Training Under Challenge: Executable Certificates and Challenge-Closed Optimality for Neural Networks

A flat training curve does not reveal whether a neural network has reached a global optimum, is locally trapped, is representation-limited, or is mismatched to its trainer. We introduce Training Under Challenge, an executable-certificate framework in which predeclared, architecture-valid procedures construct complete alternatives in the same certified class and reevaluate the same objective. Any lower-valued candidate is a replayable witness that lower-bounds the checkpoint's empirical global-optimality gap. Passing a finite suite is only suite-relative; global-gap conclusions require a separately justified coverage mechanism. We define a resource-indexed challenge-power modulus that characterizes the largest gap compatible with passage. For squared loss, current block-decrease operators make coverage checkable and yield uniform and realized-residual bounds. We prove the converse frontier: without coverage, a first-order ReLU trainer can reach infinitely many exact conditional head optima while converging to a non-global point. On a channel-gated ResNet-18 distillation problem with known optimum, eight internal challenges cover all 240 audited output directions, and realized-residual bounds lie within factors of 1.74--3.02 of the true gap. Paired predictive certificates separate decoder under-use from representation insufficiency, while quantized-denoising studies demonstrate diagnosis, repair, and current-state recertification.

Generalized Convexity and Smoothness via Conjugate Duality: Optimization Theory for Deep Neural Networks

Deep neural network (DNN) training with stochastic gradient descent (SGD) and its variants achieves strong empirical performance, yet classical optimization theory does not fully explain this success. This limitation arises because conventional analyses rely on assumptions such as differentiability, convexity, or smoothness, which are often violated by DNN objectives. In this paper, we establish a unified optimization framework for DNN training by generalizing classical convexity and smoothness through Legendre functions and convex conjugation. Specifically, we introduce $\mathcal{H}(ψ)$-convexity and $\mathcal{H}(Ψ)$-smoothness, which unify convex and non-convex as well as smooth and non-smooth objectives within a single formalism and reveal a natural duality between generalized smoothness and convexity. Building on these generalized properties, we introduce generalized gradient descent (GD) and generalized SGD through convex conjugation. We theoretically prove that generalized GD admits an optimal learning rate of exactly $1$, and derive rigorous gradient-energy-based convergence rates for both proposed optimizers. We further reformulate DNN training as a composite optimization problem, demonstrating that its convergence relies on jointly reducing the gradient energy and controlling the induced norm of the network Jacobian. To characterize the practical influences of network architectures and training configurations, we introduce the gradient correlation factor and model capacity risk, and quantitatively analyze how architectural designs, batch size, and model capacity shape training convergence. Extensive experiments across diverse network architectures, datasets, optimizers, and loss functions validate our theoretical bounds and demonstrate precise alignment between our theoretical predictions and empirical training dynamics.

Geometric--Nongeometric Optimizer Calculus: A Modular Language for Reachable Gradient Methods

Adaptive optimizers mix several mechanisms: a metric or preconditioner maps gradients to descent directions, while estimation, memory, step-size control, constraints, stochasticity, target modification, and discretization determine which directions are available and how they are used. We introduce geometric--nongeometric optimizer calculus, a modular language for auditing reachable gradient methods under explicit oracle, budget, state, and rule constraints. The geometric module is a positive cometric family that maps covectors to parameter-space directions; the nongeometric modules are information, memory, control, operator, noise, target, and discretization mechanisms. The main formal result is a direction-expressivity theorem: away from critical points, full positive-definite geometry expresses exactly the strict descent directions. We then define restricted direction residuals for admissible metric families, prove exact expressivity conditions for diagonal and block geometries, and separate this direction-level diagnostic from condition-number geometric complexity. The resulting design problem is a Pareto optimization over module budgets, not a single universal optimizer ordering. We also lift pointwise residuals to a trajectory-level residual complexity that couples direction mismatch with the variation of the explaining geometry. We include diagnostic prototypes only as evidence for the language: a high-information full-metric probe solves deterministic quadratic benchmarks to numerical precision, while a practical Muon-style PyTorch candidate gives small-scale evidence that matrix-operator updates can be audited by the calculus. The paper is a theory and benchmark-language manuscript; it does not claim large-scale optimizer state-of-the-art performance.

