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We propose a projection-free conditional gradient-type algorithm for smooth stochastic multi-level composition optimization, where the objective function is a nested composition of $T$ functions and the constraint set is a closed convex set. Our algorithm assumes access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle satisfying certain standard unbiasedness and second moment assumptions. We show that the number of calls to the stochastic first-order oracle and the linear-minimization oracle required by the proposed algorithm, to obtain an $\epsilon$-stationary solution, are of order $\mathcal{O}_T(\epsilon^{-2})$ and $\mathcal{O}_T(\epsilon^{-3})$ respectively, where $\mathcal{O}_T$ hides constants in $T$. Notably, the dependence of these complexity bounds on $\epsilon$ and $T$ are separate in the sense that changing one does not impact the dependence of the bounds on the other. Moreover, our algorithm is parameter-free and does not require any (increasing) order of mini-batches to converge unlike the common practice in the analysis of stochastic conditional gradient-type algorithms.

In this paper, we propose GT-GDA, a distributed optimization method to solve saddle point problems of the form: $\min_{\mathbf{x}} \max_{\mathbf{y}} \{F(\mathbf{x},\mathbf{y}) :=G(\mathbf{x}) + \langle \mathbf{y}, \overline{P} \mathbf{x} \rangle - H(\mathbf{y})\}$, where the functions $G(\cdot)$, $H(\cdot)$, and the the coupling matrix $\overline{P}$ are distributed over a strongly connected network of nodes. GT-GDA is a first-order method that uses gradient tracking to eliminate the dissimilarity caused by heterogeneous data distribution among the nodes. In the most general form, GT-GDA includes a consensus over the local coupling matrices to achieve the optimal (unique) saddle point, however, at the expense of increased communication. To avoid this, we propose a more efficient variant GT-GDA-Lite that does not incur the additional communication and analyze its convergence in various scenarios. We show that GT-GDA converges linearly to the unique saddle point solution when $G(\cdot)$ is smooth and convex, $H(\cdot)$ is smooth and strongly convex, and the global coupling matrix $\overline{P}$ has full column rank. We further characterize the regime under which GT-GDA exhibits a network topology-independent convergence behavior. We next show the linear convergence of GT-GDA to an error around the unique saddle point, which goes to zero when the coupling cost ${\langle \mathbf y, \overline{P} \mathbf x \rangle}$ is common to all nodes, or when $G(\cdot)$ and $H(\cdot)$ are quadratic. Numerical experiments illustrate the convergence properties and importance of GT-GDA and GT-GDA-Lite for several applications.

For the task of sampling from a density $\pi \propto \exp(-V)$ on $\mathbb{R}^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O( L^2 d^2/\varepsilon^2)$ iterations. This is the sampling analogue of complexity bounds for finding an $\varepsilon$-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincar\'e inequality.

We study iterative methods based on Krylov subspaces for low-rank approximation under any Schatten-$p$ norm. Here, given access to a matrix $A$ through matrix-vector products, an accuracy parameter $\epsilon$, and a target rank $k$, the goal is to find a rank-$k$ matrix $Z$ with orthonormal columns such that $\| A(I -ZZ^\top)\|_{S_p} \leq (1+\epsilon)\min_{U^\top U = I_k} \|A(I - U U^\top)\|_{S_p}$, where $\|M\|_{S_p}$ denotes the $\ell_p$ norm of the the singular values of $M$. For the special cases of $p=2$ (Frobenius norm) and $p = \infty$ (Spectral norm), Musco and Musco (NeurIPS 2015) obtained an algorithm based on Krylov methods that uses $\tilde{O}(k/\sqrt{\epsilon})$ matrix-vector products, improving on the na\"ive $\tilde{O}(k/\epsilon)$ dependence obtainable by the power method, where $\tilde{O}$ suppresses poly$(\log(dk/\epsilon))$ factors. Our main result is an algorithm that uses only $\tilde{O}(kp^{1/6}/\epsilon^{1/3})$ matrix-vector products, and works for all $p \geq 1$. For $p = 2$ our bound improves the previous $\tilde{O}(k/\epsilon^{1/2})$ bound to $\tilde{O}(k/\epsilon^{1/3})$. Since the Schatten-$p$ and Schatten-$\infty$ norms are the same up to a $1+ \epsilon$ factor when $p \geq (\log d)/\epsilon$, our bound recovers the result of Musco and Musco for $p = \infty$. Further, we prove a matrix-vector query lower bound of $\Omega(1/\epsilon^{1/3})$ for any fixed constant $p \geq 1$, showing that surprisingly $\tilde{\Theta}(1/\epsilon^{1/3})$ is the optimal complexity for constant~$k$. To obtain our results, we introduce several new techniques, including optimizing over multiple Krylov subspaces simultaneously, and pinching inequalities for partitioned operators. Our lower bound for $p \in [1,2]$ uses the Araki-Lieb-Thirring trace inequality, whereas for $p>2$, we appeal to a norm-compression inequality for aligned partitioned operators.

