Yann Ollivier

Benjamin Scellier, Siddhartha Mishra, Yoshua Bengio et al.

This work establishes that a physical system can perform statistical learning without gradient computations, via an Agnostic Equilibrium Propagation (Aeqprop) procedure that combines energy minimization, homeostatic control, and nudging towards the correct response. In Aeqprop, the specifics of the system do not have to be known: the procedure is based only on external manipulations, and produces a stochastic gradient descent without explicit gradient computations. Thanks to nudging, the system performs a true, order-one gradient step for each training sample, in contrast with order-zero methods like reinforcement or evolutionary strategies, which rely on trial and error. This procedure considerably widens the range of potential hardware for statistical learning to any system with enough controllable parameters, even if the details of the system are poorly known. Aeqprop also establishes that in natural (bio)physical systems, genuine gradient-based statistical learning may result from generic, relatively simple mechanisms, without backpropagation and its requirement for analytic knowledge of partial derivatives.

Ahmed Touati, Jérémy Rapin, Yann Ollivier

A zero-shot RL agent is an agent that can solve any RL task in a given environment, instantly with no additional planning or learning, after an initial reward-free learning phase. This marks a shift from the reward-centric RL paradigm towards "controllable" agents that can follow arbitrary instructions in an environment. Current RL agents can solve families of related tasks at best, or require planning anew for each task. Strategies for approximate zero-shot RL ave been suggested using successor features (SFs) [BBQ+ 18] or forward-backward (FB) representations [TO21], but testing has been limited. After clarifying the relationships between these schemes, we introduce improved losses and new SF models, and test the viability of zero-shot RL schemes systematically on tasks from the Unsupervised RL benchmark [LYL+21]. To disentangle universal representation learning from exploration, we work in an offline setting and repeat the tests on several existing replay buffers. SFs appear to suffer from the choice of the elementary state features. SFs with Laplacian eigenfunctions do well, while SFs based on auto-encoders, inverse curiosity, transition models, low-rank transition matrix, contrastive learning, or diversity (APS), perform unconsistently. In contrast, FB representations jointly learn the elementary and successor features from a single, principled criterion. They perform best and consistently across the board, reaching 85% of supervised RL performance with a good replay buffer, in a zero-shot manner.

Shobhita Sundaram, John Quan, Ariel Kwiatkowski et al.

Can a model learn to escape its own learning plateau? Reinforcement learning methods for finetuning large reasoning models stall on datasets with low initial success rates, and thus little training signal. We investigate a fundamental question: Can a pretrained LLM leverage latent knowledge to generate an automated curriculum for problems it cannot solve? To explore this, we design SOAR: A self-improvement framework designed to surface these pedagogical signals through meta-RL. A teacher copy of the model proposes synthetic problems for a student copy, and is rewarded with its improvement on a small subset of hard problems. Critically, SOAR grounds the curriculum in measured student progress rather than intrinsic proxy rewards. Our study on the hardest subsets of mathematical benchmarks (0/128 success) reveals three core findings. First, we show that it is possible to realize bi-level meta-RL that unlocks learning under sparse, binary rewards by sharpening a latent capacity of pretrained models to generate useful stepping stones. Second, grounded rewards outperform intrinsic reward schemes used in prior LLM self-play, reliably avoiding the instability and diversity collapse modes they typically exhibit. Third, analyzing the generated questions reveals that structural quality and well-posedness are more critical for learning progress than solution correctness. Our results suggest that the ability to generate useful stepping stones does not require the preexisting ability to actually solve the hard problems, paving a principled path to escape reasoning plateaus without additional curated data.

Léonard Blier, Pierre Wolinski, Yann Ollivier

Hyperparameter tuning is a bothersome step in the training of deep learning models. One of the most sensitive hyperparameters is the learning rate of the gradient descent. We present the 'All Learning Rates At Once' (Alrao) optimization method for neural networks: each unit or feature in the network gets its own learning rate sampled from a random distribution spanning several orders of magnitude. This comes at practically no computational cost. Perhaps surprisingly, stochastic gradient descent (SGD) with Alrao performs close to SGD with an optimally tuned learning rate, for various architectures and problems. Alrao could save time when testing deep learning models: a range of models could be quickly assessed with Alrao, and the most promising models could then be trained more extensively. This text comes with a PyTorch implementation of the method, which can be plugged on an existing PyTorch model: https://github.com/leonardblier/alrao .

