In modern parametric model training, full-batch gradient descent (and its variants) suffers due to progressively stronger biasing towards the exact realization of training data; this drives the systematic ``generalization gap'', where the train error becomes an unreliable proxy for test error. Existing approaches either argue this gap is benign through complex analysis or sacrifice data to a validation set. In contrast, we introduce decoupled descent (DD), a novel theory-based training algorithm that satisfies a train-test identity -- enforcing the train error to asymptotically track the test error for stylized Gaussian mixture models. Within this specific regime, leveraging approximate message passing theory, DD iteratively cancels the biases due to data reuse, rigorously demonstrating the feasibility of zero-cost validation and $100\%$ data utilization. Moreover, DD is governed by a low-dimensional state evolution recursion, rendering the dynamics of the algorithm transparent and tractable. We validate DD on XOR classification, yielding superior performance compared to GD; additionally, we implement noisy MNIST and non-linear probing of CIFAR-10, demonstrating that even when our stylized assumptions are relaxed, DD narrows the generalization gap compared to GD.

Here we provide theoretical evidence that vanilla MoE do not6 guarantee convergence when mixing multiple datasets. Consider a binary classification problem over P-patch inputs where each8 patch has d dimensions and label y = { 1}. Thus, a labeled data point (x,y) has input x =9 (x(1),x(2),x(3),...,x(P)) (Rd)P is a collection of P patch inputs with y as the data label. The10 data x is generated from K clusters.11 Chen et al. [2022] proves that in such a binary-classification problem, an MoE layer converges to an12 o(1) test loss and zero training loss.

B.3 Derivations of Eq. (19) Similar to derivation above, we give the gradient with respect to weight vector w RM+, which is given by wDKL = w log Z(U,w) wEU,w (log pθ(X |z))T1N + wEU,w (log pθ(U |z))Tw . The learning rate of each stochastic gradient descent step is γt t 1, where t {1,,T}denotes the iteration for optimization. We already report the t-SNE visualization of ByPE-VAE and standard VAE in Figure. Here we give more t-SNE visualization results. First, we randomly sample from ByPE-VAEs trained on different datasets, namely, MNIST, Fashion MNIST, and Celeba, as shown in Fig.7.

In this paper, we study the generalization properties of Model-Agnostic MetaLearning (MAML) algorithms for supervised learning problems. We focus on the setting in which we train the MAML model over mtasks, each with ndata points, and characterize its generalization error from two points of view: First, we assume the new task at test time is one of the training tasks, and we show that, for strongly convex objective functions, the expected excess population loss is bounded by O(1/mn). Second, we consider the MAML algorithm's generalization to an unseen task and show that the resulting generalization error depends on the total variation distance between the underlying distributions of the new task and the tasks observed during the training process. Our proof techniques rely on the connections between algorithmic stability and generalization bounds of algorithms. In particular, we propose a new definition of stability for meta-learning algorithms, which allows us to capture the role of both the number of tasks mand number of samples per task non the generalization error of MAML.

As modern machine learning models continue to advance the computational frontier, it has become increasingly important to develop precise estimates for expected performance improvements under different model and data scaling regimes. Currently, theoretical understanding of the learning curves that characterize how the prediction error depends on the number of samples is restricted to either largesample asymptotics (m!1) or, for certain simple data distributions, to the high-dimensional asymptotics in which the number of samples scales linearly with the dimension (m / d). There is a wide gulf between these two regimes, including all higher-order scaling relations m / dr, which are the subject of the present paper. We focus on the problem of kernel ridge regression for dot-product kernels and present precise formulas for the mean of the test error, bias, and variance, for data drawn uniformly from the sphere with isotropic random labels in the rth-order asymptotic scaling regime m!1 with m/dr held constant. We observe a peak in the learning curve whenever m dr/r! for any integer r, leading to multiple sample-wise descent and nontrivial behavior at multiple scales. We include a colab2 notebook that reproduces the essential results of the paper.