Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

=== Assistant: There are two very different issues mixed into “can we assume γ≳ρ\gamma \gtrsim \sqrt{\rho}γ≳ρ?”: ===
There are two very different issues mixed into “can we assume γ≳ρ\gamma \gtrsim \sqrt{\rho}γ≳ρ?”:
# A hard upper bound coming from the penalty weights in (3), and
# A nondegeneracy / “don’t sit at a breakpoint” condition (strict complementarity).

Let me separate them.

==== For your objective (3), the ℓ1\ell_1ℓ1 weights are ====

λi  =  ραdi.('')\lambda_i \;=\; \rho\alpha\sqrt{d_i}. \tag{''}λi=ραdi.(*)
and for every inactive coordinate i∈I⋆i\in I^\stari∈I⋆,

∣∇f(x⋆)i∣≤λi,γ=min⁡i∈I⋆(λi−∣∇f(x⋆)i∣).(''')|\nabla f(x^\star)_i|\le \lambda_i,
\qquad
\gamma = \min_{i\in I^\star}\bigl(\lambda_i-|\nabla f(x^\star)_i|\bigr). \tag{'''}∣∇f(x⋆)i∣≤λi,γ=i∈I⋆min(λi−∣∇f(x⋆)i∣).(**)
So automatically

0≤γ≤min⁡i∈I⋆λi≤ραmin⁡i∈I⋆di≤ραdmin⁡,(1)0 \le \gamma \le \min_{i\in I^\star}\lambda_i \le \rho\alpha\min_{i\in I^\star}\sqrt{d_i}
\le \rho\alpha\sqrt{d_{\min}}, \tag{1}0≤γ≤i∈I⋆minλi≤ραi∈I⋆mindi≤ραdmin,(1)
where dmin⁡d_{\min}dmin is the global minimum degree (or just the minimum degree among inactive nodes). This is immediate from the definition. open-problem-fountoulakis22a

===== If your graph has any inactive node with bounded degree (say degree 1 or 2), then =====

γ≤O(ρ).\gamma \le O(\rho).γ≤O(ρ).
And when ρ→0\rho\to 0ρ→0, O(ρ)=o(ρ)O(\rho)=o(\sqrt{\rho})O(ρ)=o(ρ). So on typical sparse graphs with leaves/whiskers, no structural assumption “around the seed” can force γ≳ρ\gamma \gtrsim \sqrt{\rho}γ≳ρ, because the bound (1) is global and purely degree-based.

This is why the star/path families are so effective as counterexamples: they contain low-degree inactive nodes, forcing γ\gammaγ to be at most O(ρ)O(\rho)O(ρ), and (by tuning ρ\rhoρ near breakpoints) even smaller.

==== To even make γ∼ρ\gamma\sim \sqrt{\rho}γ∼ρ possible when ρ\rhoρ is small, you need λi\lambda_iλi itself to be Ω(ρ)\Omega(\sqrt{\rho})Ω(ρ) for the relevant inactive coordinates. Using (∗)(*)(∗), that means: ====

ραdi  ≳  ρ⟹di  ≳  1α2ρ.(2)\rho\alpha\sqrt{d_i} \;\gtrsim\; \sqrt{\rho}
\quad\Longrightarrow\quad
d_i \;\gtrsim\; \frac{1}{\alpha^2\rho}. \tag{2}ραdi≳ρ⟹di≳α2ρ1.(2)
So a necessary (and essentially unavoidable) type of assumption is:

===== A clean version is: =====
* either the graph has minimum degree dmin⁡≳1/(α2ρ)d_{\min} \gtrsim 1/(\alpha^2\rho)dmin≳1/(α2ρ),
* or (more weakly) every node that ends up inactive has degree at least c/(α2ρ)c/(\alpha^2\rho)c/(α2ρ).

