We give explicit low-rank bilinear non-commutative schemes for multiplying structured n × n matrices with 2 ≤ n ≤ 5, which serve as building blocks for recursive algorithms with improved multiplicative factors in asymptotic complexity. Our schemes are discovered over \(\mathbb {F}_2\) or \(\mathbb {F}_3\) and lifted to \(\mathbb {Z}\) or \(\mathbb {Q}\). Using a flip graph search over tensor decompositions, we derive schemes for general, upper-triangular, lower-triangular, symmetric, and skew-symmetric inputs, as well as products of a structured matrix with its transpose. These schemes improve asymptotic constants for 13 of 15 structured formats. In particular, we obtain 4 × 4 rank-34 schemes for both multiplying a general matrix by its transpose and an upper-triangular matrix by a general matrix, improving the asymptotic factor from 26/41 (0.634) to 22/37 (0.595). Additionally, using \(\mathbb {F}_3\) flip graphs, we discover schemes over \(\mathbb {Q}\) that fundamentally require the inverse of 2, including a 2 × 2 symmetric-symmetric multiplication of rank 5 and a 3 × 3 skew-symmetric-general multiplication of rank 14 (improving upon AlphaTensor’s 15).

Abstract In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz constant. In recent literature, methods based on non-convex outer approximation are proposed for tackling one-dimensional equality constraints that are Lipschitz with respect to the maximum norm. To the best of our knowledge, however, there does not exist a non-convex outer approximation method for a general problem class. We introduce a meta-level solution framework to solve such problems and tackle the underlying theoretical foundations. Considering the feasible domain without the constraint function as manageable, our method relaxes the multidimensional constraint and iteratively refines the feasible region by means of norm-induced cuts, relying on an oracle for the resulting subproblems. We show the method’s correctness and investigate the problem complexity. In order to account for discussions about functionality, limits, and extensions, we present computational examples including illustrations.

We propose the pivoting meta algorithm (PM) to enhance optimization algorithms that generate iterates as convex combinations of vertices of a feasible region $C\subseteq \mathbb{R}^n$, including Frank-Wolfe (FW) variants. PM guarantees that the active set (the set of vertices in the convex combination) of the modified algorithm remains as small as $dim(C)+1$ as stipulated by Carath{é}odory’s theorem. PM achieves this by reformulating the active set expansion task into an equivalent linear program, which can be efficiently solved using a single pivot step akin to the primal simplex algorithm; the convergence rate of the original algorithms are maintained. Furthermore, we establish the connection between PM and active set identification, in particular showing under mild assumptions that PM applied to the away-step Frank-Wolfe algorithm or the blended pairwise Frank-Wolfe algorithm bounds the active set size by the dimension of the optimal face plus $1$. We provide numerical experiments to illustrate practicality and efficacy on active set size reduction.

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 $\mu_x$-strongly geodesically convex (g-convex) in $x$ and $\mu_y$-strongly g-concave in $y$, for $\mu_x, \mu_y \geq 0$. 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.

We study the problem of finding optimal sparse, manifold-aligned counterfactual explanations for classifiers. Canonically, this can be formulated as an optimization problem with multiple non-convex components, including classifier loss functions and manifold alignment (or \emph{plausibility}) metrics. The added complexity of enforcing \emph{sparsity}, or shorter explanations, complicates the problem further. Existing methods often focus on specific models and plausibility measures, relying on convex $\ell_1$ regularizers to enforce sparsity. In this paper, we tackle the canonical formulation using the accelerated proximal gradient (APG) method, a simple yet efficient first-order procedure capable of handling smooth non-convex objectives and non-smooth $\ell_p$ (where $0 \leq p < 1$) regularizers. This enables our approach to seamlessly incorporate various classifiers and plausibility measures while producing sparser solutions. Our algorithm only requires differentiable data-manifold regularizers and supports box constraints for bounded feature ranges, ensuring the generated counterfactuals remain \emph{actionable}. Finally, experiments on real-world datasets demonstrate that our approach effectively produces sparse, manifold-aligned counterfactual explanations while maintaining proximity to the factual data and computational efficiency.

Federated Learning (FL) algorithms using Knowledge Distillation (KD) have received increasing attention due to their favorable properties with respect to privacy, non-i.i.d. data and communication cost. These methods depart from transmitting model parameters and instead communicate information about a learning task by sharing predictions on a public dataset. In this work, we study the performance of such approaches in the byzantine setting, where a subset of the clients act in an adversarial manner aiming to disrupt the learning process. We show that KD-based FL algorithms are remarkably resilient and analyze how byzantine clients can influence the learning process. Based on these insights, we introduce two new byzantine attacks and demonstrate their ability to break existing byzantine-resilient methods. Additionally, we propose a novel defence method which enhances the byzantine resilience of KD-based FL algorithms. Finally, we provide a general framework to obfuscate attacks, making them significantly harder to detect, thereby improving their effectiveness.

Sparse adversarial attacks fool deep neural networks (DNNs) through minimal pixel perturbations, often regularized by the $\ell_0$ norm. Recent efforts have replaced this norm with a structural sparsity regularizer, such as the nuclear group norm, to craft group-wise sparse adversarial attacks. The resulting perturbations are thus explainable and hold significant practical relevance, shedding light on an even greater vulnerability of DNNs. However, crafting such attacks poses an optimization challenge, as it involves computing norms for groups of pixels within a non-convex objective. We address this by presenting a two-phase algorithm that generates group-wise sparse attacks within semantically meaningful areas of an image. Initially, we optimize a quasinorm adversarial loss using the $1/2$-quasinorm proximal operator tailored for non-convex programming. Subsequently, the algorithm transitions to a projected Nesterov's accelerated gradient descent with $2$-norm regularization applied to perturbation magnitudes. Rigorous evaluations on CIFAR-10 and ImageNet datasets demonstrate a remarkable increase in group-wise sparsity, e.g., $50.9\%$ on CIFAR-10 and $38.4\%$ on ImageNet (average case, targeted attack). This performance improvement is accompanied by significantly faster computation times, improved interpretability, and a $100\%$ attack success rate.

