In this paper, we introduce a discrete variant of the Meta-learning framework. Meta-learning aims at exploiting prior experience and data to improve performance on future tasks. By now, there exist numerous formulations for Meta-learning in the continuous domain. Notably, the Model-Agnostic Meta-Learning (MAML) formulation views each task as a continuous optimization problem and based on prior data learns a suitable initialization that can be adapted to new, unseen tasks after a few simple gradient updates. Motivated by this terminology, we propose a novel Meta-learning framework in the discrete domain where each task is equivalent to maximizing a set function under a cardinality constraint.

Modularity maximization has been a fundamental tool for understanding the community structure of a network, but the underlying optimization problem is nonconvex and NP-hard to solve. State-of-the-art algorithms like the Louvain or Leiden methods focus on different heuristics to help escape local optima, but they still depend on a greedy step that moves node assignment locally and is prone to getting trapped. In this paper, we propose a new class of low-cardinality algorithm that generalizes the local update to maximize a semidefinite relaxation derived from max-k-cut. This proposed algorithm is scalable, empirically achieves the global semidefinite optimality for small cases, and outperforms the state-of-the-art algorithms in real-world datasets with little additional time cost. From the algorithmic perspective, it also opens a new avenue for scaling-up semidefinite programming when the solutions are sparse instead of low-rank.

While policy optimization algorithms have played an important role in recent empirical success of Reinforcement Learning (RL), the existing theoretical understanding of policy optimization remains rather limited---they are either restricted to tabular MDPs or suffer from highly suboptimal sample complexity, especial in online RL where exploration is necessary. This paper proposes a simple efficient policy optimization framework---Optimistic NPG for online RL. Optimistic NPG can be viewed as simply combining of the classic natural policy gradient (NPG) algorithm [Kakade, 2001] with optimistic policy evaluation subroutines to encourage exploration. For $d$-dimensional linear MDPs, Optimistic NPG is computationally efficient, and learns an $\epsilon$-optimal policy within $\tilde{\mathcal{O}}(d^2/\epsilon^3)$ samples, which is the first computationally efficient algorithm whose sample complexity has the optimal dimension dependence $\tilde{\Theta}(d^2)$. It also improves over state-of-the-art results of policy optimization algorithms [Zanette et al., 2021] by a factor of $d$. For general function approximation that subsumes linear MDPs, Optimistic NPG, to our best knowledge, is also the first policy optimization algorithm that achieves the polynomial sample complexity for learning near-optimal policies.

Tyler's M-estimator is a well known procedure for robust and heavy-tailed covariance estimation. Tyler himself suggested an iterative fixed-point algorithm for computing his estimator however, it requires super-linear (in the size of the data) runtime per iteration, which maybe prohibitive in large scale. In this work we propose, to the best of our knowledge, the first Frank-Wolfe-based algorithms for computing Tyler's estimator. One variant uses standard Frank-Wolfe steps, the second also considers \textit{away-steps} (AFW), and the third is a \textit{geodesic} version of AFW (GAFW). AFW provably requires, up to a log factor, only linear time per iteration, while GAFW runs in linear time (up to a log factor) in a large $n$ (number of data-points) regime. All three variants are shown to provably converge to the optimal solution with sublinear rate, under standard assumptions, despite the fact that the underlying optimization problem is not convex nor smooth. Under an additional fairly mild assumption, that holds with probability 1 when the (normalized) data-points are i.i.d.

Over the last two decades, submodular function maximization has been the workhorse of many discrete optimization problems in machine learning applications. Traditionally, the study of submodular functions was based on binary function properties, but recent works began to consider continuous function properties such as the submodularity ratio and the curvature. The monotonicity property of set functions plays a central role in submodular maximization. Nevertheless, no continuous version of this property has been suggested to date (as far as we know), which is unfortunate since submoduar functions that are almost monotone often arise in machine learning applications. In this work we fill this gap by defining the monotonicity ratio, which is a continuous version of the monotonicity property. We then show that for many standard submodular maximization algorithms one can prove new approximation guarantees that depend on the monotonicity ratio; leading to improved approximation ratios for the common machine learning applications of movie recommendation, quadratic programming, image summarization and ride-share optimization.

