Itay Evron

Itay Evron, Edward Moroshko, Gon Buzaglo et al.

We analyze continual learning on a sequence of separable linear classification tasks with binary labels. We show theoretically that learning with weak regularization reduces to solving a sequential max-margin problem, corresponding to a special case of the Projection Onto Convex Sets (POCS) framework. We then develop upper bounds on the forgetting and other quantities of interest under various settings with recurring tasks, including cyclic and random orderings of tasks. We discuss several practical implications to popular training practices like regularization scheduling and weighting. We point out several theoretical differences between our continual classification setting and a recently studied continual regression setting.

Itay Evron, Edward Moroshko, Rachel Ward et al.

To better understand catastrophic forgetting, we study fitting an overparameterized linear model to a sequence of tasks with different input distributions. We analyze how much the model forgets the true labels of earlier tasks after training on subsequent tasks, obtaining exact expressions and bounds. We establish connections between continual learning in the linear setting and two other research areas: alternating projections and the Kaczmarz method. In specific settings, we highlight differences between forgetting and convergence to the offline solution as studied in those areas. In particular, when T tasks in d dimensions are presented cyclically for k iterations, we prove an upper bound of T^2 * min{1/sqrt(k), d/k} on the forgetting. This stands in contrast to the convergence to the offline solution, which can be arbitrarily slow according to existing alternating projection results. We further show that the T^2 factor can be lifted when tasks are presented in a random ordering.

Itay Evron, Ophir Onn, Tamar Weiss Orzech et al.

Error-correcting codes (ECC) are used to reduce multiclass classification tasks to multiple binary classification subproblems. In ECC, classes are represented by the rows of a binary matrix, corresponding to codewords in a codebook. Codebooks are commonly either predefined or problem dependent. Given predefined codebooks, codeword-to-class assignments are traditionally overlooked, and codewords are implicitly assigned to classes arbitrarily. Our paper shows that these assignments play a major role in the performance of ECC. Specifically, we examine similarity-preserving assignments, where similar codewords are assigned to similar classes. Addressing a controversy in existing literature, our extensive experiments confirm that similarity-preserving assignments induce easier subproblems and are superior to other assignment policies in terms of their generalization performance. We find that similarity-preserving assignments make predefined codebooks become problem-dependent, without altering other favorable codebook properties. Finally, we show that our findings can improve predefined codebooks dedicated to extreme classification.

Guy Bar-Shalom, Ami Tavory, Itay Evron et al.

Weight-space models learn directly from the parameters of neural networks, enabling tasks such as predicting their accuracy on new datasets. Naive methods -- like applying MLPs to flattened parameters -- perform poorly, making the design of better weight-space architectures a central challenge. While prior work leveraged permutation symmetries in standard networks to guide such designs, no analogous analysis or tailored architecture yet exists for Kolmogorov-Arnold Networks (KANs). In this work, we show that KANs share the same permutation symmetries as MLPs, and propose the KAN-graph, a graph representation of their computation. Building on this, we develop WS-KAN, the first weight-space architecture that learns on KANs, which naturally accounts for their symmetry. We analyze WS-KAN's expressive power, showing it can replicate an input KAN's forward pass - a standard approach for assessing expressiveness in weight-space architectures. We construct a comprehensive ``zoo'' of trained KANs spanning diverse tasks, which we use as benchmarks to empirically evaluate WS-KAN. Across all tasks, WS-KAN consistently outperforms structure-agnostic baselines, often by a substantial margin. Our code is available at https://github.com/BarSGuy/KAN-Graph-Metanetwork.

Gilad Karpel, Edward Moroshko, Ran Levinstein et al.

We study generalization in an overparameterized continual linear regression setting, where a model is trained with L2 (isotropic) regularization across a sequence of tasks. We derive a closed-form expression for the expected generalization loss in the high-dimensional regime that holds for arbitrary linear teachers. We demonstrate that isotropic regularization mitigates label noise under both single-teacher and multiple i.i.d. teacher settings, whereas prior work accommodating multiple teachers either did not employ regularization or used memory-demanding methods. Furthermore, we prove that the optimal fixed regularization strength scales nearly linearly with the number of tasks $T$, specifically as $T/\ln T$. To our knowledge, this is the first such result in theoretical continual learning. Finally, we validate our theoretical findings through experiments on linear regression and neural networks, illustrating how this scaling law affects generalization and offering a practical recipe for the design of continual learning systems.

Yochai Zur, Chaim Baskin, Evgenii Zheltonozhskii et al.

Recently, deep learning has become a de facto standard in machine learning with convolutional neural networks (CNNs) demonstrating spectacular success on a wide variety of tasks. However, CNNs are typically very demanding computationally at inference time. One of the ways to alleviate this burden on certain hardware platforms is quantization relying on the use of low-precision arithmetic representation for the weights and the activations. Another popular method is the pruning of the number of filters in each layer. While mainstream deep learning methods train the neural networks weights while keeping the network architecture fixed, the emerging neural architecture search (NAS) techniques make the latter also amenable to training. In this paper, we formulate optimal arithmetic bit length allocation and neural network pruning as a NAS problem, searching for the configurations satisfying a computational complexity budget while maximizing the accuracy. We use a differentiable search method based on the continuous relaxation of the search space proposed by Liu et al. (arXiv:1806.09055). We show, by grid search, that heterogeneous quantized networks suffer from a high variance which renders the benefit of the search questionable. For pruning, improvement over homogeneous cases is possible, but it is still challenging to find those configurations with the proposed method. The code is publicly available at https://github.com/yochaiz/Slimmable and https://github.com/yochaiz/darts-UNIQ

Daniel Goldfarb, Itay Evron, Nir Weinberger et al.

