For multivariate nonparametric regression, doubly penalized ANOVA modeling (DPAM) has recently been proposed, using hierarchical total variations (HTVs) and empirical norms as penalties on the component functions such as main effects and multi-way interactions in a functional ANOVA decomposition of the underlying regression function. The two penalties play complementary roles: the HTV penalty promotes sparsity in the selection of basis functions within each component function, whereas the empirical-norm penalty promotes sparsity in the selection of component functions. We adopt backfitting or block minimization for training DPAM, and develop two suitable primal-dual algorithms, including both batch and stochastic versions, for updating each component function in single-block optimization. Existing applications of primal-dual algorithms are intractable in our setting with both HTV and empirical-norm penalties. Through extensive numerical experiments, we demonstrate the validity and advantage of our stochastic primal-dual algorithms, compared with their batch versions and a previous active-set algorithm, in large-scale scenarios.
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We consider the constrained Linear Inverse Problem (LIP), where a certain atomic norm (like the $\ell_1 $ and the Nuclear norm) is minimized subject to a quadratic constraint. Typically, such cost functions are non-differentiable which makes them not amenable to the fast optimization methods existing in practice. We propose two equivalent reformulations of the constrained LIP with improved convex regularity: (i) a smooth convex minimization problem, and (ii) a strongly convex min-max problem. These problems could be solved by applying existing acceleration based convex optimization methods which provide better \mmode{ O \left( \nicefrac{1}{k^2} \right) } theoretical convergence guarantee. However, to fully exploit the utility of these reformulations, we also provide a novel algorithm, to which we refer as the Fast Linear Inverse Problem Solver (FLIPS), that is tailored to solve the reformulation of the LIP. We demonstrate the performance of FLIPS on the sparse coding problem arising in image processing tasks. In this setting, we observe that FLIPS consistently outperforms the Chambolle-Pock and C-SALSA algorithms--two of the current best methods in the literature.
The central question in representation learning is what constitutes a good or meaningful representation. In this work we argue that if we consider data with inherent cluster structures, where clusters can be characterized through different means and covariances, those data structures should be represented in the embedding as well. While Autoencoders (AE) are widely used in practice for unsupervised representation learning, they do not fulfil the above condition on the embedding as they obtain a single representation of the data. To overcome this we propose a meta-algorithm that can be used to extend an arbitrary AE architecture to a tensorized version (TAE) that allows for learning cluster-specific embeddings while simultaneously learning the cluster assignment. For the linear setting we prove that TAE can recover the principle components of the different clusters in contrast to principle component of the entire data recovered by a standard AE. We validated this on planted models and for general, non-linear and convolutional AEs we empirically illustrate that tensorizing the AE is beneficial in clustering and de-noising tasks.
This article develops a new algorithm named TTRISK to solve high-dimensional risk-averse optimization problems governed by differential equations (ODEs and/or PDEs) under uncertainty. As an example, we focus on the so-called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid non-smoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the KKT system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large-scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID-19. The results indicate that the risk-averse framework is feasible with the tensor approximations under tens of random variables.
The emerging modular vehicle (MV) technology possesses the ability to physically connect/disconnect with each other and thus travel in platoon for less energy consumption. Moreover, a platoon of MVs can be regarded as a new bus-like platform with expanded on-board carrying capacity and provide larger service throughput according to the demand density. This innovation concept might solve the mismatch problems between the fixed vehicle capacity and the temporal-spatial variations of demand in current transportation system. To obtain the optimal assignments and routes for the operation of MVs, a mixed integer linear programming (MILP) model is formulated to minimize the weighted total cost of vehicle travel cost and passenger service time. The temporal and spatial synchronization of vehicle platoons and passenger en-route transfers are determined and optimized by the MILP model while constructing the paths. Heuristic algorithms based on large neighborhood search are developed to solve the modular dial-a-ride problem (MDARP) for practical scenarios. A set of small-scale synthetic numerical experiments are tested to evaluate the optimality gap and computation time between our proposed MILP model and heuristic algorithms. Large-scale experiments are conducted on the Anaheim network with 378 candidate join/split nodes to further explore the potentials and identify the ideal operation scenarios of MVs. The results show that the innovative MV technology can save up to 52.0% in vehicle travel cost, 35.6% in passenger service time, and 29.4% in total cost against existing on-demand mobility services. Results suggest that MVs best benefit from platooning by serving enclave pairs as a hub-and-spoke service.
Constrained multiagent reinforcement learning (C-MARL) is gaining importance as MARL algorithms find new applications in real-world systems ranging from energy systems to drone swarms. Most C-MARL algorithms use a primal-dual approach to enforce constraints through a penalty function added to the reward. In this paper, we study the structural effects of this penalty term on the MARL problem. First, we show that the standard practice of using the constraint function as the penalty leads to a weak notion of safety. However, by making simple modifications to the penalty term, we can enforce meaningful probabilistic (chance and conditional value at risk) constraints. Second, we quantify the effect of the penalty term on the value function, uncovering an improved value estimation procedure. We use these insights to propose a constrained multiagent advantage actor critic (C-MAA2C) algorithm. Simulations in a simple constrained multiagent environment affirm that our reinterpretation of the primal-dual method in terms of probabilistic constraints is effective, and that our proposed value estimate accelerates convergence to a safe joint policy.
