DR-Submodularity: Theory and Algorithms

• DR-submodularity is a property that extends the diminishing returns principle to multi-dimensional continuous and discrete domains, emphasizing coordinate-wise concavity.
• It enables efficient greedy and double greedy algorithms with provable approximation guarantees across applications like budget allocation, sensor placement, and facility location.
• Its rigorous analysis uncovers computational hardness in constrained settings, driving the development of novel approaches for challenging non-convex optimization problems.

DR-submodularity is a generalization of the classical diminishing returns property of discrete submodular set functions to more general domains, such as the integer lattice, distributive lattices, and continuous domains. A function is DR-submodular if its marginal gain from increasing a coordinate (or “adding” an element) decreases as the current input increases, formalizing the concept that early investments or selections provide larger incremental benefits than later ones. DR-submodularity underpins the structure of many practical optimization problems in machine learning, economics, network theory, and combinatorial optimization, enabling the design of polynomial-time algorithms with provable approximation guarantees in otherwise intractable non-convex or combinatorial settings.

1. Formal Definitions and Fundamental Properties

Traditional submodularity for set functions $f: 2^N \to \mathbb{R}$ is characterized by: $f(S \cup \{e\}) - f(S) \geq f(T \cup \{e\}) - f(T)$ for all $S \subseteq T \subseteq N$ and $e \notin T$. This is the diminishing returns (DR) property: marginal gains decrease as the set grows.

DR-submodularity extends this to multivariate and continuous domains. For functions $f: \mathcal{D} \to \mathbb{R}$ where $\mathcal{D}$ is a product domain (often the integer lattice $\{0, \ldots, C\}^n$, a distributive lattice, or $[0,1]^n$), the DR property requires that for all $x \leq y$ (coordinate-wise) and every feasible coordinate $i$ and increment $k \geq 0$: $f(x + k\chi_i) - f(x) \ge f(y + k\chi_i) - f(y)$ where $\chi_i$ is the $i$th standard basis vector.

Key properties:

• For set functions, submodularity and the DR property are equivalent.
• For integer or continuous domains, submodularity (via the lattice inequality $f(x) + f(y) \ge f(x \wedge y) + f(x \vee y)$) does not imply DR-submodularity; additional coordinate-wise concavity is often required (Gottschalk et al., 2015, Bian et al., 2020).
• In the continuous setting, if $f$ is twice differentiable, DR-submodularity is equivalent to all cross-partial derivatives off the diagonal being non-positive and diagonal terms non-positive (i.e., $f$ is coordinate-wise concave) (Bian et al., 2020).

2. Algorithmic Frameworks for DR-Submodular Maximization

Efficient algorithms for DR-submodular maximization typically build on generalizations of greedy or double greedy paradigms and exploit concavity along nonnegative directions.

• Unconstrained Maximization:
  • For monotone DR-submodular functions over distributive lattices, a greedy approach yields a $1/2$-approximation (Gottschalk et al., 2015).
  • For set functions and integer lattices, double greedy frameworks achieve a $1/2$-approximation for DR-submodular objectives (randomized), and $1/3$ for general submodular (non-DR) functions (Gottschalk et al., 2015, Soma et al., 2016).
  • For continuous domains with box constraints, the DR-DoubleGreedy algorithm achieves a tight $1/2$-approximation in linear time (Bian et al., 2018).
• Constrained Maximization:
  • For matroid or poset matroid constraints and monotone DR-submodular functions, greedy selection (augmenting with feasible elements maximizing marginal gain) attains $1/2$-approximation; for cardinality (uniform matroid), it reaches $(1 - 1/e)$ (Gottschalk et al., 2015).
  • Continuous greedy algorithms and Frank-Wolfe–style projection-free optimization provide $(1 - 1/e)$-approximation for monotone DR-submodular functions under down-closed convex constraints (Bian et al., 2016, Gottschalk et al., 2015, Bian et al., 2017).
• Non-monotone Settings:
  • For non-monotone DR-submodular maximization, double greedy–type algorithms attain $1/(2+\varepsilon)$-approximation in strongly polynomial time (Soma et al., 2016), $1/4$ via a two-phase or discretized approach with convergence guarantees (Bian et al., 2017, Du et al., 2022).

3. Hardness and Complexity Landscape

While many tractable cases exist, particular constraints can render DR-submodular maximization intractable:

• Knapsack Constraints: Knapsack constraints in general distributive lattice settings cause a dramatic hardness increase—no constant-factor approximation is achievable unless $3$-SAT can be solved in sub-exponential time (Gottschalk et al., 2015). The inapproximability bound flows from a reduction to dense subhypergraph problems.
• General Continuous Nonconvexity: DR-submodular maximization is NP-hard in general, and the best possible polynomial-time approximation ratios under value-oracle access are $1 - 1/e$ for monotone and $1/2$ for non-monotone objectives unless RP = NP (Bian et al., 2020).
• Recent Advances: The best-known solver for multilinear extension maximization subject to down-closed constraints achieves a $0.401$ approximation, nearly matching the inapproximability bound $0.478$ (Buchbinder et al., 2023).