Edoardo Cetin, Ahmed Touati, Yann Ollivier

The forward-backward representation (FB) is a recently proposed framework (Touati et al., 2023; Touati & Ollivier, 2021) to train behavior foundation models (BFMs) that aim at providing zero-shot efficient policies for any new task specified in a given reinforcement learning (RL) environment, without training for each new task. Here we address two core limitations of FB model training. First, FB, like all successor-feature-based methods, relies on a linear encoding of tasks: at test time, each new reward function is linearly projected onto a fixed set of pre-trained features. This limits expressivity as well as precision of the task representation. We break the linearity limitation by introducing auto-regressive features for FB, which let finegrained task features depend on coarser-grained task information. This can represent arbitrary nonlinear task encodings, thus significantly increasing expressivity of the FB framework. Second, it is well-known that training RL agents from offline datasets often requires specific techniques.We show that FB works well together with such offline RL techniques, by adapting techniques from (Nair et al.,2020b; Cetin et al., 2024) for FB. This is necessary to get non-flatlining performance in some datasets, such as DMC Humanoid. As a result, we produce efficient FB BFMs for a number of new environments. Notably, in the D4RL locomotion benchmark, the generic FB agent matches the performance of standard single-task offline agents (IQL, XQL). In many setups, the offline techniques are needed to get any decent performance at all. The auto-regressive features have a positive but moderate impact, concentrated on tasks requiring spatial precision and task generalization beyond the behaviors represented in the trainset.

Ariel Kwiatkowski, Natasha Butt, Ismail Labiad et al.

Fine-tuning large language models (LLMs) on reasoning benchmarks via reinforcement learning requires a specific reward function, often binary, for each benchmark. This comes with two potential limitations: the need to design the reward, and the potentially sparse nature of binary rewards. Here, we systematically investigate rewards derived from the probability or log-probability of emitting the reference answer (or any other prompt continuation present in the data), which have the advantage of not relying on specific verifiers and being available at scale. Several recent works have advocated for the use of similar rewards (e.g., VeriFree, JEPO, RLPR, NOVER). We systematically compare variants of likelihood-based rewards with standard baselines, testing performance both on standard mathematical reasoning benchmarks, and on long-form answers where no external verifier is available. We find that using the log-probability of the reference answer as the reward for chain-of-thought (CoT) learning is the only option that performs well in all setups. This reward is also consistent with the next-token log-likelihood loss used during pretraining. In verifiable settings, log-probability rewards bring comparable or better success rates than reinforcing with standard binary rewards, and yield much better perplexity. In non-verifiable settings, they perform on par with SFT. On the other hand, methods based on probability, such as VeriFree, flatline on non-verifiable settings due to vanishing probabilities of getting the correct answer. Overall, this establishes log-probability rewards as a viable method for CoT fine-tuning, bridging the short, verifiable and long, non-verifiable answer settings.

Natasha Butt, Ariel Kwiatkowski, Ismail Labiad et al.

The use of continuous instead of discrete tokens during the Chain-of-Thought (CoT) phase of reasoning LLMs has garnered attention recently, based on the intuition that a continuous mixture of discrete tokens could simulate a superposition of several reasoning paths simultaneously. Theoretical results have formally proven that continuous tokens have much greater expressivity and can solve specific problems more efficiently. However, practical use of continuous tokens has been limited by strong training difficulties: previous works either just use continuous tokens at inference time on a pre-trained discrete-token model, or must distill the continuous CoT from ground-truth discrete CoTs and face computational costs that limit the CoT to very few tokens. This is the first work introducing a scalable method to learn continuous CoTs via reinforcement learning (RL), without distilling from reference discrete CoTs. We use "soft" tokens: mixtures of tokens together with noise on the input embedding to provide RL exploration. Computational overhead is minimal, enabling us to learn continuous CoTs with hundreds of tokens. On math reasoning benchmarks with Llama and Qwen models up to 8B, training with continuous CoTs match discrete-token CoTs for pass@1 and surpass them for pass@32, showing greater CoT diversity. In systematic comparisons, the best-performing scenario is to train with continuous CoT tokens then use discrete tokens for inference, meaning the "soft" models can be deployed in a standard way. Finally, we show continuous CoT RL training better preserves the predictions of the base model on out-of-domain tasks, thus providing a softer touch to the base model.

