Adam Challenge in Deep Learning

• The Adam Challenge is defined as the tension between fast adaptive optimization and generalization, where models often converge to sharp minima with suboptimal performance.
• Proposed remedies like InvAdam and DualAdam modify moment estimation and adaptive scaling to enable escape from high-curvature regions while retaining convergence.
• Experimental evaluations on benchmarks like CIFAR and ImageNet demonstrate that DualAdam achieves flatter minima and improved generalization compared to vanilla Adam.

Searching arXiv for the primary paper and closely related optimizer work on the "Adam Challenge". The Adam Challenge denotes a persistent problem in deep learning optimization: the empirical observation that Adam often converges rapidly yet tends to produce suboptimal generalization, commonly attributed to convergence toward sharp minima rather than flat minima (Shi et al., 7 Mar 2026). In the optimizer literature, the term is used to summarize a broader tension between fast adaptive optimization and the ability to reach solutions that transfer well to held-out data. One recent formulation makes this explicit by proposing inverse Adam (InvAdam) and DualAdam as direct responses to Adam’s tendency to shrink steps in high-curvature regions, thereby becoming trapped in sharp basins (Shi et al., 7 Mar 2026). Related work situates the same challenge in terms of direction distortion, ill-conditioned effective learning rates, deficient long-term memory, or limitations of coordinate-wise adaptivity (Zhang et al., 2017, Li et al., 2020, Wang et al., 2021).

1. Definition of the challenge

The Adam Challenge arises from a recurring empirical pattern: Adam’s popularity stems from its ability to converge very rapidly in practice, but models trained with Adam often settle into “sharp” minima, which are associated with poorer generalization than flatter alternatives (Shi et al., 7 Mar 2026). In this framing, the central issue is not optimization speed in isolation, but the mismatch between fast loss reduction and the geometry of the final solution.

A standard explanation is that Adam’s adaptive step-size decreases when the second-moment estimate is large. Near sharp valleys, where gradients fluctuate rapidly, this causes very small steps, making escape difficult; by contrast, standard (non-adaptive) methods such as SGD often take larger steps out of those regions and land in flatter basins (Shi et al., 7 Mar 2026). This diagnosis is consistent with earlier work arguing that Adam often generalizes worse than SGD, especially in deep neural networks (Zhang et al., 2017).

The broader literature decomposes the challenge into several related mechanisms. One line attributes the problem to a direction-missing problem, in which Adam’s per-coordinate scaling rotates the update away from the raw gradient direction, and an ill-conditioning problem, in which the effective learning rate depends on weight norms in scale-invariant architectures (Zhang et al., 2017). Another line argues that Adam’s short-term exponential averaging of squared gradients can cause premature convergence to local minima when true gradient magnitudes decay during training (Li et al., 2020). A further perspective holds that Adam’s diagonal preconditioner cannot capture off-axis structure in rotated or valley-shaped landscapes, thereby missing important cross-coordinate information (Zhang et al., 25 Nov 2025).

2. Adam, InvAdam, and DualAdam

The formulation centered on the Adam Challenge in "Combining Adam and its Inverse Counterpart to Enhance Generalization of Deep Learning Optimizers" defines Adam, InvAdam, and DualAdam in a common moment-estimation framework (Shi et al., 7 Mar 2026). Let the stochastic gradient at iteration $t$ and coordinate $i$ be

$$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$

Adam maintains exponential moving averages of the gradient and its square:

$$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$

With bias correction,

$$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$

The standard Adam update is

$$u^{\rm Adam}_{t,i} \;=\; \frac{\hat m_{t,i}} {\sqrt{\hat v_{t,i}} + \epsilon}, \qquad \theta_{t+1,i} = \theta_{t,i} - \eta_t\,u^{\rm Adam}_{t,i}.$$

The proposed InvAdam reverses the adaptive relation between first and second moments by replacing element-wise division with element-wise multiplication:

$$\tilde u^{\rm Inv}_{t,i} \;=\; \hat m_{t,i}\,\sqrt{\hat v_{t,i}}, \qquad \theta_{t+1,i} = \theta_{t,i} - \eta_t\,\tilde u^{\rm Inv}_{t,i}.$$

Under this rule, when $\hat v_{t,i}$ is large, as near a sharp minimum, the step grows rather than shrinks, which is intended to help the optimizer escape (Shi et al., 7 Mar 2026). The paper emphasizes that this benefit comes at a cost: InvAdam’s update mechanism may face challenges in convergence (Shi et al., 7 Mar 2026).

