In this paper, a Convolution-Based Converter (CBC) is proposed to develop a methodology for removing the strong or fixed priors in estimating the probability distribution of targets based on observations in the stochastic process. Traditional approaches, e.g., Markov-based and Gaussian process-based methods, typically leverage observations to estimate targets based on strong or fixed priors (such as Markov properties or Gaussian prior). However, the effectiveness of these methods depends on how well their prior assumptions align with the characteristics of the problem. When the assumed priors are not satisfied, these approaches may perform poorly or even become unusable. To overcome the above limitation, we introduce the Convolution-Based converter (CBC), which implicitly estimates the conditional probability distribution of targets without strong or fixed priors, and directly outputs the expected trajectory of the stochastic process that satisfies the constraints from observations. This approach reduces the dependence on priors, enhancing flexibility and adaptability in modeling stochastic processes when addressing different problems. Experimental results demonstrate that our method outperforms existing baselines across multiple metrics.

Symbolic regression (SR) is an emerging branch of machine learning focused on discovering simple and interpretable mathematical expressions from data. Although a wide-variety of SR methods have been developed, they often face challenges such as high computational cost, poor scalability with respect to the number of input dimensions, fragility to noise, and an inability to balance accuracy and complexity. This work introduces SyMANTIC, a novel SR algorithm that addresses these challenges. SyMANTIC efficiently identifies (potentially several) low-dimensional descriptors from a large set of candidates (from $\sim 10^5$ to $\sim 10^{10}$ or more) through a unique combination of mutual information-based feature selection, adaptive feature expansion, and recursively applied $\ell_0$-based sparse regression. In addition, it employs an information-theoretic measure to produce an approximate set of Pareto-optimal equations, each offering the best-found accuracy for a given complexity. Furthermore, our open-source implementation of SyMANTIC, built on the PyTorch ecosystem, facilitates easy installation and GPU acceleration. We demonstrate the effectiveness of SyMANTIC across a range of problems, including synthetic examples, scientific benchmarks, real-world material property predictions, and chaotic dynamical system identification from small datasets. Extensive comparisons show that SyMANTIC uncovers similar or more accurate models at a fraction of the cost of existing SR methods.

This thesis explores advanced approaches to improve explainability in computer vision by analyzing and modeling the features exploited by deep neural networks. Initially, it evaluates attribution methods, notably saliency maps, by introducing a metric based on algorithmic stability and an approach utilizing Sobol indices, which, through quasi-Monte Carlo sequences, allows a significant reduction in computation time. In addition, the EVA method offers a first formulation of attribution with formal guarantees via verified perturbation analysis. Experimental results indicate that in complex scenarios these methods do not provide sufficient understanding, particularly because they identify only "where" the model focuses without clarifying "what" it perceives. Two hypotheses are therefore examined: aligning models with human reasoning -- through the introduction of a training routine that integrates the imitation of human explanations and optimization within the space of 1-Lipschitz functions -- and adopting a conceptual explainability approach. The CRAFT method is proposed to automate the extraction of the concepts used by the model and to assess their importance, complemented by MACO, which enables their visualization. These works converge towards a unified framework, illustrated by an interactive demonstration applied to the 1000 ImageNet classes in a ResNet model.

Gaussian smoothing combined with a probabilistic framework for denoising via the empirical Bayes formalism, i.e., the Tweedie-Miyasawa formula (TMF), are the two key ingredients in the success of score-based generative models in Euclidean spaces. Smoothing holds the key for easing the problem of learning and sampling in high dimensions, denoising is needed for recovering the original signal, and TMF ties these together via the score function of noisy data. In this work, we extend this paradigm to the problem of learning and sampling the distribution of binary data on the Boolean hypercube by adopting Bernoulli noise, instead of Gaussian noise, as a smoothing device. We first derive a TMF-like expression for the optimal denoiser for the Hamming loss, where a score function naturally appears. Sampling noisy binary data is then achieved using a Langevin-like sampler which we theoretically analyze for different noise levels. At high Bernoulli noise levels sampling becomes easy, akin to log-concave sampling in Euclidean spaces. In addition, we extend the sequential multi-measurement sampling of Saremi et al. (2024) to the binary setting where we can bring the "effective noise" down by sampling multiple noisy measurements at a fixed noise level, without the need for continuous-time stochastic processes. We validate our formalism and theoretical findings by experiments on synthetic data and binarized images.

