In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on non-interpolating ones; we show that generally similar improvements are impossible in the private setting. However, when the functions exhibit quadratic growth around the optimum, we show (near) exponential improvements in the private sample complexity. In particular, we propose an adaptive algorithm that improves the sample complexity to achieve expected error $\alpha$ from $\frac{d}{\varepsilon \sqrt{\alpha}}$ to $\frac{1}{\alpha^\rho} + \frac{d}{\varepsilon} \log\left(\frac{1}{\alpha}\right)$ for any fixed $\rho >0$, while retaining the standard minimax-optimal sample complexity for non-interpolation problems. We prove a lower bound that shows the dimension-dependent term is tight. Furthermore, we provide a superefficiency result which demonstrates the necessity of the polynomial term for adaptive algorithms: any algorithm that has a polylogarithmic sample complexity for interpolation problems cannot achieve the minimax-optimal rates for the family of non-interpolation problems.

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for orthogonal input vectors, a precise description of the gradient flow dynamics of training one-hidden layer ReLU neural networks for the mean squared error at small initialisation. In this setting, despite non-convexity, we show that the gradient flow converges to zero loss and characterise its implicit bias towards minimum variation norm. Furthermore, some interesting phenomena are highlighted: a quantitative description of the initial alignment phenomenon and a proof that the process follows a specific saddle to saddle dynamics.

Adaptive gradient methods have shown their ability to adjust the stepsizes on the fly in a parameter-agnostic manner, and empirically achieve faster convergence for solving minimization problems. When it comes to nonconvex minimax optimization, however, current convergence analyses of gradient descent ascent (GDA) combined with adaptive stepsizes require careful tuning of hyper-parameters and the knowledge of problem-dependent parameters. Such a discrepancy arises from the primal-dual nature of minimax problems and the necessity of delicate time-scale separation between the primal and dual updates in attaining convergence. In this work, we propose a single-loop adaptive GDA algorithm called TiAda for nonconvex minimax optimization that automatically adapts to the time-scale separation. Our algorithm is fully parameter-agnostic and can achieve near-optimal complexities simultaneously in deterministic and stochastic settings of nonconvex-strongly-concave minimax problems. The effectiveness of the proposed method is further justified numerically for a number of machine learning applications.

We propose the first character-level white-box adversarial attack method against transformer models. The intuition of our method comes from the observation that words are split into subtokens before being fed into the transformer models and the substitution between two close subtokens has a similar effect to the character modification. Our method mainly contains three steps. First, a gradient-based method is adopted to find the most vulnerable words in the sentence. Then we split the selected words into subtokens to replace the origin tokenization result from the transformer tokenizer. Finally, we utilize an adversarial loss to guide the substitution of attachable subtokens in which the Gumbel-softmax trick is introduced to ensure gradient propagation. Meanwhile, we introduce the visual and length constraint in the optimization process to achieve minimum character modifications. Extensive experiments on both sentence-level and token-level tasks demonstrate that our method could outperform the previous attack methods in terms of success rate and edit distance. Furthermore, human evaluation verifies our adversarial examples could preserve their origin labels.

We consider linear prediction with a convex Lipschitz loss, or more generally, stochastic convex optimization problems of generalized linear form, i.e.~where each instantaneous loss is a scalar convex function of a linear function. We show that in this setting, early stopped Gradient Descent (GD), without any explicit regularization or projection, ensures excess error at most $\epsilon$ (compared to the best possible with unit Euclidean norm) with an optimal, up to logarithmic factors, sample complexity of $\tilde{O}(1/\epsilon^2)$ and only $\tilde{O}(1/\epsilon^2)$ iterations. This contrasts with general stochastic convex optimization, where $\Omega(1/\epsilon^4)$ iterations are needed Amir et al. [2021b]. The lower iteration complexity is ensured by leveraging uniform convergence rather than stability. But instead of uniform convergence in a norm ball, which we show can guarantee suboptimal learning using $\Theta(1/\epsilon^4)$ samples, we rely on uniform convergence in a distribution-dependent ball.

Neural network quantum states (NQS), incorporating with variational Monte Carlo (VMC) method, are shown to be a promising way to investigate quantum many-body physics. Whereas vanilla VMC methods perform one gradient update per sample, we introduce a novel objective function with proximal optimization (PO) that enables multiple updates via reusing the mismatched samples. Our VMC-PO method keeps the advantage of the previous importance sampling gradient optimization algorithm [L. Yang, {\it et al}, Phys. Rev. Research {\bf 2}, 012039(R)(2020)] that efficiently uses sampled states. PO mitigates the numerical instabilities during network updates, which is similar to stochastic reconfiguration (SR) methods, but achieves an alternative and simpler implement with lower computational complexity. We investigate the performance of our VMC-PO algorithm for ground-state searching with a 1-dimensional transverse-field Ising model and 2-dimensional Heisenberg antiferromagnet on a square lattice, and demonstrate that the reached ground-state energies are comparable to state-of-the-art results.

