In this study, we propose Shortcut Fine-Tuning (SFT), a new approach for addressing the challenge of fast sampling of pretrained Denoising Diffusion Probabilistic Models (DDPMs). SFT advocates for the fine-tuning of DDPM samplers through the direct minimization of Integral Probability Metrics (IPM), instead of learning the backward diffusion process. This enables samplers to discover an alternative and more efficient sampling shortcut, deviating from the backward diffusion process. Inspired by a control perspective, we propose a new algorithm SFT-PG: Shortcut Fine-Tuning with Policy Gradient, and prove that under certain assumptions, gradient descent of diffusion models with respect to IPM is equivalent to performing policy gradient. To our best knowledge, this is the first attempt to utilize reinforcement learning (RL) methods to train diffusion models. Through empirical evaluation, we demonstrate that our fine-tuning method can further enhance existing fast DDPM samplers, resulting in sample quality comparable to or even surpassing that of the full-step model across various datasets.

Adversarial training has gained great popularity as one of the most effective defenses for deep neural network and more generally for gradient-based machine learning models against adversarial perturbations on data points. This paper establishes a continuous-time approximation for the mini-max game of adversarial training. This approximation approach allows for precise and analytical comparisons between stochastic gradient descent and its adversarial training counterpart; and confirms theoretically the robustness of adversarial training from a new gradient-flow viewpoint. The analysis is then corroborated through various analytical and numerical examples.

We provide the first proof that gradient descent $\left({\color{green}\sf GD}\right)$ with greedy sparsification $\left({\color{green}\sf TopK}\right)$ and error feedback $\left({\color{green}\sf EF}\right)$ can obtain better communication complexity than vanilla ${\color{green}\sf GD}$ when solving the distributed optimization problem $\min_{x\in \mathbb{R}^d} {f(x)=\frac{1}{n}\sum_{i=1}^n f_i(x)}$, where $n$ = # of clients, $d$ = # of features, and $f_1,\dots,f_n$ are smooth nonconvex functions. Despite intensive research since 2014 when ${\color{green}\sf EF}$ was first proposed by Seide et al., this problem remained open until now. We show that ${\color{green}\sf EF}$ shines in the regime when features are rare, i.e., when each feature is present in the data owned by a small number of clients only. To illustrate our main result, we show that in order to find a random vector $\hat{x}$ such that $\lVert {\nabla f(\hat{x})} \rVert^2 \leq \varepsilon$ in expectation, ${\color{green}\sf GD}$ with the ${\color{green}\sf Top1}$ sparsifier and ${\color{green}\sf EF}$ requires ${\cal O} \left(\left( L+{\color{blue}r} \sqrt{ \frac{{\color{red}c}}{n} \min \left( \frac{{\color{red}c}}{n} \max_i L_i^2, \frac{1}{n}\sum_{i=1}^n L_i^2 \right) }\right) \frac{1}{\varepsilon} \right)$ bits to be communicated by each worker to the server only, where $L$ is the smoothness constant of $f$, $L_i$ is the smoothness constant of $f_i$, ${\color{red}c}$ is the maximal number of clients owning any feature ($1\leq {\color{red}c} \leq n$), and ${\color{blue}r}$ is the maximal number of features owned by any client ($1\leq {\color{blue}r} \leq d$). Clearly, the communication complexity improves as ${\color{red}c}$ decreases (i.e., as features become more rare), and can be much better than the ${\cal O}({\color{blue}r} L \frac{1}{\varepsilon})$ communication complexity of ${\color{green}\sf GD}$ in the same regime.

The Stochastic Gradient Langevin Dynamics (SGLD) are popularly used to approximate Bayesian posterior distributions in statistical learning procedures with large-scale data. As opposed to many usual Markov chain Monte Carlo (MCMC) algorithms, SGLD is not stationary with respect to the posterior distribution; two sources of error appear: The first error is introduced by an Euler--Maruyama discretisation of a Langevin diffusion process, the second error comes from the data subsampling that enables its use in large-scale data settings. In this work, we consider an idealised version of SGLD to analyse the method's pure subsampling error that we then see as a best-case error for diffusion-based subsampling MCMC methods. Indeed, we introduce and study the Stochastic Gradient Langevin Diffusion (SGLDiff), a continuous-time Markov process that follows the Langevin diffusion corresponding to a data subset and switches this data subset after exponential waiting times. There, we show that the Wasserstein distance between the posterior and the limiting distribution of SGLDiff is bounded above by a fractional power of the mean waiting time. Importantly, this fractional power does not depend on the dimension of the state space. We bring our results into context with other analyses of SGLD.

