Balanced Adaptive Optimizers

• Balanced adaptive optimizers are machine learning methods that interpolate between fully adaptive (Adam, RMSProp) and non-adaptive (SGD) update rules using balance parameters.
• They incorporate mechanisms like adaptive friction, Bayesian expansion, and dynamic lookahead to enhance stability, generalization, and fairness.
• Empirical and theoretical analyses show these optimizers improve convergence, reduce group disparity, and offer adjustable trade-offs between fitting speed and robustness.

Balanced adaptive optimizers comprise a class of machine learning optimization algorithms characterized by parameter update rules that combine coordinate-wise adaptivity (e.g., per-feature gradient normalization) with mechanisms for controlling fairness, stability, or generalization via explicit balance parameters or adaptive friction coefficients. These optimizers interpolate between fully adaptive schemes (such as Adam or RMSProp) and non-adaptive approaches (such as stochastic gradient descent, SGD), often providing additional levers to control fitting, generalization, convergence rate, and fairness across groups—especially under distributional or dataset imbalance (Kolahdouzi et al., 21 Apr 2025, Zhang et al., 2024, Zhang et al., 25 Nov 2025, Zheng et al., 2024, Chen et al., 2023, Chen et al., 2020).

1. Theoretical Foundations and Generalization of Update Rules

Balanced adaptive optimizers broadly rest on generalizing first-order update rules. The central formula unifies SGD, RMSProp, and Adam through the introduction of a balance exponent β, as in the Second-Moment Exponential Scaling (SMES) framework (Zhang et al., 2024):

$$\theta_{t+1} = \theta_t - \eta \frac{m_t}{(v_t)^{\beta}+\varepsilon}$$

• $m_t = \beta_1 m_{t-1} + (1-\beta_1)g_t$: momentum estimate.
• $v_t = \beta_2 v_{t-1} + (1-\beta_2)g_t^2$: second-moment estimate.
• β: scalar balance coefficient; β=0 recovers SGD, β=1 recovers RMSProp, β=1/2 yields Adam-style updates.

In HVAdam, the adaptivity parameter α explicitly interpolates between scalar and diagonal preconditioning (Zhang et al., 25 Nov 2025):

$$H_t(\alpha) = (1-\alpha)\bar s_t I + \alpha\mathrm{diag}(\hat{s}_t+\varepsilon)$$

with special cases α=0 (SGD-like), α=1 (Adam-like), and α∈(0,1) allowing fine-grained adaptation.

Theoretical analyses show that tuning β or α modifies the optimizer’s bias toward fitting versus generalization and regulates gradient flow across layers, countering vanishing or exploding gradients and mitigating overfitting on dense or sparse tasks (Zhang et al., 2024, Zhang et al., 25 Nov 2025).

2. Adaptive Mechanisms: Friction, Expansion, and Momentum

Balanced adaptive optimizers incorporate several explicit mechanisms to enhance adaptivity while maintaining convergence stability:

• Adaptive Friction: Introducing dynamic friction coefficients (S) based on recent gradient history, such as sigSignGrad and tanhSignGrad, modulates each step’s magnitude to smooth trajectories and control oscillations (Zheng et al., 2024):
  • Sₜ = Sigmoid(gₜ₋₁·gₜ), Sₜ ∈ (0,1): amplifies steps when consecutive gradients align.
  • Sₜ = Tanh(gₜ₋₁·gₜ) + 1, Sₜ ∈ (0,2): stronger self-tuning over update range.
• Bayesian Expansion: In global optimization tasks, AEBO adaptively balances exploration and exploitation by expanding the feasible search region (in function space) only when local improvement diminishes, using a GP-based uncertainty threshold τ (Chen et al., 2020):
  • The τ threshold is dynamically set by equating expected improvement at the boundary with local expected improvement.
• Bidirectional Moving Average and Dynamic Lookahead: ADmeta blends double exponential moving average (DEMA) for backward smoothing of gradients and a dynamic, decaying lookahead parameter ηₜ that transitions from aggressive exploration to stably anchored convergence (Chen et al., 2023).

3. Fairness, Imbalance, and Parameter Update Regularization

Optimizers with balanced adaptivity explicitly address group fairness and data imbalance. Stochastic differential equation (SDE) analysis demonstrates that coordinate-wise normalization in RMSProp and Adam can shrink per-step disparity between subgroup gradients, thus promoting fairer outcomes compared to SGD (Kolahdouzi et al., 21 Apr 2025):

• Under class imbalance, adaptive methods (RMSProp, Adam) converge with higher probability to fairer minima, especially when subgroup representation ratios (p₀, p₁) deviate significantly.
• Theoretical bounds show that the maximum increase in demographic parity gap 𝔽(w) per RMSProp update is strictly smaller than for SGD, due to adaptive scaling suppressing dominance by over-represented groups (Theorem 3 in (Kolahdouzi et al., 21 Apr 2025)).

These optimizers maintain predictive accuracy while reducing fairness violations on vision classification tasks across datasets and architectures (e.g., CelebA, FairFace, MS-COCO; ResNet, TinyViT) (Kolahdouzi et al., 21 Apr 2025).