To reconcile early escape behavior with eventual convergence, DualAdam uses a linear blend of Adam and InvAdam. With switching rate $\xi>0$,

$\alpha_t = \max\!\bigl(0,\;1 - \xi\;t\bigr),$

$i$0

$i$1

Initially, $i$2, so DualAdam behaves like InvAdam; later, $i$3, so it becomes standard Adam (Shi et al., 7 Mar 2026). This schedule is presented as the mechanism that ensures convergence while enhancing generalization performance (Shi et al., 7 Mar 2026).

3. Geometric and diffusion-theoretic interpretations

The sharp-minimum interpretation of the Adam Challenge is made mathematically explicit in the diffusion-theoretic analysis accompanying InvAdam and DualAdam (Shi et al., 7 Mar 2026). Under a diffusion approximation, SGD-like dynamics are modeled by the Langevin-type stochastic differential equation

$i$4

where $i$5 is the gradient-noise covariance and $i$6 is standard Brownian motion (Shi et al., 7 Mar 2026). Around a sharp local minimum $i$7, the loss is approximated quadratically:

$i$8

For a saddle $i$9 with height $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$0, the mean escape time for Adam-scale dynamics admits the asymptotic form

$$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$1

where $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$2 is the Hessian eigenvalue at $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$3 along escape direction $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$4 (Shi et al., 7 Mar 2026). For the InvAdam-style mechanism, the effective diffusion term scales differently, yielding

$$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$5

The resulting comparison,

$$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$6

implies faster escape from sharper minima under InvAdam (Shi et al., 7 Mar 2026). In this account, the Adam Challenge is not merely an empirical nuisance but a consequence of how the second-moment denominator suppresses motion precisely in high-curvature directions.

This interpretation aligns with other theoretical or semi-theoretical accounts. AdaMomentum argues that substituting the raw gradient in the second raw moment by the momentumized gradient yields smoother, more favorable learning-rate scaling and longer expected residence time in wide basins (Wang et al., 2021). AdaX attributes Adam’s failure mode to collapse of the denominator when gradients decay geometrically, causing persistent effective steps that overshoot the optimum (Li et al., 2020). ND-Adam instead emphasizes that per-coordinate scaling distorts directions and interacts poorly with scale invariances in deep networks (Zhang et al., 2017). These formulations differ in mechanism, but they converge on a common theme: Adam’s update geometry can be misaligned with the geometry of generalizable solutions.

4. Experimental evidence associated with the challenge

The DualAdam study evaluates the Adam Challenge on both image classification and large-language-model fine-tuning (Shi et al., 7 Mar 2026). The image-classification benchmarks include CIFAR-10, CIFAR-100, Tiny ImageNet, ImageNet-1k with architectures such as ResNet-18, ResNet-50, VGG-16, ViT-Small; the language-model setting uses OpenPangu-Embedded-1B on Alpaca-GPT4-CN (Shi et al., 7 Mar 2026). Metrics include test accuracy (top-1) for vision, Hessian flatness via eigenvalue density, maximum eigenvalue, trace, and for LLM fine-tuning validation perplexity (PPL) and generalization gap (PPL minus training loss) (Shi et al., 7 Mar 2026).

The reported quantitative results are representative of the stated challenge. On CIFAR-100 / ResNet-18 (200 epochs, batch 128), Adam: 72.56 ± 0.08 % and DualAdam: 75.29 ± 0.21 % (Shi et al., 7 Mar 2026). On ImageNet-1k / ResNet-18, Adam: 69.55 % and DualAdam: 72.61 % (Shi et al., 7 Mar 2026). In OpenPangu-1B fine-tuning, DualAdam achieves lower and more stable validation PPL, closing the generalization gap, whereas AdamW’s PPL rises (overfitting) (Shi et al., 7 Mar 2026).