This rules out leaves/paths/stars-with-leaves-outside-support, and it matches the algebra: then

min⁡i∈I⋆λi  ≥  ραc/(α2ρ)  =  cρ.\min_{i\in I^\star}\lambda_i \;\ge\; \rho\alpha\sqrt{c/(\alpha^2\rho)} \;=\; \sqrt{c\rho}.i∈I⋆minλi≥ραc/(α2ρ)=cρ.
Is this “reasonable”?
* On dense graphs, yes.
* On many sparse real-world networks, no—unless you pre-process (e.g., work on a core where minimum degree is large).

==== Even if λi\lambda_iλi is large, γ\gammaγ can still be tiny if some inactive coordinate is nearly tight: ====

∣∇f(x⋆)i∣≈λi.|\nabla f(x^\star)_i|\approx \lambda_i.∣∇f(x⋆)i∣≈λi.
That’s exactly what happens at (or near) breakpoints of the regularization path.

So you need a second assumption:

===== A natural version is a relative slack condition: =====

∣∇f(x⋆)i∣≤(1−η)λifor all i∈I⋆,|\nabla f(x^\star)_i| \le (1-\eta)\lambda_i \quad\text{for all } i\in I^\star,∣∇f(x⋆)i∣≤(1−η)λifor all i∈I⋆,
for some fixed η∈(0,1)\eta\in(0,1)η∈(0,1). This is stronger than just γ>0\gamma>0γ>0; it forces a uniform fraction of slack.

Then

γ=min⁡i∈I⋆(λi−∣∇f(x⋆)i∣)≥ηmin⁡i∈I⋆λi.(3)\gamma
= \min_{i\in I^\star}\bigl(\lambda_i-|\nabla f(x^\star)_i|\bigr)
\ge \eta \min_{i\in I^\star}\lambda_i. \tag{3}γ=i∈I⋆min(λi−∣∇f(x⋆)i∣)≥ηi∈I⋆minλi.(3)
Combine (3) with the degree condition (2) and you get

γ  ≳  ηρ(up to α-constants).(4)\gamma \;\gtrsim\; \eta \sqrt{\rho} \quad (\text{up to }\alpha\text{-constants}). \tag{4}γ≳ηρ(up to α-constants).(4)
Is Assumption B “reasonable”?
* Deterministically in worst case: no (adversarial ρ\rhoρ can always sit near a breakpoint; your star/path show this).
* Practically / statistically: often yes, because exact tightness is a codimension-1 event. If ρ\rhoρ is chosen from a coarse grid or with small random jitter, hitting a breakpoint “too closely” becomes unlikely.

A very concrete “algorithmic” way to justify B is:
* choose ρ\rhoρ on a multiplicative grid (e.g., ρ=2−j\rho=2^{-j}ρ=2−j) and/or
* add a tiny random perturbation to ρ\rhoρ (or to edge weights), then with high probability you are not extremely close to a breakpoint, hence γ\gammaγ is not extremely small.

==== ### ====

Then γ≳ρ\gamma \gtrsim \sqrt{\rho}γ≳ρ is not a reasonable target in general, because γ≤O(ρ)\gamma\le O(\rho)γ≤O(ρ) already from (1). The best you can hope for is γ=Θ(ρ)\gamma=\Theta(\rho)γ=Θ(ρ) (and even that needs a nondegeneracy buffer).

===== If you assume something like: =====
* all nodes outside the seed’s local region / outside the optimal support have degree ≳1/ρ\gtrsim 1/\rho≳1/ρ, and
* ρ\rhoρ is not tuned near a breakpoint (relative slack η\etaη),

then γ\gammaγ can be of order ρ\sqrt{\rho}ρ (more precisely Θ(αρ)\Theta(\alpha\sqrt{\rho})Θ(αρ) up to constants), by the simple chain (3)+(2).

If you tell me which regime you care about (typical local clustering on sparse graphs, or a dense-core model, or a probabilistic “ρ\rhoρ not adversarial” model), I can write down a crisp assumption set and show the exact inequality chain that yields γ≥c αρ\gamma \ge c\,\alpha\sqrt{\rho}γ≥cαρ (or explain why it’s impossible in that regime).