Stochastic Gradient Descent-like relaxation is equivalent to Glauber dynamics in discrete optimization and inference problems

Is Stochastic Gradient Descent (SGD) substantially different from Glauber dynamics? This is a fundamental question at the time of understanding the most used training algorithm in the field of Machine Learning, but it received no answer until now. Here we show that in discrete optimization and inference problems, the dynamics of an SGD-like algorithm resemble very closely that of Metropolis Monte Carlo with a properly chosen temperature, which depends on the mini-batch size. This quantitative matching holds both at equilibrium and in the out-of-equilibrium regime, despite the two algorithms having fundamental differences (e.g.\ SGD does not satisfy detailed balance). Such equivalence allows us to use results about performances and limits of Monte Carlo algorithms to optimize the mini-batch size in the SGD-like algorithm and make it efficient at recovering the signal in hard inference problems.

Parallel Learning by Multitasking Neural Networks

A modern challenge of Artificial Intelligence is learning multiple patterns at once (i.e.parallel learning). While this can not be accomplished by standard Hebbian associative neural networks, in this paper we show how the Multitasking Hebbian Network (a variation on theme of the Hopfield model working on sparse data-sets) is naturally able to perform this complex task. We focus on systems processing in parallel a finite (up to logarithmic growth in the size of the network) amount of patterns, mirroring the low-storage level of standard associative neural networks at work with pattern recognition. For mild dilution in the patterns, the network handles them hierarchically, distributing the amplitudes of their signals as power-laws w.r.t. their information content (hierarchical regime), while, for strong dilution, all the signals pertaining to all the patterns are raised with the same strength (parallel regime). Further, confined to the low-storage setting (i.e., far from the spin glass limit), the presence of a teacher neither alters the multitasking performances nor changes the thresholds for learning: the latter are the same whatever the training protocol is supervised or unsupervised. Results obtained through statistical mechanics, signal-to-noise technique and Monte Carlo simulations are overall in perfect agreement and carry interesting insights on multiple learning at once: for instance, whenever the cost-function of the model is minimized in parallel on several patterns (in its description via Statistical Mechanics), the same happens to the standard sum-squared error Loss function (typically used in Machine Learning).

Phase transitions in the mini-batch size for sparse and dense neural networks

The use of mini-batches of data in training artificial neural networks is nowadays very common. Despite its broad usage, theories explaining quantitatively how large or small the optimal mini-batch size should be are missing. This work presents a systematic attempt at understanding the role of the mini-batch size in training two-layer neural networks. Working in the teacher-student scenario, with a sparse teacher, and focusing on tasks of different complexity, we quantify the effects of changing the mini-batch size $m$. We find that often the generalization performances of the student strongly depend on $m$ and may undergo sharp phase transitions at a critical value $m_c$, such that for $m<m_c$ the training process fails, while for $m>m_c$ the student learns perfectly or generalizes very well the teacher. Phase transitions are induced by collective phenomena firstly discovered in statistical mechanics and later observed in many fields of science. Finding a phase transition varying the mini-batch size raises several important questions on the role of a hyperparameter which have been somehow overlooked until now.

Multi-mode fiber reservoir computing overcomes shallow neural networks classifiers

In disordered photonics, one typically tries to characterize the optically opaque material in order to be able to deliver light or perform imaging through it. Among others, multi-mode optical fibers are extensively studied because they are cheap and easy-to-handle complex devices. Here, instead, we use the reservoir computing paradigm to turn these optical tools into random projectors capable of introducing a sufficient amount of interaction to perform non-linear classification. We show that training a single logistic regression layer on the data projected by the fiber improves the accuracy with respect to learning it on the raw images. Surprisingly, the classification accuracy performed with physical measurements is higher than the one obtained using the standard transmission matrix model, a widely accepted tool to describe light transmission through disordered devices. Consistently with the current theory of deep neural networks, we also reveal that the classifier lives in a flatter region of the loss landscape when trained on fiber data. These facts suggest that multi-mode fibers exhibit robust generalization properties, thus making them promising tools for optically-aided machine learning.

Cracking nuts with a sledgehammer: when modern graph neural networks do worse than classical greedy algorithms

The recent work ``Combinatorial Optimization with Physics-Inspired Graph Neural Networks'' [Nat Mach Intell 4 (2022) 367] introduces a physics-inspired unsupervised Graph Neural Network (GNN) to solve combinatorial optimization problems on sparse graphs. To test the performances of these GNNs, the authors of the work show numerical results for two fundamental problems: maximum cut and maximum independent set (MIS). They conclude that "the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables." In this comment, we show that a simple greedy algorithm, running in almost linear time, can find solutions for the MIS problem of much better quality than the GNN. The greedy algorithm is faster by a factor of $10^4$ with respect to the GNN for problems with a million variables. We do not see any good reason for solving the MIS with these GNN, as well as for using a sledgehammer to crack nuts. In general, many claims of superiority of neural networks in solving combinatorial problems are at risk of being not solid enough, since we lack standard benchmarks based on really hard problems. We propose one of such hard benchmarks, and we hope to see future neural network optimizers tested on these problems before any claim of superiority is made.