We present a new step-size strategy based on the secant method for Frank-Wolfe algorithms. This strategy, which requires mild assumptions about the function under consideration, can be applied to any Frank-Wolfe algorithm. It is as effective as full line search and, in particular, allows for adapting to the local smoothness of the function, such as in (Pedregosa et al., 2020), but comes with a significantly reduced computational cost, leading to higher effective rates of convergence. We provide theoretical guarantees and demonstrate the effectiveness of the strategy through numerical experiments.

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

With the rise in global greenhouse gas emissions, accurate large-scale tree canopy height maps are essential for understanding forest structure, estimating above-ground biomass, and monitoring ecological disruptions. To this end, we present a novel approach to generate large-scale, high-resolution canopy height maps over time. Our model accurately predicts canopy height over multiple years given Sentinel 2 time series satellite data. Using GEDI LiDAR data as the ground truth for training the model, we present the first 10 m resolution temporal canopy height map of the European continent for the period 2019–2022. As part of this product, we also offer a detailed canopy height map for 2020, providing more precise estimates than previous studies. Our pipeline and the resulting temporal height map are publicly available, enabling comprehensive large-scale monitoring of forests and, hence, facilitating future research and ecological analyses. For an interactive viewer, see https://europetreemap.projects.earthengine.app/view/europeheight.

Deep neural networks have reshaped modern machine learning by learning powerful latent representations that often align with the manifold hypothesis: high-dimensional data lie on lower-dimensional manifolds. In this paper, we establish a connection between manifold learning and computational algebra by demonstrating how vanishing ideals can characterize the latent manifolds of deep networks. To that end, we propose a new neural architecture that (i) truncates a pretrained network at an intermediate layer, (ii) approximates each class manifold via polynomial generators of the vanishing ideal, and (iii) transforms the resulting latent space into linearly separable features through a single polynomial layer. The resulting models have significantly fewer layers than their pretrained baselines, while maintaining comparable accuracy, achieving higher throughput, and utilizing fewer parameters. Furthermore, drawing on spectral complexity analysis, we derive sharper theoretical guarantees for generalization, showing that our approach can in principle offer tighter bounds than standard deep networks. Numerical experiments confirm the effectiveness and efficiency of the proposed approach.

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.

We formalize and analyze the trade-off between backdoor-based watermarks and adversarial defenses, framing it as an interactive protocol between a verifier and a prover. While previous works have primarily focused on this trade-off, our analysis extends it by identifying transferable attacks as a third, counterintuitive but necessary option. Our main result shows that for all learning tasks, at least one of the three exists: a _watermark_, an _adversarial defense_, or a _transferable attack_. By transferable attack, we refer to an efficient algorithm that generates queries indistinguishable from the data distribution and capable of fooling _all_ efficient defenders. Using cryptographic techniques, specifically fully homomorphic encryption, we construct a transferable attack and prove its necessity in this trade-off. Finally, we show that tasks of bounded VC-dimension allow adversarial defenses against all attackers, while a subclass allows watermarks secure against fast adversaries.

We develop a Frank-Wolfe algorithm with corrective steps, generalizing previous algorithms including Blended Conditional Gradients, Blended Pairwise Conditional Gradients, and Fully-Corrective Frank-Wolfe. For this, we prove tight convergence guarantees together with an optimal face identification property. Furthermore, we propose two highly efficient corrective steps for convex quadratic objectives based on linear optimization or linear system solving, akin to Wolfe's Minimum-Norm Point algorithm, and prove finite-time convergence under suitable conditions. Beyond optimization problems that are directly quadratic, we revisit two algorithms, Split Conditional Gradient and Second-Order Conditional Gradient Sliding, which can leverage quadratic corrections to accelerate the solution of their quadratic subproblems. We show improved convergence rates for the first and prove broader applicability for the second. Finally, we demonstrate substantial computational speedups for Frank-Wolfe-based algorithms with quadratic corrections across the considered problem classes.

Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gröbner and Border bases are fundamental but suffer from high computational costs, which have motivated recent Deep Learning approaches to improve efficiency, albeit at the expense of output correctness. In this work, we introduce the Oracle Border Basis Algorithm, the first Deep Learning approach that accelerates Border basis computation while maintaining output guarantees. To this end, we design and train a Transformer-based oracle that identifies and eliminates computationally expensive reduction steps, which we find to dominate the algorithm's runtime. By selectively invoking this oracle during critical phases of computation, we achieve substantial speedup factors of up to 3.5x compared to the base algorithm, without compromising the correctness of results. To generate the training data, we develop a sampling method and provide the first sampling theorem for border bases. We construct a tokenization and embedding scheme tailored to monomial-centered algebraic computations, resulting in a compact and expressive input representation, which reduces the number of tokens to encode an $n$-variate polynomial by a factor of $O(n)$. Our learning approach is data efficient, stable, and a practical enhancement to traditional computer algebra algorithms and symbolic computation.

Constraint-based metabolic models can be used to investigate the intracellular physiology of microorganisms. These models couple genes to reactions, and typically seek to predict metabolite fluxes that optimize some biologically important metric. Classical techniques, like Flux Balance Analysis (FBA), formulate the metabolism of a microbe as an optimization problem where growth rate is maximized. While FBA has found widespread use, it often leads to thermodynamically infeasible solutions that contain internal cycles (loops). To address this shortcoming, Loopless-Flux Balance Analysis (ll-FBA) seeks to predict flux distributions that do not contain these loops. ll-FBA is a disjunctive program, usually reformulated as a mixed-integer program, and is challenging to solve for biological models that often contain thousands of reactions and metabolites. In this paper, we compare various reformulations of ll-FBA and different solution approaches. Overall, the combinatorial Benders' decomposition is the most promising of the tested approaches with which we could solve most instances. However, the model size and numerical instability pose a challenge to the combinatorial Benders' method.