Semi-supervised learning seeks to learn a machine learning model when only a small amount of the available data is labeled. The most widespread approach uses a graph prior, which encourages similar instances to have similar predictions. This has been very successful with models ranging from kernel machines to neural networks, but has remained inapplicable to decision trees, for which the optimization problem is much harder. We solve this based on a reformulation of the problem which requires iteratively solving two simpler problems: a supervised tree learning problem, which can be solved by the Tree Alternating Optimization algorithm; and a label smoothing problem, which can be solved through a sparse linear system. The algorithm is scalable and highly effective even with very few labeled instances, and makes it possible to learn accurate, interpretable models based on decision trees in such situations.

We study the problem of representation learning in stochastic contextual linear bandits. While the primary concern in this domain is usually to find \textit{realizable} representations (i.e., those that allow predicting the reward function at any context-action pair exactly), it has been recently shown that representations with certain spectral properties (called \textit{HLS}) may be more effective for the exploration-exploitation task, enabling \textit{LinUCB} to achieve constant (i.e., horizon-independent) regret. In this paper, we propose \textsc{BanditSRL}, a representation learning algorithm that combines a novel constrained optimization problem to learn a realizable representation with good spectral properties with a generalized likelihood ratio test to exploit the recovered representation and avoid excessive exploration. We prove that \textsc{BanditSRL} can be paired with any no-regret algorithm and achieve constant regret whenever an \textit{HLS} representation is available. Furthermore, \textsc{BanditSRL} can be easily combined with deep neural networks and we show how regularizing towards \textit{HLS} representations is beneficial in standard benchmarks.

In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called \COORDINATE-ASCENT+, achieves a $(\frac{e-1}{2e-1}-\eps)$-approximation guarantee while performing $O(n/\epsilon)$ iterations, where the computational complexity of each iteration is roughly $O(n/\sqrt{\epsilon}+n\log n)$ (here, $n$ denotes the dimension of the optimization problem). We then propose \COORDINATE-ASCENT++, that achieves the tight $(1-1/e-\eps)$-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly $O(n^3/\eps^{2.5}

We study the problem of learning a linear model to set the reserve price in an auction, given contextual information, in order to maximize expected revenue from the seller side. First, we show that it is not possible to solve this problem in polynomial time unless the Exponential Time Hypothesis fails. Second, we present a strong mixed-integer programming (MIP) formulation for this problem, which is capable of exactly modeling the nonconvex and discontinuous expected reward function. Moreover, we show that this MIP formulation is ideal (i.e. the strongest possible formulation) for the revenue function of a single impression. Since it can be computationally expensive to exactly solve the MIP formulation in practice, we also study the performance of its linear programming (LP) relaxation. Though it may work well in practice, we show that, unfortunately, in the worst case the optimal objective of the LP relaxation can be O(number of samples) times larger than the optimal objective of the true problem. Finally, we present computational results, showcasing that the MIP formulation, along with its LP relaxation, are able to achieve superior in-and out-of-sample performance, as compared to state-of-the-art algorithms on both real and synthetic datasets. More broadly, we believe this work offers an indication of the strength of optimization methodologies like MIP to exactly model intrinsic discontinuities in machine learning problems.

Decision-Focused Learning (DFL) is a paradigm for tailoring a predictive model to a downstream optimization task that uses its predictions in order to perform better \textit{on that specific task}. The main technical challenge associated with DFL is that it requires being able to differentiate through the optimization problem, which is difficult due to discontinuous solutions and other challenges. Past work has largely gotten around this this issue by \textit{handcrafting} task-specific surrogates to the original optimization problem that provide informative gradients when differentiated through. However, the need to handcraft surrogates for each new task limits the usability of DFL. In addition, there are often no guarantees about the convexity of the resulting surrogates and, as a result, training a predictive model using them can lead to inferior local optima. In this paper, we do away with surrogates altogether and instead \textit{learn} loss functions that capture task-specific information. To the best of our knowledge, ours is the first approach that entirely replaces the optimization component of decision-focused learning with a loss that is automatically learned. Our approach (a) only requires access to a black-box oracle that can solve the optimization problem and is thus \textit{generalizable}, and (b) can be \textit{convex by construction} and so can be easily optimized over. We evaluate our approach on three resource allocation problems from the literature and find that our approach outperforms learning without taking into account task-structure in all three domains, and even hand-crafted surrogates from the literature.