Softmax policy gradient is a popular algorithm for policy optimization in single-agent reinforcement learning, particularly since projection is not needed for each gradient update. However, in multi-agent systems, the lack of central coordination introduces significant additional difficulties in the convergence analysis. Even for a stochastic game with identical interest, there can be multiple Nash Equilibria (NEs), which disables proof techniques that rely on the existence of a unique global optimum. Moreover, the softmax parameterization introduces non-NE policies with zero gradient, making it difficult for gradient-based algorithms in seeking NEs. In this paper, we study the finite time convergence of decentralized softmax gradient play in a special form of game, Markov Potential Games (MPGs), which includes the identical interest game as a special case. We investigate both gradient play and natural gradient play, with and without $\log$-barrier regularization. The established convergence rates for the unregularized cases contain a trajectory-dependent constant that can be arbitrarily large, whereas the $\log$-barrier regularization overcomes this drawback, with the cost of slightly worse dependence on other factors such as the action set size. An empirical study on an identical interest matrix game confirms the theoretical findings.

Distributed machine learning (DML) can be an important capability for modern military to take advantage of data and devices distributed at multiple vantage points to adapt and learn. The existing distributed machine learning frameworks, however, cannot realize the full benefits of DML, because they are all based on the simple linear aggregation framework, but linear aggregation cannot handle the $\textit{divergence challenges}$ arising in military settings: the learning data at different devices can be heterogeneous ($\textit{i.e.}$, Non-IID data), leading to model divergence, but the ability for devices to communicate is substantially limited ($\textit{i.e.}$, weak connectivity due to sparse and dynamic communications), reducing the ability for devices to reconcile model divergence. In this paper, we introduce a novel DML framework called aggregation in the mirror space (AIMS) that allows a DML system to introduce a general mirror function to map a model into a mirror space to conduct aggregation and gradient descent. Adapting the convexity of the mirror function according to the divergence force, AIMS allows automatic optimization of DML. We conduct both rigorous analysis and extensive experimental evaluations to demonstrate the benefits of AIMS. For example, we prove that AIMS achieves a loss of $O\left((\frac{m^{r+1}}{T})^{\frac1r}\right)$ after $T$ network-wide updates, where $m$ is the number of devices and $r$ the convexity of the mirror function, with existing linear aggregation frameworks being a special case with $r=2$. Our experimental evaluations using EMANE (Extendable Mobile Ad-hoc Network Emulator) for military communications settings show similar results: AIMS can improve DML convergence rate by up to 57\% and scale well to more devices with weak connectivity, all with little additional computation overhead compared to traditional linear aggregation.

Deep learning experiments by Cohen et al. [2021] using deterministic Gradient Descent (GD) revealed an Edge of Stability (EoS) phase when learning rate (LR) and sharpness (i.e., the largest eigenvalue of Hessian) no longer behave as in traditional optimization. Sharpness stabilizes around $2/$LR and loss goes up and down across iterations, yet still with an overall downward trend. The current paper mathematically analyzes a new mechanism of implicit regularization in the EoS phase, whereby GD updates due to non-smooth loss landscape turn out to evolve along some deterministic flow on the manifold of minimum loss. This is in contrast to many previous results about implicit bias either relying on infinitesimal updates or noise in gradient. Formally, for any smooth function $L$ with certain regularity condition, this effect is demonstrated for (1) Normalized GD, i.e., GD with a varying LR $\eta_t =\frac{\eta}{\| \nabla L(x(t)) \|}$ and loss $L$; (2) GD with constant LR and loss $\sqrt{L- \min_x L(x)}$. Both provably enter the Edge of Stability, with the associated flow on the manifold minimizing $\lambda_{1}(\nabla^2 L)$. The above theoretical results have been corroborated by an experimental study.

We formulate the first differentiable analog quantum computing framework with a specific parameterization design at the analog signal (pulse) level to better exploit near-term quantum devices via variational methods. We further propose a scalable approach to estimate the gradients of quantum dynamics using a forward pass with Monte Carlo sampling, which leads to a quantum stochastic gradient descent algorithm for scalable gradient-based training in our framework. Applying our framework to quantum optimization and control, we observe a significant advantage of differentiable analog quantum computing against SOTAs based on parameterized digital quantum circuits by orders of magnitude.