Frank-Wolfe (FW) algorithms have emerged as an essential class of methods for constrained optimization, especially on large-scale problems. In this article, we summarize the algorithmic design choices and progress made in the last years of the development of FrankWolfe.jl, a Julia package gathering high-performance implementations of state-of-the-art FW variants. We review key use cases of the library in the recent literature, which match its original dual purpose: first, becoming the de-facto toolbox for practitioners applying FW methods to their problem, and second, offering a modular ecosystem to algorithm designers who experiment with their own variants and implementations of algorithmic blocks. Finally, we demonstrate the performance of several FW variants on important problem classes in several experiments, which we curated in a separate repository for continuous benchmarking.

We propose an interactive multi-agent classifier that provides provable interpretability guarantees even for complex agents such as neural networks. These guarantees consist of lower bounds on the mutual information between selected features and the classification decision. Our results are inspired by the Merlin-Arthur protocol from Interactive Proof Systems and express these bounds in terms of measurable metrics such as soundness and completeness. Compared to existing interactive setups, we rely neither on optimal agents nor on the assumption that features are distributed independently. Instead, we use the relative strength of the agents as well as the new concept of Asymmetric Feature Correlation which captures the precise kind of correlations that make interpretability guarantees difficult. We evaluate our results on two small-scale datasets where high mutual information can be verified explicitly.

Neural networks can be significantly compressed by pruning, yielding sparse models with reduced storage and computational demands while preserving predictive performance. Model soups (Wortsman et al., 2022) enhance generalization and out-of-distribution (OOD) performance by averaging the parameters of multiple models into a single one, without increasing inference time. However, achieving both sparsity and parameter averaging is challenging as averaging arbitrary sparse models reduces the overall sparsity due to differing sparse connectivities. This work addresses these challenges by demonstrating that exploring a single retraining phase of Iterative Magnitude Pruning (IMP) with varied hyperparameter configurations such as batch ordering or weight decay yields models suitable for averaging, sharing identical sparse connectivity by design. Averaging these models significantly enhances generalization and OOD performance over their individual counterparts. Building on this, we introduce Sparse Model Soups (SMS), a novel method for merging sparse models by initiating each prune-retrain cycle with the averaged model from the previous phase. SMS preserves sparsity, exploits sparse network benefits, is modular and fully parallelizable, and substantially improves IMP's performance. We further demonstrate that SMS can be adapted to enhance state-of-the-art pruning-during-training approaches.

In this work, we analyze two of the most fundamental algorithms in geodesically convex optimization: Riemannian gradient descent and (possibly inexact) Riemannian proximal point. We quantify their rates of convergence and produce different variants with several trade-offs. Crucially, we show the iterates naturally stay in a ball around an optimizer, of radius depending on the initial distance and, in some cases, on the curvature. Previous works simply assumed bounded iterates, resulting in rates that were not fully quantified. We also provide an implementable inexact proximal point algorithm and prove several new useful properties of Riemannian proximal methods: they work when positive curvature is present, the proximal operator does not move points away from any optimizer, and we quantify the smoothness of its induced Moreau envelope. Further, we explore beyond our theory with empirical tests.

We propose a framework for global-scale canopy height estimation based on satellite data. Our model leverages advanced data preprocessing techniques, resorts to a novel loss function designed to counter geolocation inaccuracies inherent in the ground-truth height measurements, and employs data from the Shuttle Radar Topography Mission to effectively filter out erroneous labels in mountainous regions, enhancing the reliability of our predictions in those areas. A comparison between predictions and ground-truth labels yields an MAE/RMSE of 2.43 / 4.73 (meters) overall and 4.45 / 6.72 (meters) for trees taller than five meters, which depicts a substantial improvement compared to existing global-scale products. The resulting height map as well as the underlying framework will facilitate and enhance ecological analyses at a global scale, including, but not limited to, large-scale forest and biomass monitoring.

We tackle the Optimal Experiment Design Problem, which consists of choosing experiments to run or observations to select from a finite set to estimate the parameters of a system. The objective is to maximize some measure of information gained about the system from the observations, leading to a convex integer optimization problem. We leverage Boscia.jl, a recent algorithmic framework, which is based on a nonlinear branch-and-bound algorithm with node relaxations solved to approximate optimality using Frank-Wolfe algorithms. One particular advantage of the method is its efficient utilization of the polytope formed by the original constraints which is preserved by the method, unlike alternative methods relying on epigraph-based formulations. We assess the method against both generic and specialized convex mixed-integer approaches. Computational results highlight the performance of the proposed method, especially on large and challenging instances.

Abstract. We investigate the following question: How close can two disjoint lattice polytopes contained in a fixed hypercube be? This question stems from various contexts where the minimal distance between such polytopes appears in complexity bounds of optimization algorithms. We provide nearly matching bounds on this distance and discuss its exact computation. We also give similar bounds for disjoint rational polytopes whose binary encoding length is prescribed.

This note closes a gap in the proof of Theorem VI.3 from the article “Lower Bounds on the Oracle Complexity of Nonsmooth Convex Optimization via Information Theory” (2017).

We present a fully computer-assisted proof system for solving a particular family of problems in Extremal Combinatorics. Existing techniques using Flag Algebras have proven powerful in the past, but have so far lacked a computational counterpart to derive matching constructive bounds. We demonstrate that common search heuristics are capable of finding constructions far beyond the reach of human intuition. Additionally, the most obvious downside of such heuristics, namely a missing guarantee of global optimality, can often be fully eliminated in this case through lower bounds and stability results coming from the Flag Algebra approach. To illustrate the potential of this approach, we study two related and well-known problems in Extremal Graph Theory that go back to questions of Erdős from the 60s. Most notably, we present the first major improvement in the upper bound of the Ramsey multiplicity of K_4 in 25 years, precisely determine the first off-diagonal Ramsey multiplicity number, and settle the minimum number of independent sets of size four in graphs with clique number strictly less than five.

