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+# Provable Data Subset Selection For Efficient Neural Networks Training
+
+Murad Tukan *¹ Samson Zhou² Alaa Maalouf¹³ Daniela Rus³ Vladimir Braverman² Dan Feldman⁴
+
+# Abstract
+
+Radial basis function neural networks (RBFNN) are well-known for their capability to approximate any continuous function on a closed bounded set with arbitrary precision given enough hidden neurons. In this paper, we introduce the first algorithm to construct coresets for RBFNNs, i.e., small weighted subsets that approximate the loss of the input data on any radial basis function network and thus approximate any function defined by an RBFNN on the larger input data. In particular, we construct coresets for radial basis and Laplacian loss functions. We then use our coresets to obtain a provable data subset selection algorithm for training deep neural networks. Since our coresets approximate every function, they also approximate the gradient of each weight in a neural network, which is a particular function on the input. We then perform empirical evaluations on function approximation and dataset subset selection on popular network architectures and data sets, demonstrating the efficacy and accuracy of our coreset construction.
+
+# 1. Introduction
+
+Radial basis function neural networks (RBFNNs) are artificial neural networks that generally have three layers: an input layer, a hidden layer with a radial basis function (RBF) as an activation function, and a linear output layer. In this paper, the input layer receives a  $d$ -dimensional vector  $x \in \mathbb{R}^d$  of real numbers. The hidden layer then consists of various nodes representing RBFs, to compute  $\rho(\|x - c_i\|_2) := \exp \left(-\|x - c_i\|_2^2\right)$ , where  $c_i \in \mathbb{R}^d$  is the center vector for neuron  $i$  across, say,  $N$  neurons in
+
+* This research was first carried out during the Ph.D. studies at the department of computer science of the University of Haifa  ${}^{1}$  DataHeroes,Israel.  ${}^{2}$  Department of Computer science,Rice university.  ${}^{3}$  CSAIL,MIT,Cambridge,USA.  ${}^{4}$  Department of computer science,University of Haifa,Israel. Correspondence to: Murad Tukan <murad@dataheroes.ai>.
+
+Proceedings of the  $40^{th}$  International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
+
+the hidden layer. The linear output layer then computes  $\sum_{i=1}^{N} \alpha_i \rho(\|x - c_i\|_2)$ , where  $\alpha_i$  is the weight of neuron  $i$  in the linear output neuron. Therefore, RBFNNs are feedforward neural networks because the edges between the nodes do not form a cycle, and enjoy advantages such as simplicity of analysis, faster training time, and interpretability, compared to alternatives such as convolutional neural networks (CNNs) and even multi-layer perceptrons (MLPs) (Padmavati, 2011).
+
+Function approximation via RBFNNs. RBFNNs are universal approximators in the sense that an RBFNN with a sufficient number of hidden neurons (large  $N$ ) can approximate any continuous function on a closed, bounded subset of  $\mathbb{R}^d$  with arbitrary precision (Park & Sandberg, 1991), i.e., given a sufficiently large input set  $P$  of  $n$  points in  $\mathbb{R}^d$  and given its corresponding label function  $y: P \to \mathbb{R}$ , an RBFNN, can be trained to approximate the function  $y$ . Therefore, RBFNNs are commonly used across a wide range of applications, such as function approximation (Park & Sandberg, 1991; Lu et al., 1997), time series prediction (Whitehead & Choate, 1996; Leung et al., 2001; Harpham & Dawson, 2006), classification (Leonard & Kramer, 1991; Wuxing et al., 2004; Babu & Suresh, 2012), and system control (Yu et al., 2011; Liu, 2013), due to their faster learning speed.
+
+For a given RBFNN size, i.e., the number of neurons in the hidden layer, and an input set, the aim of this paper is to compute a small weighted subset that approximates the loss of the input data on any radial basis function neural network of this size and thus approximates any function defined (approximated) by such an RBFNN on the big input data. This small weighted subset is called a coreset.
+
+Coresets. Consider a prototypical machine/deep learning problem in which we are given an input set  $P \subseteq \mathbb{R}^d$  of  $n$  points, its corresponding weights function  $w: P \to \mathbb{R}$ , a set of queries  $X$  (a set of candidate solutions for the involved optimization problem), and a loss function  $f: P \times X \to [0, \infty)$ . The tuple  $(P, w, X, f)$  is called the query space, and it defines the optimization problem at hand — where usually, the goal is to find  $x^* \in \arg \min_{x \in X} \sum_{p \in P} w(p)f(p, x)$ . Given a query space  $(P, w, X, f)$ , a coreset is a small weighted subset of the input  $P$  that can provably approximate the cost of every query  $x \in X$  on  $P$  (Feldman, 2020); see Definition 2.1. In particular, a coreset for a RBFNN can
+
+approximate the cost of an RBFNN on the original training data for every set of centers and weights that define the RBFNN (see Section 4). Hence, the coreset approximates also the centers and weights that form the optimal solution of the RBFNN (the solution that approximates the desired function). Thus a coreset for a RBFNN would facilitate training data for function approximation without reading the full training data and more generally, a strong coreset for an RBFNN with enough hidden neurons would give a strong coreset for any function that can be approximated to some precision using the RBFNN.
+
+# To this end, in this paper, we aim to provide the first coreset for RBFNNs, and thus provably approximating (providing a coreset to) any function that can be approximated by a given RBFNN.
+
+Furthermore, we can use this small weighted subset (coreset) to suggest a provable data subset selection algorithm for training deep neural networks efficiently (on the small subset). Since our coreset approximates every function that can be approximated by an RBFNN of this size, it also approximates the gradient of each weight in a neural network (if it can be approximated by the RBFNN).
