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Cedric Malherbe

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2 papers

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NeurIPS Conference 2022 Conference Paper

Optimistic Tree Searches for Combinatorial Black-Box Optimization

Cedric Malherbe
Antoine Grosnit
Rasul Tutunov
Haitham Bou Ammar
Jun Wang

The optimization of combinatorial black-box functions is pervasive in computer science and engineering. However, the combinatorial explosion of the search space and lack of natural ordering pose significant challenges for current techniques from a theoretical and practical perspective, and require new algorithmic ideas. In this paper, we propose to adapt the recent advances in tree searches and partitioning techniques to design and analyze novel black-box combinatorial solvers. A first contribution is the analysis of a first tree-search algorithm called Optimistic Lipschitz Tree Search (OLTS) which assumes the Lipschitz constant of the function to be known. Linear convergence rates are provided for this algorithm under specific conditions, improving upon the logarithmic rates of baselines. An adaptive version, called Optimistic Combinatorial Tree Search (OCTS), is then introduced for the more realistic setup where we do not have any information on the Lipschitz constant of the function. Similar theoretical guarantees are shown to hold for OCTS and a numerical assessment is provided to illustrate the potential of tree searches with respect to state-of-the-art methods over typical benchmarks.

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NeurIPS Conference 2020 Conference Paper

Robustness Analysis of Non-Convex Stochastic Gradient Descent using Biased Expectations

Kevin Scaman
Cedric Malherbe

This work proposes a novel analysis of stochastic gradient descent (SGD) for non-convex and smooth optimization. Our analysis sheds light on the impact of the probability distribution of the gradient noise on the convergence rate of the norm of the gradient. In the case of sub-Gaussian and centered noise, we prove that, with probability $1-\delta$, the number of iterations to reach a precision $\varepsilon$ for the squared gradient norm is $O(\varepsilon^{-2}\ln(1/\delta))$. In the case of centered and integrable heavy-tailed noise, we show that, while the expectation of the iterates may be infinite, the squared gradient norm still converges with probability $1-\delta$ in $O(\varepsilon^{-p}\delta^{-q})$ iterations, where $p, q > 2$. This result shows that heavy-tailed noise on the gradient slows down the convergence of SGD without preventing it, proving that SGD is robust to gradient noise with unbounded variance, a setting of interest for Deep Learning. In addition, it indicates that choosing a step size proportional to $T^{-1/b}$ where $b$ is the tail-parameter of the noise and $T$ is the number of iterations leads to the best convergence rates. Both results are simple corollaries of a unified analysis using the novel concept of biased expectations, a simple and intuitive mathematical tool to obtain concentration inequalities. Using this concept, we propose a new quantity to measure the amount of noise added to the gradient, and discuss its value in multiple scenarios.

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