Yann Ollivier

In reinforcement learning, universal successor features (SFs) are a way to provide zero-shot adaptation to new tasks at test time: they provide optimal policies for all downstream reward functions lying in the linear span of a set of base features. But it is unclear what constitutes a good set of base features, that could be useful for a wide set of downstream tasks beyond their linear span. Laplacian eigenfunctions (the eigenfunctions of $Δ+Δ^\ast$ with $Δ$ the Laplacian operator of some reference policy and $Δ^\ast$ that of the time-reversed dynamics) have been argued to play a role, and offer good empirical performance. Here, for the first time, we identify the optimal base features based on an objective criterion of downstream performance, in a non-tautological way without assuming the downstream tasks are linear in the features. We do this for three generic classes of downstream tasks: reaching a random goal state, dense random Gaussian rewards, and random ``scattered'' sparse rewards. The features yielding optimal expected downstream performance turn out to be the \emph{same} for these three task families. They do not coincide with Laplacian eigenfunctions in general, though they can be expressed from $Δ$: in the simplest case (deterministic environment and decay factor $γ$ close to $1$), they are the eigenfunctions of $Δ^{-1}+(Δ^{-1})^\ast$. We obtain these results under an assumption of large behavior cloning regularization with respect to a reference policy, a setting often used for offline RL. Along the way, we get new insights into KL-regularized\option{natural} policy gradient, and into the lack of SF information in the norm of Bellman gaps.

Yann Ollivier

Zero-shot reinforcement learning (RL) methods aim at instantly producing a behavior for an RL task in a given environment, from a description of the reward function. These methods are usually tested by evaluating their average performance on a series of downstream tasks. Yet they cannot be trained directly for that objective, unless the distribution of downstream tasks is known. Existing approaches either use other learning criteria [BBQ+ 18, TRO23, TO21, HDB+ 19], or explicitly set a prior on downstream tasks, such as reward functions given by a random neural network [FPAL24]. Here we prove that the zero-shot RL loss can be optimized directly, for a range of non-informative priors such as white noise rewards, temporally smooth rewards, ``scattered'' sparse rewards, or a combination of those. Thus, it is possible to learn the optimal zero-shot features algorithmically, for a wide mixture of priors. Surprisingly, the white noise prior leads to an objective almost identical to the one in VISR [HDB+19], via a different approach. This shows that some seemingly arbitrary choices in VISR, such as Von Mises--Fisher distributions, do maximize downstream performance. This also suggests more efficient ways to tackle the VISR objective. Finally, we discuss some consequences and limitations of the zero-shot RL objective, such as its tendency to produce narrow optimal features if only using Gaussian dense reward priors.

Edoardo Cetin, Andrea Tirinzoni, Matteo Pirotta et al.

Offline reinforcement learning algorithms have proven effective on datasets highly connected to the target downstream task. Yet, leveraging a novel testbed (MOOD) in which trajectories come from heterogeneous sources, we show that existing methods struggle with diverse data: their performance considerably deteriorates as data collected for related but different tasks is simply added to the offline buffer. In light of this finding, we conduct a large empirical study where we formulate and test several hypotheses to explain this failure. Surprisingly, we find that scale, more than algorithmic considerations, is the key factor influencing performance. We show that simple methods like AWAC and IQL with increased network size overcome the paradoxical failure modes from the inclusion of additional data in MOOD, and notably outperform prior state-of-the-art algorithms on the canonical D4RL benchmark.

Léonard Blier, Yann Ollivier

In multi-goal reinforcement learning (RL) settings, the reward for each goal is sparse, and located in a small neighborhood of the goal. In large dimension, the probability of reaching a reward vanishes and the agent receives little learning signal. Methods such as Hindsight Experience Replay (HER) tackle this issue by also learning from realized but unplanned-for goals. But HER is known to introduce bias, and can converge to low-return policies by overestimating chancy outcomes. First, we vindicate HER by proving that it is actually unbiased in deterministic environments, such as many optimal control settings. Next, for stochastic environments in continuous spaces, we tackle sparse rewards by directly taking the infinitely sparse reward limit. We fully formalize the problem of multi-goal RL with infinitely sparse Dirac rewards at each goal. We introduce unbiased deep Q-learning and actor-critic algorithms that can handle such infinitely sparse rewards, and test them in toy environments.