+
+Training neural networks on data subset. Although deep learning has become widely successful with the increasing availability of data (Krizhevsky et al., 2017; Devlin et al., 2019), modern deep learning systems have correspondingly increased in their computational resources, resulting in significantly larger training times, financial costs (Sharir et al., 2020), energy costs (Strubell et al., 2019), and carbon footprints (Strubell et al., 2019; Schwartz et al., 2020). Data subset selection (coresets) allows for efficient learning at several levels (Wei et al., 2014; Kaushal et al., 2019; Coleman et al., 2019; Har-Peled & Mazumdar, 2004; Clarkson, 2010). By employing a significantly smaller subset of the big dataset, (i) we enable learning on relatively low resource computing settings without requiring a huge number of GPU and CPU servers, (ii) we may greatly optimize the end-to-end turnaround time, which frequently necessitates many training runs for hyper-parameter tweaking, and (iii) because a large number of deep learning trials must be done in practice, we allow for considerable reductions in deep learning energy usage and  $\mathrm{CO}_{2}$  emissions (Strubell et al., 2019). Multiple efforts have recently been made to improve the efficiency of machine learning models using data subset selection (Mirzasoleiman et al., 2020a; Killamsetty et al., 2021b;a; Pooladzandi et al., 2022). However, existing techniques either (i) employ proxy functions to choose data points, (ii) are specialized to specific machine learning models, (iii) use approximations of parameters such as gradient error or generalization errors, (iv) lack provable guarantees on the approximation error, or (v) require an inefficient gradient computation of the whole data. Most importantly, all
+
+of these methods are model/network dependent, and thus computing the desired subset of the data after several training epochs (for the same network) takes a lot of time and must be repeated each time the network changes.
+
+To this end, in this paper, we introduce a provable and efficient model-independent subset selection algorithm for training neural networks. This will allow us to compute a subset of the training data, that is guaranteed to be a coreset for training multiple neural network architectures/models.
+
+# 1.1. Our Contributions
+
+In this paper, we introduce a coreset that approximates any function can be represented by an RBFNN architecture. Specifically:
+
+(i) We provide a coreset for the  $RBF$  and Laplacian cost functions; see Algorithm 1, and Section 3.1.2.  
+(ii) We generate a coreset for any RBFNN model, in turn approximating any function that can be represented by the RBFNN; see Figure 1 for illustration, and Section 4 for more details.  
+(iii) We then exploit the properties of  $RBFNNs$  to approximate the gradients of any deep neural networks ( $DNNs$ ), leading towards provable subset selection for learning/training  $DNNs$ . We also show the advantages of using our coreset against previous subset selection techniques; see Section 5 and Section 7.  
+(iv) Finally, we provide an open-source code implementation of our algorithm for reproducing our results and future research (ope, 2023).
+
+# 1.2. Related Work
+
+A long line of active work has studied efficient coreset constructions for various problems in computational geometry and machine learning, such as  $k$ -means and  $k$ -median clustering (Har-Peled & Mazumdar, 2004; Chen, 2009; Braverman et al., 2016; Huang & Vishnoi, 2020; Jubran et al., 2020; Cohen-Addad et al., 2022), regression (Dasgupta et al., 2008; Chhaya et al., 2020; Tolochinsky et al., 2022; Meyer et al., 2022; Maalouf et al., 2019; 2022b), low-rank approximation (Cohen et al., 2017; Braverman et al., 2020; Maalouf et al., 2020; 2021), volume maximization (Indyk et al., 2020; Mahabadi et al., 2020; Woodruff & Yasuda, 2022), projective clustering (Feldman et al., 2020; Tukan et al., 2022c), support vector machines (SVMs) (Clarkson, 2010; Tukan et al., 2021; Maalouf et al., 2022a), Bayesian inference (Campbell & Broderick, 2018), and sine wave fitting (Maalouf et al., 2022c). In addition, recently, coresets were leveraged for boosting sampling-based path planners
+
+![](images/b168bd525e5e467fb39604b719d4776cf738aa95a57404ffca6dbfd14f66853f.jpg)  
+Figure 1. Our contribution in a nutshell.
+
+![](images/a02a87609a193a9e508fd8336f02d1a44260bbcbe247bfb6ec384356814b66a4.jpg)  
+For  $x\in R^3$  , we have:
+
+![](images/3998dff8b993954bb4282ed31e50351b4b8b40de7c4baef4d337662dc540b208.jpg)
+
+in the field of Robotics (Tukan et al., 2022a), as well as in marine applications (Tukan et al., 2023)
+
+Baykal et al. (2022) suggested coreset-based algorithms for compressing the parameters of a trained fully-connected neural network by using sensitivity sampling on the weights of neurons after training, though without pruning full neurons. (Mussay et al., 2020; Liebenwein et al., 2019; Tukan et al., 2022b) sidestepped this issue by identifying the neurons that can be compressed regardless of their weights, due to the choice of the activation functions, thereby achieving coreset-based algorithms for neural pruning.
+
+These approaches use coresets to achieve an orthogonal goal to data subset selection in the context of deep learning – they greatly reduce the number of neurons in the network while we greatly reduce the number of samples in the dataset that need to be read by the neural network. Correspondingly, we reduce the effective size of the data that needs to be stored or even measured prior to the training stage. Moreover, we remark that even if the number of inputs to the input layer was greatly reduced by these neural compression approaches, the union of the inputs can still consist of the entire input dataset and so these approaches generally cannot guarantee any form of data distillation.
+
+Toward the goal of data subset selection, (Mirzasoleiman et al., 2020a;b) introduced algorithms for selecting representative subsets of the training data to accurately estimate the full gradient for tasks in both deep learning and classical machine learning models such as logistic regression and these approaches were subsequently refined by (Killamsetty et al., 2021a;b). Data distillation has also received a lot of attention in image classification (Bohdal et al., 2020; Nguyen et al., 2021; Dosovitskiy et al., 2021), natural language processing (Devlin et al., 2019; Brown et al., 2020), and federated learning (Ozkara et al., 2021; Zhu et al., 2021).