Frank-Wolfe algorithms (FW) are popular first-order methods for solving constrained convex optimization problems that rely on a linear minimization oracle instead of potentially expensive projection-like oracles. Many works have identified accelerated convergence rates under various structural assumptions on the optimization problem and for specific FW variants when using line-search or short-step, requiring feedback from the objective function. Little is known about accelerated convergence regimes when utilizing open-loop step-size rules, a.k.a. FW with pre-determined step-sizes, which are algorithmically extremely simple and stable. Not only is FW with open-loop step-size rules not always subject to the same convergence rate lower bounds as FW with line-search or short-step, but in some specific cases, such as kernel herding in infinite dimensions, it has been empirically observed that FW with open-loop step-size rules leads to faster convergence than FW with line-search or short-step. We propose a partial answer to this unexplained phenomenon in kernel herding, characterize a general setting for which FW with open-loop step-size rules converges non-asymptotically faster than with line-search or short-step, and derive several accelerated convergence results for FW with open-loop step-size rules.

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 (Kim and Yang, 2022) 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.

It has recently been shown that ISTA, an unaccelerated optimization method, presents sparse updates for the $\ell_1$-regularized undirected personalized PageRank problem (Fountoulakis et al., 2019), leading to cheap iteration complexity and providing the same guarantees as the approximate personalized PageRank algorithm (\appr{}), (Andersen et al., 2016). In this work, we design an accelerated optimization algorithm for this problem that also performs sparse updates, providing an affirmative answer to the COLT 2022 open question of (Fountoulakis et al., 2022). Acceleration provides a reduced dependence on the condition number, while the dependence on the sparsity in our updates differs from the ISTA approach. Further, we design another algorithm by using conjugate directions to achieve an exact solution while exploiting sparsity. Both algorithms lead to faster convergence for certain parameter regimes. Our findings apply beyond PageRank and work for any quadratic objective whose Hessian is a positive-definite $M$-matrix.

Cutting-planes are one of the most important building blocks for solving large-scale integer programming (IP) problems to (near) optimality. The majority of cutting plane approaches rely on explicit rules to derive valid inequalities that can separate the target point from the feasible set. Local cuts, on the other hand, seek to directly derive the facets of the underlying polyhedron and use them as cutting planes. However, current approaches rely on solving Linear Programming (LP) problems in order to derive such a hyperplane. In this paper, we present a novel generic approach for learning the facets of the underlying polyhedron by accessing it implicitly via an enumeration oracle in a reduced dimension. This is achieved by embedding the oracle in a variant of the Frank-Wolfe algorithm which is capable of generating strong cutting planes, effectively turning the enumeration oracle into a separation oracle. We demonstrate the effectiveness of our approach with a case study targeting the multidimensional knapsack problem (MKP).

The vanishing ideal of a set of points X is the set of polynomials that evaluate to 0 over all points x in X and admits an efficient representation by a finite set of polynomials called generators. To accommodate the noise in the data set, we introduce the Conditional Gradients Approximately Vanishing Ideal algorithm (CGAVI) for the construction of the set of generators of the approximately vanishing ideal. The constructed set of generators captures polynomial structures in data and gives rise to a feature map that can, for example, be used in combination with a linear classifier for supervised learning. In CGAVI, we construct the set of generators by solving specific instances of (constrained) convex optimization problems with the Pairwise Frank-Wolfe algorithm (PFW). Among other things, the constructed generators inherit the LASSO generalization bound and not only vanish on the training but also on out-sample data. Moreover, CGAVI admits a compact representation of the approximately vanishing ideal by constructing few generators with sparse coefficient vectors.

We study the effects of constrained optimization formulations and Frank-Wolfe algorithms for obtaining interpretable neural network predictions. Reformulating the Rate-Distortion Explanations (RDE) method for relevance attribution as a constrained optimization problem provides precise control over the sparsity of relevance maps. This enables a novel multi-rate as well as a relevance-ordering variant of RDE that both empirically outperform standard RDE and other baseline methods in a well-established comparison test. We showcase several deterministic and stochastic variants of the Frank-Wolfe algorithm and their effectiveness for RDE.

The Pairwise Conditional Gradients (PCG) algorithm is a powerful extension of the Frank-Wolfe algorithm leading to particularly sparse solutions, which makes PCG very appealing for problems such as sparse signal recovery, sparse regression, and kernel herding. Unfortunately, PCG exhibits so-called swap steps that might not provide sufficient primal progress. The number of these bad steps is bounded by a function in the dimension and as such known guarantees do not generalize to the infinite-dimensional case, which would be needed for kernel herding. We propose a new variant of PCG, the so-called Blended Pairwise Conditional Gradients (BPCG). This new algorithm does not exhibit any swap steps, is very easy to implement, and does not require any internal gradient alignment procedures. The convergence rate of BPCG is basically that of PCG if no drop steps would occur and as such is no worse than PCG but improves and provides new rates in many cases. Moreover, we observe in the numerical experiments that BPCG’s solutions are much sparser than those of PCG. We apply BPCG to the kernel herding setting, where we derive nice quadrature rules and provide numerical results demonstrating the performance of our method.

Characteristic functions (from cooperative game theory) are able to evaluate partial inputs and form the basis for attribution methods like Shapley values. These attribution methods allow us to measure how important each input component is for the function output—one of the goals of explainable AI (XAI). Given a standard classifier function, it is unclear how partial input should be realised. Instead, most XAI-methods for black-box classifiers like neural networks consider counterfactual inputs that generally lie off-manifold, which makes them hard to evaluate and easy to manipulate. We propose a setup to directly train characteristic functions in the form of neural networks to play simple two-player games. We apply this to the game of Connect Four by randomly hiding colour information from our agents during training. This has three advantages for comparing XAI-methods: It alleviates the ambiguity about how to realise partial input, makes off-manifold evaluation unnecessary and allows us to compare the methods by letting them play against each other.

