Jia Bi

Kuangdai Leng, Jia Bi, Jaehoon Cha et al.

The Moving Sofa Problem, formally proposed by Leo Moser in 1966, seeks to determine the largest area of a two-dimensional shape that can navigate through an $L$-shaped corridor with unit width. The current best lower bound is about 2.2195, achieved by Joseph Gerver in 1992, though its global optimality remains unproven. In this paper, we investigate this problem by leveraging the universal approximation strength and computational efficiency of neural networks. We report two approaches, both supporting Gerver's conjecture that his shape is the unique global maximum. Our first approach is continuous function learning. We drop Gerver's assumptions that i) the rotation of the corridor is monotonic and symmetric and, ii) the trajectory of its corner as a function of rotation is continuously differentiable. We parameterize rotation and trajectory by independent piecewise linear neural networks (with input being some pseudo time), allowing for rich movements such as backward rotation and pure translation. We then compute the sofa area as a differentiable function of rotation and trajectory using our "waterfall" algorithm. Our final loss function includes differential terms and initial conditions, leveraging the principles of physics-informed machine learning. Under such settings, extensive training starting from diverse function initialization and hyperparameters is conducted, unexceptionally showing rapid convergence to Gerver's solution. Our second approach is via discrete optimization of the Kallus-Romik upper bound, which converges to the maximum sofa area from above as the number of rotation angles increases. We uplift this number to 10000 to reveal its asymptotic behavior. It turns out that the upper bound yielded by our models does converge to Gerver's area (within an error of 0.01% when the number of angles reaches 2100). We also improve their five-angle upper bound from 2.37 to 2.3337.

Wei Lou, Lei Xun, Amin Sabet et al.

Mobile and embedded platforms are increasingly required to efficiently execute computationally demanding DNNs across heterogeneous processing elements. At runtime, the available hardware resources to DNNs can vary considerably due to other concurrently running applications. The performance requirements of the applications could also change under different scenarios. To achieve the desired performance, dynamic DNNs have been proposed in which the number of channels/layers can be scaled in real time to meet different requirements under varying resource constraints. However, the training process of such dynamic DNNs can be costly, since platform-aware models of different deployment scenarios must be retrained to become dynamic. This paper proposes Dynamic-OFA, a novel dynamic DNN approach for state-of-the-art platform-aware NAS models (i.e. Once-for-all network (OFA)). Dynamic-OFA pre-samples a family of sub-networks from a static OFA backbone model, and contains a runtime manager to choose different sub-networks under different runtime environments. As such, Dynamic-OFA does not need the traditional dynamic DNN training pipeline. Compared to the state-of-the-art, our experimental results using ImageNet on a Jetson Xavier NX show that the approach is up to 3.5x (CPU), 2.4x (GPU) faster for similar ImageNet Top-1 accuracy, or 3.8% (CPU), 5.1% (GPU) higher accuracy at similar latency.

Jia Bi, Steve R. Gunn

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 $λ$ 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. $λ$ 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. $\mathbb{E}\|\nabla{f}(x)\|^{2}\leqε$) within $\mathcal{O}(min\{1/ε^{3/2},n^{1/4}/ε\})$ number of stochastic gradient evaluations, which improves the leading gradient complexity of stochastic gradient-based method SCS $(\mathcal{O}(min\{1/ε^{5/3},n^{2/3}/ε\})$. It is shown theoretically and experimentally that VCSG can be deployed to improve convergence.

Jia Bi, Steve R. Gunn

A number of optimization approaches have been proposed for optimizing nonconvex objectives (e.g. deep learning models), such as batch gradient descent, stochastic gradient descent and stochastic variance reduced gradient descent. Theory shows these optimization methods can converge by using an unbiased gradient estimator. However, in practice biased gradient estimation can allow more efficient convergence to the vicinity since an unbiased approach is computationally more expensive. To produce fast convergence there are two trade-offs of these optimization strategies which are between stochastic/batch, and between biased/unbiased. This paper proposes an integrated approach which can control the nature of the stochastic element in the optimizer and can balance the trade-off of estimator between the biased and unbiased by using a hyper-parameter. It is shown theoretically and experimentally that this hyper-parameter can be configured to provide an effective balance to improve the convergence rate.