Across implicit-neural-representation (INR) architectures and analytic benchmarks we observe that a thoroughly tuned Adam (especially its learning rate (lr), e.g. in a hyperparameter sweep from $lr = 0.05$ to $10^{-8}$) can potentially reach a very low loss even on ill-conditioned loss landscape or converge at a plateau far above the loss attained by second-order methods. This report defines the measured metrics that help determine if Adam can mitigate the ill-conditioning on a given loss landscape. We provide the indicators by which each outcome is determined, that are: the condition number of the Hessian and of the Adam-preconditioned Hessian $D^{-1/2}HD^{-1/2}$ (with the derivation from Adam's update rule), the diagonal mass $ρ$ that distinguishes axis-aligned from cross-coupled ill-conditioning, the negative spectral mass estimated by stochastic Lanczos quadrature, and the gradient energy fractions over curvature bands, including the flat fraction that indicates the Adam stall. A worked out $2\times 2$ example and an illustration show the reasons why a diagonal preconditioning by Adam can remove axis-aligned ill-conditioning by rescaling and why it cannot do the same if the ill-conditioning is cross coupled. In addition, we present a case study of FINER image fitting architecture that goes over the whole loss landscape analysis framework: the fitting architecture description, reasons due to which its landscape stalls Adam at saddles, the measured PSNR values through our tuned baselines to the $120$--$134$\,dB results of the blockwise second order methods, the error maps behind those numbers, and description of the benefits such image fitting accuracy gives in practice.
Haichen Hu, David Simchi-Levics.LG math.OC stat.ML
We study whether stochastic nonconvex optimization can be reduced to ordinary static regret minimization in online convex optimization in a black-box manner. For smooth nonconvex objectives, our reduction maintains a predictable gradient tracker, while a black-box online learner selects a preconditioner that determines how this tracker is transformed into the update direction. The learner receives linear convex losses and is evaluated against a single fixed comparator over one undiscounted online game. For a $β$-smooth objective with range bounded by $M$ and an unbiased stochastic-gradient oracle with variance bounded by \(σ^2\), we establish $$\frac{1}{T}\sum_{t=1}^T \mathbb E\!\left[\|\nabla f(x_t)\|_2^2\right] \lesssim \frac{σ\sqrt{Mβ}}{\sqrt T} + \frac{\sqrt{Mβ}\, \mathscr R_T(\mathcal A,I_d)}{T} + \frac{Mβ}{T}.$$ Consequently, any black-box OCO algorithm with $\mathscr R_T(\mathcal A,I_d)=O(\sqrt T)$ recovers the classical $O(\frac{1}{\sqrt{T}})$ convergence rate. We further show that the same black-box framework extends beyond the smooth setting to Lipschitz nonconvex objectives without Lipschitz continuous gradients. Importantly, this extension continues to rely only on an ordinary static-regret guarantee and requires no stronger notion of online regret. When the OCO oracle admits square-root static regret, the resulting conversion achieves the optimal $O(T^{-2/7})$ convergence rate for the corresponding Goldstein stationary point. These results resolve the open problem posed by Chen and Hazan (2024). More broadly, our framework separates optimizer design into gradient prediction and online preconditioner selection, providing a principled perspective on how adaptive optimization methods may be understood through static regret and applied in nonconvex optimization.
Optimization geometrodynamics views optimizer state as evolving geometry. Its full positive-definite quadratic benchmark gives the least affine-invariant deformation needed to reduce condition number when arbitrary metrics are allowed. This paper records that benchmark in the present notation and develops restricted dynamic geometric complexity: an intrinsic certificate distance for reaching a target condition-number class when the metric is restricted to a specified family. The main proved results are monotonicity and submanifold-distance principles, diagonal and block reachability as linear matrix inequality feasibility problems, an exact two-dimensional diagonal complexity formula, and affine-invariant Kronecker projection theorems with normal equations, computable mismatch certificates, Armijo solver convergence, auxiliary self-conditioned K-target bounds, and Hessian-relative candidate certificates through an exact Kronecker Loewner-sandwich reachability condition, including a Kronecker expression threshold and a fixed-basis exact subproblem. Low-rank spectral models, curvature-proxy inflation, stochastic restricted complexity, discrete geometric length, and expression--estimation--flow--discretization accounting are presented as diagnostic interfaces rather than full optimizer characterizations. The resulting language turns structural preconditioner questions into geometric distance, reachability, and certificate problems. The repository includes deterministic toy and synthetic workflows that check diagonal expression gaps, block primal/dual certificates, Kronecker spectral width, and Hessian-relative Kronecker candidate certificates on small quadratic instances, together with low-rank spectral monotonicity.
