David Martínez-Rubio

David Martínez-Rubio, Sebastian Pokutta

We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds. We achieve the same convergence rates as Nesterov's accelerated gradient descent, up to a multiplicative geometric penalty and log factors. Crucially, we can enforce our method to stay within a compact set we define. Prior fully accelerated works \emph{resort to assuming} that the iterates of their algorithms stay in some pre-specified compact set, except for two previous methods of limited applicability. For our manifolds, this solves the open question in [KY22] about obtaining global general acceleration without iterates assumptively staying in the feasible set. In our solution, we design an accelerated Riemannian inexact proximal point algorithm, which is a result that was unknown even with exact access to the proximal operator, and is of independent interest. For smooth functions, we show we can implement the prox step inexactly with first-order methods in Riemannian balls of certain diameter that is enough for global accelerated optimization.

Kimon Fountoulakis, David Martínez-Rubio

We study the degree-weighted work required to compute $\ell_1$-regularized PageRank using the standard one-gradient-per-iteration accelerated proximal-gradient method (FISTA). For non-accelerated local methods, the best known worst-case work scales as $\widetilde{O} ((αρ)^{-1})$, where $α$ is the teleportation parameter and $ρ$ is the $\ell_1$-regularization parameter. A natural question is whether FISTA can improve the dependence on $α$ from $1/α$ to $1/\sqrtα$ while preserving the $1/ρ$ locality scaling. The challenge is that acceleration can break locality by transiently activating nodes that are zero at optimality, thereby increasing the cost of gradient evaluations. We analyze FISTA on a slightly over-regularized objective and show that, under a checkable confinement condition, all spurious activations remain inside a boundary set $\mathcal{B}$. This yields a bound consisting of an accelerated $(ρ\sqrtα)^{-1}\log(α/\varepsilon)$ term plus a boundary overhead $\sqrt{vol(\mathcal{B})}/(ρα^{3/2})$. We provide graph-structural conditions that imply such confinement. Experiments on synthetic and real graphs show the resulting speedup and slowdown regimes under the degree-weighted work model.

David Martínez-Rubio, Sebastian Pokutta

We introduce novel techniques to enhance Frank-Wolfe algorithms by leveraging function smoothness beyond traditional short steps. Our study focuses on Frank-Wolfe algorithms with step sizes that incorporate primal-dual guarantees, offering practical stopping criteria. We present a new Frank-Wolfe algorithm utilizing an optimistic framework and provide a primal-dual convergence proof. Additionally, we propose a generalized short-step strategy aimed at optimizing a computable primal-dual gap. Interestingly, this new generalized short-step strategy is also applicable to gradient descent algorithms beyond Frank-Wolfe methods. As a byproduct, our work revisits and refines primal-dual techniques for analyzing Frank-Wolfe algorithms, achieving tighter primal-dual convergence rates. Empirical results demonstrate that our optimistic algorithm outperforms existing methods, highlighting its practical advantages.

Simon Vary, David Martínez-Rubio, Patrick Rebeschini

We study first-order algorithms that are uniformly stable for empirical risk minimization (ERM) problems that are convex and smooth with respect to $p$-norms, $p \geq 1$. We propose a black-box reduction method that, by employing properties of uniformly convex regularizers, turns an optimization algorithm for Hölder smooth convex losses into a uniformly stable learning algorithm with optimal statistical risk bounds on the excess risk, up to a constant factor depending on $p$. Achieving a black-box reduction for uniform stability was posed as an open question by (Attia and Koren, 2022), which had solved the Euclidean case $p=2$. We explore applications that leverage non-Euclidean geometry in addressing binary classification problems.

David Martínez-Rubio

Quasar-convex functions form a broad nonconvex class with applications to linear dynamical systems, generalized linear models, and Riemannian optimization, among others. Current nearly optimal algorithms work only in affine spaces due to the loss of one degree of freedom when working with general convex constraints. Obtaining an accelerated algorithm that makes nearly optimal $\widetilde{O}(1/(γ\sqrtε))$ first-order queries to a $γ$-quasar convex smooth function \emph{with constraints} was independently asked as an open problem in Martínez-Rubio (2022); Lezane, Langer, and Koolen (2024). In this work, we solve this question by designing an inexact accelerated proximal point algorithm that we implement using a first-order method achieving the aforementioned rate and, as a consequence, we improve the complexity of the accelerated geodesically Riemannian optimization solution in Martínez-Rubio (2022). We also analyze projected gradient descent and Frank-Wolfe algorithms in this constrained quasar-convex setting. To the best of our knowledge, our work provides the first analyses of first-order methods for quasar-convex smooth functions with general convex constraints.