4. Empirical Performance and Benchmarks

Experiments consistently demonstrate improved balance, convergence, and generalization:

• Accuracy vs Fairness (CelebA/TinyViT):

| Optimizer | Accuracy % | Fₑₒ𝒹 | Fₑₒ𝓅 | F_{DPA} | |-----------|------------|-------|-------|---------| | SGD | 91.23 | 0.85 | 0.82 | 0.74 | | RMSProp | 91.54 | 0.92 | 0.90 | 0.88 | | Adam | 92.08 | 0.94 | 0.91 | 0.90 |

Adaptive variants outperform SGD by up to 10 percentage points in fairness metrics (equalized odds, opportunity, demographic parity) while matching or exceeding accuracy (Kolahdouzi et al., 21 Apr 2025).

• SMES on CIFAR-10/100 (Zhang et al., 2024): Negative β (anti-adaptive) yields smoothed loss curves and faster early convergence; β<0 leads to lower test error and less sensitivity to sharp minima.
• sigSignGrad/tanhSignGrad (Zheng et al., 2024): These friction-augmented optimizers improved ResNet50 and ViT test accuracy by +0.2–2.4% on CIFAR-10/100 and Mini-ImageNet, compared to diffGrad and AngularGrad.
• HVAdam (Zhang et al., 25 Nov 2025): Across deep learning tasks (image, language, diffusion, GANs), HVAdam (with tunable α and hidden-vector machinery) achieves the best or highly competitive performance with enhanced stability.
• ADmeta (Chen et al., 2023): Bidirectional averaging and dynamic lookahead yield consistent empirical gains across image, NLP, and speech benchmarks with provably optimal rates.
• AEBO (Chen et al., 2020): Adaptive expansion achieves state-of-the-art performance on synthetic and real-world black-box functions, outperforming fixed-bound Bayesian optimization and maintaining robustness to noise.

5. Convergence and Stability Guarantees

Balanced adaptive optimizers are supported by theoretical convergence results in both convex and nonconvex settings:

• HVAdam: Regret for convex functions is $O(\sqrt{T}\sum_i\sqrt{s_{T,i}} + O(\ln T))$; nonconvex rate is $O(\log(T)/\sqrt{T})$ (Zhang et al., 25 Nov 2025).
• ADmeta: Achieves $O(\sqrt{T})$ regret in convex problems and matches established rates in nonconvex problems (Chen et al., 2023).
• AEBO: Expansion strategy provably avoids pathological acquisition-to-infinity and guarantees monotonic coverage of the space (Chen et al., 2020).
• sigSignGrad/tanhSignGrad: Empirical evidence and trajectory analysis support reduced oscillations and smoother convergence, though formal regret bounds remain the same as Adam (Zheng et al., 2024).
• SMES: The balance parameter β directly affects stability; β<0 smooths validation loss and mitigates overfitting (Zhang et al., 2024).

6. Practical Guidelines and Hyperparameter Tuning

The deployment of balanced adaptive optimizers involves careful selection of balance coefficients or adaptivity parameters:

• Tune β in SMES: Start with β=0; set β>0 for fast fitting, β<0 for generalization and smooth convergence (Zhang et al., 2024).
• Set HVAdam’s α∈[0.8,1.2] for intermediate adaptivity; adjust hidden-vector learning rates and IDU restart-thresholds to trade off stability and responsiveness (Zhang et al., 25 Nov 2025).
• For RMSProp and Adam in fairness-sensitive applications, use high decay γ (≈0.9–0.99) and small ε for variance stability (Kolahdouzi et al., 21 Apr 2025).
• Leverage adaptive friction coefficients in sigSignGrad/tanhSignGrad for plug-in enhancement of Adam-family optimizers; negligible overhead (Zheng et al., 2024).
• In AEBO, dynamically solve for τ at each iteration; initialize with a conservative region and anneal exploration-exploitation (Chen et al., 2020).
• Combine adaptive optimizers with group reweighting or augmentation for severe imbalance (Kolahdouzi et al., 21 Apr 2025).

7. Current Limitations and Future Directions

Limitations include diminishing fairness improvements in mildly imbalanced settings, possible adverse effects on adversarial robustness, and increased memory/computational overhead for full-dimension adaptive schemes (e.g., HVAdam’s hidden-vector tracking) (Kolahdouzi et al., 21 Apr 2025, Zhang et al., 25 Nov 2025). Future work is proposed in optimal balance parameter selection under strongly convex regimes, efficient scaling in distributed or federated settings, and theoretical characterization in more complex or highly nonconvex landscapes (Zhang et al., 25 Nov 2025, Zhang et al., 2024).

Balanced adaptive optimizers represent a principled advance in optimizer design, offering unified frameworks that interpolate between classical and modern update regimes, control over fit versus generalization, and robust mechanisms for mitigating group disparity or instability in deep learning applications (Kolahdouzi et al., 21 Apr 2025, Zhang et al., 2024, Zhang et al., 25 Nov 2025, Chen et al., 2020, Chen et al., 2023, Zheng et al., 2024).