+
+On coresets for any function. To the best of our knowledge,
+
+the only other coresets eligible for handling a wide family of functions without the need to devise a problem-dependent sensitivity are (Claici & Solomon, 2018; Claici et al., 2018). While such coresets are interesting and related, (i) both works provide coreset constructions resulting in an additive approximation, (ii) the coresets' theoretical applications seem quite a bit restrictive as they intend to handle mainly a family of functions that are either  $k$ -Lipschitz (functions with bounded gradient, usually  $k$ ), having a bounded Dual-Sobolev distance, or functions satisfying the properties of reproducing kernel Hilbert space (RKHS). In addition, the running time in the worst-case scenario is not practical, i.e., exponential in the dimension of the points.
+
+On the other hand, our coreset under mild assumptions can satisfy any function approximated by RBFNN (Wu et al., 2012), in time that is polynomial in the dimension of the points and linear in the number of nonzero entries of the points (Clarkson & Woodruff, 2017).
+
+# 2. Preliminaries
+
+For an integer  $n > 0$ , we use  $[n]$  to denote the set  $\{1,2,\ldots,n\}$ . A weighted set of points is a pair  $(P,w)$  where  $P \subseteq \mathbb{R}^d$  is a set of points and  $w: P \to [0,\infty)$  is a weight function.
+
+We now formally provide the notion of  $\varepsilon$ -coreset for the  $RBF$  loss. This will be later extended to a coreset for  $RBFNN$ .
+
+Definition 2.1 (RBF  $\varepsilon$ -coreset). Let  $(P, w)$  be a weighted of  $n$  points in  $\mathbb{R}^d$ ,  $X \subseteq \mathbb{R}^d$  be a set of queries,  $\varepsilon \in (0, 1)$ . For every  $x \in X$  and  $p \in P$  let  $f(p, x) := \exp \left(-\| p - x \|_2^2\right)$  denote the RBF loss function between  $p$  and  $x$ . An  $\varepsilon$ -coreset for  $(P, w)$  with respect to  $f$ , is a pair  $(S, v)$  where  $S \subseteq P$ ,  $v : S \to (0, \infty)$  is a weight function, such that for every  $x \in X$ ,  $\left|1 - \frac{\sum_{q \in S} v(q) f(q, x)}{\sum_{p \in P} w(p) f(p, x)}\right| \leq \varepsilon$ .
+
+We say the RBF coreset is strong if it guarantees correctness
+
+over all  $x \in X$ . Otherwise, we say the coreset is weak if it only provides guarantees for all  $x$  only in some subset of  $X$ .
+
+Sensitivity sampling. To compute our  $RBF$ $\varepsilon$ -coreset, we utilize the sensitivity sampling framework (Braverman et al., 2016). In short, the sensitivity of a point  $p \in P$  corresponds to the "importance" of this point with respect to the other points and the problem at hand. In our context (with respect to the  $RBF$  loss), the sensitivity is defined as  $s(p) = \sup_{x \in X} \frac{w(p)f(p,x)}{\sum_{q \in P} w(q)f(q,x)}$ , where the denominator is nonzero. Once we bound the sensitivities for every  $p \in P$ , we can sample points from  $P$  according to their corresponding sensitivity bounds, and re-weight the sampled points to obtain an RBF  $\varepsilon$ -coreset as in Definition 2.1. The size of the sample (coreset) is proportional to the sum of these bounds – the tighter (smaller) these bounds, the smaller the coreset size; we refer the reader to Section A in the appendix.
+
+Sensitivity bounding. We now present our main tool for bounding the sensitivity of each input point with respect to the RBF and Laplacian loss functions.
+
+Definition 2.2 ( $\ell_1$ -SVD - Special case of Definition 4 (Tukan et al., 2020)). Let  $(P, w, \mathbb{R}^d, f)$  be a query space (see Definition A.1) where for every  $p \in P$  and  $x \in \mathbb{R}^d$ ,  $f(p, x) = |p^Tx|$ . Let  $D \in [0, \infty)^{d \times d}$  be a diagonal matrix of full rank and let  $V \in \mathbb{R}^{d \times d}$  be an orthogonal matrix, such that for every  $x \in \mathbb{R}^d$ ,  $\|DV^Tx\|_2 \leq \sum_{p \in P} w(p) |p^Tx| \leq \sqrt{d} \|DV^Tx\|_2$ . Define  $U: P \to \mathbb{R}^d$  such that  $U(p) = D^{-1}V^Tp$  for every  $p \in P$ . The tuple  $(U, D, V)$  is the  $\| \cdot \|_1$ -SVD of  $P$ .
+
+Using the above tool, the sensitivity with respect to the RBF loss function can be bounded using the following.
+
+Lemma 2.3 (Special case of Lemma 35, (Tukan et al., 2020)). Let  $(P,w,\mathbb{R}^d,f)$  be query space as in Definition A.1 where for every  $p\in P$  and  $x\in \mathbb{R}^d$ ,  $f(p,x) = |p^{T}x|$ . Let  $(U,D,V)$  be the  $\| \cdot \| _1$ -SVD of  $(P,w)$  with respect to  $|\cdot|$  (see Definition 2.2). Then: (i) for every  $p\in P$ , the sensitivity of  $p$  with respect to the query space  $(P,w,\mathbb{R}^d,|\cdot|)$  is bounded by  $s(p)\leq \| U(p)\| _1$ , and (ii) the total sensitivity is bounded by  $\sum_{p\in P}s(p)\leq d^{1.5}$ .
+
+# 3. Method
+
+In this section, we provide coresets for the Gaussian and Laplacian loss functions. We detail our coreset construction for the Gaussian loss function and Laplacian loss function in Section 3.1.2.
