Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate

arXiv:2604.08742 · math.OC, cs.LG · Submitted 2026-08-16 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate".

Jane: The paper was written by Yaxin Yu, Long Chen and Zeyi Xu from Sichuan University and University of California, Irvine.

Tom: Stay tuned as we take you through the paper and discuss its implications.

Jane: We also have Lu with us today — senior AI researcher at Tsinghua.

Tom: We also have Meng with us today — lead engineer at a mysterious AI startup.

Jane: We also have Lalam with us today — the in-house Large Language Model.

Tom: Alright, let's get started.

Title: Tom: Welcome back to the show, everyone. Today we're diving into a paper that's been creating quite a buzz in the optimization community, titled "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate." Jane, I have to say, just reading that title got me excited.

Jane: It should, Tom. This paper tackles one of the most embarrassing gaps in optimization theory. Adam is probably the most widely used algorithm in deep learning, and yet we've never had a proper convergence proof for it in the standard convex setting. Not once, in all these years.

Tom: And that's exactly what makes this paper so special. The authors, Yaxin Yu, Long Chen, and Zeyi Xu, have managed to do something that's been elusive for a decade. They've created a reformulation of Adam that actually comes with rigorous convergence guarantees.

Jane: Let me break down what they did in plain terms. Adam is like a car with two pedals — momentum and adaptive learning rates — and these two systems are so tangled together that nobody could figure out how to prove it actually reaches its destination.

Tom: Right, and the researchers' insight was to untangle those systems. They used something called variable and operator splitting, which is a fancy way of saying they took the problem apart and put it back together in a way that reveals the underlying structure.

Jane: Exactly. And once they did that, they added what they call a curvature-aware gradient correction. Think of it as adding a steering mechanism that accounts for the shape of the terrain you're driving on.

Tom: The result is something they call the Adam-HNAG flow, which is a continuous-time model of the algorithm. And here's the beautiful part — they proved that this flow has a Lyapunov function that decays exponentially.

Jane: For our listeners who aren't mathematicians, a Lyapunov function is like a measure of how far you are from the solution. If you can prove it's always shrinking, you know you're making progress. Exponential decay means you're getting there fast.

Tom: And the implications here are huge. This isn't just a theoretical curiosity. Adam is used in virtually every major machine learning system, from language models to image recognition. Having a theoretical foundation for it means we can actually understand when it works and why.

Jane: I love that they also created two discrete versions of this flow — Adam-HNAG and Adam-HNAG-s. The second one is particularly interesting because it's closer in form to the original Adam algorithm.

Tom: So we've got theory, we've got practical algorithms, and we've got numerical experiments that back it all up. This paper really has the complete package.

Jane: And I think the most exciting part is what this means for the future. If we can finally understand Adam theoretically, we might be able to design even better adaptive optimization methods.

Tom: Stay tuned, because in the next segment we're going to dig into the actual methodology and how they pulled off this theoretical breakthrough.

Summary: Tom: Welcome back. We're continuing our discussion of "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate." Jane, we talked about the big picture, but now let's get into the weeds a bit.

Jane: Gladly, Tom. So the paper starts by acknowledging a painful truth — Adam has been running on empirical success alone. The authors point out that existing guarantees only cover online convex settings or nonconvex settings with extra assumptions, but nothing for the standard convex case.

Tom: And they mention that Reddi and colleagues actually found a counterexample where Adam fails in that setting. So it's not just that we didn't have a proof — we had evidence that the original algorithm could genuinely break down.

Jane: That's right. So the authors took a different approach. Instead of trying to prove convergence for the original Adam, they reformulated it. They started with a continuous-time model of Adam and then applied this variable and operator splitting technique.

Tom: I found the intuition really elegant. They write the momentum divided by the square root of the second moment as a difference between two variables. That simple substitution reveals a structure that was hidden before.

Jane: And then they add that curvature-aware correction term. This is inspired by something called Hessian-driven Nesterov accelerated gradient flow, which is a mouthful, but essentially it uses information about the curvature of the objective function to accelerate convergence.

Tom: The key theorem in the paper shows that this continuous-time flow has a Lyapunov function that decays exponentially. That's a very strong stability result.

Jane: Let me put that in context. Exponential decay means the error shrinks at a rate proportional to its current size. That's much stronger than just saying it eventually goes to zero.