Ahmed Touati, Yann Ollivier

We introduce the forward-backward (FB) representation of the dynamics of a reward-free Markov decision process. It provides explicit near-optimal policies for any reward specified a posteriori. During an unsupervised phase, we use reward-free interactions with the environment to learn two representations via off-the-shelf deep learning methods and temporal difference (TD) learning. In the test phase, a reward representation is estimated either from observations or an explicit reward description (e.g., a target state). The optimal policy for that reward is directly obtained from these representations, with no planning. We assume access to an exploration scheme or replay buffer for the first phase. The corresponding unsupervised loss is well-principled: if training is perfect, the policies obtained are provably optimal for any reward function. With imperfect training, the sub-optimality is proportional to the unsupervised approximation error. The FB representation learns long-range relationships between states and actions, via a predictive occupancy map, without having to synthesize states as in model-based approaches. This is a step towards learning controllable agents in arbitrary black-box stochastic environments. This approach compares well to goal-oriented RL algorithms on discrete and continuous mazes, pixel-based MsPacman, and the FetchReach virtual robot arm. We also illustrate how the agent can immediately adapt to new tasks beyond goal-oriented RL.

Léonard Blier, Corentin Tallec, Yann Ollivier

In reinforcement learning, temporal difference-based algorithms can be sample-inefficient: for instance, with sparse rewards, no learning occurs until a reward is observed. This can be remedied by learning richer objects, such as a model of the environment, or successor states. Successor states model the expected future state occupancy from any given state for a given policy and are related to goal-dependent value functions, which learn how to reach arbitrary states. We formally derive the temporal difference algorithm for successor state and goal-dependent value function learning, either for discrete or for continuous environments with function approximation. Especially, we provide finite-variance estimators even in continuous environments, where the reward for exactly reaching a goal state becomes infinitely sparse. Successor states satisfy more than just the Bellman equation: a backward Bellman operator and a Bellman-Newton (BN) operator encode path compositionality in the environment. The BN operator is akin to second-order gradient descent methods and provides the true update of the value function when acquiring more observations, with explicit tabular bounds. In the tabular case and with infinitesimal learning rates, mixing the usual and backward Bellman operators provably improves eigenvalues for asymptotic convergence, and the asymptotic convergence of the BN operator is provably better than TD, with a rate independent from the environment. However, the BN method is more complex and less robust to sampling noise. Finally, a forward-backward (FB) finite-rank parameterization of successor states enjoys reduced variance and improved samplability, provides a direct model of the value function, has fully understood fixed points corresponding to long-range dependencies, approximates the BN method, and provides two canonical representations of states as a byproduct.

Pierre-Yves Massé, Yann Ollivier

We prove local convergence of several notable gradient descent algorithms used in machine learning, for which standard stochastic gradient descent theory does not apply directly. This includes, first, online algorithms for recurrent models and dynamical systems, such as \emph{Real-time recurrent learning} (RTRL) and its computationally lighter approximations NoBackTrack and UORO; second, several adaptive algorithms such as RMSProp, online natural gradient, and Adam with $β^2\to 1$.Despite local convergence being a relatively weak requirement for a new optimization algorithm, no local analysis was available for these algorithms, as far as we knew. Analysis of these algorithms does not immediately follow from standard stochastic gradient (SGD) theory. In fact, Adam has been proved to lack local convergence in some simple situations \citep{j.2018on}. For recurrent models, online algorithms modify the parameter while the model is running, which further complicates the analysis with respect to simple SGD.Local convergence for these various algorithms results from a single, more general set of assumptions, in the setup of learning dynamical systems online. Thus, these results can cover other variants of the algorithms considered.We adopt an "ergodic" rather than probabilistic viewpoint, working with empirical time averages instead of probability distributions. This is more data-agnostic and creates differences with respect to standard SGD theory, especially for the range of possible learning rates. For instance, with cycling or per-epoch reshuffling over a finite dataset instead of pure i.i.d.\ sampling with replacement, empirical averages of gradients converge at rate $1/T$ instead of $1/\sqrt{T}$ (cycling acts as a variance reduction method), theoretically allowing for larger learning rates than in SGD.

