We present a probabilistic analysis of the long-time behaviour of the nonlocal, diffusive equations with a gradient flow structure in 2-Wasserstein metric, namely, the Mean-Field Langevin Dynamics (MFLD). Our work is motivated by a desire to provide a theoretical underpinning for the convergence of stochastic gradient type algorithms widely used for non-convex learning tasks such as training of deep neural networks. The key insight is that the certain class of the finite dimensional non-convex problems becomes convex when lifted to infinite dimensional space of measures. We leverage this observation and show that the corresponding energy functional defined on the space of probability measures has a unique minimiser which can be characterised by a first order condition using the notion of linear functional derivative. Next, we show that the flow of marginal laws induced by the MFLD converges to the stationary distribution which is exactly the minimiser of the energy functional. We show that this convergence is exponential under conditions that are satisfied for highly regularised learning tasks. At the heart of our analysis is a pathwise perspective on Otto calculus used in gradient flow literature which is of independent interest. Our proof of convergence to stationary probability measure is novel and it relies on a generalisation of LaSalle's invariance principle. Importantly we do not assume that interaction potential of MFLD is of convolution type nor that has any particular symmetric structure. This is critical for applications. Finally, we show that the error between finite dimensional optimisation problem and its infinite dimensional limit is of order one over the number of parameters.

The practical success of widely used machine learning (ML) and deep learning (DL) algorithms in Artificial Intelligence (AI) community owes to availability of large datasets for training and huge computational resources. Despite the enormous practical success of AI, these algorithms are only loosely inspired from the biological brain and do not mimic any of the fundamental properties of neurons in the brain, one such property being the chaotic firing of biological neurons. This motivates us to develop a novel neuronal architecture where the individual neurons are intrinsically chaotic in nature. By making use of the topological transitivity property of chaos, our neuronal network is able to perform classification tasks with very less number of training samples. For the MNIST dataset, with as low as $0.1 \%$ of the total training data, our method outperforms ML and matches DL in classification accuracy for up to $7$ training samples/class. For the Iris dataset, our accuracy is comparable with ML algorithms, and even with just two training samples/class, we report an accuracy as high as $95.8 \%$. This work highlights the effectiveness of chaos and its properties for learning and paves the way for chaos-inspired neuronal architectures by closely mimicking the chaotic nature of neurons in the brain.

How different initializations and loss functions affect the learning of a deep neural network (DNN), specifically its generalization error, is an important problem in practice. In this work, focusing on regression problems, we develop a kernel-norm minimization framework for the analysis of DNNs in the kernel regime in which the number of neurons in each hidden layer is sufficiently large (Jacot et al. 2018, Lee et al. 2019). We find that, in the kernel regime, for any loss in a general class of functions, e.g., any Lp loss for $1 < p < \infty$, the DNN finds the same global minima-the one that is nearest to the initial value in the parameter space, or equivalently, the one that is closest to the initial DNN output in the corresponding reproducing kernel Hilbert space. With this framework, we prove that a non-zero initial output increases the generalization error of DNN. We further propose an antisymmetrical initialization (ASI) trick that eliminates this type of error and accelerates the training. We also demonstrate experimentally that even for DNNs in the non-kernel regime, our theoretical analysis and the ASI trick remain effective. Overall, our work provides insight into how initialization and loss function quantitatively affect the generalization of DNNs, and also provides guidance for the training of DNNs.

As artificial intelligence plays an increasingly important role in our society, there are ethical and moral obligations for both businesses and researchers to ensure that their machine learning models are designed, deployed, and maintained responsibly. These models need to be rigorously audited for fairness, robustness, transparency, and interpretability. A variety of methods have been developed that focus on these issues in isolation, however, managing these methods in conjunction with model development can be cumbersome and timeconsuming. In this paper, we introduce a unified and model-agnostic approach to address these issues: Counterfactual Explanations for Robustness, Transparency, Interpretability, and Fairness of Artificial Intelligence models (CERTIFAI). Unlike previous methods in this domain, CERTIFAI is a general tool that can be applied to any black-box model and any type of input data. Given a model and an input instance, CERTIFAI uses a custom genetic algorithm to generate counterfactuals: instances close to the input that change the prediction of the model. We demonstrate how these counterfactuals can be used to examine issues of robustness, interpretability, transparency, and fairness. Additionally, we introduce CERScore, the first black-box model robustness score that performs comparably to methods that have access to model internals.

