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We provide new adaptive first-order methods for constrained convex optimization. Our main algorithms AdaACSA and AdaAGD+ are accelerated methods, which are universal in the sense that they achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even when they only have access to stochastic gradients. In addition, they do not require any prior knowledge on how the objective function is parametrized, since they automatically adjust their per-coordinate learning rate. These can be seen as truly accelerated Adagrad methods for constrained optimization. We complement them with a simpler algorithm AdaGrad+ which enjoys the same features, and achieves the standard non-accelerated convergence rate. We also present a set of new results involving adaptive methods for unconstrained optimization and variational inequalities arising from monotone operators. 

We design differentially private algorithms for the bandit convex optimization problem in the projection-free setting. This setting is important whenever the decision set has a complex geometry, and access to it is done efficiently only through a linear optimization oracle, hence Euclidean projections are unavailable (e.g. matroid polytope, submodular base polytope). This is the first differentially-private algorithm for projection-free bandit optimization, and in fact our bound matches the best known non-private projection-free algorithm (Garber-Kretzu, AISTATS '20) and the best known private algorithm, even for the weaker setting when projections are available (Smith-Thakurta, NeurIPS '13). 

We design differentially private algorithms for the bandit convex optimization problem in the projection-free setting. This setting is important whenever the decision set has a complex geometry, and access to it is done efficiently only through a linear optimization oracle, hence Euclidean projections are unavailable (e.g. matroid polytope, submodular base polytope). This is the first differentially-private algorithm for projection-free bandit optimization, and in fact our bound matches the best known non-private projection-free algorithm and the best known private algorithm, even for the weaker setting when projections are available. 

We provide new adaptive first-order methods for constrained convex optimization. Our main algorithms AdaACSA and AdaAGD+ are accelerated methods, which are universal in the sense that they achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even when they only have access to stochastic gradients. In addition, they do not require any prior knowledge on how the objective function is parametrized, since they automatically adjust their per-coordinate learning rate. These can be seen as truly accelerated Adagrad methods for constrained optimization. We complement them with a simpler algorithm AdaGrad+ which enjoys the same features, and achieves the standard non-accelerated convergence rate. We also present a set of new results involving adaptive methods for unconstrained optimization and variational inequalities arising from monotone operators. 

We present an m^{4/3+o(1)} log W -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite b-matching when |b|_1 = O(m), and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very recent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on l_p-norm minimizing flows (Kyng-Peng-Sachdeva-Wang, 2019). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

We present an $m^{4/3}+o(1) \log W$ -time algorithm for solving the minimum cost flow problem in graphs with unit capacity, where W is the maximum absolute value of any edge weight. For sparse graphs, this improves over the best known running time for this problem and, by well-known reductions, also implies improved running times for the shortest path problem with negative weights, minimum cost bipartite $b$-matching when $\|b\|_1 = O(m)$, and recovers the running time of the currently fastest algorithm for maximum flow in graphs with unit capacities (Liu-Sidford, 2020). Our algorithm relies on developing an interior point method–based framework which acts on the space of circulations in the underlying graph. From the combinatorial point of view, this framework can be viewed as iteratively improving the cost of a suboptimal solution by pushing flow around circulations. These circulations are derived by computing a regularized version of the standard Newton step, which is partially inspired by previous work on the unit-capacity maximum flow problem (Liu-Sidford, 2019), and subsequently refined based on the very re- cent progress on this problem (Liu-Sidford, 2020). The resulting step problem can then be computed efficiently using the recent work on $l_p$-norm minimizing flows (Kyng-Peng-Sachdeva- Wang, 2019). We obtain our faster algorithm by combining this new step primitive with a customized preconditioning method, which aims to ensure that the graph on which these circulations are computed has sufficiently large conductance. 

The iteratively reweighted least squares method (IRLS) is a popular technique used in practice for solving regression problems. Various versions of this method have been proposed, but their theoretical analyses failed to capture the good practical performance. In this paper we propose a simple and natural version of IRLS for solving $\ell_\infty$ and $\ell_1$ regression, which provably converges to a $(1+\epsilon)$-approximate solution in $O(m^{1/3}\log(1/\epsilon)/\epsilon^{2/3} + \log m/\epsilon^2)$ iterations, where $m$ is the number of rows of the input matrix. Interestingly, this running time is independent of the conditioning of the input, and the dominant term of the running time depends sublinearly in $\epsilon^{-1}$, which is atypical for the optimization of non-smooth functions. This improves upon the more complex algorithms of Chin et al. (ITCS '12), and Christiano et al. (STOC '11) by a factor of at least $1/\epsilon^2$, and yields a truly efficient natural algorithm for the slime mold dynamics (Straszak-Vishnoi, SODA '16, ITCS '16, ITCS '17). 

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-\epsilon$ approximation for monotone functions and a $1/e-\epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/\epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-\epsilon$ approximation in $O(\log(n/\epsilon)\log(m)/\epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016). Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function. 

The algorithms in this paper (when combined with our FOCS’16 paper) allow one to, in almost linear time, compute a whole bunch of things about random walks in directed graphs. For example, one can compute the stationary distribution, hitting the time between a pair of vertices, commute times between all vertices, escape probabilities, approximations of the mixing time, and more.