One-shot controllable video editing (OCVE) is an important yet challenging task, aiming to propagate user edits that are made -- using any image editing tool -- on the first frame of a video to all subsequent frames, while ensuring content consistency between edited frames and source frames. To achieve this, prior methods employ DDIM inversion to transform source frames into latent noise, which is then fed into a pre-trained diffusion model, conditioned on the user-edited first frame, to generate the edited video. However, the DDIM inversion process accumulates errors, which hinder the latent noise from accurately reconstructing the source frames, ultimately compromising content consistency in the generated edited frames. To overcome it, our method eliminates the need for DDIM inversion by performing OCVE through a novel perspective based on visual prompting. Furthermore, inspired by consistency models that can perform multi-step consistency sampling to generate a sequence of content-consistent images, we propose a content consistency sampling (CCS) to ensure content consistency between the generated edited frames and the source frames. Moreover, we introduce a temporal-content consistency sampling (TCS) based on Stein Variational Gradient Descent to ensure temporal consistency across the edited frames. Extensive experiments validate the effectiveness of our approach.

Stochastic Gradient Descent (SGD) methods see many uses in optimization problems. Modifications to the algorithm, such as momentum-based SGD methods have been known to produce better results in certain cases. Much of this, however, is due to empirical information rather than rigorous proof. While the dynamics of gradient descent methods can be studied through continuous approximations, existing works only cover scenarios with constant learning rates or SGD without momentum terms. We present approximation results under weak assumptions for SGD that allow learning rates and momentum parameters to vary with respect to time.

Non-Intrusive Load Monitoring (NILM) has emerged as a key smart grid technology, identifying electrical device and providing detailed energy consumption data for precise demand response management. Nevertheless, NILM data suffers from missing values due to inescapable factors like sensor failure, leading to inaccuracies in non-intrusive load monitoring. A stochastic gradient descent (SGD)-based latent factorization of tensors model has proven to be effective in estimating missing data, however, it updates a latent factor solely based on the current stochastic gradient, without considering past information, which leads to slow convergence of anLFT model. To address this issue, this paper proposes a Nonlinear Proportional-integral-derivative (PID)-Incorporated Latent factorization of tensors (NPIL) model with two-fold ideas: a) rebuilding the instant learning error according to the principle of a nonlinear PID controller, thus, the past update information is efficiently incorporated into the learning scheme, and b) implementing gain parameter adaptation by utilizing particle swarm optimization (PSO) algorithm, hence, the model computational efficiency is effectively improved. Experimental results on real-world NILM datasets demonstrate that the proposed NPIL model surpasses state-of-the-art models in convergence rate and accuracy when predicting the missing NILM data.

One of the goals of language model unlearning is to reduce memorization of selected text instances while retaining the model's general abilities. Despite various proposed methods, reducing memorization of large datasets without noticeable degradation in model utility remains challenging. In this paper, we investigate the mean teacher algorithm (Tarvainen & Valpola, 2017), a simple proximal optimization method from continual learning literature that gradually modifies the teacher model. We show that the mean teacher can approximate a trajectory of a slow natural gradient descent (NGD), which inherently seeks low-curvature updates that are less likely to degrade the model utility. While slow NGD can suffer from vanishing gradients, we introduce a new unlearning loss called "negative log-unlikelihood" (NLUL) that avoids this problem. We show that the combination of mean teacher and NLUL improves some metrics on the MUSE benchmarks (Shi et al., 2024).

Decentralized Federated Learning (DFL) eliminates the reliance on the server-client architecture inherent in traditional federated learning, attracting significant research interest in recent years. Simultaneously, the objective functions in machine learning tasks are often nonconvex and frequently incorporate additional, potentially nonsmooth regularization terms to satisfy practical requirements, thereby forming nonconvex composite optimization problems. Employing DFL methods to solve such general optimization problems leads to the formulation of Decentralized Nonconvex Composite Federated Learning (DNCFL), a topic that remains largely underexplored. In this paper, we propose a novel DNCFL algorithm, termed \bf{DEPOSITUM}. Built upon proximal stochastic gradient tracking, DEPOSITUM mitigates the impact of data heterogeneity by enabling clients to approximate the global gradient. The introduction of momentums in the proximal gradient descent step, replacing tracking variables, reduces the variance introduced by stochastic gradients. Additionally, DEPOSITUM supports local updates of client variables, significantly reducing communication costs. Theoretical analysis demonstrates that DEPOSITUM achieves an expected $ε$-stationary point with an iteration complexity of $\mathcal{O}(1/ε^2)$. The proximal gradient, consensus errors, and gradient estimation errors decrease at a sublinear rate of $\mathcal{O}(1/T)$. With appropriate parameter selection, the algorithm achieves network-independent linear speedup without requiring mega-batch sampling. Finally, we apply DEPOSITUM to the training of neural networks on real-world datasets, systematically examining the influence of various hyperparameters on its performance. Comparisons with other federated composite optimization algorithms validate the effectiveness of the proposed method.