The recently proposed Lottery Ticket Hypothesis of Frankle and Carbin (2019) suggests that the performance of over-parameterized deep networks is due to the random initialization seeding the network with a small fraction of favorable weights. These weights retain their dominant status throughout training -- in a very real sense, this sub-network "won the lottery" during initialization. The authors find sub-networks via unstructured magnitude pruning with 85-95% of parameters removed that train to the same accuracy as the original network at a similar speed, which they call winning tickets. In this paper, we extend the Lottery Ticket Hypothesis to a variety of transfer learning tasks. We show that sparse sub-networks with approximately 90-95% of weights removed achieve (and often exceed) the accuracy of the original dense network in several realistic settings. We experimentally validate this by transferring the sparse representation found via pruning on CIFAR-10 to SmallNORB and FashionMNIST for object recognition tasks.

Point-of-Interest (POI) recommender systems play a vital role in people's lives by recommending unexplored POIs to users and have drawn extensive attention from both academia and industry. Despite their value, however, they still suffer from the challenges of capturing complicated user preferences and fine-grained user-POI relationship for spatio-temporal sensitive POI recommendation. Existing recommendation algorithms, including both shallow and deep approaches, usually embed the visiting records of a user into a single latent vector to model user preferences: this has limited power of representation and interpretability. In this paper, we propose a novel topic-enhanced memory network (TEMN), a deep architecture to integrate the topic model and memory network capitalising on the strengths of both the global structure of latent patterns and local neighbourhood-based features in a nonlinear fashion. We further incorporate a geographical module to exploit user-specific spatial preference and POI-specific spatial influence to enhance recommendations. The proposed unified hybrid model is widely applicable to various POI recommendation scenarios. Extensive experiments on real-world WeChat datasets demonstrate its effectiveness (improvement ratio of 3.25% and 29.95% for context-aware and sequential recommendation, respectively). Also, qualitative analysis of the attention weights and topic modeling provides insight into the model's recommendation process and results.

In this paper, we use several techniques with conventional vocal feature extraction (MFCC, STFT), along with deep-learning approaches such as CNN, and also context-level analysis, by providing the textual data, and combining different approaches for improved emotion-level classification. We explore models that have not been tested to gauge the difference in performance and accuracy. We apply hyperparameter sweeps and data augmentation to improve performance. Finally, we see if a real-time approach is feasible, and can be readily integrated into existing systems.

Class incremental learning refers to a special multi-class classification task, in which the number of classes is not fixed but is increasing with the continual arrival of new data. Existing researches mainly focused on solving catastrophic forgetting problem in class incremental learning. To this end, however, these models still require the old classes cached in the auxiliary data structure or models, which is inefficient in space or time. In this paper, it is the first time to discuss the difficulty without support of old classes in class incremental learning, which is called as softmax suppression problem. To address these challenges, we develop a new model named Label Mapping with Response Consolidation (LMRC), which need not access the old classes anymore. We propose the Label Mapping algorithm combined with the multi-head neural network for mitigating the softmax suppression problem, and propose the Response Consolidation method to overcome the catastrophic forgetting problem. Experimental results on the benchmark datasets show that our proposed method achieves much better performance compared to the related methods in different scenarios.

Let's go over the Algorithm: 1. The algorithm takes an input x and a classifier f . And the loop variable to 1. 4. Start and continue loop while the true label and the label of the adversarially perturbed image is the same. 5. Calculate the projection of the input onto the closest hyperplane. With multiclass classifiers, let's say the input is x and for each class there is a hyperplane (straight plane that divides one class from the others) and based on the place in the space where x lies it is classified into a class. Now, all this algorithm does is, it finds the closest hyperplane, and then projects x onto that hyperplane and pushes it a bit beyond, thus misclassifying it with the minimal perturbation possible.

Every environment is fixed to "CartPole-v1" from OpenAI GYM. You can just focus on the implementations.