Pierre Wolinski, Guillaume Charpiat, Yann Ollivier

In machine learning, it is common to optimize the parameters of a probabilistic model, modulated by an ad hoc regularization term that penalizes some values of the parameters. Regularization terms appear naturally in Variational Inference, a tractable way to approximate Bayesian posteriors: the loss to optimize contains a Kullback--Leibler divergence term between the approximate posterior and a Bayesian prior. We fully characterize the regularizers that can arise according to this procedure, and provide a systematic way to compute the prior corresponding to a given penalty. Such a characterization can be used to discover constraints over the penalty function, so that the overall procedure remains Bayesian.

Alexandre Sablayrolles, Matthijs Douze, Yann Ollivier et al.

Membership inference determines, given a sample and trained parameters of a machine learning model, whether the sample was part of the training set. In this paper, we derive the optimal strategy for membership inference with a few assumptions on the distribution of the parameters. We show that optimal attacks only depend on the loss function, and thus black-box attacks are as good as white-box attacks. As the optimal strategy is not tractable, we provide approximations of it leading to several inference methods, and show that existing membership inference methods are coarser approximations of this optimal strategy. Our membership attacks outperform the state of the art in various settings, ranging from a simple logistic regression to more complex architectures and datasets, such as ResNet-101 and Imagenet.

Joshua Romoff, Peter Henderson, Ahmed Touati et al.

In many finite horizon episodic reinforcement learning (RL) settings, it is desirable to optimize for the undiscounted return - in settings like Atari, for instance, the goal is to collect the most points while staying alive in the long run. Yet, it may be difficult (or even intractable) mathematically to learn with this target. As such, temporal discounting is often applied to optimize over a shorter effective planning horizon. This comes at the risk of potentially biasing the optimization target away from the undiscounted goal. In settings where this bias is unacceptable - where the system must optimize for longer horizons at higher discounts - the target of the value function approximator may increase in variance leading to difficulties in learning. We present an extension of temporal difference (TD) learning, which we call TD($Δ$), that breaks down a value function into a series of components based on the differences between value functions with smaller discount factors. The separation of a longer horizon value function into these components has useful properties in scalability and performance. We discuss these properties and show theoretic and empirical improvements over standard TD learning in certain settings.

Corentin Tallec, Léonard Blier, Yann Ollivier

Despite remarkable successes, Deep Reinforcement Learning (DRL) is not robust to hyperparameterization, implementation details, or small environment changes (Henderson et al. 2017, Zhang et al. 2018). Overcoming such sensitivity is key to making DRL applicable to real world problems. In this paper, we identify sensitivity to time discretization in near continuous-time environments as a critical factor; this covers, e.g., changing the number of frames per second, or the action frequency of the controller. Empirically, we find that Q-learning-based approaches such as Deep Q- learning (Mnih et al., 2015) and Deep Deterministic Policy Gradient (Lillicrap et al., 2015) collapse with small time steps. Formally, we prove that Q-learning does not exist in continuous time. We detail a principled way to build an off-policy RL algorithm that yields similar performances over a wide range of time discretizations, and confirm this robustness empirically.

Yann Ollivier

The extended Kalman filter is perhaps the most standard tool to estimate in real time the state of a dynamical system from noisy measurements of some function of the system, with extensive practical applications (such as position tracking via GPS). While the plain Kalman filter for linear systems is well-understood, the extended Kalman filter relies on linearizations which have been debated. We recover the exact extended Kalman filter equations from first principles in statistical learning: the extended Kalman filter is equal to Amari's online natural gradient, applied in the space of trajectories of the system. Namely, each possible trajectory of the dynamical system defines a probability law over possible observations. In principle this makes it possible to treat the underlying trajectory as the parameter of a statistical model of the observations. Then the parameter can be learned by gradient ascent on the log-likelihood of observations, as they become available. Using Amari's natural gradient from information geometry (a gradient descent preconditioned with the Fisher matrix, which provides parameterization-invariance) exactly recovers the extended Kalman filter. This applies only to a particular choice of process noise in the Kalman filter, namely, taking noise proportional to the posterior covariance - a canonical choice in the absence of specific model information.

Thomas Lucas, Corentin Tallec, Jakob Verbeek et al.