Christophe Roux, David Martínez-Rubio, Sebastian Pokutta

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.

Juan Pablo Contreras, Cristóbal Guzmán, David Martínez-Rubio

We develop algorithms for the optimization of convex objectives that have Hölder continuous $q$-th derivatives by using a $q$-th order oracle, for any $q \geq 1$. Our algorithms work for general norms under mild conditions, including the $\ell_p$-settings for $1\leq p\leq \infty$. We can also optimize structured functions that allow for inexactly implementing a non-Euclidean ball optimization oracle. We do this by developing a non-Euclidean inexact accelerated proximal point method that makes use of an \emph{inexact uniformly convex regularizer}. We show a lower bound for general norms that demonstrates our algorithms are nearly optimal in high-dimensions in the black-box oracle model for $\ell_p$-settings and all $q \geq 1$, even in randomized and parallel settings. This new lower bound, when applied to the first-order smooth case, resolves an open question in parallel convex optimization.

Nico Pelleriti, Christoph Spiegel, Shiwei Liu et al.

Certifying nonnegativity of polynomials is a well-known NP-hard problem with direct applications spanning non-convex optimization, control, robotics, and beyond. A sufficient condition for nonnegativity is the Sum of Squares (SOS) property, i.e., it can be written as a sum of squares of other polynomials. In practice, however, certifying the SOS criterion remains computationally expensive and often involves solving a Semidefinite Program (SDP), whose dimensionality grows quadratically in the size of the monomial basis of the SOS expression; hence, various methods to reduce the size of the monomial basis have been proposed. In this work, we introduce the first learning-augmented algorithm to certify the SOS criterion. To this end, we train a Transformer model that predicts an almost-minimal monomial basis for a given polynomial, thereby drastically reducing the size of the corresponding SDP. Our overall methodology comprises three key components: efficient training dataset generation of over 100 million SOS polynomials, design and training of the corresponding Transformer architecture, and a systematic fallback mechanism to ensure correct termination, which we analyze theoretically. We validate our approach on over 200 benchmark datasets, achieving speedups of over $100\times$ compared to state-of-the-art solvers and enabling the solution of instances where competing approaches fail. Our findings provide novel insights towards transforming the practical scalability of SOS programming.

Emmeran Johnson, David Martínez-Rubio, Ciara Pike-Burke et al.

We study online convex optimization on $\ell_p$-balls in $\mathbb{R}^d$ for $p > 2$. While always sub-linear, the optimal regret exhibits a shift between the high-dimensional setting ($d > T$), when the dimension $d$ is greater than the time horizon $T$ and the low-dimensional setting ($d \leq T$). We show that Follow-the-Regularised-Leader (FTRL) with time-varying regularisation which is adaptive to the dimension regime is anytime optimal for all dimension regimes. Motivated by this, we ask whether it is possible to obtain anytime optimality of FTRL with fixed non-adaptive regularisation. Our main result establishes that for separable regularisers, adaptivity in the regulariser is necessary, and that any fixed regulariser will be sub-optimal in one of the two dimension regimes. Finally, we provide lower bounds which rule out sub-linear regret bounds for the linear bandit problem in sufficiently high-dimension for all $\ell_p$-balls with $p \geq 1$.

David Martínez-Rubio, Christophe Roux, Christopher Criscitiello et al.