Stochastic Gradient Descent (SGD) is widely used in machine learning research. Previous convergence analyses of SGD under the vanishing step-size setting typically require Robbins-Monro conditions. However, in practice, a wider variety of step-size schemes are frequently employed, yet existing convergence results remain limited and often rely on strong assumptions. This paper bridges this gap by introducing a novel analytical framework based on a stopping-time method, enabling asymptotic convergence analysis of SGD under more relaxed step-size conditions and weaker assumptions. In the non-convex setting, we prove the almost sure convergence of SGD iterates for step-sizes $ \{ ε_t \}_{t \geq 1} $ satisfying $\sum_{t=1}^{+\infty} ε_t = +\infty$ and $\sum_{t=1}^{+\infty} ε_t^p < +\infty$ for some $p > 2$. Compared with previous studies, our analysis eliminates the global Lipschitz continuity assumption on the loss function and relaxes the boundedness requirements for higher-order moments of stochastic gradients. Building upon the almost sure convergence results, we further establish $L_2$ convergence. These significantly relaxed assumptions make our theoretical results more general, thereby enhancing their applicability in practical scenarios.

We consider SGD-type optimization on infinite-dimensional quadratic problems with power law spectral conditions. It is well-known that on such problems deterministic GD has loss convergence rates $L_t=O(t^{-ζ})$, which can be improved to $L_t=O(t^{-2ζ})$ by using Heavy Ball with a non-stationary Jacobi-based schedule (and the latter rate is optimal among fixed schedules). However, in the mini-batch Stochastic GD setting, the sampling noise causes the Jacobi HB to diverge; accordingly no $O(t^{-2ζ})$ algorithm is known. In this paper we show that rates up to $O(t^{-2ζ})$ can be achieved by a generalized stationary SGD with infinite memory. We start by identifying generalized (S)GD algorithms with contours in the complex plane. We then show that contours that have a corner with external angle $θπ$ accelerate the plain GD rate $O(t^{-ζ})$ to $O(t^{-θζ})$. For deterministic GD, increasing $θ$ allows to achieve rates arbitrarily close to $O(t^{-2ζ})$. However, in Stochastic GD, increasing $θ$ also amplifies the sampling noise, so in general $θ$ needs to be optimized by balancing the acceleration and noise effects. We prove that the optimal rate is given by $θ_{\max}=\min(2,ν,\tfrac{2}{ζ+1/ν})$, where $ν,ζ$ are the exponents appearing in the capacity and source spectral conditions. Furthermore, using fast rational approximations of the power functions, we show that ideal corner algorithms can be efficiently approximated by finite-memory algorithms, and demonstrate their practical efficiency on a synthetic problem and MNIST.

We study the approximation gap between the dynamics of a polynomial-width neural network and its infinite-width counterpart, both trained using projected gradient descent in the mean-field scaling regime. We demonstrate how to tightly bound this approximation gap through a differential equation governed by the mean-field dynamics. A key factor influencing the growth of this ODE is the local Hessian of each particle, defined as the derivative of the particle's velocity in the mean-field dynamics with respect to its position. We apply our results to the canonical feature learning problem of estimating a well-specified single-index model; we permit the information exponent to be arbitrarily large, leading to convergence times that grow polynomially in the ambient dimension $d$. We show that, due to a certain ``self-concordance'' property in these problems -- where the local Hessian of a particle is bounded by a constant times the particle's velocity -- polynomially many neurons are sufficient to closely approximate the mean-field dynamics throughout training.

We study $\textit{gradient descent}$ (GD) for logistic regression on linearly separable data with stepsizes that adapt to the current risk, scaled by a constant hyperparameter $\eta$. We show that after at most $1/\gamma^2$ burn-in steps, GD achieves a risk upper bounded by $\exp(-\Theta(\eta))$, where $\gamma$ is the margin of the dataset. As $\eta$ can be arbitrarily large, GD attains an arbitrarily small risk $\textit{immediately after the burn-in steps}$, though the risk evolution may be $\textit{non-monotonic}$. We further construct hard datasets with margin $\gamma$, where any batch (or online) first-order method requires $\Omega(1/\gamma^2)$ steps to find a linear separator. Thus, GD with large, adaptive stepsizes is $\textit{minimax optimal}$ among first-order batch methods. Notably, the classical $\textit{Perceptron}$ (Novikoff, 1962), a first-order online method, also achieves a step complexity of $1/\gamma^2$, matching GD even in constants. Finally, our GD analysis extends to a broad class of loss functions and certain two-layer networks.

Conformal prediction (CP) is an important tool for distribution-free predictive uncertainty quantification. Yet, a major challenge is to balance computational efficiency and prediction accuracy, particularly for multiple predictions. We propose Leave-One-Out Stable Conformal Prediction (LOO-StabCP), a novel method to speed up full conformal using algorithmic stability without sample splitting. By leveraging leave-one-out stability, our method is much faster in handling a large number of prediction requests compared to existing method RO-StabCP based on replace-one stability. We derived stability bounds for several popular machine learning tools: regularized loss minimization (RLM) and stochastic gradient descent (SGD), as well as kernel method, neural networks and bagging. Our method is theoretically justified and demonstrates superior numerical performance on synthetic and real-world data. We applied our method to a screening problem, where its effective exploitation of training data led to improved test power compared to state-of-the-art method based on split conformal.