We propose a parameter-free model for estimating the price or valuation of financial derivatives like options, forwards and futures using non-supervised learning networks and Monte Carlo. Although some arbitrage-based pricing formula performs greatly on derivatives pricing like Black-Scholes on option pricing, generative model-based Monte Carlo estimation(GAN-MC) will be more accurate and holds more generalizability when lack of training samples on derivatives, underlying asset's price dynamics are unknown or the no-arbitrage conditions can not be solved analytically. We analyze the variance reduction feature of our model and to validate the potential value of the pricing model, we collect real world market derivatives data and show that our model outperforms other arbitrage-based pricing models and non-parametric machine learning models. For comparison, we estimate the price of derivatives using Black-Scholes model, ordinary least squares, radial basis function networks, multilayer perception regression, projection pursuit regression and Monte Carlo only models.
We study sublinear time algorithms for estimating the size of maximum matching. After a long line of research, the problem was finally settled by Behnezhad [FOCS'22], in the regime where one is willing to pay an approximation factor of $2$. Very recently, Behnezhad et al.[SODA'23] improved the approximation factor to $(2-\frac{1}{2^{O(1/\gamma)}})$ using $n^{1+\gamma}$ time. This improvement over the factor $2$ is, however, minuscule and they asked if even $1.99$-approximation is possible in $n^{2-\Omega(1)}$ time. We give a strong affirmative answer to this open problem by showing $(1.5+\epsilon)$-approximation algorithms that run in $n^{2-\Theta(\epsilon^{2})}$ time. Our approach is conceptually simple and diverges from all previous sublinear-time matching algorithms: we show a sublinear time algorithm for computing a variant of the edge-degree constrained subgraph (EDCS), a concept that has previously been exploited in dynamic [Bernstein Stein ICALP'15, SODA'16], distributed [Assadi et al. SODA'19] and streaming [Bernstein ICALP'20] settings, but never before in the sublinear setting. Independent work: Behnezhad, Roghani and Rubinstein [BRR'23] independently showed sublinear algorithms similar to our Theorem 1.2 in both adjacency list and matrix models. Furthermore, in [BRR'23], they show additional results on strictly better-than-1.5 approximate matching algorithms in both upper and lower bound sides.
Recently Convolution-augmented Transformer (Conformer) has shown promising results in Automatic Speech Recognition (ASR), outperforming the previous best published Transformer Transducer. In this work, we believe that the output information of each block in the encoder and decoder is not completely inclusive, in other words, their output information may be complementary. We study how to take advantage of the complementary information of each block in a parameter-efficient way, and it is expected that this may lead to more robust performance. Therefore we propose the Block-augmented Transformer for speech recognition, named Blockformer. We have implemented two block ensemble methods: the base Weighted Sum of the Blocks Output (Base-WSBO), and the Squeeze-and-Excitation module to Weighted Sum of the Blocks Output (SE-WSBO). Experiments have proved that the Blockformer significantly outperforms the state-of-the-art Conformer-based models on AISHELL-1, our model achieves a CER of 4.29\% without using a language model and 4.05\% with an external language model on the testset.
We consider the problem of creating assistants that can help agents solve new sequential decision problems, assuming the agent is not able to specify the reward function explicitly to the assistant. Instead of acting in place of the agent as in current automation-based approaches, we give the assistant an advisory role and keep the agent in the loop as the main decision maker. The difficulty is that we must account for potential biases of the agent which may cause it to seemingly irrationally reject advice. To do this we introduce a novel formalization of assistance that models these biases, allowing the assistant to infer and adapt to them. We then introduce a new method for planning the assistant's actions which can scale to large decision making problems. We show experimentally that our approach adapts to these agent biases, and results in higher cumulative reward for the agent than automation-based alternatives. Lastly, we show that an approach combining advice and automation outperforms advice alone at the cost of losing some safety guarantees.
We study differentially private (DP) stochastic optimization (SO) with loss functions whose worst-case Lipschitz parameter over all data points may be extremely large. To date, the vast majority of work on DP SO assumes that the loss is uniformly Lipschitz continuous over data (i.e. stochastic gradients are uniformly bounded over all data points). While this assumption is convenient, it often leads to pessimistic excess risk bounds. In many practical problems, the worst-case Lipschitz parameter of the loss over all data points may be extremely large due to outliers. In such cases, the error bounds for DP SO, which scale with the worst-case Lipschitz parameter of the loss, are vacuous. To address these limitations, this work provides near-optimal excess risk bounds that do not depend on the uniform Lipschitz parameter of the loss. Building on a recent line of work [WXDX20, KLZ22], we assume that stochastic gradients have bounded $k$-th order moments for some $k \geq 2$. Compared with works on uniformly Lipschitz DP SO, our excess risk scales with the $k$-th moment bound instead of the uniform Lipschitz parameter of the loss, allowing for significantly faster rates in the presence of outliers and/or heavy-tailed data. For convex and strongly convex loss functions, we provide the first asymptotically optimal excess risk bounds (up to a logarithmic factor). In contrast to [WXDX20, KLZ22], our bounds do not require the loss function to be differentiable/smooth. We also devise an accelerated algorithm for smooth losses that runs in linear time and has excess risk that is tight in certain practical parameter regimes. Additionally, our work is the first to address non-convex non-uniformly Lipschitz loss functions satisfying the Proximal-PL inequality; this covers some practical machine learning models. Our Proximal-PL algorithm has near-optimal excess risk.
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