Bilevel optimization has recently regained interest owing to its applications in emerging machine learning fields such as hyperparameter optimization, meta-learning, and reinforcement learning. Recent results have shown that simple alternating (implicit) gradient-based algorithms can match the convergence rate of single-level gradient descent (GD) when addressing bilevel problems with a strongly convex lower-level objective. However, it remains unclear whether this result can be generalized to bilevel problems beyond this basic setting. In this paper, we first introduce a stationary metric for the considered bilevel problems, which generalizes the existing metric, for a nonconvex lower-level objective that satisfies the Polyak-{\L}ojasiewicz (PL) condition. We then propose a Generalized ALternating mEthod for bilevel opTimization (GALET) tailored to BLO with convex PL LL problem and establish that GALET achieves an $\epsilon$-stationary point for the considered problem within $\tilde{\cal O}(\epsilon^{-1})$ iterations, which matches the iteration complexity of GD for single-level smooth nonconvex problems.

Differentiable logics (DL) have recently been proposed as a method of training neural networks to satisfy logical specifications. A DL consists of a syntax in which specifications are stated and an interpretation function that translates expressions in the syntax into loss functions. These loss functions can then be used during training with standard gradient descent algorithms. The variety of existing DLs and the differing levels of formality with which they are treated makes a systematic comparative study of their properties and implementations difficult. This paper remedies this problem by suggesting a meta-language for defining DLs that we call the Logic of Differentiable Logics, or LDL. Syntactically, it generalises the syntax of existing DLs to FOL, and for the first time introduces the formalism for reasoning about vectors and learners. Semantically, it introduces a general interpretation function that can be instantiated to define loss functions arising from different existing DLs. We use LDL to establish several theoretical properties of existing DLs, and to conduct their empirical study in neural network verification.

We show that the canonical approach for training differentially private GANs -- updating the discriminator with differentially private stochastic gradient descent (DPSGD) -- can yield significantly improved results after modifications to training. Specifically, we propose that existing instantiations of this approach neglect to consider how adding noise only to discriminator updates inhibits discriminator training, disrupting the balance between the generator and discriminator necessary for successful GAN training. We show that a simple fix -- taking more discriminator steps between generator steps -- restores parity between the generator and discriminator and improves results. Additionally, with the goal of restoring parity, we experiment with other modifications -- namely, large batch sizes and adaptive discriminator update frequency -- to improve discriminator training and see further improvements in generation quality. Our results demonstrate that on standard image synthesis benchmarks, DPSGD outperforms all alternative GAN privatization schemes. Code: https://github.com/alexbie98/dpgan-revisit.

We propose a new gradient descent algorithm with added stochastic terms for finding the global optimizers of nonconvex optimization problems. A key component in the algorithm is the adaptive tuning of the randomness based on the value of the objective function. In the language of simulated annealing, the temperature is state-dependent. With this, we prove the global convergence of the algorithm with an algebraic rate both in probability and in the parameter space. This is a significant improvement over the classical rate from using a more straightforward control of the noise term. The convergence proof is based on the actual discrete setup of the algorithm, not just its continuous limit as often done in the literature. We also present several numerical examples to demonstrate the efficiency and robustness of the algorithm for reasonably complex objective functions.

In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms appearing in the ODEs penalize the two-norm of the loss gradients. We prove that the existence of similar implicit regularization in RMSProp and Adam depends on their hyperparameters and the training stage, but with a different "norm" involved: the corresponding ODE terms either penalize the (perturbed) one-norm of the loss gradients or, on the contrary, hinder its decrease (the latter case being typical). We also conduct numerical experiments and discuss how the proven facts can influence generalization.