+
+Overview of Algorithm 1. Algorithm 1 receives as input, a set  $P$  of  $n$  points in  $\mathbb{R}^d$ , a weight function  $w: P \to [0, \infty)$ , a bound  $R$  on the radius of the ball containing query space  $X$ , and a sample size  $m > 0$ . If the sample size  $m$  is
+
+# Algorithm 1 CORESET  $(P, w, R, m)$
+
+input A set  $P\subseteq \mathbb{R}^d$  of  $n$  points, a weight function  $w:$ $P\to [0,\infty)$  , a bound on radius of the query space  $X$  , and a sample size  $m\geq 1$    
+output A pair  $(S,v)$  that satisfies Theorem 3.2  
+1: Set  $d' \coloneqq$  the VC dimension of quadruple  $(P, w, X, \rho(\cdot))$  {See Definition A.2}  
+2:  $P^{\prime}:= \left\{q_{p} = \left[\| p\|_{2}^{2}, - 2p^{T},1\right]^{T}\mid \forall p\in P\right\}$  
+3:  $(U, D, V) := \text{the } f\text{-SVD of } (P', w, |\cdot|) \{\text{See Definition 2.2}\}$  
+4: for every  $p \in P$  do  
+5:  $s(p) \coloneqq \eta \cdot \left(\frac{w(p)}{\sum_{q \in P} w(q)} + w(p)\|U(q_p)\|_1\right)$  {bound on the sensitivity of  $p$  as in Lemma B.1 in the appendix, where  $\eta \coloneqq e^{12R^2}\left(1 + 8R^2\right)$ }  
+6: end for  
+7:  $t\coloneqq \sum_{p\in P}s(p)$  
+8: Set  $\tilde{c} \geq 1$  to be a sufficiently large constant {Can be determined from Theorem 3.2}  
+9: Pick an i.i.d sample  $S$  of  $m$  points from  $P$ , where each  $p \in P$  is sampled with probability  $\frac{s(p)}{t}$  
+10: set  $v: \mathbb{R}^d \to [0, \infty]$  to be a weight function such that for every  $q \in S$ ,  $v(q) = \frac{t}{s(q) \cdot m}$ .
+
+return  $(S,v)$
+
+sufficiently large, then Algorithm 1 outputs a pair  $(S,v)$  that is an  $\varepsilon$ -coreset for RBF cost function; see Theorem 3.2.
+
+First,  $d'$  is set to be the VC dimension of the quadruple  $(P, w, X, \rho(\cdot))$ ; see Definition A.2. The heart of our algorithm lies in formalizing the RBF loss function as a variant of the regression problem, specifically, a variant of the  $\ell_1$ -regression problem. The conversion requires manipulation of the input data as presented at Line 2. We then compute the  $f$ -SVD of the new input data with respect to the  $\ell_1$ -regression problem followed by bounded the sensitivity of such points (Lines 3-5). Now we have all the needed ingredients to obtain an  $\varepsilon$ -coreset (see Theorem A.3), i.e., we sample i.i.d m points from P based on their sensitivity bounds (see Line 9), followed by assigning a new weight for every sampled point at Line 10.
+
+# 3.1. Analysis
+
+# 3.1.1. LOWER BOUND ON THE CORESET SIZE FOR THE GAUSSIAN LOSS FUNCTION
+
+We first show the lower bound on the size of coresets, to emphasize the need for assumptions on the data and the query space.
+
+Theorem 3.1. There exists a set of  $n$  points  $P \subseteq \mathbb{R}^d$  such that  $\sum_{p \in P} s(p) = \Omega(n)$ .
+
+Proof. Let  $d \geq 3$  and let  $P \subseteq \mathbb{R}^d$  be a set of  $n$  points distributed evenly on a 2-dimensional sphere of radius  $\sqrt{\frac{\ln n}{2\cos\left(\frac{\pi}{n}\right)}}$ . In other words, using the law of cosines, every  $p \in P$ ,  $\sqrt{\ln n} = \min_{q \in P \setminus \{p\}} \| p - q \|_2$ ; see Figure C. Observe that for every  $p \in P$ ,
+
+$$
+\begin{array}{l} s(p):= \max_{x\in \mathbb{R}^{d}}\frac{e^{-\|p - x\|_{2}^{2}}}{\sum_{q\in P}e^{-\|q - x\|_{2}^{2}}}\geq \frac{e^{-\|p - p\|_{2}^{2}}}{\sum_{q\in P}e^{-\|p - q\|_{2}^{2}}} \\ = \frac {1}{1 + \sum_ {q \in P \backslash \{p \}} e ^ {- \| p - q \| _ {2} ^ {2}}} \geq \frac {1}{1 + \sum_ {q \in P \backslash \{p \}} \frac {1}{n}} \geq \frac {1}{2}, \\ \end{array}
+$$
+
+where the first equality holds by definition of the sensitivity, the first inequality and second equality hold trivially, the second inequality follows from the assumption that  $\sqrt{\ln n} \leq \min_{q \in P \setminus \{p\}} \| p - q \|_2$ , and finally the last inequality holds since  $\sum_{q \in P \setminus \{p\}} \frac{1}{n} \leq 1$ .
+
+# 3.1.2. REASONABLE ASSUMPTIONS LEAD TO EXISTENCE OF CORESETS
+
+Unfortunately, it is not immediately straightforward to bound the sensitivities of either the Gaussian loss function or the Laplacian loss function. Therefore, we first require the following structural properties in order to relate the Gaussian and Laplacian loss functions into more manageable quantities. We shall ultimately relate the function  $e^{-|p^T x|}$  to both the Gaussian and Laplacian loss functions. Thus, we first relate the function  $e^{-|p^Tx|}$  to the function  $|p^Tx| + 1$ .
+
+Let  $p \in \mathbb{R}^d$  such that  $\| p \|_2 \leq 1$ , and let  $R > 0$  be positive real number. Then for every  $x \in \left\{ x \in \mathbb{R}^d \big| \| x \|_2 \leq R \right\}$ ,  $\frac{1}{e^R(1 + R)} \left( 1 + |p^T x | \right) \leq e^{-|p^Tx|} \leq |p^Tx| + 1$ .
