Selecting a small subset of items that represent the diversity of a larger population lies at the heart of many data analysis and machine learning applications. However, when it comes to items described by discrete features, the lack of natural ordering and the combinatorial nature of the search space pose significant challenges to the current selection techniques and make existing methods ill-suited. In this paper, we propose to make a step in that direction by proposing novel methods to select subsets of diverse categorical data based on the advances in combinatorial optimization. First, we start to cast the subset selection problem through the lens of the optimization of three diversity metrics. We then provide novel bounds for this problem and present exact solvers that unfortunately come with a high computational cost. To overcome this bottleneck, we go on and show how to employ tools from linear programming and submodular optimization by introducing two computationally plausible methods that still present approximation guarantees about the diversity metrics. Finally, a numerical assessment is provided to illustrate the potential of the designs with respect to state-of-the-art methods.

Optimising the quality-of-results (QoR) of circuits during logic synthesis is a formidable challenge necessitating the exploration of exponentially sized search spaces. While expert-designed operations aid in uncovering effective sequences, the increase in complexity of logic circuits favours automated procedures. To enable efficient and scalable solvers, we propose BOiLS, the first algorithm adapting Bayesian optimisation to navigate the space of synthesis operations. BOiLS requires no human intervention and trades-off exploration versus exploitation through novel Gaussian process kernels and trust-region constrained acquisitions. In a set of experiments on EPFL benchmarks, we demonstrate BOiLS's superior performance compared to state-of-the-art in terms of both sample efficiency and QoR values.

The optimization of multi-objective submodular systems appears in a wide variety of applications. However, there are currently very few techniques which are able to provide a robust allocation to such systems. In this work, we propose to design and analyse novel algorithms for the robust allocation of submodular systems through lens of quantile maximization. We start by observing that identifying an exact solution for this problem is computationally intractable. To tackle this issue, we propose a proxy for the quantile function using a softmax formulation, and show that this proxy is well suited to submodular optimization. Based on this relaxation, we propose a novel and simple algorithm called SOFTSAT. Theoretical properties are provided for this algorithm as well as novel approximation guarantees. Finally, we provide numerical experiments showing the efficiency of our algorithm with regards to state-of-the-art methods in a test bed of real-world applications, and show that SOFTSAT is particularly robust and well-suited to online scenarios.

Training over-parameterized neural networks involves the empirical minimization of highly non-convex objective functions. Recently, a large body of works provided theoretical evidence that, despite this non-convexity, properly initialized over-parameterized networks can converge to a zero training loss through the introduction of the Polyak-Lojasiewicz condition. However, these analyses are restricted to quadratic losses such as mean square error, and tend to indicate fast exponential convergence rates that are seldom observed in practice. In this work, we propose to extend these results by analyzing stochastic gradient descent under more generic Lojasiewicz conditions that are applicable to any convex loss function, thus extending the current theory to a larger panel of losses commonly used in practice such as cross-entropy. Moreover, our analysis provides high-probability bounds on the approximation error under sub-Gaussian gradient noise and only requires the local smoothness of the objective function, thus making it applicable to deep neural networks in realistic settings.

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.

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.

The goal of the paper is to design sequential strategies which lead to efficient optimization of an unknown function under the only assumption that it has a finite Lipschitz constant. We first identify sufficient conditions for the consistency of generic sequential algorithms and formulate the expected minimax rate for their performance. We introduce and analyze a first algorithm called LIPO which assumes the Lipschitz constant to be known. Consistency, minimax rates for LIPO are proved, as well as fast rates under an additional Hölder like condition. An adaptive version of LIPO is also introduced for the more realistic setup where Lipschitz constant is unknown and has to be estimated along with the optimization. Similar theoretical guarantees are shown to hold for the adaptive LIPO algorithm and a numerical assessment is provided at the end of the paper to illustrate the potential of this strategy with respect to state-of-the-art methods over typical benchmark problems for global optimization.

We consider the problem of maximizing an unknown function f over a compact and convex set using as few observations f(x) as possible. We observe that the optimization of the function f essentially relies on learning the induced bipartite ranking rule of f. Based on this idea, we relate global optimization to bipartite ranking which allows to address problems with high dimensional input space, as well as cases of functions with weak regularity properties. The paper introduces novel meta-algorithms for global optimization which rely on the choice of any bipartite ranking method. Theoretical properties are provided as well as convergence guarantees and equivalences between various optimization methods are obtained as a by-product. Eventually, numerical evidence is given to show that the main algorithm of the paper which adapts empirically to the underlying ranking structure essentially outperforms existing state-of-the-art global optimization algorithms in typical benchmarks.