Current state-of-the-art analyses on the convergence of gradient descent for training neural networks focus on characterizing properties of the loss landscape, such as the Polyak-Lojaciewicz (PL) condition and the restricted strong convexity. While gradient descent converges linearly under such conditions, it remains an open question whether Nesterov's momentum enjoys accelerated convergence under similar settings and assumptions. In this work, we consider a new class of objective functions, where only a subset of the parameters satisfies strong convexity, and show Nesterov's momentum achieves acceleration in theory for this objective class. We provide two realizations of the problem class, one of which is deep ReLU networks, which --to the best of our knowledge--constitutes this work the first that proves accelerated convergence rate for non-trivial neural network architectures.

In this paper, we provide the first convergence guarantee for the factorization approach. Specifically, to avoid the scaling ambiguity and to facilitate theoretical analysis, we optimize over the so-called left-orthogonal TT format which enforces orthonormality among most of the factors. To ensure the orthonormal structure, we utilize the Riemannian gradient descent (RGD) for optimizing those factors over the Stiefel manifold. We first delve into the TT factorization problem and establish the local linear convergence of RGD. Notably, the rate of convergence only experiences a linear decline as the tensor order increases. We then study the sensing problem that aims to recover a TT format tensor from linear measurements. Assuming the sensing operator satisfies the restricted isometry property (RIP), we show that with a proper initialization, which could be obtained through spectral initialization, RGD also converges to the ground-truth tensor at a linear rate. Furthermore, we expand our analysis to encompass scenarios involving Gaussian noise in the measurements. We prove that RGD can reliably recover the ground truth at a linear rate, with the recovery error exhibiting only polynomial growth in relation to the tensor order. We conduct various experiments to validate our theoretical findings.

This paper investigates group distributionally robust optimization (GDRO), with the purpose to learn a model that performs well over $m$ different distributions. First, we formulate GDRO as a stochastic convex-concave saddle-point problem, and demonstrate that stochastic mirror descent (SMD), using $m$ samples in each iteration, achieves an $O(m (\log m)/\epsilon^2)$ sample complexity for finding an $\epsilon$-optimal solution, which matches the $\Omega(m/\epsilon^2)$ lower bound up to a logarithmic factor. Then, we make use of techniques from online learning to reduce the number of samples required in each round from $m$ to $1$, keeping the same sample complexity. Specifically, we cast GDRO as a two-players game where one player simply performs SMD and the other executes an online algorithm for non-oblivious multi-armed bandits. Next, we consider a more practical scenario where the number of samples that can be drawn from each distribution is different, and propose a novel formulation of weighted GDRO, which allows us to derive distribution-dependent convergence rates. Denote by $n_i$ the sample budget for the $i$-th distribution, and assume $n_1 \geq n_2 \geq \cdots \geq n_m$. In the first approach, we incorporate non-uniform sampling into SMD such that the sample budget is satisfied in expectation, and prove that the excess risk of the $i$-th distribution decreases at an $O(\sqrt{n_1 \log m}/n_i)$ rate. In the second approach, we use mini-batches to meet the budget exactly and also reduce the variance in stochastic gradients, and then leverage stochastic mirror-prox algorithm, which can exploit small variances, to optimize a carefully designed weighted GDRO problem. Under appropriate conditions, it attains an $O((\log m)/\sqrt{n_i})$ convergence rate, which almost matches the optimal $O(\sqrt{1/n_i})$ rate of only learning from the $i$-th distribution with $n_i$ samples.

We present a new method that includes three key components of distributed optimization and federated learning: variance reduction of stochastic gradients, partial participation, and compressed communication. We prove that the new method has optimal oracle complexity and state-of-the-art communication complexity in the partial participation setting. Regardless of the communication compression feature, our method successfully combines variance reduction and partial participation: we get the optimal oracle complexity, never need the participation of all nodes, and do not require the bounded gradients (dissimilarity) assumption.

