arXiv subjects

Franck Gabriel

Publications and source records attributed to Franck Gabriel.

At least 19 recordsLinked to original sources

arXiv2026

Cross-Entropy Games and Frost Training

We present Frost Training, a method for improving Monte Carlo-based policy optimization for a large family of LLM-as-a-judge tasks called Cross-Entropy Games. The key idea is to exploit the gradient of the reward function in embedding space. This signal is used in the Greedy Coordinate Gradient (GCG) jailbreaking technique; we demonstrate for the first time that it can also be used to boost model training. We validate our method using GRPO training for maximum-likelihood infilling. Frost Training improves the model's ability to generate high-scoring outputs, reaching higher maximum scores in a best-of-k setting, and does so at an increased speed.

cs.AI↗

arXiv2026

Cognitive Training for Language Models: Towards General Capabilities via Cross-Entropy Games

Defining a constructive process to build general capabilities for language models in an automatic manner is considered an open problem in artificial intelligence. Towards this, we consider the problem of building a curriculum of tasks that grows a model via relevant skill discovery. We provide a concrete framework for this task, using a family of tasks called Cross-Entropy Games, which we postulate is universal in a suitable sense. We show that if it is possible to grow the curriculum for relevant skill discovery by iterating a greedy optimization algorithm, then, under natural assumptions, there is essentially only one meta-objective possible (up to a few hyper-parameters). We call the resulting process cognitive training. We postulate that, given sufficiently capable language models as players and meta-samplers, cognitive training provides a principled way to relevant skill discovery; and hence to the extent general capabilities are achievable via greedy curriculum learning, cognitive training would be a solution.

math.OC↗

arXiv2025

Kernel-Smoothed Scores for Denoising Diffusion: A Bias-Variance Study

Diffusion models now set the benchmark in high-fidelity generative sampling, yet they can, in principle, be prone to memorization. In this case, their learned score overfits the finite dataset so that the reverse-time SDE samples are mostly training points. In this paper, we interpret the empirical score as a noisy version of the true score and show that its covariance matrix is asymptotically a re-weighted data PCA. In large dimension, the small time limit makes the noise variance blow up while simultaneously reducing spatial correlation. To reduce this variance, we introduce a kernel-smoothed empirical score and analyze its bias-variance trade-off. We derive asymptotic bounds on the Kullback-Leibler divergence between the true distribution and the one generated by the modified reverse SDE. Regularization on the score has the same effect as increasing the size of the training dataset, and thus helps prevent memorization. A spectral decomposition of the forward diffusion suggests better variance control under some regularity conditions of the true data distribution. Reverse diffusion with kernel-smoothed empirical score can be reformulated as a gradient descent drifted toward a Log-Exponential Double-Kernel Density Estimator (LED-KDE). This perspective highlights two regularization mechanisms taking place in denoising diffusions: an initial Gaussian kernel first diffuses mass isotropically in the ambient space, while a second kernel applied in score space concentrates and spreads that mass along the data manifold. Hence, even a straightforward regularization-without any learning-already mitigates memorization and enhances generalization. Numerically, we illustrate our results with several experiments on synthetic and MNIST datasets.

cs.LG↗

arXiv2022

Feature Learning in $L_{2}$-regularized DNNs: Attraction/Repulsion and Sparsity

We study the loss surface of DNNs with $L_{2}$ regularization. We show that the loss in terms of the parameters can be reformulated into a loss in terms of the layerwise activations $Z_{\ell}$ of the training set. This reformulation reveals the dynamics behind feature learning: each hidden representations $Z_{\ell}$ are optimal w.r.t. to an attraction/repulsion problem and interpolate between the input and output representations, keeping as little information from the input as necessary to construct the activation of the next layer. For positively homogeneous non-linearities, the loss can be further reformulated in terms of the covariances of the hidden representations, which takes the form of a partially convex optimization over a convex cone. This second reformulation allows us to prove a sparsity result for homogeneous DNNs: any local minimum of the $L_{2}$-regularized loss can be achieved with at most $N(N+1)$ neurons in each hidden layer (where $N$ is the size of the training set). We show that this bound is tight by giving an example of a local minimum that requires $N^{2}/4$ hidden neurons. But we also observe numerically that in more traditional settings much less than $N^{2}$ neurons are required to reach the minima.

stat.ML↗

arXiv2022

Freeness of type $B$ and conditional freeness for random matrices

The asymptotic freeness of independent unitarily invariant $N\times N$ random matrices holds in expectation up to $O(N^{-2})$. An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type $B$. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type $B$, conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles.