We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness proposed recently by [CST21]. (1) Separable minimax optimization. We study separable minimax optimization problems $\min_x \max_y f(x) - g(y) + h(x, y)$, where $f$ and $g$ have smoothness and strong convexity parameters $(L^x, \mu^x)$, $(L^y, \mu^y)$, and $h$ is convex-concave with a $(\Lambda^{xx}, \Lambda^{xy}, \Lambda^{yy})$-blockwise operator norm bounded Hessian. We provide an algorithm with gradient query complexity $\tilde{O}\left(\sqrt{\frac{L^{x}}{\mu^{x}}} + \sqrt{\frac{L^{y}}{\mu^{y}}} + \frac{\Lambda^{xx}}{\mu^{x}} + \frac{\Lambda^{xy}}{\sqrt{\mu^{x}\mu^{y}}} + \frac{\Lambda^{yy}}{\mu^{y}}\right)$. Notably, for convex-concave minimax problems with bilinear coupling (e.g.\ quadratics), where $\Lambda^{xx} = \Lambda^{yy} = 0$, our rate matches a lower bound of [ZHZ19]. (2) Finite sum optimization. We study finite sum optimization problems $\min_x \frac{1}{n}\sum_{i\in[n]} f_i(x)$, where each $f_i$ is $L_i$-smooth and the overall problem is $\mu$-strongly convex. We provide an algorithm with gradient query complexity $\tilde{O}\left(n + \sum_{i\in[n]} \sqrt{\frac{L_i}{n\mu}} \right)$. Notably, when the smoothness bounds $\{L_i\}_{i\in[n]}$ are non-uniform, our rate improves upon accelerated SVRG [LMH15, FGKS15] and Katyusha [All17] by up to a $\sqrt{n}$ factor. (3) Minimax finite sums. We generalize our algorithms for minimax and finite sum optimization to solve a natural family of minimax finite sum optimization problems at an accelerated rate, encapsulating both above results up to a logarithmic factor.

We resolve an open problem posed by Joswig et al. by providing an $\tilde{O}(N)$ time, $O(\log^2(N))$-factor approximation algorithm for the min-Morse unmatched problem (MMUP) Let $\Lambda$ be the no. of critical cells of the optimal discrete Morse function and $N$ be the total no. of cells of a regular cell complex K. The goal of MMUP is to find $\Lambda$ for a given complex K. To begin with, we apply an approx. preserving graph reduction on MMUP to obtain a new problem namely the min-partial order problem (min-POP)(a strict generalization of the min-feedback arc set problem). The reduction involves introduction of rigid edges which are edges that demand strict inclusion in output solution. To solve min-POP, we use the Leighton- Rao divide-&-conquer paradigm that provides solutions to SDP-formulated instances of min-directed balanced cut with rigid edges (min-DBCRE). Our first algorithm for min-DBCRE extends Agarwal et al.'s rounding procedure for digraph formulation of ARV-algorithm to handle rigid edges. Our second algorithm to solve min-DBCRE SDP, adapts Arora et al.'s primal dual MWUM. In terms of applications, under the mild assumption1 of the size of topological features being significantly smaller compared to the size of the complex, we obtain an (a) $\tilde{O}(N)$ algorithm for computing homology groups $H_i(K,A)$ of a simplicial complex K, (where A is an arbitrary Abelian group.) (b) an $\tilde{O}(N^2)$ algorithm for computing persistent homology and (c) an $\tilde{O}(N)$ algorithm for computing the optimal discrete Morse-Witten function compatible with input scalar function as simple consequences of our approximation algorithm for MMUP thereby giving us the best known complexity bounds for each of these applications under the aforementioned assumption. Such an assumption is realistic in applied settings, and often a characteristic of modern massive datasets.

We consider variants of the classical Frank-Wolfe algorithm for constrained smooth convex minimization, that instead of access to the standard oracle for minimizing a linear function over the feasible set, have access to an oracle that can find an extreme point of the feasible set that is closest in Euclidean distance to a given vector. We first show that for many feasible sets of interest, such an oracle can be implemented with the same complexity as the standard linear optimization oracle. We then show that with such an oracle we can design new Frank-Wolfe variants which enjoy significantly improved complexity bounds in case the set of optimal solutions lies in the convex hull of a subset of extreme points with small diameter (e.g., a low-dimensional face of a polytope). In particular, for many $0\text{--}1$ polytopes, under quadratic growth and strict complementarity conditions, we obtain the first linearly convergent variant with rate that depends only on the dimension of the optimal face and not on the ambient dimension.

Escaping saddle points is a central research topic in nonconvex optimization. In this paper, we propose a simple gradient-based algorithm such that for a smooth function $f\colon\mathbb{R}^n\to\mathbb{R}$, it outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}(\log n/\epsilon^{1.75})$ iterations. Compared to the previous state-of-the-art algorithms by Jin et al. with $\tilde{O}((\log n)^{4}/\epsilon^{2})$ or $\tilde{O}((\log n)^{6}/\epsilon^{1.75})$ iterations, our algorithm is polynomially better in terms of $\log n$ and matches their complexities in terms of $1/\epsilon$. For the stochastic setting, our algorithm outputs an $\epsilon$-approximate second-order stationary point in $\tilde{O}((\log n)^{2}/\epsilon^{4})$ iterations. Technically, our main contribution is an idea of implementing a robust Hessian power method using only gradients, which can find negative curvature near saddle points and achieve the polynomial speedup in $\log n$ compared to the perturbed gradient descent methods. Finally, we also perform numerical experiments that support our results.

We present an end-to-end framework for solving the Vehicle Routing Problem (VRP) using reinforcement learning. In this approach, we train a single model that finds near-optimal solutions for problem instances sampled from a given distribution, only by observing the reward signals and following feasibility rules. Our model represents a parameterized stochastic policy, and by applying a policy gradient algorithm to optimize its parameters, the trained model produces the solution as a sequence of consecutive actions in real time, without the need to re-train for every new problem instance. On capacitated VRP, our approach outperforms classical heuristics and Google's OR-Tools on medium-sized instances in solution quality with comparable computation time (after training). We demonstrate how our approach can handle problems with split delivery and explore the effect of such deliveries on the solution quality. Our proposed framework can be applied to other variants of the VRP such as the stochastic VRP, and has the potential to be applied more generally to combinatorial optimization problems.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.