+
+In what follows, we provide the analysis of coreset construction for the RBF and Laplacian loss functions, considering an input set of points lying in the unit ball. We refer the reader to the supplementary material for generalization of our approaches towards general input set of points.
+
+Theorem 3.2 (Coreset for RBF). Let  $R \geq 1$  be a positive real number,  $X = \{x \in \mathbb{R}^d \mid \| x \|_2 \leq R\}$ , and let  $\varepsilon, \delta \in (0,1)$ . Let  $(P, w, X, f)$  be query space as in Definition A.1 such that every  $p \in P$  satisfies  $\| p \|_2 \leq 1$ . For every  $x \in X$  and  $p \in P$ , let  $f(p, x) := \rho(\| p - x \|_2)$ . Let  $(S, v)$  be a call to CORESET $(P, w, R, m)$  where  $S \subseteq P$  and  $v: S \to [0, \infty)$ . Then  $(S, v)$  is coreset of  $(P, w)$  with probability at least  $1 - \delta$ , if  $m = O\left(\frac{e^{12R^2} R^2 d^{1.5}}{\varepsilon^2} \left(R^2 + \log d + \log \frac{1}{\delta}\right)\right)$ .
+
+**Coreset for Laplacian loss function.** In what follows, we provide a coreset for the Laplacian loss function. Intuitively speaking, leveraging the properties of the Laplacian loss function, we were able to construct a coreset that holds for every vector  $x \in \mathbb{R}^d$  unlike the RBF case where the coreset
+
+holds for a ball of radius  $R$ . We emphasize that the reason for this is due to the fact that the Laplacian loss function is less sensitive than the RBF.
+
+Theorem 3.3 (Coreset for the Laplacian loss function). Let  $(P, w, \mathbb{R}^d, f)$  be query space as in Definition A.1 such that every  $p \in P$  satisfies  $\| p \|_2 \leq 1$ . For  $x \in \mathbb{R}^d$  and  $p \in P$ , let  $f(p, x) := e^{-\| p - x \|_2}$ . Let  $\varepsilon, \delta \in (0, 1)$ . Then there exists an algorithm which given  $P, w, \varepsilon, \delta$  return a weighted set  $(S, v)$  where  $S \subseteq P$  of size  $O\left(\frac{\sqrt{n} d^{1.25}}{\varepsilon^2} \left( \log n + \log d + \log \frac{1}{\delta} \right)\right)$  and a weight function  $v: S \to [0, \infty)$  such that  $(S, v)$  is an  $\varepsilon$ -coreset of  $(P, w)$  with probability at least  $1 - \delta$ .
+
+# 4. Radial Basis Function Networks
+
+In this section, we consider coresets for RBFNNs. Consider an RBFNN with  $L$  neurons in the hidden layer and a single output neuron. First note that the hidden layer uses radial basis functions as activation functions so that the output is a scalar function of the input layer,  $\phi : \mathbb{R}^d \to \mathbb{R}$  defined by  $\phi(x) = \sum_{i=1}^{L} \alpha_i \rho(\|x - c^{(i)}\|_2)$ , where  $c^{(i)} \in \mathbb{R}^n$  for each  $i \in [d]$ .
+
+For an input dataset  $P$  and a corresponding desired output function  $y: P \to \mathbb{R}$ , RBFNNs aim to optimize  $\sum_{p \in P} \left(y(p) - \sum_{i=1}^{L} \alpha_i e^{-\left\| p - c^{(i)} \right\|^2_2}\right)^2$ . Expanding the cost function, we obtain that RBFNNs aim to optimize
+
+$$
+\begin{array}{l} \sum_ {p \in P} y (p) ^ {2} - 2 \sum_ {i = 1} ^ {L} \alpha_ {i} \overbrace {\left(\sum_ {p \in P} y (p) e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}}\right)} ^ {\alpha} \tag {2} \\ + \underbrace {\sum_ {p \in P} \left(\sum_ {i = 1} ^ {L} \alpha_ {i} e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}}\right) ^ {2}} _ {\beta}. \\ \end{array}
+$$
+
+Bounding the  $\alpha$  term in equation 2. We first define for every  $x\in \mathbb{R}^d$
+
+$$
+\phi^ {+} (x) = \sum_ {p \in P, y (p) > 0} y (p) e ^ {- \| p - x \| _ {2} ^ {2}}
+$$
+
+$$
+\phi^ {-} (x) = \sum_ {p \in P, y (p) <   0} | y (p) | e ^ {- \| p - x \| _ {2} ^ {2}}.
+$$
+
+Observe that  $\sum_{p\in P}y(p)\rho (\| p - c^{(i)}\| _2) = \phi^+ (c^{(i)}) - \phi^{-}(c^{(i)})$ . Thus the  $\alpha$  term in equation 2 can be approximated using the following.
+
+Theorem 4.1. There exists an algorithm that samples  $O\left(\frac{e^{8R^2}R^2d^{1.5}}{\varepsilon^2}\left(R^2 + \log d + \log \frac{2}{\delta}\right)\right)$  points to form weighted sets  $(S_1, w_1)$  and  $(S_2, w_2)$  such that with proba
+
+bility at least  $1 - 2\delta$
+
+$$
+\frac{\left|\sum_{p\in P}y(p)\phi(p) - \left(\sum_{\substack{i\in[L]\\ \alpha_{i} > 0}}\alpha_{i}\gamma_{S_{1}} + \sum_{\substack{j\in[L]\\ \alpha_{j} <   0}}\alpha_{j}\gamma_{S_{2}}\right)\right|}{\sum_{i\in[L]}|\alpha_{i}|  \bigl(\phi^{+}  \bigl(c^{(i)}\bigr) + \phi^{-}  \bigl(c^{(i)}\bigr)\bigr)}\leq \varepsilon ,
+$$
+
+where  $\gamma_{S_1} := \sum_{p \in S_1} w_1(p) e^{-\left\| p - c^{(i)} \right\|_2^2}$  and  $\gamma_{S_2} := \sum_{q \in S_2} w_2(q) e^{-\left\| q - c^{(j)} \right\|_2^2}$ .