math.PR↗

arXiv2021

Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity

The dynamics of Deep Linear Networks (DLNs) is dramatically affected by the variance $\sigma^2$ of the parameters at initialization $\theta_0$. For DLNs of width $w$, we show a phase transition w.r.t. the scaling $\gamma$ of the variance $\sigma^2=w^{-\gamma}$ as $w\to\infty$: for large variance ($\gamma<1$), $\theta_0$ is very close to a global minimum but far from any saddle point, and for small variance ($\gamma>1$), $\theta_0$ is close to a saddle point and far from any global minimum. While the first case corresponds to the well-studied NTK regime, the second case is less understood. This motivates the study of the case $\gamma \to +\infty$, where we conjecture a Saddle-to-Saddle dynamics: throughout training, gradient descent visits the neighborhoods of a sequence of saddles, each corresponding to linear maps of increasing rank, until reaching a sparse global minimum. We support this conjecture with a theorem for the dynamics between the first two saddles, as well as some numerical experiments.

stat.ML↗

arXiv2021

Smart Proofs via Smart Contracts: Succinct and Informative Mathematical Derivations via Decentralized Markets

Modern mathematics is built on the idea that proofs should be translatable into formal proofs, whose validity is an objective question, decidable by a computer. Yet, in practice, proofs are informal and may omit many details. An agent considers a proof valid if they trust that it could be expanded into a machine-verifiable proof. A proof's validity can thus become a subjective matter and lead to a debate, which may be difficult to settle. Hence, while the concept of valid proof is well-defined, the process to establish validity is itself a complex multi-agent problem. We introduce the SPRIG protocol. SPRIG allows agents to propose and verify succinct and informative proofs in a decentralized fashion; the trust is established by agents being able to request more details in the proof steps; debates, if they arise, must isolate details of proofs and, if they persist, go down to machine-level details, where they are automatically settled. A structure of bounties and stakes is set to incentivize agents to act in good faith. We propose a game-theoretic discussion of SPRIG, showing how agents with various types of information interact, leading to a proof tree with an appropriate level of detail and to the invalidation of wrong proofs, and we discuss resilience against various attacks. We then analyze a simplified model, characterize its equilibria and compute the agents' level of trust. SPRIG is designed to run as a smart contract on a blockchain platform. This allows anonymous agents to participate in the verification debate, and to contribute with their information. The smart contract mediates the interactions, settles debates, and guarantees that bounties and stakes are paid as specified. SPRIG enables new applications, such as the issuance of bounties for open problems, and the creation of derivatives markets, allowing agents to inject more information pertaining to proofs.

cs.GT↗

arXiv2020

Order and Chaos: NTK views on DNN Normalization, Checkerboard and Boundary Artifacts

We analyze architectural features of Deep Neural Networks (DNNs) using the so-called Neural Tangent Kernel (NTK), which describes the training and generalization of DNNs in the infinite-width setting. In this setting, we show that for fully-connected DNNs, as the depth grows, two regimes appear: "order", where the (scaled) NTK converges to a constant, and "chaos", where it converges to a Kronecker delta. Extreme order slows down training while extreme chaos hinders generalization. Using the scaled ReLU as a nonlinearity, we end up in the ordered regime. In contrast, Layer Normalization brings the network into the chaotic regime. We observe a similar effect for Batch Normalization (BN) applied after the last nonlinearity. We uncover the same order and chaos modes in Deep Deconvolutional Networks (DC-NNs). Our analysis explains the appearance of so-called checkerboard patterns and border artifacts. Moving the network into the chaotic regime prevents checkerboard patterns; we propose a graph-based parametrization which eliminates border artifacts; finally, we introduce a new layer-dependent learning rate to improve the convergence of DC-NNs. We illustrate our findings on DCGANs: the ordered regime leads to a collapse of the generator to a checkerboard mode, which can be avoided by tuning the nonlinearity to reach the chaotic regime. As a result, we are able to obtain good quality samples for DCGANs without BN.