The flatness analyses are integral to the paper’s claim. It reports that Hessian spectra for DualAdam solutions are more concentrated near zero, with smaller maximum eigenvalue and trace, and that 1D loss-landscape visualizations show a visibly flatter basin around the DualAdam optimum (Shi et al., 7 Mar 2026). These observations support the sharp-versus-flat minimum interpretation rather than treating generalization purely as an optimization-speed phenomenon.

Comparable empirical patterns appear in related proposals that target the same optimizer pathology. ND-Adam reports that on CIFAR-10 and CIFAR-100 with Wide ResNet 22-7.5, vanilla Adam underperforms both SGD and ND-Adam, while ND-Adam closes or removes the generalization gap (Zhang et al., 2017). AdaX-W matches SGDM on CIFAR-10 and ImageNet while outperforming AdamW and AMSGrad (Li et al., 2020). AdaMomentum reports that it trains as fast as Adam, approaches or exceeds SGD on vision, and improves performance on language modeling, translation, and GAN training (Wang et al., 2021). Taken together, these results reinforce the view that the Adam Challenge is broad rather than specific to one benchmark family.

5. Alternative diagnoses and proposed remedies

A substantial literature treats the Adam Challenge as a family of related deficiencies rather than a single defect. The main proposals differ in where they intervene in Adam’s update rule.

ND-Adam addresses two identified causes of poor generalization: the direction-missing problem and the ill-conditioning problem (Zhang et al., 2017). It separates parameters into per-unit weight vectors and scalars, projects gradients onto the sphere tangent to each unit-norm weight vector, uses vector-wise moments, and renormalizes each vector after update (Zhang et al., 2017). This restores an SGD-like directional constraint while keeping adaptive normalization. The same work further proposes Batch-normalized softmax (BN-Softmax) and $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$7 regularization on logits to stabilize the scale of the classifier head (Zhang et al., 2017).

AdaX modifies the second-moment memory itself. Instead of Adam’s decaying-memory rule, it uses

$$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$8

with bias-corrected $$g_{t,i} \;=\;\frac{\partial L(\theta_{t})}{\partial \theta_{t,i}}.$$9 (Li et al., 2020). The stated purpose is to accumulate exponential long-term memory so that the denominator does not collapse when gradients decrease (Li et al., 2020). This change is presented as preventing Adam’s tendency to be driven into suboptimal local minima by overly aggressive normalization late in training.

AdaMomentum replaces $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$0 with $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$1 in the second raw moment estimate:

$$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$2

in its bias-uncorrected form (Wang et al., 2021). The rationale is that momentumized gradients contain more accurate directional information than raw gradients, so their second moment should be a more favorable basis for learning-rate scaling (Wang et al., 2021).

More recent proposals extend the challenge to settings beyond classical CNN training. Coupled Adam argues that Adam’s second moment is a cause of anisotropic embeddings in LLMs because embedding coordinates associated with frequent tokens acquire larger second moments and are updated more slowly (Stollenwerk et al., 12 Feb 2025). It replaces each individual embedding-layer $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$3 with the vocabulary-mean second moment $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$4, thereby restoring a shared denominator and the vanishing-sum property of embedding updates (Stollenwerk et al., 12 Feb 2025). AlphaAdam targets large-language-model efficiency and stability through intra-layer asynchronous masked adaptive updates, constructing binary masks from momentum–gradient sign consistency and compensating the masked update norm with a dynamic $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$5 (Chang et al., 30 Jan 2025). HVAdam introduces a full-dimension hidden vector to recover cross-coordinate information that diagonal preconditioners cannot represent (Zhang et al., 25 Nov 2025).

A plausible implication is that the Adam Challenge is best viewed as a design-space problem concerning how adaptivity should be structured, rather than a binary question of whether adaptivity itself is beneficial.

6. Convergence, divergence, and the status of vanilla Adam

The Adam Challenge has often been conflated with a separate but related issue: whether vanilla Adam converges without modification. The contemporary literature distinguishes the two. Poor generalization and sharp-minimum convergence do not by themselves imply divergence, and divergence examples do not exhaust Adam’s practical behavior.