Generative adversarial networks (GANs) are pow- erful generative models based on providing feed- back to a generative network via a discriminator network. However, the discriminator usually as- sesses individual samples. This prevents the dis- criminator from accessing global distributional statistics of generated samples, and often leads to mode dropping: the generator models only part of the target distribution. We propose to feed the discriminator with mixed batches of true and fake samples, and train it to predict the ratio of true samples in the batch. The latter score does not depend on the order of samples in a batch. Rather than learning this invariance, we introduce a generic permutation-invariant discriminator ar- chitecture. This architecture is provably a uni- versal approximator of all symmetric functions. Experimentally, our approach reduces mode col- lapse in GANs on two synthetic datasets, and obtains good results on the CIFAR10 and CelebA datasets, both qualitatively and quantitatively.

Yann Ollivier

In reinforcement learning, temporal difference (TD) is the most direct algorithm to learn the value function of a policy. For large or infinite state spaces, exact representations of the value function are usually not available, and it must be approximated by a function in some parametric family. However, with \emph{nonlinear} parametric approximations (such as neural networks), TD is not guaranteed to converge to a good approximation of the true value function within the family, and is known to diverge even in relatively simple cases. TD lacks an interpretation as a stochastic gradient descent of an error between the true and approximate value functions, which would provide such guarantees. We prove that approximate TD is a gradient descent provided the current policy is \emph{reversible}. This holds even with nonlinear approximations. A policy with transition probabilities $P(s,s')$ between states is reversible if there exists a function $μ$ over states such that $\frac{P(s,s')}{P(s',s)}=\frac{μ(s')}{μ(s)}$. In particular, every move can be undone with some probability. This condition is restrictive; it is satisfied, for instance, for a navigation problem in any unoriented graph. In this case, approximate TD is exactly a gradient descent of the \emph{Dirichlet norm}, the norm of the difference of \emph{gradients} between the true and approximate value functions. The Dirichlet norm also controls the bias of approximate policy gradient. These results hold even with no decay factor ($γ=1$) and do not rely on contractivity of the Bellman operator, thus proving stability of TD even with $γ=1$ for reversible policies.

Corentin Tallec, Yann Ollivier

Successful recurrent models such as long short-term memories (LSTMs) and gated recurrent units (GRUs) use ad hoc gating mechanisms. Empirically these models have been found to improve the learning of medium to long term temporal dependencies and to help with vanishing gradient issues. We prove that learnable gates in a recurrent model formally provide quasi- invariance to general time transformations in the input data. We recover part of the LSTM architecture from a simple axiomatic approach. This result leads to a new way of initializing gate biases in LSTMs and GRUs. Ex- perimentally, this new chrono initialization is shown to greatly improve learning of long term dependencies, with minimal implementation effort.

Léonard Blier, Yann Ollivier

Solomonoff's general theory of inference and the Minimum Description Length principle formalize Occam's razor, and hold that a good model of data is a model that is good at losslessly compressing the data, including the cost of describing the model itself. Deep neural networks might seem to go against this principle given the large number of parameters to be encoded. We demonstrate experimentally the ability of deep neural networks to compress the training data even when accounting for parameter encoding. The compression viewpoint originally motivated the use of variational methods in neural networks. Unexpectedly, we found that these variational methods provide surprisingly poor compression bounds, despite being explicitly built to minimize such bounds. This might explain the relatively poor practical performance of variational methods in deep learning. On the other hand, simple incremental encoding methods yield excellent compression values on deep networks, vindicating Solomonoff's approach.

Carl-Johann Simon-Gabriel, Yann Ollivier, Léon Bottou et al.

Over the past few years, neural networks were proven vulnerable to adversarial images: targeted but imperceptible image perturbations lead to drastically different predictions. We show that adversarial vulnerability increases with the gradients of the training objective when viewed as a function of the inputs. Surprisingly, vulnerability does not depend on network topology: for many standard network architectures, we prove that at initialization, the $\ell_1$-norm of these gradients grows as the square root of the input dimension, leaving the networks increasingly vulnerable with growing image size. We empirically show that this dimension dependence persists after either usual or robust training, but gets attenuated with higher regularization.