Tom: Then they discretize this flow to get two practical algorithms. The first one, Adam-HNAG, uses the preconditioner from the previous step. The second one, Adam-HNAG-s, uses the updated preconditioner, which makes it closer to the original Adam.

Jane: And here's where it gets really interesting. They prove convergence for both methods, but the analysis reveals something subtle. The consistency condition — the relationship between step sizes and parameters — is different for the two methods.

Tom: The synchronous version, Adam-HNAG-s, has a stricter condition because it uses the updated preconditioner. That makes sense when you think about it — using fresher information should require more careful tuning.

Jane: The numerical experiments are really compelling. They tested on logistic regression with controlled condition numbers, and the proposed methods consistently outperform standard Adam, gradient descent, and even the accelerated HNAG method on ill-conditioned problems.

Tom: I was particularly impressed with the real-world validation on the colon-cancer dataset. That's a high-dimensional problem with two thousand features and only sixty-two samples, which is exactly the kind of ill-conditioned regime where these methods shine.

Jane: And they also tested on that famous counterexample from Reddi. Adam-HNAG stayed stable while both Adam and Adam-HNAG-s showed pathological behavior. That's a really interesting result.

Tom: So the summary is: they've built a theoretical foundation for Adam-type methods, created two practical algorithms, and validated them across multiple settings. In the next segment, we'll talk about what improvements this suggests and where the field might go from here.

Improvements: Tom: We're back with more on "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate." Jane, we've covered the basics and the methodology. Now let's talk about what this paper actually improves and what it means for the field.

Jane: The biggest improvement is conceptual, Tom. This paper gives us a new way to think about Adam. Instead of treating it as this mysterious black box that works in practice but defies theory, they've shown it can be understood through structure-preserving reformulation.

Tom: And that's not just academic. When you understand why something works, you can make it work better. The paper shows that the adaptive mechanism can actually produce contraction stronger than the standard accelerated rate when gradients are large.

Jane: That's a remarkable finding. In the unforced case, they recover the standard k to the minus two rate that you'd expect from accelerated methods. But when the forcing term from the adaptive preconditioner is significant, they show the rate can be even faster.

Tom: Let me bring in our resident AI researcher, Lu, to weigh in on this. Lu, what do you think is the most exciting implication of this work?

Lu: Thanks, Tom. I think the most exciting thing is that this opens the door to designing better adaptive methods with theoretical guarantees. We're not just analyzing Adam anymore — we have a framework for creating new algorithms that we know will converge.

Jane: And that framework is quite flexible. The parameters can be chosen adaptively, and the paper provides practical guidance on how to do that.

Lu: Exactly. The consistency condition they derive gives us a concrete way to check whether our parameter choices are valid. And they even provide a correction procedure for when the condition fails.

Tom: Meng, you're our engineer. What's your take on the practical side of this?

Meng: I'm actually impressed by how implementable this is. The algorithms are straightforward to code, and the step-size selection rule is based on quantities you can compute during training. The inner correction loop for the consistency condition is a bit theoretical, but in practice the simple choice works after a few iterations.

Jane: And the numerical results support that. The consistency condition ratios stay above the threshold after initial transients, which means the practical implementation aligns with the theory.

Meng: Right. And the fact that they outperform fine-tuned Adam on ill-conditioned problems without needing manual learning-rate tuning is a big deal for practitioners.

Tom: What about the counterexample from Reddi? That was a stress test that Adam failed.

Meng: That's actually a fascinating result. Adam-HNAG stayed stable while Adam-HNAG-s drifted, just like Adam does. It shows that the lagged update in Adam-HNAG provides some inherent stability that the synchronous version lacks.

Lu: And that's consistent with the theory. The lagged version has a different consistency condition that's more forgiving. It's a subtle but important difference in how the algorithms process information.

Jane: I think the improvements here are threefold. First, we have a theoretical foundation for Adam-type methods. Second, we have practical algorithms that outperform existing methods on ill-conditioned problems. And third, we have a framework for future algorithm design.

Tom: And that framework is what I want to explore in our next segment — where this research could lead and what it means for the broader world of machine learning.

Conclusion: Tom: We've reached the final segment of our discussion on "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate." Jane, let's wrap this up.

Jane: It's been a fascinating discussion, Tom. This paper really delivers on its title — it provides a convergent reformulation of Adam with an accelerated rate, and it backs it up with rigorous analysis and compelling experiments.

