We consider an on-line least squares regression problem with optimal solution $\theta^*$ and Hessian matrix H, and study a time-average stochastic gradient descent estimator of $\theta^*$. For $k\ge2$, we provide an unbiased estimator of $\theta^*$ that is a modification of the time-average estimator, runs with an expected number of time-steps of order k, with O(1/k) expected excess risk. The constant behind the O notation depends on parameters of the regression and is a poly-logarithmic function of the smallest eigenvalue of H. We provide both a biased and unbiased estimator of the expected excess risk of the time-average estimator and of its unbiased counterpart, without requiring knowledge of either H or $\theta^*$. We describe an "average-start" version of our estimators with similar properties. Our approach is based on randomized multilevel Monte Carlo. Our numerical experiments confirm our theoretical findings.

The typical training of neural networks using large stepsize gradient descent (GD) under the logistic loss often involves two distinct phases, where the empirical risk oscillates in the first phase but decreases monotonically in the second phase. We investigate this phenomenon in two-layer networks that satisfy a near-homogeneity condition. We show that the second phase begins once the empirical risk falls below a certain threshold, dependent on the stepsize. Additionally, we show that the normalized margin grows nearly monotonically in the second phase, demonstrating an implicit bias of GD in training non-homogeneous predictors. If the dataset is linearly separable and the derivative of the activation function is bounded away from zero, we show that the average empirical risk decreases, implying that the first phase must stop in finite steps. Finally, we demonstrate that by choosing a suitably large stepsize, GD that undergoes this phase transition is more efficient than GD that monotonically decreases the risk. Our analysis applies to networks of any width, beyond the well-known neural tangent kernel and mean-field regimes.

This paper introduces ZIP-DL, a novel privacy-aware decentralized learning (DL) algorithm that exploits correlated noise to provide strong privacy protection against a local adversary while yielding efficient convergence guarantees for a low communication cost. The progressive neutralization of the added noise during the distributed aggregation process results in ZIP-DL fostering a high model accuracy under privacy guarantees. ZIP-DL further uses a single communication round between each gradient descent, thus minimizing communication overhead. We provide theoretical guarantees for both convergence speed and privacy guarantees, thereby making ZIP-DL applicable to practical scenarios. Our extensive experimental study shows that ZIP-DL significantly outperforms the state-of-the-art in terms of vulnerability/accuracy trade-off. In particular, ZIP-DL (i) reduces the efficacy of linkability attacks by up to 52 percentage points compared to baseline DL, (ii) improves accuracy by up to 37 percent w.r.t. the state-of-the-art privacy-preserving mechanism operating under the same threat model as ours, when configured to provide the same protection against membership inference attacks, and (iii) reduces communication by up to 10.5x against the same competitor for the same level of protection.

Efficient training and inference algorithms, such as low-rank adaption and model pruning, have shown impressive performance for learning Transformer-based large foundation models. However, due to the technical challenges of the non-convex optimization caused by the complicated architecture of Transformers, the theoretical study of why these methods can be applied to learn Transformers is mostly elusive. To the best of our knowledge, this paper shows the first theoretical analysis of the property of low-rank and sparsity of one-layer Transformers by characterizing the trained model after convergence using stochastic gradient descent. By focusing on a data model based on label-relevant and label-irrelevant patterns, we quantify that the gradient updates of trainable parameters are low-rank, which depends on the number of label-relevant patterns. We also analyze how model pruning affects the generalization while improving computation efficiency and conclude that proper magnitude-based pruning has a slight effect on the testing performance. We implement numerical experiments to support our findings.

In the context of machine unlearning, the primary challenge lies in effectively removing traces of private data from trained models while maintaining model performance and security against privacy attacks like membership inference attacks. Traditional gradient-based unlearning methods often rely on extensive historical gradients, which becomes impractical with high unlearning ratios and may reduce the effectiveness of unlearning. Addressing these limitations, we introduce Mini-Unlearning, a novel approach that capitalizes on a critical observation: unlearned parameters correlate with retrained parameters through contraction mapping. Our method, Mini-Unlearning, utilizes a minimal subset of historical gradients and leverages this contraction mapping to facilitate scalable, efficient unlearning. This lightweight, scalable method significantly enhances model accuracy and strengthens resistance to membership inference attacks. Our experiments demonstrate that Mini-Unlearning not only works under higher unlearning ratios but also outperforms existing techniques in both accuracy and security, offering a promising solution for applications requiring robust unlearning capabilities.