Nonequilibrium Monte Carlo for unfreezing variables in hard combinatorial optimization

Optimizing highly complex cost/energy functions over discrete variables is at the heart of many open problems across different scientific disciplines and industries. A major obstacle is the emergence of many-body effects among certain subsets of variables in hard instances leading to critical slowing down or collective freezing for known stochastic local search strategies. An exponential computational effort is generally required to unfreeze such variables and explore other unseen regions of the configuration space. Here, we introduce a quantum-inspired family of nonlocal Nonequilibrium Monte Carlo (NMC) algorithms by developing an adaptive gradient-free strategy that can efficiently learn key instance-wise geometrical features of the cost function. That information is employed on-the-fly to construct spatially inhomogeneous thermal fluctuations for collectively unfreezing variables at various length scales, circumventing costly exploration versus exploitation trade-offs. We apply our algorithm to two of the most challenging combinatorial optimization problems: random k-satisfiability (k-SAT) near the computational phase transitions and Quadratic Assignment Problems (QAP). We observe significant speedup and robustness over both specialized deterministic solvers and generic stochastic solvers. In particular, for 90% of random 4-SAT instances we find solutions that are inaccessible for the best specialized deterministic algorithm known as Survey Propagation (SP) with an order of magnitude improvement in the quality of solutions for the hardest 10% instances. We also demonstrate two orders of magnitude improvement in time-to-solution over the state-of-the-art generic stochastic solver known as Adaptive Parallel Tempering (APT).

How to iron out rough landscapes and get optimal performances: Replicated Gradient Descent and its application to tensor PCA

In many high-dimensional estimation problems the main task consists in minimizing a cost function, which is often strongly non-convex when scanned in the space of parameters to be estimated. A standard solution to flatten the corresponding rough landscape consists in summing the losses associated to different data points and obtain a smoother empirical risk. Here we propose a complementary method that works for a single data point. The main idea is that a large amount of the roughness is uncorrelated in different parts of the landscape. One can then substantially reduce the noise by evaluating an empirical average of the gradient obtained as a sum over many random independent positions in the space of parameters to be optimized. We present an algorithm, called Replicated Gradient Descent, based on this idea and we apply it to tensor PCA, which is a very hard estimation problem. We show that Replicated Gradient Descent over-performs physical algorithms such as gradient descent and approximate message passing and matches the best algorithmic thresholds known so far, obtained by tensor unfolding and methods based on sum-of-squares.

SpaRTA - Tracking across occlusions via global partitioning of 3D clouds of points

Any 3D tracking algorithm has to deal with occlusions: multiple targets get so close to each other that the loss of their identities becomes likely. In the best case scenario, trajectories are interrupted, thus curbing the completeness of the data-set; in the worse case scenario, identity switches arise, potentially affecting in severe ways the very quality of the data. Here, we present a novel tracking method that addresses the problem of occlusions within large groups of featureless objects by means of three steps: i) it represents each target as a cloud of points in 3D; ii) once a 3D cluster corresponding to an occlusion occurs, it defines a partitioning problem by introducing a cost function that uses both attractive and repulsive spatio-temporal proximity links; iii) it minimizes the cost function through a semi-definite optimization technique specifically designed to cope with link frustration. The algorithm is independent of the specific experimental method used to collect the data. By performing tests on public data-sets, we show that the new algorithm produces a significant improvement over the state-of-the-art tracking methods, both by reducing the number of identity switches and by increasing the accuracy of the actual positions of the targets in real space.

Improving variational methods via pairwise linear response identities

Inference methods are often formulated as variational approximations: these approximations allow easy evaluation of statistics by marginalization or linear response, but these estimates can be inconsistent. We show that by introducing constraints on covariance, one can ensure consistency of linear response with the variational parameters, and in so doing inference of marginal probability distributions is improved. For the Bethe approximation and its generalizations, improvements are achieved with simple choices of the constraints. The approximations are presented as variational frameworks; iterative procedures related to message passing are provided for finding the minima.

The backtracking survey propagation algorithm for solving random K-SAT problems

Discrete combinatorial optimization has a central role in many scientific disciplines, however, for hard problems we lack linear time algorithms that would allow us to solve very large instances. Moreover, it is still unclear what are the key features that make a discrete combinatorial optimization problem hard to solve. Here we study random K-satisfiability problems with $K=3,4$, which are known to be very hard close to the SAT-UNSAT threshold, where problems stop having solutions. We show that the backtracking survey propagation algorithm, in a time practically linear in the problem size, is able to find solutions very close to the threshold, in a region unreachable by any other algorithm. All solutions found have no frozen variables, thus supporting the conjecture that only unfrozen solutions can be found in linear time, and that a problem becomes impossible to solve in linear time when all solutions contain frozen variables.