The proportional fair resource allocation problem is a major problem studied in flow control of networks, operations research, and economic theory, where it has found numerous applications. This problem, defined as the constrained maximization of $\sum_i \log x_i$, is known as the packing proportional fairness problem when the feasible set is defined by positive linear constraints and $x \in \mathbb{R}_{\geq 0}^n$. In this work, we present a distributed accelerated first-order method for this problem which improves upon previous approaches. We also design an algorithm for the optimization of its dual problem. Both algorithms are width-independent.

We present FrankWolfe.jl, an open-source implementation of several popular Frank–Wolfe and conditional gradients variants for first-order constrained optimization. The package is designed with flexibility and high performance in mind, allowing for easy extension and relying on few assumptions regarding the user-provided functions. It supports Julia’s unique multiple dispatch feature, and it interfaces smoothly with generic linear optimization formulations using MathOptInterface.jl.

The Frank-Wolfe method solves smooth constrained convex optimization problems at a generic sublinear rate of $\mathcal{O}(1/T)$, and it (or its variants) enjoys accelerated convergence rates for two fundamental classes of constraints: polytopes and strongly-convex sets. Uniformly convex sets non-trivially subsume strongly convex sets and form a large variety of \textit{curved} convex sets commonly encountered in machine learning and signal processing. For instance, the $\ell_p$-balls are uniformly convex for all $p > 1$, but strongly convex for $p\in]1,2]$ only. We show that these sets systematically induce accelerated convergence rates for the original Frank-Wolfe algorithm, which continuously interpolate between known rates. Our accelerated convergence rates emphasize that it is the curvature of the constraint sets – not just their strong convexity – that leads to accelerated convergence rates. These results also importantly highlight that the Frank-Wolfe algorithm is adaptive to much more generic constraint set structures, thus explaining faster empirical convergence. Finally, we also show accelerated convergence rates when the set is only locally uniformly convex around the optima and provide similar results in online linear optimization.

Propagation of linear constraints has become a crucial sub-routine in modern Mixed-Integer Programming (MIP) solvers. In practice, iterative algorithms with tolerance-based stopping criteria are used to avoid problems with slow or infinite convergence. However, these heuristic stopping criteria can pose difficulties for fairly comparing the efficiency of different implementations of iterative propagation algorithms in a real-world setting. Most significantly, the presence of unbounded variable domains in the problem formulation makes it difficult to quantify the relative size of reductions performed on them. In this work, we develop a method to measure - independently of the algorithmic design - the progress that a given iterative propagation procedure has made at a given point in time during its execution. Our measure makes it possible to study and better compare the behavior of bounds propagation algorithms for linear constraints. We apply the new measure to answer two questions of practical relevance: (i) We investigate to what extent heuristic stopping criteria can lead to premature termination on real-world MIP instances. (ii) We compare a GPU-parallel propagation algorithm against a sequential state-of-the-art implementation and show that the parallel version is even more competitive in a real-world setting than originally reported.

Projection-free conditional gradient (CG) methods are the algorithms of choice for constrained optimization setups in which projections are often computationally prohibitive but linear optimization over the constraint set remains computationally feasible. Unlike in projection-based methods, globally accelerated convergence rates are in general unattainable for CG. However, a very recent work on Locally accelerated CG (LaCG) has demonstrated that local acceleration for CG is possible for many settings of interest. The main downside of LaCG is that it requires knowledge of the smoothness and strong convexity parameters of the objective function. We remove this limitation by introducing a novel, Parameter-Free Locally accelerated CG (PF-LaCG) algorithm, for which we provide rigorous convergence guarantees. Our theoretical results are complemented by numerical experiments, which demonstrate local acceleration and showcase the practical improvements of PF-LaCG over non-accelerated algorithms, both in terms of iteration count and wall-clock time.

Generalized self-concordance is a key property present in the objective function of many important learning problems. We establish the convergence rate of a simple Frank-Wolfe variant that uses the open-loop step size strategy $\gamma_t = 2/(t+2)$, obtaining a $\mathcal{O}(1/t)$ convergence rate for this class of functions in terms of primal gap and Frank-Wolfe gap, where $t$ is the iteration count. This avoids the use of second-order information or the need to estimate local smoothness parameters of previous work. We also show improved convergence rates for various common cases, e.g., when the feasible region under consideration is uniformly convex or polyhedral.

Primal heuristics play a crucial role in exact solvers for Mixed Integer Programming (MIP). While solvers are guaranteed to find optimal solutions given sufficient time, real-world applications typically require finding good solutions early on in the search to enable fast decision-making. While much of MIP research focuses on designing effective heuristics, the question of how to manage multiple MIP heuristics in a solver has not received equal attention. Generally, solvers follow hard-coded rules derived from empirical testing on broad sets of instances. Since the performance of heuristics is problem-dependent, using these general rules for a particular problem might not yield the best performance. In this work, we propose the first data-driven framework for scheduling heuristics in an exact MIP solver. By learning from data describing the performance of primal heuristics, we obtain a problem-specific schedule of heuristics that collectively find many solutions at minimal cost. We formalize the learning task and propose an efficient algorithm for computing such a schedule. Compared to the default settings of a state-of-the-art academic MIP solver, we are able to reduce the average primal integral by up to 49% on two classes of challenging instances.

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. Although these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids and knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem are known in advance and the online setting where the input data are revealed over time. For the offline setting, we give a general (nearly) optimal bicriteria approximation algorithm that relies on new extensions of classical algorithms for submodular maximization. For the online version of the problem, we give an algorithm that returns a bicriteria solution with sublinear regret. Summary of Contribution: Constrained submodular maximization is one of the core areas in combinatorial optimization with a wide variety of applications in operations research and computer science. Over the last decades, both communities have been interested on the design and analysis of new algorithms with provable guarantees. Sensor location, influence maximization and data summarization are some of the applications of submodular optimization that lie at the intersection of the aforementioned communities. Particularly, our work focuses on optimizing several submodular functions simultaneously. We provide new insights and algorithms to the offline and online variants of the problem which significantly expand the related literature. At the same time, we provide a computational study that supports our theoretical results.

