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Experimental results have shown that curriculum learning, i.e., presenting simpler examples before more complex ones, can improve the efficiency of learning. Some recent theoretical results also showed that changing the sampling distribution can help neural networks learn parities, with formal results only for large learning rates and one-step arguments. Here we show a separation result in the number of training steps with standard (bounded) learning rates on a common sample distribution: if the data distribution is a mixture of sparse and dense inputs, there exists a regime in which a 2-layer ReLU neural network trained by a curriculum noisy-GD (or SGD) algorithm that uses sparse examples first, can learn parities of sufficiently large degree, while any fully connected neural network of possibly larger width or depth trained by noisy-GD on the unordered samples cannot learn without additional steps. We also provide experimental results supporting the qualitative separation beyond the specific regime of the theoretical results., Comment: 34 pages, 8 figures

The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using reproducing kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function., Comment: These are notes from the lecture of Francis Bach given at the summer school "Statistical Physics & Machine Learning", that took place in Les Houches School of Physics in France from 4th to 29th July 2022. The school was organized by Florent Krzakala and Lenka Zdeborov\'a from EPFL. 19 pages, 4 figures

Curriculum learning (CL) - training using samples that are generated and presented in a meaningful order - was introduced in the machine learning context around a decade ago. While CL has been extensively used and analysed empirically, there has been very little mathematical justification for its advantages. We introduce a CL model for learning the class of k-parities on d bits of a binary string with a neural network trained by stochastic gradient descent (SGD). We show that a wise choice of training examples involving two or more product distributions, allows to reduce significantly the computational cost of learning this class of functions, compared to learning under the uniform distribution. Furthermore, we show that for another class of functions - namely the `Hamming mixtures' - CL strategies involving a bounded number of product distributions are not beneficial.

This paper considers the Pointer Value Retrieval (PVR) benchmark introduced in [ZRKB21], where a 'reasoning' function acts on a string of digits to produce the label. More generally, the paper considers the learning of logical functions with gradient descent (GD) on neural networks. It is first shown that in order to learn logical functions with gradient descent on symmetric neural networks, the generalization error can be lower-bounded in terms of the noise-stability of the target function, supporting a conjecture made in [ZRKB21]. It is then shown that in the distribution shift setting, when the data withholding corresponds to freezing a single feature (referred to as canonical holdout), the generalization error of gradient descent admits a tight characterization in terms of the Boolean influence for several relevant architectures. This is shown on linear models and supported experimentally on other models such as MLPs and Transformers. In particular, this puts forward the hypothesis that for such architectures and for learning logical functions such as PVR functions, GD tends to have an implicit bias towards low-degree representations, which in turn gives the Boolean influence for the generalization error under quadratic loss., Comment: To appear in NeurIPS 2022

One of the most classical results in high-dimensional learning theory provides a closed-form expression for the generalisation error of binary classification with the single-layer teacher-student perceptron on i.i.d. Gaussian inputs. Both Bayes-optimal estimation and empirical risk minimisation (ERM) were extensively analysed for this setting. At the same time, a considerable part of modern machine learning practice concerns multi-class classification. Yet, an analogous analysis for the corresponding multi-class teacher-student perceptron was missing. In this manuscript we fill this gap by deriving and evaluating asymptotic expressions for both the Bayes-optimal and ERM generalisation errors in the high-dimensional regime. For Gaussian teacher weights, we investigate the performance of ERM with both cross-entropy and square losses, and explore the role of ridge regularisation in approaching Bayes-optimality. In particular, we observe that regularised cross-entropy minimisation yields close-to-optimal accuracy. Instead, for a binary teacher we show that a first-order phase transition arises in the Bayes-optimal performance., Comment: 14 pages + appendix

This paper introduces the notion of ``Initial Alignment'' (INAL) between a neural network at initialization and a target function. It is proved that if a network and a Boolean target function do not have a noticeable INAL, then noisy gradient descent on a fully connected network with normalized i.i.d. initialization will not learn in polynomial time. Thus a certain amount of knowledge about the target (measured by the INAL) is needed in the architecture design. This also provides an answer to an open problem posed in [AS20]. The results are based on deriving lower-bounds for descent algorithms on symmetric neural networks without explicit knowledge of the target function beyond its INAL.

We study the implicit bias of ReLU neural networks trained by a variant of SGD where at each step, the label is changed with probability $p$ to a random label (label smoothing being a close variant of this procedure). Our experiments demonstrate that label noise propels the network to a sparse solution in the following sense: for a typical input, a small fraction of neurons are active, and the firing pattern of the hidden layers is sparser. In fact, for some instances, an appropriate amount of label noise does not only sparsify the network but further reduces the test error. We then turn to the theoretical analysis of such sparsification mechanisms, focusing on the extremal case of $p=1$. We show that in this case, the network withers as anticipated from experiments, but surprisingly, in different ways that depend on the learning rate and the presence of bias, with either weights vanishing or neurons ceasing to fire.

The limit of the entropy in the stochastic block model (SBM) has been characterized in the sparse regime for the special case of disassortative communities [COKPZ17] and for the classical case of assortative communities but in the dense regime [DAM16]. The problem has not been closed in the classical sparse and assortative case. This paper establishes the result in this case for any SNR besides for the interval (1, 3.513). It further gives an approximation to the limit in this window. The result is obtained by expressing the global SBM entropy as an integral of local tree entropies in a broadcasting on tree model with erasure side-information. The main technical advancement then relies on showing the irrelevance of the boundary in such a model, also studied with variants in [KMS16], [MNS16] and [MX15]. In particular, we establish the uniqueness of the BP fixed point in the survey model for any SNR above 3.513 or below 1. This only leaves a narrow region in the plane between SNR and survey strength where the uniqueness of BP conjectured in these papers remains unproved.

We settle a version of the conjecture about intransitive dice posed by Conrey, Gabbard, Grant, Liu and Morrison in 2016 and Polymath in 2017. We consider generalized dice with $n$ faces and we say that a die $A$ beats $B$ if a random face of $A$ is more likely to show a higher number than a random face of $B$. We study random dice with faces drawn iid from the uniform distribution on $[0,1]$ and conditioned on the sum of the faces equal to $n/2$. Considering the "beats" relation for three such random dice, Polymath showed that each of eight possible tournaments between them is asymptotically equally likely. In particular, three dice form an intransitive cycle with probability converging to $1/4$. In this paper we prove that for four random dice not all tournaments are equally likely and the probability of a transitive tournament is strictly higher than $3/8$.

This paper introduces a model for opinion dynamics, where at each time step, randomly selected agents see their opinions - modeled as scalars in [0,1] - evolve depending on a local interaction function. In the classical Bounded Confidence Model, agents opinions get attracted when they are close enough. The proposed model extends this by adding a repulsion component, which models the effect of opinions getting further pushed away when dissimilar enough. With this repulsion component added, and under a repulsion-attraction cleavage assumption, it is shown that a new stable configuration emerges beyond the classical consensus configuration, namely the polarization configuration. More specifically, it is shown that total consensus and total polarization are the only two possible limiting configurations. The paper further provides an analysis of the infinite population regime in dimension 1 and higher, with a phase transition phenomenon conjectured and backed heuristically.