A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable. We build DDC, a Dead-Direction Conditioner that lifts a base optimizer into a $G$-equivariant one: it conditions the optimizer's state in the orbit decomposition of a $G$-invariant metric, so the trajectory stays a preconditioned gradient flow on the quotient $\barΘ= Θ/G$. The construction carries four architectural gauges (cross-entropy shift, ReLU and SwiGLU rescaling, LayerNorm and RMSNorm scale, and a per-head $O(d_{\rm head})$ attention rotation matched to RoPE), proves exactly equivariant on an Adam base, and composes with a Muon base through a gauge-equivariant orthogonaliser. Respecting the symmetry changes both the minimum the optimizer reaches and what it leaves measurable there. On a language model trained past the point of fit, DDCAdam resists the over-training collapse AdamW falls into, holding a validation-train loss gap of 0.67 against 5.88, and reads the dead-direction rate in 32 of 65 layer-by-observable cells where AdamW reads it in 7. A vision transformer trained from scratch reaches lower validation loss (1.71 against 2.12) while compressing spare feed-forward capacity a matched AdamW leaves intact. On a Muon base, where the rotation gauge composes exactly, DDCMuon groks ten of eleven seeds at depth 24 that a plain Muon never reaches. Built into the optimizer, a network's gauge symmetry sharpens the minimum it finds and turns that minimum's geometry into something the trajectory can measure.
Thomas T. Zhang, Alok Shah, Yifei Zhang +3cs.LG cs.AI eess.SY
Many modern applications of deep learning involve training a neural network via a one-step prediction loss (e.g., $L^2$ regression, cross-entropy), but deploy the network by rolling out along its own predictions. Key examples include autoregressive language modeling, flow-based generative modeling, and robot policy learning. It is well-documented that these settings induce a phenomenon we call test-time feedback (TTF): the mismatch between the training/validation loss and downstream metrics of interest, such as task success rate and generation quality, which grows with task length. While data curation, architecture, and objective design have been proposed to combat train-test shift in TTF settings, this paper proposes optimization as a new design axis to mitigate error accumulation. Specifically, we introduce a new optimization paradigm called double-preconditioning (DoPr) uniquely tailored to the challenges of TTF. DoPr combines gradient-wise preconditioning, as in Adam and Muon, with activation-wise preconditioning (AP), such as in KFAC. We show that the addition of AP yields a drop-in intervention for increasing downstream model performance across a range of TTF settings. Interestingly, these gains in test-time performance do not consistently accompany improvements in validation loss, opening new questions about how to properly evaluate models trained with one-step supervised objectives.
Shampoo is attracting considerable attention for its superior performance on large-scale optimization benchmarks; yet it faces a significant practical bottleneck: the prohibitive computational overhead of matrix inversion. To mitigate this, practitioners typically rely on stale preconditioner updates, creating a fundamental trade-off between computational efficiency and optimization fidelity. In this work, we provide a theoretical study of staleness through the complementary lenses of convergence and stability. While staleness improves computational efficiency, it inherently degrades performance and introduces numerical instability. Crucially, we identify that damping, acting as a numerical stabilizer, can effectively suppress these negative effects. Guided by this analysis, we propose FOAM, an adaptive algorithm that stabilizes training by dynamically controlling both the damping factor and the eigendecomposition frequency based on an approximation of the staleness-oriented error. Experimental results demonstrate that FOAM reduces wall-clock time compared to standard Shampoo while maintaining robust convergence.