Yann Ollivier

We introduce a simple algorithm, True Asymptotic Natural Gradient Optimization (TANGO), that converges to a true natural gradient descent in the limit of small learning rates, without explicit Fisher matrix estimation. For quadratic models the algorithm is also an instance of averaged stochastic gradient, where the parameter is a moving average of a "fast", constant-rate gradient descent. TANGO appears as a particular de-linearization of averaged SGD, and is sometimes quite different on non-quadratic models. This further connects averaged SGD and natural gradient, both of which are arguably optimal asymptotically. In large dimension, small learning rates will be required to approximate the natural gradient well. Still, this shows it is possible to get arbitrarily close to exact natural gradient descent with a lightweight algorithm.

Gaétan Marceau-Caron, Yann Ollivier

One way to avoid overfitting in machine learning is to use model parameters distributed according to a Bayesian posterior given the data, rather than the maximum likelihood estimator. Stochastic gradient Langevin dynamics (SGLD) is one algorithm to approximate such Bayesian posteriors for large models and datasets. SGLD is a standard stochastic gradient descent to which is added a controlled amount of noise, specifically scaled so that the parameter converges in law to the posterior distribution [WT11, TTV16]. The posterior predictive distribution can be approximated by an ensemble of samples from the trajectory. Choice of the variance of the noise is known to impact the practical behavior of SGLD: for instance, noise should be smaller for sensitive parameter directions. Theoretically, it has been suggested to use the inverse Fisher information matrix of the model as the variance of the noise, since it is also the variance of the Bayesian posterior [PT13, AKW12, GC11]. But the Fisher matrix is costly to compute for large- dimensional models. Here we use the easily computed Fisher matrix approximations for deep neural networks from [MO16, Oll15]. The resulting natural Langevin dynamics combines the advantages of Amari's natural gradient descent and Fisher-preconditioned Langevin dynamics for large neural networks. Small-scale experiments on MNIST show that Fisher matrix preconditioning brings SGLD close to dropout as a regularizing technique.

Corentin Tallec, Yann Ollivier

Truncated Backpropagation Through Time (truncated BPTT) is a widespread method for learning recurrent computational graphs. Truncated BPTT keeps the computational benefits of Backpropagation Through Time (BPTT) while relieving the need for a complete backtrack through the whole data sequence at every step. However, truncation favors short-term dependencies: the gradient estimate of truncated BPTT is biased, so that it does not benefit from the convergence guarantees from stochastic gradient theory. We introduce Anticipated Reweighted Truncated Backpropagation (ARTBP), an algorithm that keeps the computational benefits of truncated BPTT, while providing unbiasedness. ARTBP works by using variable truncation lengths together with carefully chosen compensation factors in the backpropagation equation. We check the viability of ARTBP on two tasks. First, a simple synthetic task where careful balancing of temporal dependencies at different scales is needed: truncated BPTT displays unreliable performance, and in worst case scenarios, divergence, while ARTBP converges reliably. Second, on Penn Treebank character-level language modelling, ARTBP slightly outperforms truncated BPTT.

Yann Ollivier

We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural gradient descent on the log-likelihood of the observations. In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states and parameters is a natural gradient on top of real-time recurrent learning (RTRL), a classical algorithm to train recurrent models. This exact algebraic correspondence provides relevant interpretations for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.

Corentin Tallec, Yann Ollivier

The novel Unbiased Online Recurrent Optimization (UORO) algorithm allows for online learning of general recurrent computational graphs such as recurrent network models. It works in a streaming fashion and avoids backtracking through past activations and inputs. UORO is computationally as costly as Truncated Backpropagation Through Time (truncated BPTT), a widespread algorithm for online learning of recurrent networks. UORO is a modification of NoBackTrack that bypasses the need for model sparsity and makes implementation easy in current deep learning frameworks, even for complex models. Like NoBackTrack, UORO provides unbiased gradient estimates; unbiasedness is the core hypothesis in stochastic gradient descent theory, without which convergence to a local optimum is not guaranteed. On the contrary, truncated BPTT does not provide this property, leading to possible divergence. On synthetic tasks where truncated BPTT is shown to diverge, UORO converges. For instance, when a parameter has a positive short-term but negative long-term influence, truncated BPTT diverges unless the truncation span is very significantly longer than the intrinsic temporal range of the interactions, while UORO performs well thanks to the unbiasedness of its gradients.

