HomeAdam: Graph, Optimization & Medical Perspectives

• HomeAdam in graph theory is defined by the ‘find Adam’ problem, recovering the initial vertex in a preferential attachment tree using renormalized degree strategies.
• In optimization, HomeAdam modifies Adam/AdamW by switching to momentum SGD when adaptive estimates are unreliable to secure better generalization.
• In medical self-supervision, HomeAdam (or Adam-v2) employs a hierarchy-aware pretraining framework to model explicit anatomical part-whole relationships.

Searching arXiv for papers on "HomeAdam" and closely related usages to ground the article in current literature. HomeAdam is an overloaded research term that denotes distinct objects in different literatures. In random graph theory, it names the problem of identifying Adam, the initial/root vertex of a Barabási–Albert preferential attachment tree from the unlabeled final tree. In optimization, it names HomeAdam and HomeAdamW, Adam/AdamW variants that sometimes return to momentum SGD in order to obtain better provable generalization. In one medical self-supervision usage, “HomeAdam / Adam-v2” denotes a hierarchy-aware pretraining framework that makes anatomical part-whole relations explicit in the embedding space (Contat et al., 2023, Huang et al., 3 Mar 2026, Taher et al., 2024).

1. Terminological scope

The term appears in at least three technically unrelated senses in the supplied literature.

Usage	Domain	Meaning
HomeAdam	Preferential attachment trees	The “find Adam” problem: recover the initial/root vertex from the unlabeled tree
HomeAdam / HomeAdamW	Stochastic optimization	Adam/AdamW variants that sometimes switch to momentum SGD
HomeAdam / Adam-v2	Medical self-supervision	Anatomy-driven foundation-model pretraining with explicit part-whole hierarchy learning

The shared label “Adam” does not indicate a common mathematical framework. In the preferential-attachment setting, Adam and Eve are the first two vertices of the growing tree. In the optimization setting, Adam refers to adaptive moment estimation. In the anatomy setting, Adam-v2 extends an earlier anatomy-aware self-supervised learning framework by adding explicit hierarchy objectives (Contat et al., 2023, Huang et al., 3 Mar 2026, Taher et al., 2024).

2. HomeAdam in preferential attachment trees

In the graph-theoretic usage, HomeAdam is the reconstruction problem studied in “Eve, Adam and the Preferential Attachment Tree” (Contat et al., 2023). The model is the Barabási–Albert tree process $(\mathcal{T}(n): n \ge 1)$ built by starting from a single vertex $\mathbf{1}$ at time $1$, attaching $\mathbf{2}$ to $\mathbf{1}$ at time $2$, and for each $n \ge 3$ adding vertex $\mathbf{n}$ and connecting it to an existing vertex $\mathbf{i}\in\{1,\dots,n-1\}$ with probability

$$\Pr(\mathbf{n}\to \mathbf{i}\mid \mathcal{T}(n-1))=\frac{d_i(n-1)}{2(n-2)}.$$

Here $\mathbf{1}$0 is the degree of vertex $\mathbf{1}$1 in $\mathbf{1}$2. The initial vertex $\mathbf{1}$3 is called Adam and $\mathbf{1}$4 is called Eve.

The formal task is: given only the unlabeled tree structure of $\mathbf{1}$5 and a target error tolerance $\mathbf{1}$6, construct a subset

$\mathbf{1}$7

depending only on the tree structure such that, for large $\mathbf{1}$8,

$\mathbf{1}$9

Earlier work by Bubeck, Devroye, and Lugosi established that any such unlabeled confidence set must satisfy

$1$0

and also gave an upper bound of order

$1$1

later refined by Banerjee and Huang via selection of the vertices of largest degree. The central contribution of (Contat et al., 2023) is to prove that the lower-bound exponent is sharp. For any fixed $1$2 and all sufficiently small $1$3, the authors construct a label-free confidence set $1$4 such that

$1$5

Accordingly, the candidate-set size is essentially optimal: $1$6

3. Eve-assisted localization of Adam

The key heuristic of the graph-theoretic HomeAdam problem is that Adam is either a large-degree vertex, or a neighbor of a large-degree vertex (Contat et al., 2023). If Adam has unusually small degree, then Eve may have inherited the attachment of vertex $1$7, and that event tends to force Eve’s degree to be large. The constructive confidence set therefore searches not only for high-degree vertices, but also for vertices adjacent to sufficiently heavy neighbors.