Given the success of model-free methods for control design in many problem settings, it is natural to ask how things will change if realistic communication channels are utilized for the transmission of gradients or policies. While the resulting problem has analogies with the formulations studied under the rubric of networked control systems, the rich literature in that area has typically assumed that the model of the system is known. As a step towards bridging the fields of model-free control design and networked control systems, we ask: \textit{Is it possible to solve basic control problems - such as the linear quadratic regulator (LQR) problem - in a model-free manner over a rate-limited channel?} Toward answering this question, we study a setting where a worker agent transmits quantized policy gradients (of the LQR cost) to a server over a noiseless channel with a finite bit-rate. We propose a new algorithm titled Adaptively Quantized Gradient Descent (\texttt{AQGD}), and prove that above a certain finite threshold bit-rate, \texttt{AQGD} guarantees exponentially fast convergence to the globally optimal policy, with \textit{no deterioration of the exponent relative to the unquantized setting}. More generally, our approach reveals the benefits of adaptive quantization in preserving fast linear convergence rates, and, as such, may be of independent interest to the literature on compressed optimization.

The goal of Continual Learning (CL) is to continuously learn from new data streams and accomplish the corresponding tasks. Previously studied CL assumes that data are given in sequence nose-to-tail for different tasks, thus indeed belonging to Serial Continual Learning (SCL). This paper studies the novel paradigm of Parallel Continual Learning (PCL) in dynamic multi-task scenarios, where a diverse set of tasks is encountered at different time points. PCL presents challenges due to the training of an unspecified number of tasks with varying learning progress, leading to the difficulty of guaranteeing effective model updates for all encountered tasks. In our previous conference work, we focused on measuring and reducing the discrepancy among gradients in a multi-objective optimization problem, which, however, may still contain negative transfers in every model update. To address this issue, in the dynamic multi-objective optimization problem, we introduce task-specific elastic factors to adjust the descent direction towards the Pareto front. The proposed method, called Elastic Multi-Gradient Descent (EMGD), ensures that each update follows an appropriate Pareto descent direction, minimizing any negative impact on previously learned tasks. To balance the training between old and new tasks, we also propose a memory editing mechanism guided by the gradient computed using EMGD. This editing process updates the stored data points, reducing interference in the Pareto descent direction from previous tasks. Experiments on public datasets validate the effectiveness of our EMGD in the PCL setting.

This paper introduces an iterative algorithm designed to train additive models with favorable memory storage and computational requirements. The algorithm can be viewed as the functional counterpart of stochastic gradient descent, applied to the coefficients of a truncated basis expansion of the component functions. We show that the resulting estimator satisfies an oracle inequality that allows for model mispecification. In the well-specified setting, by choosing the learning rate carefully across three distinct stages of training, we prove that its risk is minimax optimal in terms of the dependence on the dimensionality of the data and the size of the training sample.

We consider the gradient descent flow widely used for the minimization of the $\mathcal{L}^2$ cost function in Deep Learning networks, and introduce two modified versions; one adapted for the overparametrized setting, and the other for the underparametrized setting. Both have a clear and natural invariant geometric meaning, taking into account the pullback vector bundle structure in the overparametrized, and the pushforward vector bundle structure in the underparametrized setting. In the overparametrized case, we prove that, provided that a rank condition holds, all orbits of the modified gradient descent drive the $\mathcal{L}^2$ cost to its global minimum at a uniform exponential convergence rate; one thereby obtains an a priori stopping time for any prescribed proximity to the global minimum. We point out relations of the latter to sub-Riemannian geometry.

In the arena of privacy-preserving machine learning, differentially private stochastic gradient descent (DP-SGD) has outstripped the objective perturbation mechanism in popularity and interest. Though unrivaled in versatility, DP-SGD requires a non-trivial privacy overhead (for privately tuning the model's hyperparameters) and a computational complexity which might be extravagant for simple models such as linear and logistic regression. This paper revamps the objective perturbation mechanism with tighter privacy analyses and new computational tools that boost it to perform competitively with DP-SGD on unconstrained convex generalized linear problems.

While modern machine learning models rely on increasingly large training datasets, data is often limited in privacy-sensitive domains. Generative models trained with differential privacy (DP) on sensitive data can sidestep this challenge, providing access to synthetic data instead. We build on the recent success of diffusion models (DMs) and introduce Differentially Private Diffusion Models (DPDMs), which enforce privacy using differentially private stochastic gradient descent (DP-SGD). We investigate the DM parameterization and the sampling algorithm, which turn out to be crucial ingredients in DPDMs, and propose noise multiplicity, a powerful modification of DP-SGD tailored to the training of DMs. We validate our novel DPDMs on image generation benchmarks and achieve state-of-the-art performance in all experiments. Moreover, on standard benchmarks, classifiers trained on DPDM-generated synthetic data perform on par with task-specific DP-SGD-trained classifiers, which has not been demonstrated before for DP generative models.