+
+Bounding the  $\beta$  term in equation 2. By Cauchy's inequality, it holds that
+
+$$
+\begin{array}{l} \sum_ {p \in P} \left(\sum_ {i = 1} ^ {L} \alpha_ {i} e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}}\right) ^ {2} \leq L \sum_ {p \in P} \sum_ {i = 1} ^ {L} \alpha_ {i} ^ {2} e ^ {- 2 \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}} \\ = L \sum_ {i = 1} ^ {L} \alpha_ {i} ^ {2} \sum_ {p \in P} e ^ {- 2 \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}}, \\ \end{array}
+$$
+
+where the equality holds by simple rearrangement.
+
+Using Theorem 3.2, we can approximate the upper bound on  $\beta$  with an approximation of  $L(1 + \varepsilon)$ . However, if for every  $i\in [L]$  it holds that  $\alpha_{i}\geq 0$ , then we also have the lower bound  $\sum_{i = 1}^{L}\alpha_{i}^{2}\sum_{p\in P}e^{-2\left\| p - c^{(i)}\right\|_{2}^{2}}\leq$
+
+$$
+\sum_ {p \in P} \left(\sum_ {i = 1} ^ {L} \alpha_ {i} e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}}\right) ^ {2}.
+$$
+
+Since we can generate a coreset that yields a multiplicative approximation error to the full data for the left-hand side term of the above inequality, then we obtain also a coreset with multiplicative approximation error in a sense for  $\beta$  as well.
+
+Thus by combining all of the above, we obtain the following corollary.
+
+Corollary 4.2. Let RFBNN be a sufficiently large (RBF) neural network,  $P\subset \mathbb{R}^d$  , and let  $f(p):P\to \mathbb{R}$  be a function that can be approximated by a set of centers and weights  $q$  (the learned parameters that define the RBFNN) assigned to RFBNN such that, for every  $p\in P$  ..  $RFBNN(p,q)\approx$ $f(p)$  , and  $\left\| \sum_{p\in P}RFBNN(p,q) - \sum_{p\in P}f(p)\right\| \leq \varepsilon$  , where each center of the RBFNN lies in ball centered at the origin with radius  $R$  (to satisfy our coreset's assumption). Let  $(S,v)$  be the output of a call to Algorithm 1, i.e.,  $(S,v)$  is coreset for  $P$  is an RBF coreset; see Definition 2.1. Then,
+
+$$
+\left| \sum_ {p \in P} f (p) - \sum_ {s \in S} v (s) R B F N N (s, q) \right| \leq \varepsilon \sum_ {p \in P} f (p) + \varepsilon .
+$$
+
+# 5. Advantages of our Methods
+
+One coreset for all networks. Our coreset is model-independent, i.e., we aim at improving the running time of multiple neural networks. Contrary to other methods that need to compute the coreset after each gradient update to support their theoretical proofs, our method gives the advantage of computing the sensitivity (or the coreset) only once, for all of the required networks. This is because our coreset can approximate any function that can be defined (approximated) using a RBFNN model.
+
+Efficient coreset per epoch. Practically, our competing methods for data selection are not applied before each epoch, but every  $\Upsilon$  epochs. This is since the competing methods require a lot of time to compute a new coreset since they compute the gradients of the network with respect to each input training data. However, our coreset can be computed before each epoch in a negligible time ( $\sim 0$  seconds), since we compute the sensitivity of each point (image) in the data once at the beginning, and then whenever we need to create a new coreset, we simply sample from the input data according to the sensitivity distribution.
+
+# 6. On the Novelty of Our Approach
+
+First, note that we have established a lower bound of  $\Omega(n)$  for the coreset size when the problem of RBF fitting is handled with no assumptions; see Theorem 3.1, which consequently means that the RBF fitting problem is considered hard from the perspective of coresets. With this in mind, the assumption of boundness on the query space  $(R)$  aims to eliminate cases similar depicted by Theorem 3.1 that are deemed redundant since almost all points are needed. Our results were generalized to account for any general set of points  $P \subset \mathbb{R}^d$  with modified bounds; see Section B. On the positive side of things, these bounds are usually pessimistic, and such a phenomenon can be seen in Figure 3.