The analysis uses renormalized degrees. The sequence $1$8 is defined by

$1$9

so that

$\mathbf{2}$0

The renormalized degrees are

$\mathbf{2}$1

and for each fixed $\mathbf{2}$2,

$\mathbf{2}$3

The whole vector converges almost surely in $\mathbf{2}$4: $\mathbf{2}$5

Using these quantities, the paper defines the $\mathbf{2}$6-packet

$\mathbf{2}$7

This construction is local, purely graph-theoretic, depends only on the unlabeled tree, and is not necessarily connected. A key proposition shows that for any fixed $\mathbf{2}$8, for $\mathbf{2}$9 small enough,

$\mathbf{1}$0

A central ingredient is the explicit joint distribution of the limiting degrees of Adam and Eve. Conditionally on $\mathbf{1}$1,

$\mathbf{1}$2

where

$\mathbf{1}$3

all independent. From this, the paper derives bounds such as

$\mathbf{1}$4

and

$\mathbf{1}$5

The size bound is proved using uniform tail estimates for renormalized degrees, including

$\mathbf{1}$6

On a suitable good event, one may treat

$\mathbf{1}$7

uniformly, after which the packet condition becomes roughly

$\mathbf{1}$8

for adjacent vertices $\mathbf{1}$9. The resulting theorem is

$2$0

4. HomeAdam and HomeAdamW as optimization algorithms

In optimization, HomeAdam is introduced in “HomeAdam: Adam and AdamW Algorithms Sometimes Go Home to Obtain Better Provable Generalization” (Huang et al., 3 Mar 2026). The paper begins from the observation that Adam and AdamW are default optimizers, converge faster, but generalize worse than SGD, and that prior theory typically gives Adam/AdamW generalization error on the order of

$2$1

whereas SGD / SGDM attain

$2$2

under the stability framework.

The paper first defines the square-root-free baseline Adam(W)-srf: $2$3 If $2$4, this is Adam-srf; if $2$5, this is AdamW-srf. The standard Adam(W) denominator

$2$6

is replaced by

$2$7

HomeAdam(W) uses the same moments $2$8, but switches updates according to the threshold condition

$2$9

If the condition holds, the method uses the adaptive update

$n \ge 3$0

Otherwise, it uses the SGDM-like update

$n \ge 3$1

Equivalently,

$n \ge 3$2

where

$n \ge 3$3

This switching rule motivates the name. When the adaptive second-moment estimate is reliable and large enough, the method behaves like AdamW-srf; otherwise, it returns to momentum SGD. The paper explicitly contrasts this with SWATS: SWATS switches from Adam to SGD at a fixed switchover stage, whereas HomeAdam(W) can switch back and forth at any time based on the threshold on $n \ge 3$4.

5. Stability, convergence, and provable generalization

The theoretical analysis of optimizer-side HomeAdam is based on algorithmic stability (Huang et al., 3 Mar 2026). For generalization, the paper assumes component smoothness,

$n \ge 3$5

Lipschitz loss,

$n \ge 3$6

and unbiased stochastic gradients with bounded variance,

$n \ge 3$7

For Adam(W)-srf, the key quantities are

$n \ge 3$8

and

$n \ge 3$9

The resulting generalization bound is

$\mathbf{n}$0

more specifically

$\mathbf{n}$1

Because $\mathbf{n}$2 is generally very small, the paper treats this bound as weak.

HomeAdam(W) replaces these small-denominator effects by the thresholded behavior. The theorem states

$\mathbf{n}$3

This applies to both HomeAdam ($\mathbf{n}$4) and HomeAdamW ($\mathbf{n}$5). The paper attributes the improvement to the fact that when $\mathbf{n}$6 is small, HomeAdam(W) stops using the potentially explosive adaptive scaling and switches to SGDM-like updates, thereby keeping the learning rate from becoming too large. It also proves that the weight-decay version is better, with inequalities including

$\mathbf{n}$7

so HomeAdamW has smaller generalization error than HomeAdam.

For convergence on smooth nonconvex objectives with lower bounded objective value $\mathbf{n}$8, Adam(W)-srf satisfies

$\mathbf{n}$9

whereas HomeAdam(W) satisfies

$\mathbf{i}\in\{1,\dots,n-1\}$0

Thus the exponent in $\mathbf{i}\in\{1,\dots,n-1\}$1 is unchanged, but the unfavorable small factor $\mathbf{i}\in\{1,\dots,n-1\}$2 disappears.