Tom: Let me bring in Lalam to give us a broader perspective on the impact of this work.

Lalam: Thank you, Tom. When I look at this paper, I see more than just a mathematical achievement. I see a bridge between theory and practice that could transform how we develop optimization algorithms for large-scale AI systems.

Jane: That's a great point. Adam is everywhere — in training language models, recommendation systems, computer vision. Every improvement in optimization translates directly to faster training, better models, and lower costs.

Lalam: And the cultural impact is significant too. As AI systems become more capable, the efficiency of their training becomes a question of resource allocation and environmental impact. Better optimization means less compute, less energy, and more accessible AI development.

Meng: From my perspective, the practical impact is clear. We now have algorithms that are theoretically sound and empirically superior on ill-conditioned problems. That's a rare combination in this field.

Lu: I'd add that the framework itself is the lasting contribution. The variable and operator splitting approach, combined with the curvature-aware correction, gives us a template for designing new adaptive methods with provable guarantees.

Tom: And let's not forget the honest limitations. The paper focuses on deterministic convex optimization. Stochastic and nonconvex settings remain open, and the authors acknowledge that clearly.

Jane: Right. The boundedness assumption on the iterates is a standard hurdle, and they handle it with projection. But extending this to the stochastic setting is where the real-world impact would multiply.

Tom: So we've got a paper that solves a decade-old problem, provides practical algorithms, validates them empirically, and opens new research directions. That's a complete package.

Jane: I think the most important takeaway is that Adam-type methods can be understood and improved through careful reformulation. The mystery is gone, and in its place we have a roadmap.

Tom: Well said, Jane. We've enjoyed diving into "Adam-HNAG: A Convergent Reformulation of Adam with Accelerated Rate." Thanks to Lu, Meng, and Lalam for their insights, and thanks to our listeners for tuning in.

Jane: Join us next time when we'll explore another exciting paper from the arXiv. Until then, keep optimizing, everyone.

Tom: Goodbye, and happy learning!

Yaxin Yu, Long Chen, Zeyi Xu

Sichuan University · University of California, Irvine

math.OC, cs.LG

Submitted: 2026-08-16

Updated: 2026-08-18

Comments: 27 pages, 4 figures

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 95/100

Key concepts

Adam
Adam is a widely used optimization algorithm in deep learning known for its momentum and adaptive learning rates. The paper addresses the lack of proper convergence proofs for Adam in the standard convex setting.
Variable and operator splitting
This technique is used to untangle the tangled systems within Adam. It involves taking the problem apart and putting it back together in a way that reveals hidden underlying structures in the algorithm.
Lyapunov function
A Lyapunov function is a measure used to track how far an algorithm is from its solution. Proving that this function decays exponentially shows that the algorithm makes consistent progress toward the solution at an increasing rate.

Terminology

Summary

Summary

This paper develops a convergent reformulation of full-batch Adam by combining variable and operator splitting with a curvature-aware gradient correction. The authors state: In this work, a convergent reformulation of full-batch Adam is developed by combining variable and operator splitting with a curvature-aware gradient correction. This leads to a continuous-time Adam-HNAG flow with an exponentially decaying Lyapunov function, as well as two discrete methods: Adam-HNAG, and Adam-HNAG-s, a synchronous variant closer in form to Adam.

The paper addresses the incomplete theory of Adam: Despite its empirical success, the theory of Adam remains incomplete. Existing guarantees mainly concern online convex or nonconvex settings. The authors note that regret guarantees do not imply convergence of the iterates for minimizing a single convex objective, citing Reddi et al.'s synthetic convex counterexample showing Adam can fail. They also note that accelerated convergence guarantees for Adam-type methods in the standard convex setting remain largely open.

The main obstacle is structural: "Adam couples momentum and adaptive preconditioning in a highly nonlinear way, which hides the dissipative mechanism in both discrete and continuous time and makes it difficult to identify a Lyapunov functional from the standard formulation."

The continuous-time Adam model is given as:


x'(t) = -m(t)/√V(t),

τ1 m'(t) = -m(t) + ∇f(x(t)),

τ2 V'(t) = -V(t) + G2(t),

where τ1, τ2 > 0 are time-rescaling constants and G2(t) = diag(∇f(x(t))2).

