---
title: Non-monotone DR-Submodular Maximization
url: https://www.emergentmind.com/topics/non-monotone-dr-submodular-maximization
type: topic
---

# Non-monotone DR-Submodular Maximization

Non-monotone DR-submodular maximization concerns the optimization of functions that generalize discrete submodularity (diminishing returns) to the continuous domain, encompassing non-monotonic behavior and non-down-closed convex constraints. This class unifies and extends classical set-function submodular maximization and covers a diversity of problems in machine learning, economics, and network optimization. The area is notable for a sequence of impossibility results, breakthroughs on tight polynomial-time approximability, and the interplay between constraint geometry and achievable guarantees [2210.05965].

## 1. DR-Submodularity and Problem Formulation

Let $F: [0,1]^n \to \mathbb{R}_{\geq 0}$ be a continuously differentiable function, with feasible set $K \subseteq [0,1]^n$ convex (not necessarily down-closed). $F$ is called DR-submodular if, for all $x \leq y$ (coordinate-wise), every $i \in [n]$, and all $\delta \geq 0$ with $y + \delta e_i \in [0,1]^n$,
$$
F(x + \delta e_i) - F(x) \geq F(y + \delta e_i) - F(y).
$$
Equivalently, the gradient is coordinate-wise anti-tone: $\nabla F(x) \geq \nabla F(y)$ when $x \leq y$, and all mixed Hessians $\frac{\partial^2 F}{\partial x_i \partial x_j} \leq 0$.

A function is non-monotone DR-submodular if the above holds but monotonicity ($\nabla F(x) \geq 0$ everywhere) is not assumed. Maximization of such functions over convex sets is NP-hard even in simple cases [2210.05965].

**Illustrative Example.** The function $F(x) = -\|x - \frac12 \mathbf{1}\|^2 + \frac{n}{4}$ on $[0,1]^n$ is DR-submodular but non-monotone: $F$ is initially increasing in each $x_i < 1/2$, then decreasing for $x_i > 1/2$.

## 2. Approximability Barriers and the Minimum-Norm Parameter

A central negative result (Vondrák 2013) establishes that for non-monotone DR-submodular maximization over a general convex set $K$, *no* algorithm running in sub-exponential time can achieve a constant-factor approximation in the worst case. The source of this hardness is the so-called symmetry-gap constructed by adversarially symmetrical feasible regions and objectives.

A key technique to bypass this barrier is to parameterize approximation in terms of the "minimum $\ell_\infty$-norm"
$$
m = \min_{x \in K} \|x\|_\infty.
$$
When $m < 1$, the feasible set $K$ stays "interior," breaking full symmetry and permitting nontrivial bounds. Sub-exponential-time methods achieve approximation ratios that scale as $1 - m$, gracefully degrading as $K$ approaches the cube's boundary [2210.05965].

## 3. Polynomial-time Algorithms: The $(1/4)(1-m)$ Guarantee

Du (2022) discovered the first polynomial-time, information-theoretically optimal algorithm for non-monotone DR-submodular maximization over general convex constraints, achieving a guarantee of
$$
F(w) \geq \frac14 (1 - m) \max_{x \in K} F(x) - o(1),
$$
where $w \in K$ is the output after $T \sim O(1/\epsilon)$ iterations for small $\epsilon$ [2210.05965].

**Algorithm—Non-monotone Frank-Wolfe:**
- Start from $y^{(0)} = \arg\min_{x \in K} \|x\|_\infty$.
- For $i=1, \dots, T$:
    1. $s^{(i)} \in \arg\max_{x \in K} \langle \nabla F(y^{(i-1)}), x \rangle$.
    2. $y^{(i)} = (1-\epsilon) y^{(i-1)} + \epsilon s^{(i)}$.
- Output the best $y^{(i)}$.

**Analysis:** By DR-submodularity, the Frank-Wolfe direction ensures a margin on the directional derivative related to global optimum via $F(o \vee y^{(i-1)}) + F(o \wedge y^{(i-1)}) - 2 F(y^{(i-1)})$. The iterative process contracts away from the boundary, ensuring the approximation factor dependently degrades as $m \to 1$ (when $K$ is almost fully boundary, e.g. a vertex).

This is provably information-theoretically sharp; no sub-exponential-time (let alone polynomial-time) algorithm can beat $(1/4)(1-m) + o(1)$ in worst case [2210.05965].

## 4. Online Maximization and Regret: Matching Tight Ratios

For the online version (sequentially revealed DR-submodular objectives $F_t$), a matching $(1/4)(1-m)$-approximation is obtained with $O(\sqrt{T})$ regret.

