The measured metrics that help determine if Adam can mitigate the ill-conditioning on a given loss landscape are defined.
Abstract
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 $\rho$ 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.
Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence--convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-$\beta_2$ regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50\% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.
Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not. We trace the difference to the gauge symmetry of the loss, its invariance under $(U, V) \mapsto (UQ, VQ)$. Gradient flow's low-rank mechanism is available to an optimizer only if that optimizer is gauge-equivariant, a condition necessary for the transfer but not sufficient for low-rank recovery. Gradient descent, momentum,"shared-scalar"Adam, Muon, and Shampoo satisfy it. Adam, RMSProp, and the other coordinate-wise methods do not. A structure theorem characterizes the memoryless equivariant rules as exactly the Gram-determined left preconditioners, and a transfer theorem carries gradient flow's pathwise properties to common-scalar flows. We then sort nine update rules on underdetermined matrix sensing by recovery error against the planted ground truth. A one-parameter family from coordinate-wise to shared-scalar preconditioning restores the bias monotonically, isolating anisotropy as the cause. A"spectral schedule"reconciles two opposing reports about Muon: equal-rate updates recover exactly low-rank targets but lose their edge as the spectral tail grows. In transformers, Adam separates two gauge-equivalent initializations at the first step, where the equivariant optimizers stay at float precision, and ends with the per-head invariants $W_Q^\top W_K$ 56% apart in relative Frobenius distance, a gap no per-head rotation can close. On two hyperspectral datasets at matched training loss, gradient descent cuts held-out error by 43-44% at the lowest sampling density, and at lower effective rank. Basis choice is therefore not a tuning detail but a decision about which interpolant the optimizer selects.
Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW optimizers, where the adaptivity parts in Adam and AdamW basically just coincide with RMSprop. Such adaptive methods involve several hyperparameters including the regularization parameter $\epsilon$ (which ensures that one does not divide by 0 and is often chosen to be very close to zero such as $10^{-8}$ in PyTorch by default) and the second moment decay parameter $\beta$ (which is often chosen to be very close to $1$ such as 0.99 (RMSprop) and 0.999 (Adam and AdamW) in PyTorch by default). Despite the high relevance of such methods, it remains an open research problem to provide error estimates for such methods with the error constants being not exploding but uniformly bounded with the respect to the hyperparameters, even in the situation of convex stochastic optimization problems. It is the key contribution of this work to essentially solve this problem for RMSprop. Specifically, we bound the expectation of the stopped evaluation of the objective function at the RMSprop process from above by the sum of an initialization term that decays exponentially in the training time, a stochastic approximation remainder of order $\gamma_n$, and a memory error of order $( 1 - \beta)^2$ with the error constants being uniformly controlled over all admissible choices of the step sizes, the second moment decay parameter $\beta$ and the regularization parameter $\epsilon\in[0,1]$ (also covering $\epsilon=0$). Our non-asymptotic error estimates hold not just for all sufficiently large n but hold for every gradient step $n=1,2,3,...$ with all error constants being explicitly specified. The key innovative new feature in the proof of our analysis are suitable inverse moment bounds for the second moment process in RMSprop.
Understanding deep neural networks remains a central challenge in machine learning. In particular, the theoretical properties of even two-layer ReLU networks, especially in the presence of weight decay, remain poorly understood. To this end, we derive a sufficient condition on the hyperparameter settings under which the global minima collapse to the zero solution. Interestingly, our experiments reveal that using AdamW as an optimizer prevents the collapse of the learned parameters, whereas using SGD does not, which may help explain the success of AdamW in deep learning training. In addition, when restricting the input dimension to one, we derive an analytical solution for the globally optimal parameter sets of two-layer ReLU networks and show that $\ell_2$-regularization has a width-invariant effect on connectivity, but its dimensionality-reducing effect becomes stronger as the network width increases. These results provide insight into how width-dependent hyperparameters influence the geometry of regularized loss landscapes.
Equivariant networks are commonly trained with Adam, yet recent work reports that matrix-structured optimizers such as Muon can perform better on these architectures without explaining why. We identify one source of this difference inside equivariant linear layers. Each irrep block learns a channel-mixing matrix $W_l$ shared across its $2l+1$ components, giving the expanded map $W_l \otimes I_{2l+1}$. For a single application of the layer, the gradient of $W_l$ sums $2l+1$ outer product contributions and has rank at most $2l+1$. Adam rescales stored weights individually without using the irrep boundaries, so one learning rate can produce different spectral step sizes across blocks within a layer. We address this mismatch by normalizing each block update separately, without introducing a new hyperparameter. This changes only the scale of the update, leaving Adam's moment estimates and its direction within each block unchanged. We evaluate the mechanism in a controlled $\mathrm{SO}(3)$-equivariant model with a matched dense control and in an e3nn interatomic potential model trained on rMD17 and MD22. The toy setup isolates a mismatch that grows with width while the dense control shows no corresponding growth. In the interatomic potential model, block normalization and tuning Adam's momentum coefficients independently improve performance, but neither alone matches Muon. Combined, they make Adam competitive with Muon on all datasets, indicating that blockwise step control and momentum accumulation account for much of Muon's advantage.
Deep neural networks generalize well despite their highly nonconvex, overparameterized loss landscapes, a phenomenon often associated with the geometry of the minima found by stochastic optimization. We study how incremental grow-and-optimize strategies bias training toward flatter regions by viewing growth as progressive constraint relaxation. Starting from a low-dimensional submodel, we iteratively expand the trainable parameters by unlocking nested random subspaces while freezing the orthogonal complement at the network initialization, re-optimizing after each expansion until the full architecture is reached. Under standard local regularity conditions around non-degenerate minima, we prove that local sublevel sets are well approximated by ellipsoids and that basin accessibility under frozen constraints can be characterized by an explicit effective curvature in the frozen directions. This leads to an explanation of the bias: progressive growth increases the relative weight of wide basins and suppresses sharp ones through a volume effect induced by the frozen constraints. We empirically validate these predictions in controlled toy landscapes and in a realistic ResNet/CIFAR-100 setting and confirm that although progressive subspace growth reliably produces flatter solutions, curvature reductions do not universally translate into improved test performance, highlighting subtleties in the flatness-generalization connection. The code is available at https://github.com/p0lcAi/Across-the-Loss-Landscape.
Paul Caillon, Christophe Cerisara, Alexandre Allauzen· 0 citations
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