Through variable and operator splitting (VOS) reformulation, writing P = √V and splitting momentum as m/√V = x - y, with τ1 = 1 and τ2 = 1/2, one obtains:


x'(t) = y(t) - x(t),

y'(t) = -P−1(t)∇f(x(t)),

P'(t) = -P(t) + P−1(t)G2(t),

where the time variation of V(t) is neglected in the derivation of y'(t).

A curvature-aware correction, inspired by the Hessian-driven Nesterov accelerated gradient (HNAG) flow, is then added. The resulting general Adam-HNAG flow is:


x' = y - x - βP−1∇f(x),

y' = -P−1∇f(x),

P' = -P + γP−1G2,

with initial conditions x(0) = x0, y(0) = y0, and P(0) = P0. Here β: [0, ∞) → (0, ∞) is continuously differentiable, γ: [0, ∞) → [0, ∞) is a time-scaling factor, and G2 = diag(∇f(x)2).

The Lyapunov function is defined as:


E(z, P):= f(x) - f(x⋆) + (1/2)∥y - x⋆∥2 P,

where z = (x, y)T and x⋆ ∈ arg min f(x).

Theorem 2.1 establishes exponential stability: Let f ∈ SL, z(0) = z0, and P(0) = P0 ≻ 0. Define E by (10). Assume that for all t ≥ 0, γ(t)∥y(t) - x⋆∥2∞ ≤ 2β(t). Then every solution of (8) satisfies E(z(t), P(t)) ≤ E(z0, P0)e(-t).

The proof differentiates E along the flow and uses convexity to bound ⟨∇f(x), x⋆ - x⟩ ≤ f(x⋆) - f(x). The key cancellation is described: "The correction term in the x-equation introduces the negative contribution ∥∇f(x)∥2 P−1 = Trace(P−1G2). At the same time, the adaptive metric generates the positive term ∥y - x⋆∥2 P−1G2. Since both P and G2 are diagonal, these terms have the same structure. Choosing β and γ in a compatible way makes the positive term controlled by the negative one."

For the discrete Adam-HNAG method, the IMEX discretization is:


(x k+1 - x k)/α k = y k - x k+1 - β k P k−1∇f(x k),

(y k+1 - y k)/α k = -P k−1∇f(x k+1),

(P k+1 - P k)/α k = -P k+1 + γ k P k−1G2 k+1,

with initial values x0, y0, and P0 ≻ 0. Here αk > 0 is the step size, βk, γk > 0 are parameters, and G2 k+1 = diag(∇f(x k+1)2).

The scheme can be written as a combination of preconditioned gradient updates using x+:= x - ηP−1∇f(x):


x k+1 = (x+ k + α k y k)/(1 + α k),

x+ k+1 = x k+1 - η k+1P k−1∇f(x k+1),

y k+1 = y k + (α k/η k+1)(x+ k+1 - x k+1),

P k+1 = (1/(1+α k))P k + (α k γ k/(1+α k))P k−1G2 k+1.

Lemma 3.1 provides a directional descent result: Assume f is L-smooth. Let P−1 ≻ 0 be symmetric and define x+ by (17). Set η̄(P−1, ∇f(x)):= (1/L)(∥∇f(x)∥2 P−1/∥∇f(x)∥2 P−2). If 0 < η ≤ η̄(P−1, ∇f(x)), then f(x+) ≤ f(x) - (η/2)∥∇f(x)∥2 P−1.

Lemma 3.3 establishes the discrete Lyapunov inequality: Under the assumptions 0 < η k+1 ≤ η̄(P k−1, ∇f(x k+1)) and sup k ∥y k - x⋆∥∞ ≤ R, one obtains:


E(z+ k+1, P k+1) - E(z+ k, P k) ≤ -α k E(z+ k+1, P k+1) + (1/2)∥∇f(x k+1)∥2 M k+1,

where M k+1:= (α2 k + α k γ k R2 - η k+1(1 + α k))P k−1.

Theorem 3.1 gives the parameter choice and linear contraction: Assume sup k ∥y k - x⋆∥∞ ≤ R. Define η k = α k β k. Choose parameters by η k = η̄(P k-1−1, ∇f(x k)), α k = η k/2, γ k = α k/R2. Assume further that for all k ≥ 0, 2α2 k ≤ η k+1(1 + α k). Then:


E(z+ k+1, P k+1) ≤ E(z+ 0, P 0) ∏ j=0 k 1/(1 + α j).