In this work, we study optimization problems of the form $\min_x \max_y f(x, y)$, where $f(x, y)$ is defined on a product Riemannian manifold $\mathcal{M} \times \mathcal{N}$ and is $μ_x$-strongly geodesically convex (g-convex) in $x$ and $μ_y$-strongly g-concave in $y$, for $μ_x, μ_y \geq 0$. We design accelerated methods when $f$ is $(L_x, L_y, L_{xy})$-smooth and $\mathcal{M}$, $\mathcal{N}$ are Hadamard. To that aim we introduce new g-convex optimization results, of independent interest: we show global linear convergence for metric-projected Riemannian gradient descent and improve existing accelerated methods by reducing geometric constants. Additionally, we complete the analysis of two previous works applying to the Riemannian min-max case by removing an assumption about iterates staying in a pre-specified compact set.

David Martínez-Rubio

We further research on the accelerated optimization phenomenon on Riemannian manifolds by introducing accelerated global first-order methods for the optimization of $L$-smooth and geodesically convex (g-convex) or $μ$-strongly g-convex functions defined on the hyperbolic space or a subset of the sphere. For a manifold other than the Euclidean space, these are the first methods to \emph{globally} achieve the same rates as accelerated gradient descent in the Euclidean space with respect to $L$ and $ε$ (and $μ$ if it applies), up to log factors. Due to the geometric deformations, our rates have an extra factor, depending on the initial distance $R$ to a minimizer and the curvature $K$, with respect to Euclidean accelerated algorithms As a proxy for our solution, we solve a constrained non-convex Euclidean problem, under a condition between convexity and \emph{quasar-convexity}, of independent interest. Additionally, for any Riemannian manifold of bounded sectional curvature, we provide reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa. We also reduce global optimization to optimization over bounded balls where the effect of the curvature is reduced.

Chris Mingard, Joar Skalse, Guillermo Valle-Pérez et al.

Understanding the inductive bias of neural networks is critical to explaining their ability to generalise. Here, for one of the simplest neural networks -- a single-layer perceptron with n input neurons, one output neuron, and no threshold bias term -- we prove that upon random initialisation of weights, the a priori probability $P(t)$ that it represents a Boolean function that classifies t points in ${0,1}^n$ as 1 has a remarkably simple form: $P(t) = 2^{-n}$ for $0\leq t < 2^n$. Since a perceptron can express far fewer Boolean functions with small or large values of t (low entropy) than with intermediate values of t (high entropy) there is, on average, a strong intrinsic a-priori bias towards individual functions with low entropy. Furthermore, within a class of functions with fixed t, we often observe a further intrinsic bias towards functions of lower complexity. Finally, we prove that, regardless of the distribution of inputs, the bias towards low entropy becomes monotonically stronger upon adding ReLU layers, and empirically show that increasing the variance of the bias term has a similar effect.

Mario Lezcano-Casado, David Martínez-Rubio

We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization transforms the constrained optimization problem into an unconstrained one over a Euclidean space, for which common first-order optimization methods can be used. The theoretical results presented are general enough to cover the special orthogonal group, the unitary group and, in general, any connected compact Lie group. We discuss how this and other parametrizations can be computed efficiently through an implementation trick, making numerically complex parametrizations usable at a negligible runtime cost in neural networks. In particular, we apply our results to RNNs with orthogonal recurrent weights, yielding a new architecture called expRNN. We demonstrate how our method constitutes a more robust approach to optimization with orthogonal constraints, showing faster, accurate, and more stable convergence in several tasks designed to test RNNs.

David Martínez-Rubio, Varun Kanade, Patrick Rebeschini

We study a decentralized cooperative stochastic multi-armed bandit problem with $K$ arms on a network of $N$ agents. In our model, the reward distribution of each arm is the same for each agent and rewards are drawn independently across agents and time steps. In each round, each agent chooses an arm to play and subsequently sends a message to her neighbors. The goal is to minimize the overall regret of the entire network. We design a fully decentralized algorithm that uses an accelerated consensus procedure to compute (delayed) estimates of the average of rewards obtained by all the agents for each arm, and then uses an upper confidence bound (UCB) algorithm that accounts for the delay and error of the estimates. We analyze the regret of our algorithm and also provide a lower bound. The regret is bounded by the optimal centralized regret plus a natural and simple term depending on the spectral gap of the communication matrix. Our algorithm is simpler to analyze than those proposed in prior work and it achieves better regret bounds, while requiring less information about the underlying network. It also performs better empirically.