Analysis of Adam Algorithms for Stochastic Dynamic Systems

The adaptive moment estimation algorithm, known as Adam, is widely used in modern machine learning, owing to its low per-iteration complexity and strong empirical performance. Despite its prevalent use, the theoretical foundation of Adam remains largely unexplored for time-varying and nonstationary systems. In fact, the existing theoretical analyses of Adam-type algorithms are primarily concerned with time-invariant model parameters and explicitly or implicitly rely on independent and identically distributed (i.i.d.) data assumptions, under which the learning taskcan be formulated as minimizing a fixed expected objective with a static minimizer. However, such assumptions are often violated in time-varying and nonstationary systems, thereby calling for a theoretical investigation beyond the conventional yet idealized i.i.d. setting. The main objective of this paper is to solve this challenging problem by establishing a general theory of Adam for time-varying and nonstationary stochastic systems. We will introduce some new techniques for analyzing the products of nonstationary and dependent random matrices induced by Adam's coupled first- and second-moment recursions, and will construct a new stochastic Lyapunov function that blends these two moment dynamics. Under a stochastic excitation condition that allows nonstationary and dependent data, we will derive both parameter tracking and output prediction error bounds explicitly, quantifying the effects of stepsize, first- and second-momentum parameters, gradient noise and parameter drift. These bounds not only provide guarantees for Adam performance, but also provide guidelines for hyperparameter selection. Experiments on both synthetic and real-world data validate our theory and design guidelines.

Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Compute Efficiency and Serial Runtime Tradeoffs for Stochastic Momentum Methods

Stochastic momentum methods such as heavy ball (HB), Nesterov momentum, and variants of Accelerated SGD (ASGD) [Kidambi et al., 2018] are widely used in modern training, but their stochastic benefits depend on two distinct quantities: serial runtime, the number of iterations needed to reach a target accuracy, and compute efficiency (CE), the inverse total gradient-query or FLOP cost. Larger batches reduce serial runtime without hurting CE only when the contraction gap grows linearly with batch size. We study stochastic HB and ASGD for consistent linear regression with Gaussian covariates and prove finite-dimensional, discrete-time lower bounds on their batch-size tradeoffs. Our first result shows that HB does not improve the CE frontier over SGD for arbitrary spectra; rather, it preserves SGD-level CE over a larger batch-size window, allowing larger batches to reduce serial runtime until HB reaches its deterministic accelerated scale. This window can be a factor $\sqrtκ$ larger than the SGD critical batch size. For ASGD, the picture is more spectrum-dependent: for rapidly decaying power-law spectra, ASGD improves small-batch CE over HB/SGD, but as batch size grows it trades this CE advantage for improved serial runtime. Synthetic linear-regression experiments verify these qualitative regimes, including near-overlap of ASGD and HB for slowly decaying spectra and the predicted CE--serial tradeoff for rapidly decaying spectra.

A Resolution of the SS--RS--GD Inequalities

Yun, Sra, and Jadbabaie (COLT 2021, open question) conjectured the SS--RS--GD inequalities: for well-conditioned symmetric matrices $A_1,\dots,A_n$, the operators $W_{ss}$, $W_{rs}$, and $W_{gd}$ that encode the expected iterate of single-shuffle SGD, random-reshuffle SGD, and gradient descent on a quadratic finite sum should satisfy \[ \|W_{ss}\|\le \| W_{rs}\|\le \|W_{gd}\|. \] The conjecture is resolved, $\bullet$ SS-RS inequality fails. Already for $n=3$, $K=2$, and $d=4$, we exhibit explicit PSD matrices whose condition number is arbitrarily close to $1$, yet $\|W_{ss}\|>\|W_{rs}\|$. $\bullet$ RS-GD inequality holds. For every symmetric $A_i$ with $\bigl(1-\frac1{4n^2+1}\bigr)I\preceq A_i\preceq I$, one has $\|W_{rs}\|\le\|W_{gd}\|$. The proof was found via GPT-5.5 Pro extended prompted by the author.

The Role of Symmetry in Optimizing Overparameterized Networks

Overparameterization is central to the success of deep learning, yet the mechanisms by which it improves optimization remain incompletely understood. We analyze weight-space symmetries in neural networks and show that overparameterization introduces additional symmetries that benefit optimization in two distinct ways. First, we prove that these symmetries act as a form of diagonal preconditioning on the Hessian, enabling the existence of better-conditioned minima within each equivalence class of functionally identical solutions. Second, we show that overparameterization increases the probability mass of global minima near typical initializations, making these favorable solutions more reachable. Teacher-student network experiments validate our theoretical predictions: as width increases, the Hessian trace decreases, condition numbers improve, and convergence accelerates. Our analysis provides a unified framework for understanding overparameterization and width growth as a geometric transformation of the loss landscape.