Continuous-time Riemannian SGD and SVRG Flows on Wasserstein Probabilistic Space

Recently, optimization on the Riemannian manifold has provided new insights to the optimization community. In this regard, the manifold taken as the probability measure metric space equipped with the second-order Wasserstein distance is of particular interest, since optimization on it can be linked to practical sampling processes. In general, the oracle (continuous) optimization method on Wasserstein space is Riemannian gradient flow (i.e., Langevin dynamics when minimizing KL divergence). In this paper, we aim to enrich the continuous optimization methods in the Wasserstein space by extending the gradient flow into the stochastic gradient descent (SGD) flow and stochastic variance reduction gradient (SVRG) flow. The two flows on Euclidean space are standard stochastic optimization methods, while their Riemannian counterparts are not explored yet. By leveraging the structures in Wasserstein space, we construct a stochastic differential equation (SDE) to approximate the discrete dynamics of desired stochastic methods in the corresponded random vector space. Then, the flows of probability measures are naturally obtained by applying Fokker-Planck equation to such SDE. Furthermore, the convergence rates of the proposed Riemannian stochastic flows are proven, and they match the results in Euclidean space.

SA-Solver: Stochastic Adams Solver for Fast Sampling of Diffusion Models

Diffusion Probabilistic Models (DPMs) have achieved considerable success in generation tasks. As sampling from DPMs is equivalent to solving diffusion SDE or ODE which is time-consuming, numerous fast sampling methods built upon improved differential equation solvers are proposed. The majority of such techniques consider solving the diffusion ODE due to its superior efficiency. However, stochastic sampling could offer additional advantages in generating diverse and high-quality data. In this work, we engage in a comprehensive analysis of stochastic sampling from two aspects: variance-controlled diffusion SDE and linear multi-step SDE solver. Based on our analysis, we propose SA-Solver, which is an improved efficient stochastic Adams method for solving diffusion SDE to generate data with high quality. Our experiments show that SA-Solver achieves: 1) improved or comparable performance compared with the existing state-of-the-art sampling methods for few-step sampling; 2) SOTA FID scores on substantial benchmark datasets under a suitable number of function evaluations (NFEs).

On the Generalization of Diffusion Model

The diffusion probabilistic generative models are widely used to generate high-quality data. Though they can synthetic data that does not exist in the training set, the rationale behind such generalization is still unexplored. In this paper, we formally define the generalization of the generative model, which is measured by the mutual information between the generated data and the training set. The definition originates from the intuition that the model which generates data with less correlation to the training set exhibits better generalization ability. Meanwhile, we show that for the empirical optimal diffusion model, the data generated by a deterministic sampler are all highly related to the training set, thus poor generalization. This result contradicts the observation of the trained diffusion model's (approximating empirical optima) extrapolation ability (generating unseen data). To understand this contradiction, we empirically verify the difference between the sufficiently trained diffusion model and the empirical optima. We found, though obtained through sufficient training, there still exists a slight difference between them, which is critical to making the diffusion model generalizable. Moreover, we propose another training objective whose empirical optimal solution has no potential generalization problem. We empirically show that the proposed training objective returns a similar model to the original one, which further verifies the generalization ability of the trained diffusion model.

Improved OOD Generalization via Conditional Invariant Regularizer

Recently, generalization on out-of-distribution (OOD) data with correlation shift has attracted great attention. The correlation shift is caused by the spurious attributes that correlate to the class label, as the correlation between them may vary in training and test data. For such a problem, we show that given the class label, the conditionally independent models of spurious attributes are OOD generalizable. Based on this, a metric Conditional Spurious Variation (CSV) which controls OOD generalization error, is proposed to measure such conditional independence. To improve the OOD generalization, we regularize the training process with the proposed CSV. Under mild assumptions, our training objective can be formulated as a nonconvex-concave mini-max problem. An algorithm with provable convergence rate is proposed to solve the problem. Extensive empirical results verify our algorithm's efficacy in improving OOD generalization.

Out-of-distribution Generalization with Causal Invariant Transformations

In real-world applications, it is important and desirable to learn a model that performs well on out-of-distribution (OOD) data. Recently, causality has become a powerful tool to tackle the OOD generalization problem, with the idea resting on the causal mechanism that is invariant across domains of interest. To leverage the generally unknown causal mechanism, existing works assume a linear form of causal feature or require sufficiently many and diverse training domains, which are usually restrictive in practice. In this work, we obviate these assumptions and tackle the OOD problem without explicitly recovering the causal feature. Our approach is based on transformations that modify the non-causal feature but leave the causal part unchanged, which can be either obtained from prior knowledge or learned from the training data in the multi-domain scenario. Under the setting of invariant causal mechanism, we theoretically show that if all such transformations are available, then we can learn a minimax optimal model across the domains using only single domain data. Noticing that knowing a complete set of these causal invariant transformations may be impractical, we further show that it suffices to know only a subset of these transformations. Based on the theoretical findings, a regularized training procedure is proposed to improve the OOD generalization capability. Extensive experimental results on both synthetic and real datasets verify the effectiveness of the proposed algorithm, even with only a few causal invariant transformations.

