David Martı́nez-Rubio

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.

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

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.