cs.LG↗

arXiv2020

Kernel Alignment Risk Estimator: Risk Prediction from Training Data

We study the risk (i.e. generalization error) of Kernel Ridge Regression (KRR) for a kernel $K$ with ridge $λ>0$ and i.i.d. observations. For this, we introduce two objects: the Signal Capture Threshold (SCT) and the Kernel Alignment Risk Estimator (KARE). The SCT $\vartheta_{K,λ}$ is a function of the data distribution: it can be used to identify the components of the data that the KRR predictor captures, and to approximate the (expected) KRR risk. This then leads to a KRR risk approximation by the KARE $ρ_{K, λ}$, an explicit function of the training data, agnostic of the true data distribution. We phrase the regression problem in a functional setting. The key results then follow from a finite-size analysis of the Stieltjes transform of general Wishart random matrices. Under a natural universality assumption (that the KRR moments depend asymptotically on the first two moments of the observations) we capture the mean and variance of the KRR predictor. We numerically investigate our findings on the Higgs and MNIST datasets for various classical kernels: the KARE gives an excellent approximation of the risk, thus supporting our universality assumption. Using the KARE, one can compare choices of Kernels and hyperparameters directly from the training set. The KARE thus provides a promising data-dependent procedure to select Kernels that generalize well.

stat.ML↗

arXiv2020

Implicit Regularization of Random Feature Models

Random Feature (RF) models are used as efficient parametric approximations of kernel methods. We investigate, by means of random matrix theory, the connection between Gaussian RF models and Kernel Ridge Regression (KRR). For a Gaussian RF model with $P$ features, $N$ data points, and a ridge $\lambda$, we show that the average (i.e. expected) RF predictor is close to a KRR predictor with an effective ridge $\tilde{\lambda}$. We show that $\tilde{\lambda} > \lambda$ and $\tilde{\lambda} \searrow \lambda$ monotonically as $P$ grows, thus revealing the implicit regularization effect of finite RF sampling. We then compare the risk (i.e. test error) of the $\tilde{\lambda}$-KRR predictor with the average risk of the $\lambda$-RF predictor and obtain a precise and explicit bound on their difference. Finally, we empirically find an extremely good agreement between the test errors of the average $\lambda$-RF predictor and $\tilde{\lambda}$-KRR predictor.

stat.ML↗

arXiv2020

Neural Tangent Kernel: Convergence and Generalization in Neural Networks

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function $f_θ$ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function $f_θ$ follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

cs.LG↗

arXiv2020

The asymptotic spectrum of the Hessian of DNN throughout training

The dynamics of DNNs during gradient descent is described by the so-called Neural Tangent Kernel (NTK). In this article, we show that the NTK allows one to gain precise insight into the Hessian of the cost of DNNs. When the NTK is fixed during training, we obtain a full characterization of the asymptotics of the spectrum of the Hessian, at initialization and during training. In the so-called mean-field limit, where the NTK is not fixed during training, we describe the first two moments of the Hessian at initialization.

cs.LG↗

arXiv2019

Scaling description of generalization with number of parameters in deep learning

Supervised deep learning involves the training of neural networks with a large number $N$ of parameters. For large enough $N$, in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsity-based arguments would suggest that the generalization error increases as $N$ grows past a certain threshold $N^{*}$. Instead, empirical studies have shown that in the over-parametrized regime, generalization error keeps decreasing with $N$. We resolve this paradox through a new framework. We rely on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations $\|f_{N}-\bar{f}_{N}\|\sim N^{-1/4}$ of the neural net output function $f_{N}$ around its expectation $\bar{f}_{N}$. These affect the generalization error $ε_{N}$ for classification: under natural assumptions, it decays to a plateau value $ε_{\infty}$ in a power-law fashion $\sim N^{-1/2}$. This description breaks down at a so-called jamming transition $N=N^{*}$. At this threshold, we argue that $\|f_{N}\|$ diverges. This result leads to a plausible explanation for the cusp in test error known to occur at $N^{*}$. Our results are confirmed by extensive empirical observations on the MNIST and CIFAR image datasets. Our analysis finally suggests that, given a computational envelope, the smallest generalization error is obtained using several networks of intermediate sizes, just beyond $N^{*}$, and averaging their outputs.