6. Position within the broader Adam literature

The optimizer-side meaning of HomeAdam sits within a broader line of work that modifies Adam while preserving some part of its adaptive moment structure. HyperAdam is a learnable, task-adaptive extension of Adam in which the parameter update at each iteration is an adaptive combination of multiple Adam-style updates with varying decay rates. It is modeled as a recurrent neural network with StateCell, AdamCell, and WeightCell, and the final update is

$\mathbf{i}\in\{1,\dots,n-1\}$3

(Wang et al., 2018).

Recent theory has also addressed Adam without introducing switching. “Uniform a priori bounds and error analysis for the Adam stochastic gradient descent optimization method” establishes pathwise uniform a priori bounds for Adam and uses them to provide, for the first time, an unconditional error analysis for a large class of strongly convex stochastic optimization problems. The update studied there is the bias-corrected Adam recursion

$\mathbf{i}\in\{1,\dots,n-1\}$4

under restrictions including

$\mathbf{i}\in\{1,\dots,n-1\}$5

in the main theorem (Dereich et al., 19 Mar 2026).

Adam-type methods have also been extended beyond single-level optimization. AdamBO is a single-loop Adam-type method for stochastic bilevel optimization with a $\mathbf{i}\in\{1,\dots,n-1\}$6-strongly convex lower-level problem and a nonconvex upper-level objective under unbounded smoothness. Its upper-level step is

$\mathbf{i}\in\{1,\dots,n-1\}$7

and the method achieves $\mathbf{i}\in\{1,\dots,n-1\}$8 oracle complexity to find $\mathbf{i}\in\{1,\dots,n-1\}$9-stationary points (Gong et al., 5 Mar 2025).

In multi-objective optimization, MAdam is presented as a drop-in wrapper for Adam-based multi-objective learning. Its core claim is that Adam introduces a weighting mismatch and a geometric mismatch relative to the solver’s intended Euclidean direction. MAdam therefore preconditions the reconciled direction by a preference-conditioned curvature estimate,

$$\Pr(\mathbf{n}\to \mathbf{i}\mid \mathcal{T}(n-1))=\frac{d_i(n-1)}{2(n-2)}.$$0

so that Adam’s second moment on the whitened input approximately collapses to identity (Liu et al., 2 Jun 2026).

A separate naming overlap occurs in medical imaging. “Representing Part-Whole Hierarchies in Foundation Models by Learning Localizability, Composability, and Decomposability from Anatomy via Self-Supervision” introduces Adam-v2, described in the supplied material as “HomeAdam / Adam-v2,” with three self-supervised branches—Localizability, Composability, and Decomposability—and total loss

$$\Pr(\mathbf{n}\to \mathbf{i}\mid \mathcal{T}(n-1))=\frac{d_i(n-1)}{2(n-2)}.$$1

(Taher et al., 2024).

7. Conceptual distinctions and recurring themes

The principal misconception surrounding HomeAdam is that the term names a single method. The literature instead uses it for distinct technical objects. In (Contat et al., 2023), HomeAdam is a graph reconstruction problem on preferential attachment trees, solved by a label-free confidence set based on renormalized degree products and the “Adam or Eve” heuristic. In (Huang et al., 3 Mar 2026), HomeAdam is an optimizer that sometimes returns from Adam-style preconditioning to SGDM-like updates in order to improve provable generalization.

The recurring theme is not a shared algorithmic substrate but a shared naming convention centered on “Adam.” In the tree-reconstruction problem, Adam is a distinguished vertex and Eve is the second vertex. In optimization, Adam is the adaptive moment method, and “going home” denotes a return to momentum SGD. In the anatomy literature, Adam-v2 uses the name to designate an anatomy-driven self-supervised framework rather than an optimizer or a graph-theoretic reconstruction task. For arXiv-reading audiences, precise disambiguation is therefore essential: the meaning of HomeAdam is determined entirely by domain, and the associated mathematics ranges from preferential attachment martingales to algorithmic stability and self-supervised representation learning (Contat et al., 2023, Huang et al., 3 Mar 2026, Taher et al., 2024).