In continual learning, catastrophic forgetting is affected by multiple aspects of the tasks. Previous works have analyzed separately how forgetting is affected by either task similarity or overparameterization. In contrast, our paper examines how task similarity and overparameterization jointly affect forgetting in an analyzable model. Specifically, we focus on two-task continual linear regression, where the second task is a random orthogonal transformation of an arbitrary first task (an abstraction of random permutation tasks). We derive an exact analytical expression for the expected forgetting - and uncover a nuanced pattern. In highly overparameterized models, intermediate task similarity causes the most forgetting. However, near the interpolation threshold, forgetting decreases monotonically with the expected task similarity. We validate our findings with linear regression on synthetic data, and with neural networks on established permutation task benchmarks.

Itamar Harel, William M. Hoza, Gal Vardi et al.

We study the overfitting behavior of fully connected deep Neural Networks (NNs) with binary weights fitted to perfectly classify a noisy training set. We consider interpolation using both the smallest NN (having the minimal number of weights) and a random interpolating NN. For both learning rules, we prove overfitting is tempered. Our analysis rests on a new bound on the size of a threshold circuit consistent with a partial function. To the best of our knowledge, ours are the first theoretical results on benign or tempered overfitting that: (1) apply to deep NNs, and (2) do not require a very high or very low input dimension.

Itay Evron, Ran Levinstein, Matan Schliserman et al.

We theoretically study the common continual learning setup where an overparameterized model is sequentially fitted to a set of jointly realizable tasks. We analyze the forgetting, i.e., loss on previously seen tasks, after $k$ iterations. For continual linear models, we prove that fitting a task is equivalent to a single stochastic gradient descent (SGD) step on a modified objective. We develop novel last-iterate SGD upper bounds in the realizable least squares setup, which we then leverage to derive new results for continual learning. Focusing on random orderings over $T$ tasks, we establish universal forgetting rates, whereas existing rates depend on the problem dimensionality or complexity. Specifically, in continual regression with replacement, we improve the best existing rate from $O((d-r)/k)$ to $O(\min(k^{-1/4}, \sqrt{d-r}/k, \sqrt{Tr}/k))$, where $d$ is the dimensionality and $r$ the average task rank. Furthermore, we establish the first rate for random task orderings without replacement. The obtained rate of $O(\min(T^{-1/4}, (d-r)/T))$ proves for the first time that randomization alone, with no task repetition, can prevent catastrophic forgetting in sufficiently long task sequences. Finally, we prove a matching $O(k^{-1/4})$ forgetting rate for continual linear classification on separable data. Our universal rates apply for broader projection methods, such as block Kaczmarz and POCS, illuminating their loss convergence under i.i.d. and one-pass orderings.

Matan Tsipory, Ran Levinstein, Itay Evron et al.

We analyze task orderings in continual learning for linear regression, assuming joint realizability of training data. We focus on orderings that greedily maximize dissimilarity between consecutive tasks, a concept briefly explored in prior work but still surrounded by open questions. Using tools from the Kaczmarz method literature, we formalize such orderings and develop geometric and algebraic intuitions around them. Empirically, we demonstrate that greedy orderings converge faster than random ones in terms of the average loss across tasks, both for linear regression with random data and for linear probing on CIFAR-100 classification tasks. Analytically, in a high-rank regression setting, we prove a loss bound for greedy orderings analogous to that of random ones. However, under general rank, we establish a repetition-dependent separation. Specifically, while prior work showed that for random orderings, with or without replacement, the average loss after $k$ iterations is bounded by $\mathcal{O}(1/\sqrt{k})$, we prove that single-pass greedy orderings may fail catastrophically, whereas those allowing repetition converge at rate $\mathcal{O}(1/\sqrt[3]{k})$. Overall, we reveal nuances within and between greedy and random orderings.

Pedro Savarese, Itay Evron, Daniel Soudry et al.

We consider the question of what functions can be captured by ReLU networks with an unbounded number of units (infinite width), but where the overall network Euclidean norm (sum of squares of all weights in the system, except for an unregularized bias term for each unit) is bounded; or equivalently what is the minimal norm required to approximate a given function. For functions $f : \mathbb R \rightarrow \mathbb R$ and a single hidden layer, we show that the minimal network norm for representing $f$ is $\max(\int |f''(x)| dx, |f'(-\infty) + f'(+\infty)|)$, and hence the minimal norm fit for a sample is given by a linear spline interpolation.

Itay Evron, Edward Moroshko, Koby Crammer

In extreme classification problems, learning algorithms are required to map instances to labels from an extremely large label set. We build on a recent extreme classification framework with logarithmic time and space, and on a general approach for error correcting output coding (ECOC) with loss-based decoding, and introduce a flexible and efficient approach accompanied by theoretical bounds. Our framework employs output codes induced by graphs, for which we show how to perform efficient loss-based decoding to potentially improve accuracy. In addition, our framework offers a tradeoff between accuracy, model size and prediction time. We show how to find the sweet spot of this tradeoff using only the training data. Our experimental study demonstrates the validity of our assumptions and claims, and shows that our method is competitive with state-of-the-art algorithms.