cond-mat.dis-nn↗

arXiv2019

Geometric stochastic heat equations

We consider a natural class of $\mathbf{R}^d$-valued one-dimensional stochastic PDEs driven by space-time white noise that is formally invariant under the action of the diffeomorphism group on $\mathbf{R}^d$. This class contains in particular the KPZ equation, the multiplicative stochastic heat equation, the additive stochastic heat equation, and rough Burgers-type equations. We exhibit a one-parameter family of solution theories with the following properties: - For all SPDEs in our class for which a solution was previously available, every solution in our family coincides with the previously constructed solution, whether that was obtained using It\^o calculus (additive and multiplicative stochastic heat equation), rough path theory (rough Burgers-type equations), or the Hopf-Cole transform (KPZ equation). - Every solution theory is equivariant under the action of the diffeomorphism group, i.e. identities obtained by formal calculations treating the noise as a smooth function are valid. - Every solution theory satisfies an analogue of It\^o's isometry. - The counterterms leading to our solution theories vanish at points where the equation agrees to leading order with the additive stochastic heat equation. In particular, points 2 and 3 show that, surprisingly, our solution theories enjoy properties analogous to those holding for both the Stratonovich and It\^o interpretations of SDEs simultaneously. For the natural noisy perturbation of the harmonic map flow with values in an arbitrary Riemannian manifold, we show that all these solution theories coincide. In particular, this allows us to conjecturally identify the process associated to the Markov extension of the Dirichlet form corresponding to the $L^2$-gradient flow for the Brownian loop measure.

math.PR↗

arXiv2018

Insider Trading with Penalties

We consider a one-period Kyle (1985) framework where the insider can be subject to a penalty if she trades. We establish existence and uniqueness of equilibrium for virtually any penalty function when noise is uniform. In equilibrium, the demand of the insider and the price functions are in general non-linear and remain analytically tractable because the expected price function is linear. We use this result to investigate the trade off between price efficiency and 'fairness': we consider a regulator that wants to minimise post-trade standard deviation for a given level of uninformed traders' losses. The minimisation is over the function space of penalties; for each possible penalty, our existence and uniqueness theorem allows to define unambiguously the post-trade standard deviation and the uninformed traders' losses that prevail in equilibrium.Optimal penalties are characterized in closed-form. They must increase quickly with the magnitude of the insider's order for small orders and become flat for large orders: in cases where the fundamental realizes at very high or very low values, the insider finds it optimal to trade despite the high penalty. Although such trades-if they occur-are costly for liquidity traders, they signal extreme events and therefore incorporate a lot of information into prices. We generalize this result in two directions by imposing a budget constraint on the regulator and considering the cases of either non-pecuniary or pecuniary penalties. In the first case, we establish that optimal penalties are a subset of the previously optimal penalties: the patterns of equilibrium trade volumes and prices is unchanged. In the second case, we also fully characterize the constrained efficient points and penalties and show that new patterns emerge in the demand schedules of the insider trader and the associated price functions.

q-fin.TR↗

arXiv2018

Large permutation invariant random matrices are asymptotically free over the diagonal

We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the operator norm assumption to an estimate on sums associated to graphs of matrices, further extending the range of applications (for example, to Wigner matrices with exploding moments and so the sparse regime of the Erd\H{o}s-R\'{e}nyi model). The result still holds even if the matrices are multiplied entrywise by bounded random variables (for example, as in the case of matrices with a variance profile and percolation models).

math.PR↗

arXiv2017

The Makeenko-Migdal equation for Yang-Mills theory on compact surfaces

We prove the Makeenko-Migdal equation for two-dimensional Euclidean Yang-Mills theory on an arbitrary compact surface, possibly with boundary. In particular, we show that two of the proofs given by the first, third, and fourth authors for the plane case extend essentially without change to compact surfaces.

math-ph↗

arXiv2016

Combinatorial theory of permutation-invariant random matrices II: cumulants, freeness and Levy processes

The $\mathcal{A}$-tracial algebras are algebras endowed with multi-linear forms, compatible with the product, and indexed by partitions. Using the notion of $\mathcal{A}$-cumulants, we define and study the $\mathcal{A}$-freeness property which generalizes the independence and freeness properties, and some invariance properties which model the invariance by conjugation for random matrices. A central limit theorem is given in the setting of $\mathcal{A}$-tracial algebras. A generalization of the normalized moments for random matrices is used to define convergence in $\mathcal{A}$-distribution: this allows us to apply the theory of $\mathcal{A}$-tracial algebras to random matrices. This study is deepened with the use of $\mathcal{A}$-finite dimensional cumulants which are related to some dualities as the Schur-Weyl's duality. This gives a unified and simple framework in order to understand families of random matrices which are invariant by conjugation in law by any group whose associated tensor category is spanned by partitions, this includes for example the unitary groups or the symmetric groups. Among the various by-products, we prove that unitary invariance and convergence in distribution implies convergence in $\mathcal{P}$-distribution. Besides, a new notion of strong asymptotic invariance and independence are shown to imply $\mathcal{A}$-freeness. Finally, we prove general theorems about convergence of matrix-valued additive and multiplicative L{é}vy processes which are invariant in law by conjugation by the symmetric group. Using these results, a unified point of view on the study of matricial L{é}vy processes is given.

math.PR↗