Linear bandit algorithms yield $\tilde{\mathcal{O}}(n\sqrt{T})$ pseudo-regret bounds on compact convex action sets $\mathcal{K}\subset\mathbb{R}^n$ and two types of structural assumptions lead to better pseudo-regret bounds. When $\mathcal{K}$ is the simplex or an $\ell_p$ ball with $p\in]1,2]$, there exist bandits algorithms with $\tilde{\mathcal{O}}(\sqrt{nT})$ pseudo-regret bounds. Here, we derive bandit algorithms for some strongly convex sets beyond $\ell_p$ balls that enjoy pseudo-regret bounds of $\tilde{\mathcal{O}}(\sqrt{nT})$. This result provides new elements for the open question in Bubeck and Cesa-Bianchi, 2012. When the action set is $q$-uniformly convex but not necessarily strongly convex ($q >2$), we obtain pseudo-regret bounds $\tilde{\mathcal{O}}(n^{1/q}T^{1/p})$ with $p$ s.t. $1/p + 1/q=1$. These pseudo-regret bounds are competitive with the general $\tilde{\mathcal{O}}(n\sqrt{T})$ for a time horizon range that depends on the degree $q>2$ of the set's uniform convexity and the dimension $n$ of the problem.

Conditional gradients constitute a class of projection-free first-order algorithms for smooth convex optimization. As such, they are frequently used in solving smooth convex optimization problems over polytopes, for which the computational cost of projections is prohibitive. However, they do not enjoy the optimal convergence rates achieved by projection-based accelerated methods; moreover, achieving such globally-accelerated rates is information-theoretically impossible. To address this issue, we present Locally Accelerated Conditional Gradients – an algorithmic framework that couples accelerated steps with conditional gradient steps to achieve \emph{local} acceleration on smooth strongly convex problems. Our approach does not require projections onto the feasible set, but only on (typically low-dimensional) simplices, thus keeping the computational cost of projections at bay. Further, it achieves optimal accelerated local convergence. Our theoretical results are supported by numerical experiments, which demonstrate significant speedups over state of the art methods in both per-iteration progress and wall-clock time.

The Frank-Wolfe algorithm has become a popular first-order optimization algorithm for it is simple and projection-free, and it has been successfully applied to a variety of real-world problems. Its main drawback however lies in its convergence rate, which can be excessively slow due to naive descent directions. We propose to speed up the Frank-Wolfe algorithm by better aligning the descent direction with that of the negative gradient via a subroutine. This subroutine chases the negative gradient direction in a matching pursuit-style while still preserving the projection-free property. Although the approach is reasonably natural, it produces very significant results. We derive convergence rates $\mathcal{O}(1/t)$ to $\mathcal{O}(e^{-\omega t})$ of our method and we demonstrate its competitive advantage both per iteration and in CPU time over the state-of-the-art in a series of computational experiments.

Recently non-convex optimization approaches for solving machine learning problems have gained significant attention. In this paper we explore non-convex boosting in classification by means of integer programming and demonstrate real-world practicability of the approach while circumvent- ing shortcomings of convex boosting approaches. We report results that are comparable to or better than the current state-of-the-art.

It is well known that the standard greedy algorithm guarantees a worst-case approximation factor of $1-1/e$ when maximizing a monotone submodular function under a cardinality constraint. However, empirical studies show that its performance is substantially better in practice. This raises a natural question of explaining this improved performance of the greedy algorithm. In this work, we define sharpness for submodular functions as a candidate explanation for this phenomenon. We show that the greedy algorithm provably performs better as the sharpness of the submodular function increases. This improvement ties in closely with the faster convergence rates of first order methods for sharp functions in convex optimization.

Descent directions such as movement towards Frank-Wolfe vertices, away steps, in-face away steps and pairwise directions have been an important design consideration in conditional gradient descent (CGD) variants. In this work, we attempt to demystify the impact of movement in these directions towards attaining constrained minimizers. The best local direction of descent is the directional derivative of the projection of the gradient, which we refer to as the "shadow" of the gradient. We show that the continuous-time dynamics of moving in the shadow are equivalent to those of PGD however non-trivial to discretize. By projecting gradients in PGD, one not only ensures feasibility but also is able to "wrap" around the convex region. We show that Frank-Wolfe (FW) vertices in fact recover the maximal wrap one can obtain by projecting gradients, thus providing a new perspective to these steps. We also claim that the shadow steps give the best direction of descent emanating from the convex hull of all possible away-vertices. Opening up the PGD movements in terms of shadow steps gives linear convergence, dependent on the number of faces. We combine these insights into a novel Shadow-CG method that uses FW steps (i.e., wrap around the polytope) and shadow steps (i.e., optimal local descent direction), while enjoying linear convergence. Our analysis develops properties of directional derivatives of projections (which may be of independent interest), while providing a unifying view of various descent directions in the CGD literature.

Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions obtained via this approach are unstable to perturbations in data defining the submodular functions. Robust submodular maximization has been proposed as a richer model that aims to overcome this discrepancy as well as increase the modeling scope of submodular optimization. In this work, we consider robust submodular maximization with structured combinatorial constraints and give efficient algorithms with provable guarantees. Our approach is applicable to constraints defined by single or multiple matroids, knapsack as well as distributionally robust criteria. We consider both the offline setting where the data defining the problem is known in advance as well as the online setting where the input data is revealed over time. For the offline setting, we give a nearly optimal bi-criteria approximation algorithm that relies on new extensions of the classical greedy algorithm. For the online version of the problem, we give an algorithm that returns a bi-criteria solution with sub-linear regret.