Byzantine-robust distributed learning (BRDL), in which computing devices are likely to behave abnormally due to accidental failures or malicious attacks, has recently become a hot research topic. However, even in the independent and identically distributed (i.i.d.) case, existing BRDL methods will suffer from a significant drop on model accuracy due to the large variance of stochastic gradients. Increasing batch sizes is a simple yet effective way to reduce the variance. However, when the total number of gradient computation is fixed, a too-large batch size will lead to a too-small iteration number (update number), which may also degrade the model accuracy. In view of this challenge, we mainly study the optimal batch size when the total number of gradient computation is fixed in this work. In particular, we theoretically and empirically show that when the total number of gradient computation is fixed, the optimal batch size in BRDL increases with the fraction of Byzantine workers. Therefore, compared to the case without attacks, the batch size should be set larger when under Byzantine attacks. However, for existing BRDL methods, large batch sizes will lead to a drop on model accuracy, even if there is no Byzantine attack. To deal with this problem, we propose a novel BRDL method, called Byzantine-robust stochastic gradient descent with normalized momentum (ByzSGDnm), which can alleviate the drop on model accuracy in large-batch cases. Moreover, we theoretically prove the convergence of ByzSGDnm for general non-convex cases under Byzantine attacks. Empirical results show that ByzSGDnm has a comparable performance to existing BRDL methods under bit-flipping failure, but can outperform existing BRDL methods under deliberately crafted attacks.

Modern health care systems are conducting continuous, automated surveillance of the electronic medical record (EMR) to identify adverse events with increasing frequency; however, many events such as sepsis do not have elucidated prodromes (i.e., event chains) that can be used to identify and intercept the adverse event early in its course. Clinically relevant and interpretable results require a framework that can (i) infer temporal interactions across multiple patient features found in EMR data (e.g., Labs, vital signs, etc.) and (ii) identify patterns that precede and are specific to an impending adverse event (e.g., sepsis). In this work, we propose a linear multivariate Hawkes process model, coupled with ReLU link function, to recover a Granger Causal (GC) graph with both exciting and inhibiting effects. We develop a scalable two-phase gradient-based method to obtain a maximum surrogate-likelihood estimator, which is shown to be effective via extensive numerical simulation. Our method is subsequently extended to a data set of patients admitted to Grady hospital system in Atlanta, GA, USA, where the estimated GC graph identifies several highly interpretable GC chains that precede sepsis. The code is available at \url{https://github.com/SongWei-GT/two-phase-MHP}.

Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the global effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments.

We consider the problem of few-shot spoken word classification in a setting where a model is incrementally introduced to new word classes. This would occur in a user-defined keyword system where new words can be added as the system is used. In such a continual learning scenario, a model might start to misclassify earlier words as newer classes are added, i.e. catastrophic forgetting. To address this, we propose an extension to model-agnostic meta-learning (MAML): each inner learning loop, where a model "learns how to learn'' new classes, ends with a single gradient update using stored templates from all the classes that the model has already seen (one template per class). We compare this method to OML (another extension of MAML) in few-shot isolated-word classification experiments on Google Commands and FACC. Our method consistently outperforms OML in experiments where the number of shots and the final number of classes are varied.

Recent research shows that when Gradient Descent (GD) is applied to neural networks, the loss almost never decreases monotonically. Instead, the loss oscillates as gradient descent converges to its ''Edge of Stability'' (EoS). Here, we find a quantity that does decrease monotonically throughout GD training: the sharpness attained by the gradient flow solution (GFS)-the solution that would be obtained if, from now until convergence, we train with an infinitesimal step size. Theoretically, we analyze scalar neural networks with the squared loss, perhaps the simplest setting where the EoS phenomena still occur. In this model, we prove that the GFS sharpness decreases monotonically. Using this result, we characterize settings where GD provably converges to the EoS in scalar networks. Empirically, we show that GD monotonically decreases the GFS sharpness in a squared regression model as well as practical neural network architectures.

Neural collapse (NC) refers to the surprising structure of the last layer of deep neural networks in the terminal phase of gradient descent training. Recently, an increasing amount of experimental evidence has pointed to the propagation of NC to earlier layers of neural networks. However, while the NC in the last layer is well studied theoretically, much less is known about its multi-layered counterpart - deep neural collapse (DNC). In particular, existing work focuses either on linear layers or only on the last two layers at the price of an extra assumption. Our paper fills this gap by generalizing the established analytical framework for NC - the unconstrained features model - to multiple non-linear layers. Our key technical contribution is to show that, in a deep unconstrained features model, the unique global optimum for binary classification exhibits all the properties typical of DNC. This explains the existing experimental evidence of DNC. We also empirically show that (i) by optimizing deep unconstrained features models via gradient descent, the resulting solution agrees well with our theory, and (ii) trained networks recover the unconstrained features suitable for the occurrence of DNC, thus supporting the validity of this modeling principle.