We study nonmonotone games satisfying the weak Minty variational inequality (MVI) with parameter $\rho \in (-\tfrac{1}{L}, \infty)$, where $L$ is the Lipschitz constant of the gradient operator. An error corrected version of the inexact proximal point algorithm is proposed, with which we establish the first $\mathcal O(1/\epsilon)$ rate for the entire range $\rho \in (-\tfrac{1}{L}, \infty)$, thus removing a logarithmic factor compared with the complexity of existing methods. The scheme automatically selects the needed accuracy for the proximal computation, and can recover the relaxed extragradient method when $\rho > -\tfrac{1}{2L}$ and the relaxed proximal point algorithm (rPPA) when $\rho > -\tfrac{1}{L}$. Due to the error correction, the scheme inherits the strong properties of the _exact_ rPPA. Specifically, we show that linear convergence is automatically achieved under appropriate conditions. Tightness for the range of $\rho$ is established through a lower bound for rPPA. Central to the algorithmic construction is a halfspace projection, where the key insight is that the allowed error tolerance can both be used to correct for the proximal approximation and to enlarge the problem class.

In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework. Furthermore, we propose an explicit choice of norm for deep architectures, which, as a side benefit, leads to the transferability of hyperparameters across model sizes. Experimentally, we demonstrate significant speedups on nanoGPT training without any reliance on Adam. The proposed method is memory-efficient, requiring only one set of model weights and one set of gradients, which can be stored in half-precision.

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$,$L_1$)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal $O(n^{-1/4})$ convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https://github.com/LIONS-EPFL/ClippedScion.

This paper rethinks Sharpness-Aware Minimization (SAM), which is originally formulated as a zero-sum game where the weights of a network and a bounded perturbation try to minimize/maximize, respectively, the same differentiable loss. To fundamentally improve this design, we argue that SAM should instead be reformulated using the 0-1 loss. As a continuous relaxation, we follow the simple conventional approach where the minimizing (maximizing) player uses an upper bound (lower bound) surrogate to the 0-1 loss. This leads to a novel formulation of SAM as a bilevel optimization problem, dubbed as BiSAM. BiSAM with newly designed lower-bound surrogate loss indeed constructs stronger perturbation. Through numerical evidence, we show that BiSAM consistently results in improved performance when compared to the original SAM and variants, while enjoying similar computational complexity. Our code is available at https://github.com/LIONS-EPFL/BiSAM.

Sharpness-aware minimization (SAM) has been shown to improve the generalization of neural networks. However, each SAM update requires _sequentially_ computing two gradients, effectively doubling the per-iteration cost compared to base optimizers like SGD. We propose a simple modification of SAM, termed SAMPa, which allows us to fully parallelize the two gradient computations. SAMPa achieves a twofold speedup of SAM under the assumption that communication costs between devices are negligible. Empirical results show that SAMPa ranks among the most efficient variants of SAM in terms of computational time. Additionally, our method consistently outperforms SAM across both vision and language tasks. Notably, SAMPa theoretically maintains convergence guarantees even for _fixed_ perturbation sizes, which is established through a novel Lyapunov function. We in fact arrive at SAMPa by treating this convergence guarantee as a hard requirement---an approach we believe is promising for developing SAM-based methods in general. Our code is available at https://github.com/LIONS-EPFL/SAMPa.

This paper presents a theoretical analysis of linear interpolation as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear interpolation can help by leveraging the theory of nonexpansive operators. We construct a new optimization scheme called relaxed approximate proximal point (RAPP), which is the first 1-SCLI method to achieve last iterate convergence rates for $\rho$-comonotone problems while only requiring $\rho > -\tfrac{1}{2L}$. The construction extends to constrained and regularized settings. By replacing the inner optimizer in RAPP we rediscover the family of Lookahead algorithms for which we establish convergence in cohypomonotone problems even when the base optimizer is taken to be gradient descent ascent. The range of cohypomonotone problems in which Lookahead converges is further expanded by exploiting that Lookahead inherits the properties of the base optimizer. We corroborate the results with experiments on generative adversarial networks which demonstrates the benefits of the linear interpolation present in both RAPP and Lookahead.

We examine a flexible algorithmic framework for solving monotone variational inequalities in the presence of randomness and uncertainty. The proposed template encompasses a wide range of popular first-order methods, including dual averaging, dual extrapolation and optimistic gradient algorithms – both adaptive and non-adaptive. Our first result is that the algorithm achieves the optimal rates of convergence for cocoercive problems when the profile of the randomness is known to the optimizer: $\mathcal{O}(1/\sqrt{T})$ for absolute noise profiles, and $\mathcal{O}(1/T)$ for relative ones. Subsequently, we drop all prior knowledge requirements (the absolute/relative variance of the randomness affecting the problem, the operator's cocoercivity constant, etc.), and we analyze an adaptive instance of the method that gracefully interpolates between the above rates – i.e. it achieves $\mathcal{O}(1/\sqrt{T})$ and $\mathcal{O}(1/T)$ in the absolute and relative cases, respectively. To our knowledge, this is the first universality result of its kind in the literature and, somewhat surprisingly, it shows that an extra-gradient proxy step is not required to achieve optimal rates.

Overparameterization refers to the important phenomenon where the width of a neural network is chosen such that learning algorithms can provably attain zero loss in nonconvex training. The existing theory establishes such global convergence using various initialization strategies, training modifications, and width scalings. In particular, the state-of-the-art results require the width to scale quadratically with the number of training data under standard initialization strategies used in practice for best generalization performance. In contrast, the most recent results obtain linear scaling either with requiring initializations that lead to the "lazy-training", or training only a single layer. In this work, we provide an analytical framework that allows us to adopt standard initialization strategies, possibly avoid lazy training, and train all layers simultaneously in basic shallow neural networks while attaining a desirable subquadratic scaling on the network width. We achieve the desiderata via Polyak-Lojasiewicz condition, smoothness, and standard assumptions on data, and use tools from random matrix theory.

We describe a logical calculus for process synthesis applying reaction rules for consuming, transforming and producing products at costs. Unlike common logical calculi, which deal with propositions supported by truth semantics, the process synthesis calculus deals with products (in a concrete or abstract sense) and their costs, and creation of products by the composing of appropriate reaction assemblies from a database of reactions or transactions. As such the calculus may be understood as a resource logic akin to linear logics. The calculus appeals to means-end backwards reasoning using a derivative of definite clause logic. Although rather general in scope, as an interesting case in this paper the calculus is applied to chemical retro-synthesis – in particular with the notoriously difficult Solvay cluster designs.