Stein Variational Gradient Descent (SVGD) is a popular variational inference algorithm which simulates an interacting particle system to approximately sample from a target distribution, with impressive empirical performance across various domains. Theoretically, its population (i.e, infinite-particle) limit dynamics is well studied but the behavior of SVGD in the finite-particle regime is much less understood. In this work, we design two computationally efficient variants of SVGD, namely VP-SVGD and GB-SVGD, with provably fast finite-particle convergence rates. We introduce the notion of virtual particles and develop novel stochastic approximations of population-limit SVGD dynamics in the space of probability measures, which are exactly implementable using a finite number of particles. Our algorithms can be viewed as specific random-batch approximations of SVGD, which are computationally more efficient than ordinary SVGD. We show that the $n$ particles output by VP-SVGD and GB-SVGD, run for $T$ steps with batch-size $K$, are at-least as good as i.i.d samples from a distribution whose Kernel Stein Discrepancy to the target is at most $O\left(\tfrac{d^{1/3}}{(KT)^{1/6}}\right)$ under standard assumptions. Our results also hold under a mild growth condition on the potential function, which is much weaker than the isoperimetric (e.g. Poincare Inequality) or information-transport conditions (e.g. Talagrand's Inequality $\mathsf{T}_1$) generally considered in prior works. As a corollary, we consider the convergence of the empirical measure (of the particles output by VP-SVGD and GB-SVGD) to the target distribution and demonstrate a double exponential improvement over the best known finite-particle analysis of SVGD. Beyond this, our results present the first known oracle complexities for this setting with polynomial dimension dependence.

Diffusion models (DMs) have been adopted across diverse fields with its remarkable abilities in capturing intricate data distributions. In this paper, we propose a Fast Diffusion Model (FDM) to significantly speed up DMs from a stochastic optimization perspective for both faster training and sampling. We first find that the diffusion process of DMs accords with the stochastic optimization process of stochastic gradient descent (SGD) on a stochastic time-variant problem. Then, inspired by momentum SGD that uses both gradient and an extra momentum to achieve faster and more stable convergence than SGD, we integrate momentum into the diffusion process of DMs. This comes with a unique challenge of deriving the noise perturbation kernel from the momentum-based diffusion process. To this end, we frame the process as a Damped Oscillation system whose critically damped state -- the kernel solution -- avoids oscillation and yields a faster convergence speed of the diffusion process. Empirical results show that our FDM can be applied to several popular DM frameworks, e.g., VP, VE, and EDM, and reduces their training cost by about 50% with comparable image synthesis performance on CIFAR-10, FFHQ, and AFHQv2 datasets. Moreover, FDM decreases their sampling steps by about 3x to achieve similar performance under the same samplers. The code is available at https://github.com/sail-sg/FDM.

We develop a re-weighted gradient descent technique for boosting the performance of deep neural networks, which involves importance weighting of data points during each optimization step. Our approach is inspired by distributionally robust optimization with f-divergences, which has been known to result in models with improved generalization guarantees. Our re-weighting scheme is simple, computationally efficient, and can be combined with many popular optimization algorithms such as SGD and Adam. Empirically, we demonstrate the superiority of our approach on various tasks, including supervised learning, domain adaptation. Notably, we obtain improvements of +0.7% and +1.44% over SOTA on DomainBed and Tabular classification benchmarks, respectively. Moreover, our algorithm boosts the performance of BERT on GLUE benchmarks by +1.94%, and ViT on ImageNet-1K by +1.01%. These results demonstrate the effectiveness of the proposed approach, indicating its potential for improving performance in diverse domains.

Humans sometimes show sudden improvements in task performance that have been linked to moments of insight. Such insight-related performance improvements appear special because they are preceded by an extended period of impasse, are unusually abrupt, and occur only in some, but not all, learners. Here, we ask whether insight-like behaviour also occurs in artificial neural networks trained with gradient descent algorithms. We compared learning dynamics in humans and regularised neural networks in a perceptual decision task that provided a hidden opportunity which allowed to solve the task more efficiently. We show that humans tend to discover this regularity through insight, rather than gradually. Notably, neural networks with regularised gate modulation closely mimicked behavioural characteristics of human insights, exhibiting delay of insight, suddenness and selective occurrence. Analyses of network learning dynamics revealed that insight-like behaviour crucially depended on noise added to gradient updates, and was preceded by ``silent knowledge'' that is initially suppressed by regularised (attentional) gating. This suggests that insights can arise naturally from gradual learning, where they reflect the combined influences of noise, attentional gating and regularisation.

A well-known algorithm in privacy-preserving ML is differentially private stochastic gradient descent (DP-SGD). While this algorithm has been evaluated on text and image data, it has not been previously applied to ads data, which are notorious for their high class imbalance and sparse gradient updates. In this work we apply DP-SGD to several ad modeling tasks including predicting click-through rates, conversion rates, and number of conversion events, and evaluate their privacy-utility trade-off on real-world datasets. Our work is the first to empirically demonstrate that DP-SGD can provide both privacy and utility for ad modeling tasks.