Riccardo Della Vecchia

Timothée Mathieu, Riccardo Della Vecchia, Alena Shilova et al.

Recently, the scientific community has questioned the statistical reproducibility of many empirical results, especially in the field of machine learning. To contribute to the resolution of this reproducibility crisis, we propose a theoretically sound methodology for comparing the performance of a set of algorithms. We exemplify our methodology in Deep Reinforcement Learning (Deep RL). The performance of one execution of a Deep RL algorithm is a random variable. Therefore, several independent executions are needed to evaluate its performance. When comparing algorithms with random performance, a major question concerns the number of executions to perform to ensure that the result of the comparison is theoretically sound. Researchers in Deep RL often use less than 5 independent executions to compare algorithms: we claim that this is not enough in general. Moreover, when comparing more than 2 algorithms at once, we have to use a multiple tests procedure to preserve low error guarantees. We introduce AdaStop, a new statistical test based on multiple group sequential tests. When used to compare algorithms, AdaStop adapts the number of executions to stop as early as possible while ensuring that enough information has been collected to distinguish algorithms that have different score distributions. We prove theoretically that AdaStop has a low probability of making a (family-wise) error. We illustrate the effectiveness of AdaStop in various use-cases, including toy examples and Deep RL algorithms on challenging Mujoco environments. AdaStop is the first statistical test fitted to this sort of comparisons: it is both a significant contribution to statistics, and an important contribution to computational studies performed in reinforcement learning and in other domains.

Riccardo Della Vecchia, Alena Shilova, Philippe Preux et al.

Deep Reinforcement Learning (Deep RL) has had incredible achievements on high dimensional problems, yet its learning process remains unstable even on the simplest tasks. Deep RL uses neural networks as function approximators. These neural models are largely inspired by developments in the (un)supervised machine learning community. Compared to these learning frameworks, one of the major difficulties of RL is the absence of i.i.d. data. One way to cope with this difficulty is to control the rate of change of the policy at every iteration. In this work, we challenge the common practices of the (un)supervised learning community of using a fixed neural architecture, by having a neural model that grows in size at each policy update. This allows a closed form entropy regularized policy update, which leads to a better control of the rate of change of the policy at each iteration and help cope with the non i.i.d. nature of RL. Initial experiments on classical RL benchmarks show promising results with remarkable convergence on some RL tasks when compared to other deep RL baselines, while exhibiting limitations on others.

Riccardo Della Vecchia, Debabrota Basu

Endogeneity, i.e. the dependence of noise and covariates, is a common phenomenon in real data due to omitted variables, strategic behaviours, measurement errors etc. In contrast, the existing analyses of stochastic online linear regression with unbounded noise and linear bandits depend heavily on exogeneity, i.e. the independence of noise and covariates. Motivated by this gap, we study the over- and just-identified Instrumental Variable (IV) regression, specifically Two-Stage Least Squares, for stochastic online learning, and propose to use an online variant of Two-Stage Least Squares, namely O2SLS. We show that O2SLS achieves $\mathcal O(d_{x}d_{z}\log^2 T)$ identification and $\widetilde{\mathcal O}(γ\sqrt{d_{z} T})$ oracle regret after $T$ interactions, where $d_{x}$ and $d_{z}$ are the dimensions of covariates and IVs, and $γ$ is the bias due to endogeneity. For $γ=0$, i.e. under exogeneity, O2SLS exhibits $\mathcal O(d_{x}^2 \log^2 T)$ oracle regret, which is of the same order as that of the stochastic online ridge. Then, we leverage O2SLS as an oracle to design OFUL-IV, a stochastic linear bandit algorithm to tackle endogeneity. OFUL-IV yields $\widetilde{\mathcal O}(\sqrt{d_{x}d_{z}T})$ regret that matches the regret lower bound under exogeneity. For different datasets with endogeneity, we experimentally show efficiencies of O2SLS and OFUL-IV.

Nicolò Cesa-Bianchi, Tommaso R. Cesari, Riccardo Della Vecchia

We study the interplay between communication and feedback in a cooperative online learning setting, where a network of communicating agents learn a common sequential decision-making task through a feedback graph. We bound the network regret in terms of the independence number of the strong product between the communication network and the feedback graph. Our analysis recovers as special cases many previously known bounds for cooperative online learning with expert or bandit feedback. We also prove an instance-based lower bound, demonstrating that our positive results are not improvable except in pathological cases. Experiments on synthetic data confirm our theoretical findings.

Riccardo Della Vecchia, Tommaso Cesari

We consider the problem of asynchronous online combinatorial optimization on a network of communicating agents. At each time step, some of the agents are stochastically activated, requested to make a prediction, and the system pays the corresponding loss. Then, neighbors of active agents receive semi-bandit feedback and exchange some succinct local information. The goal is to minimize the network regret, defined as the difference between the cumulative loss of the predictions of active agents and that of the best action in hindsight, selected from a combinatorial decision set. The main challenge in such a context is to control the computational complexity of the resulting algorithm while retaining minimax optimal regret guarantees. We introduce Coop-FTPL, a cooperative version of the well-known Follow The Perturbed Leader algorithm, that implements a new loss estimation procedure generalizing the Geometric Resampling of Neu and Bart{ó}k [2013] to our setting. Assuming that the elements of the decision set are k-dimensional binary vectors with at most m non-zero entries and $α$ 1 is the independence number of the network, we show that the expected regret of our algorithm after T time steps is of order Q mkT log(k)(k$α$ 1 /Q + m), where Q is the total activation probability mass. Furthermore, we prove that this is only $\sqrt$ k log k-away from the best achievable rate and that Coop-FTPL has a state-of-the-art T 3/2 worst-case computational complexity.

Carlo Baldassi, Riccardo Della Vecchia, Carlo Lucibello et al.

The geometrical features of the (non-convex) loss landscape of neural network models are crucial in ensuring successful optimization and, most importantly, the capability to generalize well. While minimizers' flatness consistently correlates with good generalization, there has been little rigorous work in exploring the condition of existence of such minimizers, even in toy models. Here we consider a simple neural network model, the symmetric perceptron, with binary weights. Phrasing the learning problem as a constraint satisfaction problem, the analogous of a flat minimizer becomes a large and dense cluster of solutions, while the narrowest minimizers are isolated solutions. We perform the first steps toward the rigorous proof of the existence of a dense cluster in certain regimes of the parameters, by computing the first and second moment upper bounds for the existence of pairs of arbitrarily close solutions. Moreover, we present a non rigorous derivation of the same bounds for sets of $y$ solutions at fixed pairwise distances.