We consider a network inventory problem motivated by one-way, on-demand vehicle sharing services. Due to uncertainties in both demand and returns, as well as a fixed number of rental units across an $n$-location network, the service provider must periodically reposition vehicles to match supply with demand spatially while minimizing costs. The optimal repositioning policy under a general $n$-location network is intractable without knowing the optimal value function. We introduce the best base-stock repositioning policy as a generalization of the classical inventory control policy to $n$ dimensions, and establish its asymptotic optimality in two distinct limiting regimes under general network structures. We present reformulations to efficiently compute this best base-stock policy in an offline setting with pre-collected data. In the online setting, we show that a natural Lipschitz-bandit approach achieves a regret guarantee of $\widetilde{O}(T^{\frac{n}{n+1}})$, which suffers from the exponential dependence on $n$. We illustrate the challenges of learning with censored data in networked systems through a regret lower bound analysis and by demonstrating the suboptimality of alternative algorithmic approaches. Motivated by these challenges, we propose an Online Gradient Repositioning algorithm that relies solely on censored demand. Under a mild cost-structure assumption, we prove that it attains an optimal regret of $O(n^{2.5} \sqrt{T})$, which matches the regret lower bound in $T$ and achieves only polynomial dependence on $n$. The key algorithmic innovation involves proposing surrogate costs to disentangle intertemporal dependencies and leveraging dual solutions to find the gradient of policy change. Numerical experiments demonstrate the effectiveness of our proposed methods.

Federated averaging (FedAvg) is a popular algorithm for horizontal federated learning (FL), where samples are gathered across different clients and are not shared with each other or a central server. Extensive convergence analysis of FedAvg exists for the discrete iteration setting, guaranteeing convergence for a range of loss functions and varying levels of data heterogeneity. We extend this analysis to the continuous-time setting where the global weights evolve according to a multivariate stochastic differential equation (SDE), which is the first time FedAvg has been studied from the continuous-time perspective. We use techniques from stochastic processes to establish convergence guarantees under different loss functions, some of which are more general than existing work in the discrete setting. We also provide conditions for which FedAvg updates to the server weights can be approximated as normal random variables. Finally, we use the continuous-time formulation to reveal generalization properties of FedAvg.

Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in finite horizon scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for backward induction, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs density ratio estimation and convex function learning to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is publicly available at https://github.com/Akinori-F-Ebihara/FIRMBOUND

A.1 Stochastic approximation Stochastic approximation [Benveniste et al., 1990] provides a standard framework for the development of adaptive algorithms. Given a random field function H(θ, x), the goal of the stochastic approximation algorithm is to find the solution to the mean-field equation h(θ) = 0, i.e., solving h(θ) = H(θ, x)ϖ

My main concern is that using a flattened surrogate energy in this fashion is suitable for most sampling situations. The main reason is, by construction our iterates are not following the true distribution particularly closely; for example a plot of the samples obtained in the synthetic experiments (figs 2c--d) would look quite different from the original. While this does allow the algorithm to bounce out of local optima, the deviance from the true energy would make samples obtained after convergence to not be super useful. For point estimation situations, we might be able to get away with these samples for cases where the multiple modes of the real energy are sort of symmetric (as in the synthetic Gaussian experiments); it seems that even if we use a'flattened' energy (can be thought of as lower peaks with higher elevation between them), the original distribution's symmetry would be essentially preserved and the mean / other point estimates would be close enough. But flattening energies with skewed distribution of modes might not be as accurate, as the flattened version might have a mean closer to the'center' of the space, but the original would be closer to one of the modes near the periphery (am visualizing a simple 2-d space).

The paper presents valuable theoretical and empirical evidence for a novel algorithm. The AC is confident this represents valuable work but was a bit torn about the acceptance decision, as the reviewers point out several important avenues where improvement in the paper is needed.