+
+In addition, from a practical point of view, the centers that the RBFNN aims to find are usually contained in (or in the vicinity of) the convex hull of the points (when setting for example the standard deviation of each neuron to 1). Combining this with the insights above, to ensure the existence of small coresets (in general), we need to lower the diameter of the data (the maximal distance between any pair of input points), which can be done using normalization/standardization. Thus,  $R$  can be chosen to be small such that the optimal RBFNN centers with respect to fitting the entire data are contained in a ball of radius  $R$ . Hence, our coreset will have a small size in such settings. We note that in machine learning, normalization, and standardization techniques are very common preprocessing used for improving the quality of the classifier/predictor. Thus, the centers will be contained in a small ball around the origin
+
+Table 1. Data Selection Results for CIFAR10 using ResNet-18.  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Top-1 Test accuracy of the Model(%)</td><td colspan="4">Model Training time(in hrs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>95.09</td><td>95.09</td><td>95.09</td><td>95.09</td><td>4.34</td><td>4.34</td><td>4.34</td><td>4.34</td></tr><tr><td>RANDOM (skyline for training time)</td><td>71.2</td><td>80.8</td><td>86.98</td><td>87.6</td><td>0.22</td><td>0.46</td><td>0.92</td><td>1.38</td></tr><tr><td>GLISTER</td><td>85.5</td><td>91.92</td><td>92.78</td><td>93.63</td><td>0.43</td><td>0.91</td><td>1.13</td><td>1.46</td></tr><tr><td>CRAIG</td><td>82.74</td><td>87.49</td><td>90.79</td><td>92.53</td><td>0.81</td><td>1.08</td><td>1.45</td><td>2.399</td></tr><tr><td>CRAIGPB</td><td>83.56</td><td>88.77</td><td>92.24</td><td>93.58</td><td>0.4466</td><td>0.70</td><td>1.13</td><td>2.07</td></tr><tr><td>GRADMATCH</td><td>86.7</td><td>90.9</td><td>91.67</td><td>91.89</td><td>0.40</td><td>0.84</td><td>1.42</td><td>1.52</td></tr><tr><td>GRADMATCHPB</td><td>85.4</td><td>90.01</td><td>93.34</td><td>93.75</td><td>0.36</td><td>0.69</td><td>1.09</td><td>1.38</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>86.9</td><td>91.4</td><td>93.61</td><td>94.44</td><td>0.28</td><td>0.52</td><td>0.98</td><td>1.38</td></tr></table>
+
+Table 2. Data Selection Results for CIFAR10 using ResNet-18 with warm start  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Top-1 Test accuracy of the Model(%)</td><td colspan="4">Model Training time(in hrs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>95.09</td><td>95.09</td><td>95.09</td><td>95.09</td><td>4.34</td><td>4.34</td><td>4.34</td><td>4.34</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>83.2</td><td>87.8</td><td>90.9</td><td>92.6</td><td>0.21</td><td>0.42</td><td>0.915</td><td>1.376</td></tr><tr><td>GLISTER-WARM</td><td>86.57</td><td>91.56</td><td>92.98</td><td>94.09</td><td>0.42</td><td>0.88</td><td>1.08</td><td>1.40</td></tr><tr><td>CRAIG-WARM</td><td>84.48</td><td>89.28</td><td>92.01</td><td>92.82</td><td>0.6636</td><td>0.91</td><td>1.31</td><td>2.20</td></tr><tr><td>CRAIGPB-WARM</td><td>86.28</td><td>90.07</td><td>93.06</td><td>93.8</td><td>0.4143</td><td>0.647</td><td>1.07</td><td>2.06</td></tr><tr><td>GRADMATCH-WARM</td><td>87.2</td><td>92.15</td><td>92.11</td><td>92.01</td><td>0.38</td><td>0.73</td><td>1.24</td><td>1.41</td></tr><tr><td>GRADMATCHPB-WARM</td><td>86.37</td><td>92.26</td><td>93.59</td><td>94.17</td><td>0.32</td><td>0.62</td><td>1.05</td><td>1.36</td></tr><tr><td>RBFNN CORESET-WARM (OURS)</td><td>87.82</td><td>91.44</td><td>93.81</td><td>94.6</td><td>0.27</td><td>0.51</td><td>0.99</td><td>1.36</td></tr></table>
+
+Table 3. Data Selection Results for CIFAR100 using ResNet-18  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Top-1 Test accuracy of the Model(%)</td><td colspan="4">Model Training time(in hrs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>75.37</td><td>75.37</td><td>75.37</td><td>75.37</td><td>4.871</td><td>4.871</td><td>4.871</td><td>4.871</td></tr><tr><td>RANDOM (skyline for training time)</td><td>19.02</td><td>31.56</td><td>49.6</td><td>58.56</td><td>0.2475</td><td>0.4699</td><td>0.92</td><td>1.453</td></tr><tr><td>GLISTER</td><td>29.94</td><td>44.03</td><td>61.56</td><td>70.49</td><td>0.3536</td><td>0.6456</td><td>1.11</td><td>1.5255</td></tr><tr><td>CRAIG</td><td>36.61</td><td>55.19</td><td>66.24</td><td>70.01</td><td>1.354</td><td>1.785</td><td>1.91</td><td>2.654</td></tr><tr><td>CRAIGPB</td><td>38.95</td><td>54.59</td><td>67.12</td><td>70.61</td><td>0.4489</td><td>0.6564</td><td>1.15</td><td>1.540</td></tr><tr><td>GRADMATCH</td><td>41.01</td><td>59.88</td><td>68.25</td><td>71.5</td><td>0.5143</td><td>0.8114</td><td>1.40</td><td>2.002</td></tr><tr><td>GRADMATCHPB</td><td>40.53</td><td>60.39</td><td>70.88</td><td>72.57</td><td>0.3797</td><td>0.6115</td><td>1.09</td><td>1.56</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>54.17</td><td>64.59</td><td>71.17</td><td>73.58</td><td>0.346</td><td>0.5699</td><td>1.01</td><td>1.552</td></tr></table>
+
+Table 4. Data Selection Results for CIFAR100 using ResNet-18 with warm start  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Top-1 Test accuracy of the Model(%)</td><td colspan="4">Model Training time(in hrs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>75.37</td><td>75.37</td><td>75.37</td><td>75.37</td><td>4.871</td><td>4.871</td><td>4.871</td><td>4.871</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>58.2</td><td>65.95</td><td>70.3</td><td>72.4</td><td>0.242</td><td>0.468</td><td>0.921</td><td>1.43</td></tr><tr><td>GLISTER-WARM</td><td>57.17</td><td>64.95</td><td>62.14</td><td>72.43</td><td>0.3185</td><td>0.6059</td><td>1.06</td><td>1.452</td></tr><tr><td>CRAIG-WARM</td><td>57.44</td><td>67.3</td><td>69.76</td><td>72.77</td><td>1.09</td><td>1.48</td><td>1.81</td><td>2.4112</td></tr><tr><td>CRAIGPB-WARM</td><td>57.66</td><td>67.8</td><td>70.84</td><td>73.79</td><td>0.394</td><td>0.6030</td><td>1.10</td><td>1.5567</td></tr><tr><td>GRADMATCH-WARM</td><td>57.72</td><td>68.23</td><td>71.34</td><td>74.06</td><td>0.3788</td><td>0.7165</td><td>1.30</td><td>1.985</td></tr><tr><td>GRADMATCHPB-WARM</td><td>58.26</td><td>69.58</td><td>73.2</td><td>74.62</td><td>0.300</td><td>0.5744</td><td>1.01</td><td>1.5683</td></tr><tr><td>RBFNN CORESET-WARM (OURS)</td><td>59.22</td><td>67.8</td><td>72.79</td><td>75.04</td><td>0.352</td><td>0.5710</td><td>1.03</td><td>1.56</td></tr></table>
+
+when using these preprocessing tools.