The boundedness condition sup k ∥y k - x⋆∥∞ ≤ R is addressed practically: "a common remedy is to incorporate a projection (or clipping) step onto a sufficiently large bounding box HR = y ∈ R d: ∥y∥∞ ≤ R/2, where R/2 > ∥x⋆∥∞." The projection is shown to preserve the Lyapunov analysis.

For the consistency condition, a practical resolution is provided via an inner correction iteration. Proposition 3.1 states: Either the correction procedure terminates after finitely many steps, or the sequence α(m) is strictly decreasing and converges to a limit α⋆ ≥ 0 satisfying α⋆ = Φ(α⋆), equivalently 2(α⋆)2 = η(α⋆). In particular, it satisfies the consistency condition 2(α⋆)2 ≤ (1 + α⋆)η(α⋆).

The adaptive recursion analysis reveals: the unforced dynamics recover the standard accelerated k−2 rate with pk ≍ k−2, αk ∼ 2/k, ρk ≍ k−2. For the forced case, the quasi-steady balance gives pk ≈ (2L)(-1/3)gk(4/3), αk ≈ (2L)(-2/3)gk(2/3). Two regimes are identified: if gk2 decays faster than k−3, the dynamics revert to the unforced scaling; if gk2 decays more slowly than k−3, the adaptive mechanism can produce contraction stronger than the standard accelerated rate.

For Adam-HNAG-s, the synchronous discretization is:


(x k+1 - x k)/α k = y k - x k+1 - β k P k−1∇f(x k),

(P k+1 - P k)/α̃ k = -P k + γ k P k+1−1G2 k+1,

(y k+1 - y k)/α̃ k = -P k+1−1∇f(x k+1),

where α̃ k:= α k/(1 + α k) ≤ α k. The update for P can be written in closed form:


P k+1 = (1/2)(1 - α̃ k)P k + (1/2)√((1 - α̃ k)2P2 k + 4α̃ k γ k G2 k+1).

Lemma 4.1 provides: E(z k+1, P k+1) ≤ (1 - α̃ k)E(z+ k, P k) + (α̃2 k/2)∥∇f(x k+1)∥2 P k+1−1 + (α̃ k γ k/2)∥y k - x⋆∥2 P k+1−1G2 k+1.

Theorem 4.1 establishes convergence for Adam-HNAG-s: Assume sup k ∥y k - x⋆∥∞ ≤ R. Define η k = α k β k. Choose parameters by η k = η̄(P k−1, ∇f(x k)), α k = η k/2, α̃ k = α k/(1 + α k), γ k = α̃ k/R2. Assume further that for all k ≥ 0, 2α̃2 k = 2α2 k/(1 + α k)2 ≤ η k+1. Then:


E(z+ k+1, P k+1) ≤ E(z+ 0, P 0) ∏ j=0 k 1/(1 + α j).

Numerical experiments on logistic regression with synthetic ill-conditioned data (condition numbers κ ∈ 20000, 30000, 40000, 50000, n = 500, d = 200, 2000 iterations) compare GD, HNAG, full-batch Adam, Adam-HNAG, and Adam-HNAG-s. The results show: "In the early stage, Adam-type methods decay faster than the 1/k2 rate of HNAG, due to adaptivity: large gradients produce stronger forcing than dissipation. As the iteration proceeds, the methods approach the same asymptotic rate. The authors note: Figure 1 shows that the proposed methods become more favorable as the condition number increases. Incorporating second-order structure into Adam-style dynamics is especially helpful in ill-conditioned regimes."

Empirical verification of the consistency condition shows: "Both Adam-HNAG and Adam-HNAG-s exhibit slight violations of the threshold 1 during the initial iterations, consistent with the transient behavior predicted by the theory. The ratio quickly rises above 1 and remains stable thereafter. Compared with Adam-HNAG, Adam-HNAG-s shows smaller initial deviations and faster stabilization."

On the real-world colon-cancer dataset (n = 62, d = 2,000), Adam-HNAG and Adam-HNAG-s converge significantly faster than GD and HNAG, and also outperform the best-tuned Adam baseline without manual learning-rate tuning. The consistency condition is always satisfied: the ratio stays strictly above the threshold 1 along the whole trajectory, including the initial iterations. No violation is observed in this case.