4. Mathematical Formulations and Approximation Guarantees

Foundational inequalities:

Problem/domain	Approximation Ratio	Reference
Unconstrained integer lattice	$1/3$ (general submod.)	(Gottschalk et al., 2015)
Unconstrained DR-submodular	$1/2$	(Gottschalk et al., 2015, Bian et al., 2018)
Monotone, card. constraint	$1 - 1/e$	(Gottschalk et al., 2015, Bian et al., 2020)
Monotone, poset matroid	$1/2$	(Gottschalk et al., 2015)
Non-monotone continuous	$1/3$ (FW/DoubleGreedy)	(Bian et al., 2016)
Non-monotone contin. (box)	$1/2$ (DR-DoubleGreedy)	(Bian et al., 2018)
Down-closed constraint (ML ext.)	$0.401$	(Buchbinder et al., 2023)
Knapsack/distributive lattice	no const. approx.	(Gottschalk et al., 2015)

Representative formulas:

• Lattice DR-submodularity:

$f(x) + f(y) \geq f(x \wedge y) + f(x \vee y)$ $f(x + \chi_i) - f(x) \geq f(y + \chi_i) - f(y)$ for $x \leq y$ (integer lattice).

• Continuous DR-submodularity (gradient characterization):

$\nabla_i f(x) \geq \nabla_i f(y)$ for $x \leq y$.

• Multilinear extension (randomization for sets):

$F(x) = \mathbb{E}_{S \sim x}[f(S)]$ (where $S$ is a random set, including each element independently with probability $x_i$).

• History-aware lower bound (Buchbinder et al., 2023):

$$F\left(1-\mathbf{a}\odot e^{-\int_0^t\mathbf{x}(\tau)d\tau}\right) \geq e^{-t}\left[F(1-\mathbf{a}) + \sum_{i=1}^\infty \frac{1}{i!}\int_{[0,t]^i}F\left((1-\mathbf{a})\oplus_{j=1}^i \mathbf{x}(\tau_j)\right) d\tau\right]$$

5. Applications in Optimization and Machine Learning

DR-submodular maximization arises in a wide variety of resource allocation, combinatorial, and statistical problems:

• Budget allocation and marketing, where additional budget yields diminishing growth in influence or revenue (Bian et al., 2016, Gu et al., 2022).
• Sensor placement, for coverage with redundancy and reliability (Sessa et al., 2019, Maehara et al., 2019).
• Facility location, where cost-benefit of new facilities is decreasing with existing coverage (Bian et al., 2016).
• Mean field inference and MAP for DPPs: maximizing expected value or log-likelihood functions with structured repulsion (Bian et al., 2017, Bian et al., 2018, Bian et al., 2020).
• Profit maximization in social networks: allocating repeated trials or investments with diminishing conversion rates (Gu et al., 2022).
• Online and game-theoretical frameworks where distributed agents maximize a monotone DR-submodular social function with proven bounds on equilibria inefficiency, related to price of anarchy (Sessa et al., 2019, Sadeghi et al., 2019, Sadeghi et al., 2019).

6. Advanced Algorithmic Developments and Open Problems

Innovations in DR-submodular optimization span multiple algorithmic fronts:

• Continuous relaxation and rounding: Multilinear extension maximization with randomized rounding is central to approaching combinatorial constraints (Buchbinder et al., 2023).
• Derivative-free and noisy optimization: Black-box methods such as LDGM yield robustness to non-differentiability and noise, matching gradient-based methods in approximation quality (Zhang et al., 2018).
• Projection-free and bandit algorithms: Recent frameworks achieve first regret guarantees for stochastic DR-submodular maximization under bandit feedback, exploiting smoothing and momentum techniques (Pedramfar et al., 2023, Pedramfar et al., 27 Apr 2024).
• Strong/curved DR-submodularity: When the objective enjoys strong concavity along nonnegative directions, fast algorithms with improved approximation and linear convergence can be realized (Sadeghi et al., 2021).
• Oracle complexity: For general convex constraints, stochastic value oracle models require $O(1/\varepsilon^5)$ calls for $O(\varepsilon)$-approximation in the worst case (Pedramfar et al., 2023).

Key open questions remain:

• Can the $0.401$ approximation for multilinear extension maximization under down-closed constraints be further improved, as the inapproximability barrier is $0.478$ (Buchbinder et al., 2023)?
• Do adaptive or history-dependent strategies exploiting the new “history-aware” bounds enable further progress (Buchbinder et al., 2023)?
• Under which settings (e.g., additional structure, dynamic/adversarial, composite constraints) can the known performance gaps be narrowed?

7. Impact and Broader Significance

DR-submodularity has fundamentally reshaped understanding of non-convex optimization in both discrete and continuous settings. The extension of the diminishing returns paradigm to richer domains has allowed for algorithmic advances in areas previously considered intractable:

• DR-submodularity underlies algorithms that efficiently bridge the gap between combinatorial and convex optimization, leveraging multilinear relaxations and randomized rounding.
• The clear separation between submodularity and DR-submodularity on lattices has elucidated sources of algorithmic hardness, indicating the necessity of the diminishing returns property for tractability especially outside the Boolean cube (Gottschalk et al., 2015).
• Theoretical results on inapproximability, tight lower bounds, and oracle complexity expose intrinsic barriers and guide algorithm development.

Recent frameworks unify diverse settings—monotone/non-monotone, continuous/lattice, deterministic/stochastic, full-information/bandit/zero-order feedback—offering a comprehensive, modular toolbox for non-convex and non-monotone optimization with rigorous guarantees (Pedramfar et al., 2023, Pedramfar et al., 27 Apr 2024).

This synthesis reflects the depth and diversity of DR-submodular optimization, encompassing rigorous theoretical analysis, algorithmic innovation, and practical applications across machine learning, data science, and operations research.