An increasing number of machine learning applications employ flexible overparameterized models to fit the training data. Theoretical analysis of such'overfitted' solutions has been a recent focus of the learning community. It is conjectured that the use of large overparameterized neural networks makes the loss landscape amenable to optimization through local search methods, such as stochastic gradient descent. It is also hypothesized that implicit regularization, arising from the choice of the optimization algorithm and the neural network architecture, mitigates the large complexity and ensures that the'overfitted' solutions generalize. Suppose a'simple' class H of models captures the relationship between the covariates X and the response variable Y. Inspired by the use of overparameterized models, we may take a much larger class F H for computational or other purposes (such as lack of explicit description of H) and minimize training loss over this larger class.

Recent studies have provided both empirical and theoretical evidence illustrating that heavy tails can emerge in stochastic gradient descent (SGD) in various scenarios. Such heavy tails potentially result in iterates with diverging variance, which hinders the use of conventional convergence analysis techniques that rely on the existence of the second-order moments. In this paper, we provide convergence guarantees for SGD under a state-dependent and heavy-tailed noise with a potentially infinite variance, for a class of strongly convex objectives. In the case where the $p$-th moment of the noise exists for some $p\in [1,2)$, we first identify a condition on the Hessian, coined '$p$-positive (semi-)definiteness', that leads to an interesting interpolation between positive semi-definite matrices ($p=2$) and diagonally dominant matrices with non-negative diagonal entries ($p=1$). Under this condition, we then provide a convergence rate for the distance to the global optimum in $L^p$. Furthermore, we provide a generalized central limit theorem, which shows that the properly scaled Polyak-Ruppert averaging converges weakly to a multivariate $\alpha$-stable random vector. Our results indicate that even under heavy-tailed noise with infinite variance, SGD can converge to the global optimum without necessitating any modification neither to the loss function or to the algorithm itself, as typically required in robust statistics. We demonstrate the implications of our results to applications such as linear regression and generalized linear models subject to heavy-tailed data.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to 0. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

The Kalman filter is a fundamental filtering algorithm that fuses noisy sensory data, a previous state estimate, and a dynamics model to produce a principled estimate of the current state. It assumes, and is optimal for, linear models and white Gaussian noise. Due to its relative simplicity and general effectiveness, the Kalman filter is widely used in engineering applications. Since many sensory problems the brain faces are, at their core, filtering problems, it is possible that the brain possesses neural circuitry that implements equivalent computations to the Kalman filter. The standard approach to Kalman filtering requires complex matrix computations that are unlikely to be directly implementable in neural circuits. In this paper, we show that a gradient-descent approximation to the Kalman filter requires only local computations with variance weighted prediction errors. Moreover, we show that it is possible under the same scheme to adaptively learn the dynamics model with a learning rule that corresponds directly to Hebbian plasticity. We demonstrate the performance of our method on a simple Kalman filtering task, and propose a neural implementation of the required equations. The Bayesian Brain hypothesis has gained significant traction in cognitive neuroscience over the last two decades (Knill and Pouget, 2004; Doya et al., 2007; Pouget et al., 2013).

In this paper, we proposed a new technique, {\em variance controlled stochastic gradient} (VCSG), to improve the performance of the stochastic variance reduced gradient (SVRG) algorithm. To avoid over-reducing the variance of gradient by SVRG, a hyper-parameter $\lambda$ is introduced in VCSG that is able to control the reduced variance of SVRG. Theory shows that the optimization method can converge by using an unbiased gradient estimator, but in practice, biased gradient estimation can allow more efficient convergence to the vicinity since an unbiased approach is computationally more expensive. $\lambda$ also has the effect of balancing the trade-off between unbiased and biased estimations. Secondly, to minimize the number of full gradient calculations in SVRG, a variance-bounded batch is introduced to reduce the number of gradient calculations required in each iteration. For smooth non-convex functions, the proposed algorithm converges to an approximate first-order stationary point (i.e. $\mathbb{E}\|\nabla{f}(x)\|^{2}\leq\epsilon$) within $\mathcal{O}(min\{1/\epsilon^{3/2},n^{1/4}/\epsilon\})$ number of stochastic gradient evaluations, which improves the leading gradient complexity of stochastic gradient-based method SCS $(\mathcal{O}(min\{1/\epsilon^{5/3},n^{2/3}/\epsilon\})$. It is shown theoretically and experimentally that VCSG can be deployed to improve convergence.