Conditional Gradients (aka Frank-Wolfe algorithms) form a classical set of methods for constrained smooth convex minimization due to their simplicity, the absence of projection step, and competitive numerical performance. While the vanilla Frank-Wolfe algorithm only ensures a worst-case rate of $O(1/\epsilon)$, various recent results have shown that for strongly convex functions, the method can be slightly modified to achieve linear convergence. However, this still leaves a huge gap between sublinear $O(1/\epsilon)$ convergence and linear $O(\log 1/\epsilon)$ convergence to reach an $\epsilon$-approximate solution. Here, we present a new variant of Conditional Gradients, that can dynamically adapt to the function’s geometric properties using restarts and thus smoothly interpolates between the sublinear and linear regimes. Furthermore, our results apply to generic compact convex constraint sets.

We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank{–}Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of Braun et al. 2017. We also present a streamlined version of the algorithm that applies when P is the probability simplex.

Matching pursuit algorithms are an important class of algorithms in signal processing and machine learning. We present a blended matching pursuit algorithm, combining coordinate descent-like steps with stronger gradient descent steps, for minimizing a smooth convex function over a linear space spanned by a set of atoms. We derive sublinear to linear convergence rates according to the smoothness and sharpness orders of the function and demonstrate computational superiority of our approach. In particular, we derive linear rates for a large class of non-strongly convex functions, and we demonstrate in experiments that our algorithm enjoys very fast rates of convergence and wall-clock speed while maintaining a sparsity of iterates very comparable to that of the (much slower) orthogonal matching pursuit.

Conditional gradient algorithms (also often called Frank-Wolfe algorithms) are popular due to their simplicity of only requiring a linear optimization oracle and more recently they also gained significant traction for online learning. While simple in principle, in many cases the actual implementation of the linear optimization oracle is costly. We show a general method to lazify various conditional gradient algorithms, which in actual computations leads to several orders of magnitude of speedup in wall-clock time. This is achieved by using a faster separation oracle instead of a linear optimization oracle, relying only on few linear optimization oracle calls.

In this paper, we demonstrate how to learn the objective function of a decision maker while only observing the problem input data and the decision maker’s corresponding decisions over multiple rounds. Our approach is based on online learning techniques and works for linear objectives over arbitrary sets for which we have a linear optimization oracle and as such generalizes previous work based on KKT-system decomposition and dualization approaches. The applicability of our framework for learning linear constraints is also discussed briefly. Our algorithm converges at a rate of O(1/sqrt(T)), and we demonstrate its effectiveness and applications in preliminary computational results.

Conditional gradient algorithms (also often called Frank-Wolfe algorithms) are popular due to their simplicity of only requiring a linear optimization oracle and more recently they also gained significant traction for online learning. While simple in principle, in many cases the actual implementation of the linear optimization oracle is costly. We show a general method to lazify various conditional gradient algorithms, which in actual computations leads to several orders of magnitude of speedup in wall-clock time. This is achieved by using a faster separation oracle instead of a linear optimization oracle, relying only on few linear optimization oracle calls.

In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate $O(1/\epsilon^2)$ improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of (Hazan and Kale, 2012) with convergence rate $O(1/\epsilon^4)$.

We study reinforcement learning under model misspecification, where we do not have access to the true environment but only to a reasonably close approximation to it. We address this problem by extending the framework of robust MDPs to the model-free Reinforcement Learning setting, where we do not have access to the model parameters, but can only sample states from it. We define robust versions of Q-learning, Sarsa, and TD-learning and prove convergence to an approximately optimal robust policy and approximate value function respectively. We scale up the robust algorithms to large MDPs via function approximation and prove convergence under two different settings. We prove convergence of robust approximate policy iteration and robust approximate value iteration for linear architectures (under mild assumptions). We also define a robust loss function, the mean squared robust projected Bellman error and give stochastic gradient descent algorithms that are guaranteed to converge to a local minimum.

We study the cost function for hierarchical clusterings introduced by (Dasgupta, 2016) where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown in (Dasgupta, 2016) that a top-down algorithm based on the uniform Sparsest Cut problem returns a hierarchical clustering of cost at most $O\left(\alpha_n \log n\right)$ times the cost of the optimal hierarchical clustering, where $\alpha_n$ is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao- Vazirani, the top-down algorithm returns a hierarchical clustering of cost at most $O\left(\log^{3/2} n\right)$ times the cost of the optimal solution. We improve this by giving an $O(\log{n})$-approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an $O(\log{n})$- approximate hierarchical clustering for a generalization of this cost function also studied in (Dasgupta, 2016). Experiments show that the hierarchies found by using the ILP formulation as well as our rounding algorithm often have better projections into flat clusters than the standard linkage based algorithms. We conclude with constant factor inapproximability results for this problem: 1) no polynomial size LP or SDP can achieve a constant factor approximation for this problem and 2) no polynomial time algorithm can achieve a constant factor approximation under the Small Set Expansion hypothesis.

We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a single source of hardness. As a measure of complexity, we use distributional oracle complexity, which subsumes randomized oracle complexity as well as worst case oracle complexity. We obtain strong lower bounds on distributional oracle complexity for the box [-1, 1]n, as well as for the L p-ball for p ≥ 1 (for both low-scale and large-scale regimes), matching worst case upper bounds, and hence we close the gap between distributional complexity, and in particular, randomized complexity and worst case complexity. Furthermore, the bounds remain essentially the same for high-probability and bounded-error oracle complexity, and even for combination of the two, i.e., bounded-error highprobability oracle complexity. This considerably extends the applicability of known bounds.