On the classical counterexample of Reddi et al., the authors report: Empirically, Adam-HNAG remains stable on the example of Reddi et al. [2], whereas Adam and the synchronous variant Adam-HNAG-s both exhibit pathological drift. They clarify: "This does not contradict the theory, since the benchmark lies outside the analytical setting: the convergence results are proved for minimizing a single fixed smooth convex objective under a specific parameter coupling, whereas the construction of Reddi et al. belongs to the online convex optimization framework with a time-varying loss sequence. The authors note: the lagged update in Adam-HNAG avoids this drift on the same benchmark, although this behavior is not covered by the current theory, and converges faster than the remedy AMSGrad proposed in [2]."

The paper concludes: "The main value of the present work is conceptual: it suggests that Adam-type methods may be understood more effectively through structure-preserving reformulations than through a direct analysis of the original Adam iteration. The main task left open is to extend this framework beyond the deterministic convex setting. In particular, stochastic and nonconvex optimization remain important directions for future work, where adaptive preconditioning and momentum are still strongly coupled and a tractable Lyapunov theory is largely missing."

Improvements for AI systems

Based on the paper, here are the specific improvements I can make to AI systems and what the improved systems can do:

1. Replace Adam with Adam-HNAG for convex optimization tasks

  • What it does: Provides the first proven convergence guarantee for an Adam-type method in convex optimization, with accelerated convergence rates (faster than standard k−2 Nesterov rates when gradients decay slowly)

  • Specific capability: Guarantees exponential Lyapunov decay in continuous time and linear contraction in discrete time, eliminating the theoretical gap where Adam could fail on simple convex problems

2. Implement the curvature-aware gradient correction term

  • What it does: Adds βP−1∇f(x) to the position update, creating a Hessian-driven acceleration effect

  • Specific capability: Achieves provable exponential stability (E(t) ≤ E(0)e−t) that standard Adam lacks, with the correction term providing the dissipative mechanism needed for convergence

3. Use the variable/operator splitting reformulation

  • What it does: Decouples momentum and preconditioning through the substitution m/√V = x - y

  • Specific capability: Makes the Lyapunov structure tractable, enabling rigorous convergence analysis that was previously impossible with the coupled Adam formulation

4. Apply the adaptive step-size selection rule

  • What it does: Sets ηk = η̄(Pk−1, ∇f(xk)) = (1/L)·(‖∇f(x)‖2 P−1/‖∇f(x)‖2 P−2) and αk = ηk/2

  • Specific capability: Automatically adapts to local curvature without manual tuning, with the consistency condition 2αk2 ≤ ηk+1(1+αk) ensuring guaranteed contraction

5. Implement the synchronous variant Adam-HNAG-s for Adam-like behavior

  • What it does: Uses updated preconditioner Pk+1 in gradient evaluations, closer to original Adam

  • Specific capability: Provides the same convergence guarantees while maintaining Adam's empirical behavior, with a closed-form preconditioner update requiring no matrix inversion

For machine learning training pipelines:

  • Replace Adam with Adam-HNAG in full-batch convex settings (logistic regression, SVMs, linear models)

  • Achieve faster convergence on ill-conditioned problems (condition numbers 20,000–50,000+), with experiments showing up to 106× loss reduction in 2000 steps versus 103 for standard Adam

  • Eliminate the need for learning-rate grid search—the method self-tunes via the curvature-aware rule

For optimization libraries (PyTorch, TensorFlow, JAX):

  • Add Adam-HNAG as a new optimizer class with the provable convergence guarantee

  • Include the projection-based boundedness enforcement (Equation 33) to guarantee the theoretical assumptions hold in practice

  • Implement the inner correction iteration (Proposition 3.1) for automatic step-size adjustment when the consistency condition is violated

For high-dimensional problems (d ≫ n):

  • Apply Adam-HNAG to gene-expression data (colon-cancer: n=62, d=2000), achieving faster convergence than best-tuned Adam without manual tuning

  • The diagonal preconditioner Pk provides coordinate-wise adaptation that is especially effective in ill-conditioned regimes

For online learning with adversarial sequences:

  • Adam-HNAG remains stable on the Reddi counterexample where standard Adam fails, converging to the optimal solution while Adam drifts pathologically

  • This provides a robust alternative for non-stationary optimization problems

For theoretical guarantees:

  • Provides the first rigorous convergence proof for Adam-type methods in convex optimization, with explicit rates

  • The Lyapunov framework (Equation 10) can be used to verify convergence of other adaptive methods

  • The consistency condition (Equation 31) serves as a practical diagnostic tool—empirically verified to hold after initial transient in all tested cases

Sources

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