Gaétan Marceau-Caron, Yann Ollivier

We provide the first experimental results on non-synthetic datasets for the quasi-diagonal Riemannian gradient descents for neural networks introduced in [Ollivier, 2015]. These include the MNIST, SVHN, and FACE datasets as well as a previously unpublished electroencephalogram dataset. The quasi-diagonal Riemannian algorithms consistently beat simple stochastic gradient gradient descents by a varying margin. The computational overhead with respect to simple backpropagation is around a factor $2$. Perhaps more interestingly, these methods also reach their final performance quickly, thus requiring fewer training epochs and a smaller total computation time. We also present an implementation guide to these Riemannian gradient descents for neural networks, showing how the quasi-diagonal versions can be implemented with minimal effort on top of existing routines which compute gradients.

Pierre-Yves Massé, Yann Ollivier

The practical performance of online stochastic gradient descent algorithms is highly dependent on the chosen step size, which must be tediously hand-tuned in many applications. The same is true for more advanced variants of stochastic gradients, such as SAGA, SVRG, or AdaGrad. Here we propose to adapt the step size by performing a gradient descent on the step size itself, viewing the whole performance of the learning trajectory as a function of step size. Importantly, this adaptation can be computed online at little cost, without having to iterate backward passes over the full data.

Yann Ollivier, Corentin Tallec, Guillaume Charpiat

We introduce the "NoBackTrack" algorithm to train the parameters of dynamical systems such as recurrent neural networks. This algorithm works in an online, memoryless setting, thus requiring no backpropagation through time, and is scalable, avoiding the large computational and memory cost of maintaining the full gradient of the current state with respect to the parameters. The algorithm essentially maintains, at each time, a single search direction in parameter space. The evolution of this search direction is partly stochastic and is constructed in such a way to provide, at every time, an unbiased random estimate of the gradient of the loss function with respect to the parameters. Because the gradient estimate is unbiased, on average over time the parameter is updated as it should. The resulting gradient estimate can then be fed to a lightweight Kalman-like filter to yield an improved algorithm. For recurrent neural networks, the resulting algorithms scale linearly with the number of parameters. Small-scale experiments confirm the suitability of the approach, showing that the stochastic approximation of the gradient introduced in the algorithm is not detrimental to learning. In particular, the Kalman-like version of NoBackTrack is superior to backpropagation through time (BPTT) when the time span of dependencies in the data is longer than the truncation span for BPTT.

Yann Ollivier

We discuss the similarities and differences between training an auto-encoder to minimize the reconstruction error, and training the same auto-encoder to compress the data via a generative model. Minimizing a codelength for the data using an auto-encoder is equivalent to minimizing the reconstruction error plus some correcting terms which have an interpretation as either a denoising or contractive property of the decoding function. These terms are related but not identical to those used in denoising or contractive auto-encoders [Vincent et al. 2010, Rifai et al. 2011]. In particular, the codelength viewpoint fully determines an optimal noise level for the denoising criterion.

Yann Ollivier

Recurrent neural networks are powerful models for sequential data, able to represent complex dependencies in the sequence that simpler models such as hidden Markov models cannot handle. Yet they are notoriously hard to train. Here we introduce a training procedure using a gradient ascent in a Riemannian metric: this produces an algorithm independent from design choices such as the encoding of parameters and unit activities. This metric gradient ascent is designed to have an algorithmic cost close to backpropagation through time for sparsely connected networks. We use this procedure on gated leaky neural networks (GLNNs), a variant of recurrent neural networks with an architecture inspired by finite automata and an evolution equation inspired by continuous-time networks. GLNNs trained with a Riemannian gradient are demonstrated to effectively capture a variety of structures in synthetic problems: basic block nesting as in context-free grammars (an important feature of natural languages, but difficult to learn), intersections of multiple independent Markov-type relations, or long-distance relationships such as the distant-XOR problem. This method does not require adjusting the network structure or initial parameters: the network used is a sparse random graph and the initialization is identical for all problems considered.

Yann Ollivier

We describe four algorithms for neural network training, each adapted to different scalability constraints. These algorithms are mathematically principled and invariant under a number of transformations in data and network representation, from which performance is thus independent. These algorithms are obtained from the setting of differential geometry, and are based on either the natural gradient using the Fisher information matrix, or on Hessian methods, scaled down in a specific way to allow for scalability while keeping some of their key mathematical properties.