We characterize weakly regularized continual classification in homogeneous models as sequential projections onto task margin sets. This result generalizes prior analyses restricted to either stationary (single-task) deep models or continual linear models. We show that global convergence generally fails, even for simple models linear in data but nonlinear in parameters. Nevertheless, by leveraging results from nonconvex projection theory, we identify regularity properties of homogeneous deep networks that guarantee local linear convergence under random and cyclic task sequences. Finally, we extend our analysis to continual regression, unifying the framework for homogeneous models.

We study the generalization performance of gradient methods in the fundamental stochastic convex optimization setting, focusing on its dimension dependence. First, for full-batch gradient descent (GD) we give a construction of a learning problem in dimension $d=O(n^2)$, where the canonical version of GD (tuned for optimal performance on the empirical risk) trained with $n$ training examples converges, with constant probability, to an approximate empirical risk minimizer with $\Omega(1)$ population excess risk. Our bound translates to a lower bound of $\smash{\Omega(\sqrt{d})}$ on the number of training examples required for standard GD to reach a non-trivial test error, answering an open question raised by Feldman (2016) and Amir, Koren and Livni (2021) and showing that a non-trivial dimension dependence is unavoidable. Furthermore, for standard one-pass stochastic gradient descent (SGD), we show that an application of the same construction technique provides a similar $\smash{\Omega(\sqrt{d})}$ lower bound for the sample complexity of SGD to reach a non-trivial empirical error, despite achieving optimal test performance. This again provides for an exponential improvement in the dimension dependence compared to previous work (Koren et al., 2022), resolving an open question left therein.

We study population convergence guarantees of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, where the noise at optimum is zero or near zero. The behavior of the last iterate of SGD in this setting---particularly with large (constant) stepsizes---has received growing attention in recent years due to implications for the training of over-parameterized models, as well as to analyzing forgetting in continual learning and to understanding the convergence of the randomized Kaczmarz method for solving linear systems. We establish that after $T$ steps of SGD on $\beta$-smooth convex loss functions with stepsize $0 < \eta < 2/\beta$, the last iterate exhibits expected excess risk $\widetilde{O}(\tfrac{1}{\eta (2-\beta \eta) T^{1-\beta\eta/2}} + \tfrac{\eta}{(2-\beta\eta)^2} T^{\beta\eta/2} \sigma_\star^2)$, where $\sigma_\star^2$ denotes the variance of the stochastic gradients at the optimum. In particular, for a well-tuned stepsize we obtain a near optimal $\widetilde{O}(1/T + \sigma_\star/\sqrt T)$ rate for the last iterate, extending the results of Varre et al. (2021) beyond least squares regression; and when $\sigma_\star=0$ we obtain a rate of $O(1/\sqrt T)$ with $\eta=1/\beta$, improving upon the best-known $O(T^{-1/4})$ rate recently established by Evron et al. (2025) in the special case of realizable linear regression.

We study realizable continual linear regression under random task orderings, a common setting for developing continual learning theory. In this setup, the worst-case expected loss after $k$ learning iterations admits a lower bound of $\Omega(1/k)$. However, prior work using an unregularized scheme has only established an upper bound of $O(1/k^{1/4})$, leaving a significant gap. Our paper proves that this gap can be narrowed, or even closed, using two frequently used regularization schemes: (1) explicit isotropic $\ell_2$ regularization, and (2) implicit regularization via finite step budgets. We show that these approaches, which are used in practice to mitigate forgetting, reduce to stochastic gradient descent (SGD) on carefully defined surrogate losses. Through this lens, we identify a fixed regularization strength that yields a near-optimal rate of $O(\log k / k)$. Formalizing and analyzing a generalized variant of SGD for time-varying functions, we derive an increasing regularization strength schedule that provably achieves an optimal rate of $O(1/k)$. This suggests that schedules that increase the regularization coefficient or decrease the number of steps per task are beneficial, at least in the worst case.

We study the generalization performance of unregularized gradient methods for separable linear classification. While previous work mostly deal with the binary case, we focus on the multiclass setting with $k$ classes and establish novel population risk bounds for Gradient Descent for loss functions that decay to zero. In this setting, we show risk bounds that reveal that convergence rates are crucially influenced by the geometry of the loss template, as formalized by Wang and Scott (2024), rather than of the loss function itself. Particularly, we establish risk upper bounds that holds for any decay rate of the loss whose template is smooth with respect to the $p$-norm. In the case of exponentially decaying losses, our results indicates a contrast between the $p=\infty$ case, where the risk exhibits a logarithmic dependence on $k$, and $p=2$ where the risk scales linearly with $k$. To establish this separation formally, we also prove a lower bound in the latter scenario, demonstrating that the polynomial dependence on $k$ is unavoidable. Central to our analysis is a novel bound on the Rademacher complexity of low-noise vector-valued linear predictors with a loss template smooth w.r.t. general $p$-norms.

We study the generalization properties of unregularized gradient methods applied to separable linear classification---a setting that has received considerable attention since the pioneering work of Soudry et al. (2018). We establish tight upper and lower (population) risk bounds for gradient descent in this setting, for any smooth loss function, expressed in terms of its tail decay rate. Our bounds take the form $\Theta(r_{\ell,T}^2 / \gamma^2 T + r_{\ell,T}^2 / \gamma^2 n)$, where $T$ is the number of gradient steps, $n$ is size of the training set, $\gamma$ is the data margin, and $r_{\ell,T}$ is a complexity term that depends on the tail decay rate of the loss function (and on $T$). Our upper bound greatly improves the existing risk bounds due to Shamir (2021) and Schliserman and Koren (2022), that either applied to specific loss functions or imposed extraneous technical assumptions, and applies to virtually any convex and smooth loss function. Our risk lower bound is the first in this context and establish the tightness of our general upper bound for any given tail decay rate and in all parameter regimes. The proof technique used to show these results is also markedly simpler compared to previous work, and is straightforward to extend to other gradient methods; we illustrate this by providing analogous results for Stochastic Gradient Descent.

An influential line of recent work has focused on the generalization properties of unregularized gradient-based learning procedures applied to separable linear classification with exponentially-tailed loss functions. The ability of such methods to generalize well has been attributed to their implicit bias towards large margin predictors, both asymptotically as well as in finite time. We give an additional unified explanation for this generalization and relate it to two simple properties of the optimization objective, that we refer to as realizability and self-boundedness. We introduce a general setting of unconstrained stochastic convex optimization with these properties, and analyze generalization of gradient methods through the lens of algorithmic stability. In this broader setting, we obtain sharp stability bounds for gradient descent and stochastic gradient descent which apply even for a very large number of gradient steps, and use them to derive general generalization bounds for these algorithms. Finally, as direct applications of the general bounds, we return to the setting of linear classification with separable data and establish several novel test loss and test accuracy bounds for gradient descent and stochastic gradient descent for a variety of loss functions with different tail decay rates. In some of these cases, our bounds significantly improve upon the existing generalization error bounds in the literature.