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The concept of Nash equilibrium enlightens the structure of rational behavior in multi-agent settings. However, the concept is as helpful as one may compute it efficiently. We introduce the Cut-and-Play, an algorithm to compute Nash equilibria for non-cooperative simultaneous games where each player's objective is linear in their variables and bilinear in the other players' variables. Using the rich theory of integer programming, we alternate between constructing (i.) increasingly tighter outer approximations of the convex hull of each player's feasible set -- by using branching and cutting plane methods -- and (ii.) increasingly better inner approximations of these hulls -- by finding extreme points and rays of the convex hulls. In particular, we prove the correctness of our algorithm when these convex hulls are polyhedra. Our algorithm allows us to leverage the mixed integer programming technology to compute equilibria for a large class of games. Further, we integrate existing cutting plane families inside the algorithm, significantly speeding up equilibria computation. We showcase a set of extensive computational results for Integer Programming Games and simultaneous games among bilevel leaders. In both cases, our framework outperforms the state-of-the-art in computing time and solution quality.

Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $\mathsf{P}_\star\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sqrt{k} o_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, i.e., graphs whose girth becomes $L$ after removing a small fraction of vertices. Earlier work established that, for random $k$-regular graphs, the typical max-cut value is $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+o_n(n)+no_k(\sqrt{k})$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max-cut, and nearly maximum min-bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near-Ramanujan property of random regular graphs.

Optimal transport (OT) naturally arises in a wide range of machine learning applications but may often become the computational bottleneck. Recently, one line of works propose to solve OT approximately by searching the \emph{transport plan} in a low-rank subspace. However, the optimal transport plan is often not low-rank, which tends to yield large approximation errors. For example, when Monge's \emph{transport map} exists, the transport plan is full rank. This paper concerns the computation of the OT distance with adequate accuracy and efficiency. A novel approximation for OT is proposed, in which the transport plan can be decomposed into the sum of a low-rank matrix and a sparse one. We theoretically analyze the approximation error. An augmented Lagrangian method is then designed to efficiently calculate the transport plan.

In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank matrix, its best low-rank approximations may still be reliably computed or estimated from a small number of its entries. This is especially relevant from a statistical viewpoint: the best low-rank approximations to a data matrix are often of more interest than itself because they capture the more stable and oftentimes more reproducible properties of an otherwise complicated data-generating model. In particular, we investigate two agnostic approaches: the first is based on spectral truncation; and the second is a projected gradient descent based optimization procedure. We argue that, while the first approach is intuitive and reasonably effective, the latter has far superior performance in general. We show that the error depends on how close the matrix is to being of low rank. Both theoretical and numerical evidence is presented to demonstrate the effectiveness of the proposed approaches.

The fundamental sparsest cut problem takes as input a graph $G$ together with the edge costs and demands, and seeks a cut that minimizes the ratio between the costs and demands across the cuts. For $n$-node graphs~$G$ of treewidth~$k$, \chlamtac, Krauthgamer, and Raghavendra (APPROX 2010) presented an algorithm that yields a factor-$2^{2^k}$ approximation in time $2^{O(k)} \cdot \operatorname{poly}(n)$. Later, Gupta, Talwar and Witmer (STOC 2013) showed how to obtain a $2$-approximation algorithm with a blown-up run time of $n^{O(k)}$. An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-$2$ approximation in time $2^{O(k)} \cdot \operatorname{poly}(n)$. In this paper, we make significant progress towards this goal, via the following results: (i) A factor-$O(k^2)$ approximation that runs in time $2^{O(k)} \cdot \operatorname{poly}(n)$, directly improving the work of Chlamt\'a\v{c} et al. while keeping the run time single-exponential in $k$. (ii) For any $\varepsilon>0$, a factor-$O(1/\varepsilon^2)$ approximation whose run time is $2^{O(k^{1+\varepsilon}/\varepsilon)} \cdot \operatorname{poly}(n)$, implying a constant-factor approximation whose run time is nearly single-exponential in $k$ and a factor-$O(\log^2 k)$ approximation in time $k^{O(k)} \cdot \operatorname{poly}(n)$. Key to these results is a new measure of a tree decomposition that we call combinatorial diameter, which may be of independent interest.

In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta, Talwar and Witmer [STOC 2013] obtained a 2-approximation with polynomial running time for fixed k, and the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in FPT time, remained open. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in FPT time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in FPT time.

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.

We consider the exploration-exploitation trade-off in reinforcement learning and we show that an agent imbued with a risk-seeking utility function is able to explore efficiently, as measured by regret. The parameter that controls how risk-seeking the agent is can be optimized exactly, or annealed according to a schedule. We call the resulting algorithm K-learning and show that the corresponding K-values are optimistic for the expected Q-values at each state-action pair. The K-values induce a natural Boltzmann exploration policy for which the `temperature' parameter is equal to the risk-seeking parameter. This policy achieves an expected regret bound of $\tilde O(L^{3/2} \sqrt{S A T})$, where $L$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the total number of elapsed time-steps. This bound is only a factor of $L$ larger than the established lower bound. K-learning can be interpreted as mirror descent in the policy space, and it is similar to other well-known methods in the literature, including Q-learning, soft-Q-learning, and maximum entropy policy gradient, and is closely related to optimism and count based exploration methods. K-learning is simple to implement, as it only requires adding a bonus to the reward at each state-action and then solving a Bellman equation. We conclude with a numerical example demonstrating that K-learning is competitive with other state-of-the-art algorithms in practice.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.