A modern theoretical result establishes that Adam converges with proper problem-dependent hyperparameters and identifies a phase transition in the $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$6-plane (Zhang et al., 2 Mar 2026). In that analysis, for sufficiently large $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$7 and $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$8, Adam provably converges, whereas for small $$m_{t,i} \;=\;\beta_1\,m_{t-1,i} + (1-\beta_1)\,g_{t,i},\quad v_{t,i} \;=\;\beta_2\,v_{t-1,i} + (1-\beta_2)\,g_{t,i}^2.$$9 there is a region where it can diverge to infinity (Zhang et al., 2 Mar 2026). The paper emphasizes that the critical boundary is problem-dependent and in particular dependent on batch size, suggesting that when Adam performs poorly one should tune up $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$0 inversely with batch size and then try $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$1 (Zhang et al., 2 Mar 2026).

Another convergence analysis proves that under coordinate-wise generalized smoothness and affine noise variance, Adam reaches an $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$2-stationary point with iteration complexity $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$3 when hyperparameters are chosen appropriately (Zhang et al., 2024). This result is notable because it gives a tight rate under relaxed assumptions and addresses the mismatch between the gradient and first-order momentum through a refined descent analysis (Zhang et al., 2024).

These works complicate a common misconception that the Adam Challenge is simply “Adam does not converge.” A more accurate statement is that the challenge concerns a combination of generalization behavior, landscape geometry, and hyperparameter-sensitive convergence properties. Some proposals, such as DualAdam, seek to improve generalization while retaining Adam-like convergence behavior (Shi et al., 7 Mar 2026). Others, such as AdaX, WADA, or AMSGrad-like approaches, are motivated primarily by stability and regret guarantees (Li et al., 2020, Zhong et al., 2021).

7. Trade-offs, practical significance, and open questions

The Adam Challenge remains active because no single remedy resolves all observed deficiencies without introducing new trade-offs. In the DualAdam formulation, InvAdam trades robust convergence for stronger escape from sharp minima, and in isolation, InvAdam can oscillate and even diverge in late training (Shi et al., 7 Mar 2026). DualAdam’s linear schedule, controlled by $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$4, therefore becomes central: too large → premature switching, too small → late convergence (Shi et al., 7 Mar 2026). The computational overhead is reported as roughly +4 FLOPs per parameter per iteration, and only during the early phase until $$\hat m_{t,i} = \frac{m_{t,i}}{1-\beta_1^t},\qquad \hat v_{t,i} = \frac{v_{t,i}}{1-\beta_2^t}.$$5 (Shi et al., 7 Mar 2026).

Several open questions are explicitly identified. For DualAdam, these include whether linear blending is optimal, whether adaptive/blended schemes based on sharpness estimates would improve performance, how the method interacts with weight decay, batch-norm, or other regularizers, and whether similar inverse ideas can be applied to other adaptive optimizers (e.g. Adagrad, Adan) or combined with SAM-style sharpness regularization (Shi et al., 7 Mar 2026). Related proposals raise their own extensions: AdaX asks about alternative memory kernels (Li et al., 2020); AlphaAdam highlights the need for stochastic-gradient theory beyond the current full-batch treatment (Chang et al., 30 Jan 2025); HVAdam suggests that full-dimension adaptive structure may bridge the gap between SGD-like and Adam-like behavior (Zhang et al., 25 Nov 2025).

In research practice, the Adam Challenge has become a unifying label for a recurring observation: adaptive normalization that accelerates optimization can also bias the optimizer toward less generalizable regions of the loss landscape (Shi et al., 7 Mar 2026). The main significance of the recent literature is not the replacement of Adam by a single superior method, but the clarification that optimizer behavior depends on the interaction among moment estimation, curvature-sensitive scaling, direction preservation, memory depth, and problem-dependent hyperparameters. This suggests that future work is likely to focus less on universal defaults and more on adaptive mechanisms whose geometry is explicitly matched to architecture, data regime, and training stage.