During the past few years, there has been considerable success of applying deep learning to complex problems ranging from speech synthesis (cite WaveNet) to deep reinforcement learning achieving victories in complex games such as Go [19]. This, in turn, has fueled massive interest in the field. At the core of deep learning are training algorithms which allow neural networks to learn from the data they are presented or learn how to extremize a target quantity. One such algorithm is the stochastic gradient descent (SGD) (cite) and it is one of most widely used training algorithms. However, this does not come without issues as many training algorithms suffer from being slow at converging to the optimal state and lack stability when approaching the optimum (citation needed).One common approach is to transform the data into a manageable form through centring and scaling the data and it is the base of many normalization techniques. In fact, the success of "Batch Normalization" by Sgzedy and Ioffe [8] sparked an interest in such techniques. In [8], they noticed that the learning procedure, which uses stochastic gradient descent (SGD), or derived adaptive algorithms like Adam [12], can be hindered and they claimed that a phenomenon known as internal covariance shift is the cause. This phenomenon depends on the change of the network's weights during backpropagation that can lead to the shift of data-distribution towards the saturation regime of the activation

Variance reduction (VR) methods for finite-sum minimization typically require the knowledge of problem-dependent constants that are often unknown and difficult to estimate. To address this, we use ideas from adaptive gradient methods to propose AdaSVRG, which is a fully adaptive variant of SVRG, a common VR method. AdaSVRG uses AdaGrad in the inner loop of SVRG, making it robust to the choice of step-size, and allowing it to adaptively determine the length of each inner-loop. When minimizing a sum of $n$ smooth convex functions, we prove that AdaSVRG requires $O(n + 1/\epsilon)$ gradient evaluations to achieve an $\epsilon$-suboptimality, matching the typical rate, but without needing to know problem-dependent constants. However, VR methods including AdaSVRG are slower than SGD when used with over-parameterized models capable of interpolating the training data. Hence, we also propose a hybrid algorithm that can adaptively switch from AdaGrad to AdaSVRG, achieving the best of both stochastic gradient and VR methods, but without needing to tune their step-sizes. Via experiments on synthetic and standard real-world datasets, we validate the robustness and effectiveness of AdaSVRG, demonstrating its superior performance over other "tune-free" VR methods.

Minimax optimization has recently gained a lot of attention as adversarial architectures and algorithms proliferate. Often, smooth minimax games proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice for many applications (e.g., GAN training), the majority of existing theoretical analyses focus on simultaneous algorithms. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. In particular, we prove that Alt-GDA achieves a near-optimal local convergence rate for strongly-convex strongly-concave problems while Sim-GDA converges with a much slower rate. Moreover, we show that the acceleration effect of alternating updates remains when the minimax problem has only strong concavity in the dual variables. Numerical experiments on quadratic minimax games validate our claims. Additionally, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.

The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice because of their excellent convergence and generalization qualities, but their theoretical properties are not yet well understood. We provide the convergence results for step decay in the non-convex regime, ensuring that the gradient norm vanishes at an $\mathcal{O}(\ln T/\sqrt{T})$ rate. We also provide the convergence guarantees for general (possibly non-smooth) convex problems, ensuring an $\mathcal{O}(\ln T/\sqrt{T})$ convergence rate. Finally, in the strongly convex case, we establish an $\mathcal{O}(\ln T/T)$ rate for smooth problems, which we also prove to be tight, and an $\mathcal{O}(\ln^2 T /T)$ rate without the smoothness assumption. We illustrate the practical efficiency of the step decay step-size in several large scale deep neural network training tasks.

We propose dynamic curriculum learning via data parameters for noise robust keyword spotting. Data parameter learning has recently been introduced for image processing, where weight parameters, so-called data parameters, for target classes and instances are introduced and optimized along with model parameters. The data parameters scale logits and control importance over classes and instances during training, which enables automatic curriculum learning without additional annotations for training data. Similarly, in this paper, we propose using this curriculum learning approach for acoustic modeling, and train an acoustic model on clean and noisy utterances with the data parameters. The proposed approach automatically learns the difficulty of the classes and instances, e.g. due to low speech to noise ratio (SNR), in the gradient descent optimization and performs curriculum learning. This curriculum learning leads to overall improvement of the accuracy of the acoustic model. We evaluate the effectiveness of the proposed approach on a keyword spotting task. Experimental results show 7.7% relative reduction in false reject ratio with the data parameters compared to a baseline model which is simply trained on the multiconditioned dataset.