Towards the Generalization of Contrastive Self-Supervised Learning

Recently, self-supervised learning has attracted great attention since it only requires unlabeled data for training. Contrastive learning is a popular approach for self-supervised learning and empirically performs well in practice. However, the theoretical understanding of its generalization ability on downstream tasks is not well studied. To this end, we present a theoretical explanation of how contrastive self-supervised pre-trained models generalize to downstream tasks. Concretely, we quantitatively show that the self-supervised model has generalization ability on downstream classification tasks if it embeds input data into a feature space with distinguishing centers of classes and closely clustered intra-class samples. With the above conclusion, we further explore SimCLR and Barlow Twins, which are two canonical contrastive self-supervised methods. We prove that the aforementioned feature space can be obtained via any of the methods, and thus explain their success on the generalization on downstream classification tasks. Finally, various experiments are also conducted to verify our theoretical findings.

Improved OOD Generalization via Adversarial Training and Pre-training

Recently, learning a model that generalizes well on out-of-distribution (OOD) data has attracted great attention in the machine learning community. In this paper, after defining OOD generalization via Wasserstein distance, we theoretically show that a model robust to input perturbation generalizes well on OOD data. Inspired by previous findings that adversarial training helps improve input-robustness, we theoretically show that adversarially trained models have converged excess risk on OOD data, and empirically verify it on both image classification and natural language understanding tasks. Besides, in the paradigm of first pre-training and then fine-tuning, we theoretically show that a pre-trained model that is more robust to input perturbation provides a better initialization for generalization on downstream OOD data. Empirically, after fine-tuning, this better-initialized model from adversarial pre-training also has better OOD generalization.

Reweighting Augmented Samples by Minimizing the Maximal Expected Loss

Data augmentation is an effective technique to improve the generalization of deep neural networks. However, previous data augmentation methods usually treat the augmented samples equally without considering their individual impacts on the model. To address this, for the augmented samples from the same training example, we propose to assign different weights to them. We construct the maximal expected loss which is the supremum over any reweighted loss on augmented samples. Inspired by adversarial training, we minimize this maximal expected loss (MMEL) and obtain a simple and interpretable closed-form solution: more attention should be paid to augmented samples with large loss values (i.e., harder examples). Minimizing this maximal expected loss enables the model to perform well under any reweighting strategy. The proposed method can generally be applied on top of any data augmentation methods. Experiments are conducted on both natural language understanding tasks with token-level data augmentation, and image classification tasks with commonly-used image augmentation techniques like random crop and horizontal flip. Empirical results show that the proposed method improves the generalization performance of the model.

BN-invariant sharpness regularizes the training model to better generalization

It is arguably believed that flatter minima can generalize better. However, it has been pointed out that the usual definitions of sharpness, which consider either the maxima or the integral of loss over a $\delta$ ball of parameters around minima, cannot give consistent measurement for scale invariant neural networks, e.g., networks with batch normalization layer. In this paper, we first propose a measure of sharpness, BN-Sharpness, which gives consistent value for equivalent networks under BN. It achieves the property of scale invariance by connecting the integral diameter with the scale of parameter. Then we present a computation-efficient way to calculate the BN-sharpness approximately i.e., one dimensional integral along the "sharpest" direction. Furthermore, we use the BN-sharpness to regularize the training and design an algorithm to minimize the new regularized objective. Our algorithm achieves considerably better performance than vanilla SGD over various experiment settings.

Towards Accelerating Training of Batch Normalization: A Manifold Perspective

Batch normalization (BN) has become a crucial component across diverse deep neural networks. The network with BN is invariant to positively linear re-scaling of weights, which makes there exist infinite functionally equivalent networks with various scales of weights. However, optimizing these equivalent networks with the first-order method such as stochastic gradient descent will converge to different local optima owing to different gradients across training. To alleviate this, we propose a quotient manifold \emph{PSI manifold}, in which all the equivalent weights of the network with BN are regarded as the same one element. Then, gradient descent and stochastic gradient descent on the PSI manifold are also constructed. The two algorithms guarantee that every group of equivalent weights (caused by positively re-scaling) converge to the equivalent optima. Besides that, we give the convergence rate of the proposed algorithms on PSI manifold and justify that they accelerate training compared with the algorithms on the Euclidean weight space. Empirical studies show that our algorithms can consistently achieve better performances over various experimental settings.