Below are details of our results.

Determinantal point processes (DPPs) offer a popular diversification model in machine learning.

In practice, perfect knowledge of M {} { atural} -concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M {} { atural} -concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). A key to proving these results is the robustness of the greedy algorithm to local errors in M {} { atural} -concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve O(T {1-c}) regret for any constant c 0 unless \mathsf{P} \mathsf{NP} .

M${}^{\natural}$-concave functions, a.k.a. gross substitute valuation functions, play a fundamental role in many fields, including discrete mathematics and economics. In practice, perfect knowledge of M${}^{\natural}$-concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M${}^{\natural}$-concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). For the stochastic bandit setting, we present $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms under $T$ times access to unbiased noisy value oracles of M${}^{\natural}$-concave functions. A key to proving these results is the robustness of the greedy algorithm to local errors in M${}^{\natural}$-concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve $O(T^{1-c})$ regret for any constant $c > 0$ unless $\mathsf{P} = \mathsf{NP}$. Our proof is based on a reduction from the matroid intersection problem for three matroids, which would be a novel idea in the context of online learning.

We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, one formulation of which is $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. Krause et al. (2008) showed that when the number of functions $m$ grows as the cardinality $k$ i.e., $m=\Omega(k)$, the problem is inapproximable (unless $P=NP$). For the more general case of matroid constraint, Chekuri et al. (2010) gave a randomized $(1-1/e)-\epsilon$ approximation for constant $m$. The runtime (number of queries to function oracle) scales exponentially as $n^{m/\epsilon^3}$. We give the first polynomial time asymptotically constant factor approximations for $m=o(\frac{k}{\log^3 k})$: $(i)$ A randomized $(1-1/e)$ algorithm based on Chekuri et al. (2010). $(ii)$ A faster and more practical $\tilde{O}(n/\delta^3)$ time, randomized $(1-1/e)^2-\delta$ approximation based on Multiplicative-Weight-Updates. Finally, we characterize the variation in optimal solution value as a function of the cardinality $k$, leading to a derandomized approximation for constant $m$.

Submodular function maximization has numerous applications in machine learning and artificial intelligence. Many real applications require multiple submodular objective func-tions to be maximized, and which function is regarded as important by a user is not known in advance. In such cases, it is desirable to have a small family of representative solutions that would satisfy any user’s preference. A traditional approach for solving such a problem is to enumerate the Pareto optimal solutions. However, owing to the massive number of Pareto optimal solutions (possibly exponentially many), it is difficult for a user to select a solution. In this paper, we propose two efficient methods for finding a small family of representative solutions, based on the notion of regret ratio. The first method outputs a family of fixed size with a nontrivial regret ratio. The second method enables us to choose the size of the output family, and in the biobjective case, it has a provable trade-off between the size and the regret ratio. Using real and synthetic data, we empirically demonstrate that our methods achieve a small regret ratio.

We consider non-monotone DR-submodular function maximization, where DR-submodularity (diminishing return submodularity) is an extension of submodularity for functions over the integer lattice based on the concept of the diminishing return property. Maximizing non-monotone DR-submodular functions has many applications in machine learning that cannot be captured by submodular set functions. In this paper, we present a 1/(2+ε)-approximation algorithm with a running time of roughly O( n /ε log 2 B ), where n is the size of the ground set, B is the maximum value of a coordinate, and ε > 0 is a parameter. The approximation ratio is almost tight and the dependency of running time on B is exponentially smaller than the naive greedy algorithm. Experiments on synthetic and real-world datasets demonstrate that our algorithm outputs almost the best solution compared to other baseline algorithms, whereas its running time is several orders of magnitude faster.