We study the cost function for hierarchical clusterings introduced by [Dasgupta, 2015] where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown in [Dasgupta, 2015] that a top-down algorithm returns a hierarchical clustering of cost at most (O\left(\alphan \log n\right)) times the cost of the optimal hierarchical clustering, where (\alphan) is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top down algorithm returns a hierarchical clustering of cost at most (O\left(\log^{3/2} n\right)) times the cost of the optimal solution. We improve this by giving an (O(\log{n}))-approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of \emph{sphere growing} which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an (O(\log{n}))-approximate hierarchical clustering for a generalization of this cost function also studied in [Dasgupta, 2015]. Experiments show that the hierarchies found by using the ILP formulation as well as our rounding algorithm often have better projections into flat clusters than the standard linkage based algorithms. We conclude with an inapproximability result for this problem, namely that no polynomial sized LP or SDP can be used to obtain a constant factor approximation for this problem.

Yannakakis [27, 26] showed that the matching problem does not have a small symmetric linear program. Rothvoß [23] recently proved that any, not necessarily symmetric, linear program also has exponential size. It is natural to ask whether the matching problem can be expressed compactly in a framework such as semidefinite programming (SDP) that is more powerful than linear programming but still allows efficient optimization. We answer this question negatively for symmetric SDPs: any symmetric SDP for the matching problem has exponential size. We also show that an O(k)-round Lasserre SDP relaxation for the asymmetric metric traveling salesperson problem yields at least as good an approximation as any symmetric SDP relaxation of size nk. The key technical ingredient underlying both these results is an upper bound on the degree needed to derive polynomial identities that hold over the space of matchings or traveling salesperson tours.

Border bases arise as a canonical generalization of Gröbner bases, using order ideals instead of term orderings. We provide a polyhedral characterization of all order ideals (and hence all border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we also provide an adaptation of the border basis algorithm of Kehrein and Kreuzer [J. Pure Appl. Algebra, 205 (2006), pp. 279--295] to allow for computing border bases for arbitrary order ideals, given implicitly via maximizing a preference on monomials (variable selection problem), independent of term orderings. The algorithm requires the same size of resources as the border basis algorithm except for some minor overhead. We also show that the underlying variable selection problem of finding an order ideal that supports a border basis is NP-hard and that any linear description of the associated convex hull of all order ideals requires a superpolynomial number of inequalities.

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [30], [31] proved that the problem is NP-hard to approximate within a factor 2 - ε, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best in approximability result for the problem is due to Dinur and Safra [16], [17]: vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor of 2 - ε has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomially many inequalities.

We characterize the performance of sequential information guided sensing, Info-Greedy Sensing [1], when there is a mismatch between the true signal model and the assumed model, which may be a sample estimate. In particular, we consider a setup where the signal is low-rank Gaussian and the measurements are taken in the directions of eigenvectors of the covariance matrix Σ in a decreasing order of eigenvalues. We establish a set of performance bounds when a mismatched covariance matrix equation is used, in terms of the gap of signal posterior entropy, as well as the additional amount of power required to achieve the same signal recovery precision. Based on this, we further study how to choose an initialization for Info-Greedy Sensing using the sample covariance matrix, or using an efficient covariance sketching scheme.

The groundbreaking work of Rothvoß [2014] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed 0 < ε < 1, every polyhedral (1 + ε/n)-approximation requires an exponential number of inequalities, where n is the number of vertices. This is sharp given the well-known ρ-approximation of size provided by the odd-sets of size up to ρ/(ρ — 1). Thus matching is the first problem in P, whose natural linear encoding does not admit a fully polynomial-size relaxation scheme (the polyhedral equivalent of an FPTAS), which provides a sharp separation from the polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets mentioned above. Our approach reuses ideas from Rothvoß [2014], however the main lower bounding technique is different. While the original proof is based on the hyperplane separation bound (also called the rectangle corruption bound), we employ the information-theoretic notion of common information as introduced in Braun and Pokutta [2013], which allows to analyze perturbations of slack matrices. It turns out that the high extension complexity for the matching polytope stems from the same source of hardness as for the correlation polytope: a direct sum structure.

Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3/2-epsilon inapproximability for Vertex Cover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1/2+ε for bounded degree IndependentSet, and we establish inapproximability of MaxMULTICUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

We solve a 20-year old problem posed by Yannakakis and prove that no polynomial-size linear program (LP) exists whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

Recently, Rothvoß established that every linear program (LP) expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed 01-ρn)) provided by the odd-sets of size up to 1/(1-ρ). Thus, matching is the first problem in F, which does not admit a fully polynomial-size LP relaxation scheme (the LP equivalent of an Fully Polynomial-Time Approximation Scheme), which provides a sharp separation from the polynomial-size LP relaxation scheme obtained, e.g., through constant-sized odd-sets mentioned above. Analyzing the size of LP formulations is equivalent to examining the nonnegative rank of matrices. We study the nonnegative rank through an information-theoretic approach; while it reuses key ideas from Rothvoß, the main lower bounding technique is different: we employ the information-theoretic notion of Wyner's common information used for studying LP formulations. This allows us to analyze the nonnegative rank of perturbations of slack matrices, e.g., the approximations of the matching polytope. It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via LPs (more precisely, linear extended formulations), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a super-polynomial lower bound with overwhelming probability for every LP that is exact for a randomly selected set of instances with a natural distribution. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the Erdös-Renyi model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from essentially p = polylog(n) / n to p = 1 / log n. Our upper bound is close as there is only an essentially quadratic gap in the exponent, which also exists in the worst case model. Finally, we state a conjecture to close the gap both in the average-case and worst-case models.

We provide a new framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in [1]. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with He linger distance estimations we can compute the (almost) exact common information of UDISJ partial matrix. We also establish robustness of this estimation under various perturbations of the UDISJ partial matrix, where rows and columns are randomly or adversarially removed or where entries are randomly or adversarially altered. This robustness translates, via a variant of Yannakakis' Factorization Theorem, to lower bounds on the average case and adversarial approximate extension complexity. We present the first family of polytopes, the hard pair introduced in [2] related to the CLIQUE problem, with high average case and adversarial approximate extension complexity. We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that quadratic approximations for CLIQUE require linear programs of exponential size. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem) Moreover, we establish a similar result for approximations of semi definite programs by linear programs. Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjoint ness matrix.

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.