**Algorithm—Non-monotone Meta-Frank-Wolfe:**
- At each round $t$, initialize $y^{(0), t}$ at minimum $\ell_\infty$ norm in $K$.
- Execute $L \sim O(\ln 2/(2\epsilon))$ Frank-Wolfe steps with independent online linear-optimization subroutines $\mathcal{A}_i$.
- For each $i=1, \dots, L$:
    - Receive $s^{(i), t}$ from $\mathcal{A}_i$.
    - $y^{(i), t} = (1-\epsilon) y^{(i-1), t} + \epsilon s^{(i), t}$.
    - Receive/estimate an unbiased $\hat{g}^{(i), t}$ and feed as the loss vector to $\mathcal{A}_i$.
- Play $y^t = y^{(L), t}$.

The expected average reward over $T$ rounds satisfies
$$
\frac{1}{T} \sum_{t=1}^T \mathbb{E}[F_t(y^t)] \geq \frac14(1-m)\max_{x \in K} \frac{1}{T} \sum_{t=1}^T F_t(x) - O(T^{-1/2}).
$$
This guarantee, both offline and online, is proven optimal [2210.05965].

## 5. Information-theoretic Hardness

A symmetry-gap argument demonstrates that for any $h \in [0,1)$ and $\delta > 0$, there is no sub-exponential-time algorithm that achieves
$$
\frac14(1-h) + \delta
$$
approximation for maximizing non-negative, $L$-smooth DR-submodular $F$ over any polytope $K$ with $\min_{x \in K} \|x\|_\infty = h$. The construction involves adversarial, high-dimensional instances where distinguishing optimal from near-optimal regions is exponentially hard due to function symmetry.

This implies the $(1/4)(1-m)$ factor achieved by Du (2022) and in the presented online method is not improvable short of exponential time, for general $K$.

## 6. Extensions: Comparison to Other Settings and Interpolated Guarantees

The $(1/4)(1-m)$ bound specializes as follows:
- For $m=0$ (e.g., $0 \in K$, "fully down-closed"), the approximation is tight at $1/4$.
- For $m \uparrow 1$ (e.g., $K$ shrinks to a singleton or a low-dimensional facet), the guarantee vanishes, as expected.
This characterizes a smooth transition between the easy (down-closed) and hard (general) cases.

Intermediate approximation ratios $1/e$, $0.385$, and $1/3$ arise in the down-closed, box, or other special settings, addressed in the literature by continuous greedy, measured continuous greedy, double-greedy, and hybrid approaches [2307.09616][1711.02515][1805.09480].

## 7. Empirical Performance across Applications

The Du (2022) and matching online algorithms were tested in several domains:
- **Revenue Maximization (Social Networks):** On datasets such as Facebook (64K nodes) and Advogato (6.5K nodes) with box+budget constraints, the method converges substantially faster and reaches higher rewards than competing algorithms (e.g., [Thắng & Srivastav 2021]).
- **Location Summarization:** For tasks on the Yelp Charlotte dataset, the method outperforms others in longitudinal objective improvement.
- **Quadratic Programming with DR-negative-definite matrices:** Varied $K$ (down-closed and non-down-closed) were used, and the polynomial-time Non-monotone Frank-Wolfe outperforms previous sub-exponential algorithms even in down-closed cases when all methods are run under the same time budget.

These results validate both the tightness and practical strength of the $(1/4)(1-m)$ class for both offline and online settings [2210.05965].

---

**Summary Table: Offline Approximability by Constraint Type**

| Constraint Type $K$        | Best Achievable Ratio    | Achieved by      | Complexity        |
|----------------------------|------------------------|------------------|-------------------|
| Down-closed (e.g. box)     | $1/e$                  | [Bian et al.], [Dürr et al.] | poly-time |
| General, $m = \min \|x\|_\infty$ | $(1/4)(1-m)$         | Du (2022) offline; [2210.05965] online | poly-time |
| General, $m \approx 1$     | $\to 0$ (hard)         | — (no c.a.r.)    | — (hardness)      |

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## 8. Concluding Remarks

Non-monotone DR-submodular maximization over general convex sets is now fully characterized with respect to worst-case polynomial-time and sub-exponential-time approximability, with the $(1/4)(1-m)$ bound being sharp. Algorithmic frameworks (Frank-Wolfe variants, online meta-FW) are efficient, general, and empirically dominant, making the area a canonical example of tight complexity-theoretic and practical trade-off in non-convex continuous optimization. Advances in constraint-specific interpolation (e.g., via convex body decomposition) and specialized oracles further expand the landscape, but the inapproximability barrier sets a final limit without additional structure [2210.05965][2401.09251].

Source: https://www.emergentmind.com/topics/non-monotone-dr-submodular-maximization