the traditional visual imperceptibility but ignore the new perceptible problem coming from measures such as Precision@k and mAP@k. Specifically, when a well-trained multi-label classifier performs far below the expectation on some samples, the victim can easily realize that this performance degeneration stems from attack, rather than the model itself. Therefore, an ideal multi-labeling adversarial attack should manage to not only deceive visual perception but also evade monitoring of measures. To this end, this paper first proposes the concept of measure imperceptibility. Then, a novel loss function is devised to generate such adversarial perturbations that could achieve both visual and measure imperceptibility. Furthermore, an efficient algorithm, which enjoys a convex objective, is established to optimize this objective. Finally, extensive experiments on large-scale benchmark datasets, such as PASCAL VOC 2012, MS COCO, and NUS WIDE, demonstrate the superiority of our proposed method in attacking the top-k multi-label systems.
Recent works on personalized text-to-image generation usually learn to bind a special token with specific subjects or styles of a few given images by tuning its embedding through gradient descent. It is natural to question whether we can optimize the textual inversions by only accessing the process of model inference. As only requiring the forward computation to determine the textual inversion retains the benefits of less GPU memory, simple deployment, and secure access for scalable models. In this paper, we introduce a gradient-free framework to optimize the continuous textual inversion in an iterative evolutionary strategy. Specifically, we first initialize an appropriate token embedding for textual inversion with the consideration of visual and text vocabulary information. Then, we decompose the optimization of evolutionary strategy into dimension reduction of searching space and non-convex gradient-free optimization in subspace, which significantly accelerates the optimization process with negligible performance loss. Experiments in several creative applications demonstrate that the performance of text-to-image model equipped with our proposed gradient-free method is comparable to that of gradient-based counterparts with variant GPU/CPU platforms, flexible employment, as well as computational efficiency.
Semi-parametric models exhibit the properties of both parametric and non-parametric modeling and have been shown to be effective in the next-word prediction language modeling task. However, there is a lack of studies on the text-discriminating properties of such models. We propose an inference-phase approach—k-Nearest Neighbor Classification Model (kNN-CM)—that enhances the capacity of a pre-trained parametric text classifier by incorporating a simple neighborhood search through the representation space of (memorized) training samples. The final class prediction of kNN-CM is based on the convex combination of probabilities obtained from kNN search and prediction of the classifier. Our experiments show consistent performance improvements on eight SuperGLUE tasks, three adversarial natural language inference (ANLI) datasets, 11 question-answering (QA) datasets, and two sentiment classification datasets.
Over recent decades, significant advancements in cross-modal retrieval is mainly driven by breakthroughs in visual and linguistic modeling. However, a recent study shows that multi-modal data representations tend to cluster within a limited convex cone (as representation degeneration problem), which hinders retrieval performance due to the inseparability of these representations. In our study, we first empirically validate the presence of the representation degeneration problem across multiple cross-modal benchmarks and methods. Next, to address it, we introduce a novel method, called InvGC, a post-processing technique inspired by graph convolution and average pooling. Specifically, InvGC defines the graph topology within the datasets and then applies graph convolution in a subtractive manner. This method effectively separates representations by increasing the distances between data points. To improve the efficiency and effectiveness of InvGC, we propose an advanced graph topology, LocalAdj, which only aims to increase the distances between each data point and its nearest neighbors. To understand why InvGC works, we present a detailed theoretical analysis, proving that the lower bound of recall will be improved after deploying InvGC. Extensive empirical results show that InvGC and InvGC w/LocalAdj significantly mitigate the representation degeneration problem, thereby enhancing retrieval performance.
Quantization is a promising approach for reducing memory overhead and accelerating inference, especially in large pre-trained language model (PLM) scenarios. While having no access to original training data due to security and privacy concerns has emerged the demand for zero-shot quantization. Most of the cutting-edge zero-shot quantization methods primarily 1) apply to computer vision tasks, and 2) neglect of overfitting problem in the generative adversarial learning process, leading to sub-optimal performance. Motivated by this, we propose a novel zero-shot sharpness-aware quantization (ZSAQ) framework for the zero-shot quantization of various PLMs. The key algorithm in solving ZSAQ is the SAM-SGA optimization, which aims to improve the quantization accuracy and model generalization via optimizing a minimax problem. We theoretically prove the convergence rate for the minimax optimization problem and this result can be applied to other nonconvex-PL minimax optimization frameworks. Extensive experiments on 11 tasks demonstrate that our method brings consistent and significant performance gains on both discriminative and generative PLMs, i.e., up to +6.98 average score. Furthermore, we empirically validate that our method can effectively improve the model generalization.
Social Aspects/Privacy-preserving Statistics and Machine Learning
While personalized recommendations systems have become increasingly popular, ensuring user data protection remains a top concern in the development of these learning systems. A common approach to enhancing privacy involves training models using anonymous data rather than individual data. In this paper, we explore a natural technique called "look-alike clustering", which involves replacing sensitive features of individuals with the cluster's average values. We provide a precise analysis of how training models using anonymous cluster centers affects their generalization capabilities. We focus on an asymptotic regime where the size of the training set grows in proportion to the features dimension. Our analysis is based on the Convex Gaussian Minimax Theorem (CGMT) and allows us to theoretically understand the role of different model components on the generalization error. In addition, we demonstrate that in certain high-dimensional regimes, training over anonymous cluster centers acts as a regularization and improves generalization error of the trained models. Finally, we corroborate our asymptotic theory with finite-sample numerical experiments where we observe a perfect match when the sample size is only of order of a few hundreds.
Probabilistic Methods/Monte Carlo and Sampling Methods
Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density $\pi(\theta)\propto \exp(-U(\theta)) $, LMC iteratively generates the next sample by taking a step in the gradient direction $\nabla U$ with added Gaussian perturbations. Expectations w.r.t. the target distribution $\pi$ are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasi-random samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.