We propose a novel time window-based analysis technique to investigate the convergence properties of the stochastic gradient descent method with momentum (SGDM) in nonconvex settings. Despite its popularity, the convergence behavior of SGDM remains less understood in nonconvex scenarios. This is primarily due to the absence of a sufficient descent property and challenges in simultaneously controlling the momentum and stochastic errors in an almost sure sense. To address these challenges, we investigate the behavior of SGDM over specific time windows, rather than examining the descent of consecutive iterates as in traditional studies. This time window-based approach simplifies the convergence analysis and enables us to establish the first iterate convergence result for SGDM under the Kurdyka-Lojasiewicz (KL) property. We further provide local convergence rates which depend on the underlying KL exponent and the utilized step size schemes.

We consider a variant of the stochastic gradient descent (SGD) with a random learning rate and reveal its convergence properties. SGD is a widely used stochastic optimization algorithm in machine learning, especially deep learning. Numerous studies reveal the convergence properties of SGD and its simplified variants. Among these, the analysis of convergence using a stationary distribution of updated parameters provides generalizable results. However, to obtain a stationary distribution, the update direction of the parameters must not degenerate, which limits the applicable variants of SGD. In this study, we consider a novel SGD variant, Poisson SGD, which has degenerated parameter update directions and instead utilizes a random learning rate. Consequently, we demonstrate that a distribution of a parameter updated by Poisson SGD converges to a stationary distribution under weak assumptions on a loss function. Based on this, we further show that Poisson SGD finds global minima in non-convex optimization problems and also evaluate the generalization error using this method. As a proof technique, we approximate the distribution by Poisson SGD with that of the bouncy particle sampler (BPS) and derive its stationary distribution, using the theoretical advance of the piece-wise deterministic Markov process (PDMP).

In the context of average-reward reinforcement learning, the requirement for oracle knowledge of the mixing time, a measure of the duration a Markov chain under a fixed policy needs to achieve its stationary distribution, poses a significant challenge for the global convergence of policy gradient methods. This requirement is particularly problematic due to the difficulty and expense of estimating mixing time in environments with large state spaces, leading to the necessity of impractically long trajectories for effective gradient estimation in practical applications. To address this limitation, we consider the Multi-level Actor-Critic (MAC) framework, which incorporates a Multi-level Monte-Carlo (MLMC) gradient estimator. With our approach, we effectively alleviate the dependency on mixing time knowledge, a first for average-reward MDPs global convergence. Furthermore, our approach exhibits the tightest available dependence of $\mathcal{O}\left( \sqrt{\tau_{mix}} \right)$known from prior work. With a 2D grid world goal-reaching navigation experiment, we demonstrate that MAC outperforms the existing state-of-the-art policy gradient-based method for average reward settings.

Proper data stewardship requires that model owners protect the privacy of individuals' data used during training. Whether through anonymization with differential privacy or the use of unlearning in non-anonymized settings, the gold-standard techniques for providing privacy guarantees can come with significant performance penalties or be too weak to provide practical assurances. In part, this is due to the fact that the guarantee provided by differential privacy represents the worst-case privacy leakage for any individual, while the true privacy leakage of releasing the prediction for a given individual might be substantially smaller or even, as we show, non-existent. This work provides a novel framework based on convex relaxations and bounds propagation that can compute formal guarantees (certificates) that releasing specific predictions satisfies $\epsilon=0$ privacy guarantees or do not depend on data that is subject to an unlearning request. Our framework offers a new verification-centric approach to privacy and unlearning guarantees, that can be used to further engender user trust with tighter privacy guarantees, provide formal proofs of robustness to certain membership inference attacks, identify potentially vulnerable records, and enhance current unlearning approaches. We validate the effectiveness of our approach on tasks from financial services, medical imaging, and natural language processing.

With the rapid development of artificial intelligence technology, the field of reinforcement learning has continuously achieved breakthroughs in both theory and practice. However, traditional reinforcement learning algorithms often entail high energy consumption during interactions with the environment. Spiking Neural Network (SNN), with their low energy consumption characteristics and performance comparable to deep neural networks, have garnered widespread attention. To reduce the energy consumption of practical applications of reinforcement learning, researchers have successively proposed the Pop-SAN and MDC-SAN algorithms. Nonetheless, these algorithms use rectangular functions to approximate the spike network during the training process, resulting in low sensitivity, thus indicating room for improvement in the training effectiveness of SNN. Based on this, we propose a trapezoidal approximation gradient method to replace the spike network, which not only preserves the original stable learning state but also enhances the model's adaptability and response sensitivity under various signal dynamics. Simulation results show that the improved algorithm, using the trapezoidal approximation gradient to replace the spike network, achieves better convergence speed and performance compared to the original algorithm and demonstrates good training stability.