+
+Table 5. Data Selection Results for ImageNet2012 using ResNet-18  
+
+<table><tr><td>Budget(%)</td><td>Top-1 Test accuracy of the Model(%)5%</td><td>Model Training time(in hrs)5%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>70.36</td><td>276.28</td></tr><tr><td>RANDOM (skyline for training time)</td><td>21.124</td><td>14.12</td></tr><tr><td>CRAIGPB</td><td>44.28</td><td>22.24</td></tr><tr><td>GRADMATCH</td><td>47.24</td><td>18.24</td></tr><tr><td>GRADMATCHPB</td><td>45.15</td><td>16.12</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>47.26</td><td>15.24</td></tr></table>
+
+![](images/97bc08a494e389f8dfa6be0cd2141d0702b5ef4b627cae9385ada95f509b04d8.jpg)  
+(a)
+
+![](images/9e403b825704b33d285afefefe2fbcc97339b9c5bac7182f99c1b2762dc96da5.jpg)  
+(b)
+
+![](images/c471b604acd6173bb004938646e7682721a247f4b5957abb772f38b56fa04eb1.jpg)  
+(c)
+
+![](images/06f885037c68f3a502cff2b209a77b742d96bf99acf2ce3669d2e623a7dd733f.jpg)  
+Figure 2. (a) is the function we wish to approximate by training an RBFNN on: (b) a uniformly sampled subset, and (c) our coreset.  
+(a)  
+Figure 3. (a) and (b) are depictions of the additive and relative error with respect to RBF fitting problem concerning the dataset, respectively. (c) and (d) re depictions of the additive and relative error with respect to RBF fitting problem concerning the dataset, respectively.
+
+![](images/e134c9e50e539510ed321769d356a285e850dcf9ee9c2f0e2e003d6e8ca9a9ff.jpg)  
+(b)
+
+![](images/8da0cd21275800601320da0e5bb935f2a39716b2b00ac4a268c45a318fad2ccc.jpg)  
+(c)
+
+![](images/beb8fc00926ce12fcc0d0d05b13a38b52e070f68870540bd971d6fa3f3ecd00d.jpg)  
+(d)
+
+# 7. Experimental Results
+
+In this section, we practically demonstrate the efficiency and stability of our RBFNN coreset approach for training deep neural networks via data subset selection. We mainly study the trade-off between accuracy and efficiency.
+
+Competing methods. We compare our method against many variants of the proposed algorithms in (Killamsetty et al., 2021a) (denoted by, GRAD-MATCH), in (Mirza-soleiman et al., 2020a) (denoted by CRAIG), and in (Kil-
+
+lamsetty et al., 2021b) (denoted by GLISTER). For each of these methods, we report the results for 4 variants: (i) the "vanilla" method, denoted by its original name, (ii) applying a warm start i.e., training on the whole data for  $50\%$  of the training time before training the other  $50\%$  on the coreset, where such methods are denoted by adding the suffix -WARM. (iii) a more efficient version of each of the competing methods denoted by adding the suffix PB (more details are given at (Killamsetty et al., 2021a)), and finally, a combination of both (ii) and (iii). In other
+
+words, the competing methods are GRAD-MATCH, GRAD-MATCHPB, GRAD-MATCH-WARM, GRAD-MATCHPB-WARM, CRAIG, CRAIGPB, CRAIG-WARM, CRAIGPB-WARM, and GLISTER-WARM. We also compare against randomly selecting points (denoted by RANDOM).
+
+Datasets and model architecture. We performed our experiments for training CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) on ResNet18 (He et al., 2016), MNIST (LeCun et al., 1998) on LeNet, and ImageNet-2012 (Deng et al., 2009) on Resnet18 (He et al., 2016).
+
+The setting. We adapted the same setting of (Killamsetty et al., 2021a), where we used SGD optimizer for training initial learning rate equal to 0.01, a momentum of 0.9, and a weight decay of  $5e - 4$ . We decay the learning rate using cosine annealing (Loshchilov & Hutter, 2016) for each epoch. For MNIST, we trained the LeNet model for 200 epochs. For CIFAR10 and CIFAR100, we trained the ResNet18 for 300 epochs - all on batches of size 20 for the subset selection training versions. We train the data selection methods and the entire data training with the same number of epochs; the main difference is the number of samples used for training a single epoch. All experiments were executed on V100 GPUs. The reported test accuracy in the results is after averaging across five runs.
+
+Subset sizes and the  $\Upsilon$  parameter. For MNIST, we use sizes of  $\{1\%, 3\%, 5\%, 10\}$ , while for CIFAR10 and CIFAR100, we use  $\{5\%, 10\%, 20\%, 30\}$ , and for ImageNet we use  $5\%$ . Since the competing methods require a lot of time to compute the gradients, we set  $\Upsilon = 20$ . We note that for our coreset we can test it with  $\Upsilon = 1$  without adding run-time since once the sensitivity vector is defined, commuting a new coreset requires  $\sim 0$  seconds. However, we test it with  $\Upsilon = 20$ , to show its robustness.
+
+Discussion. Tables 1-4 report the results for CIFAR10 and CIFAR100. It is clear from Tables 1 and 2 that our method achieves the best accuracy, with and without warm start, for  $5\%$ ,  $20\%$ , and  $30\%$  subset selection on CIFAR10. For CIFAR100, our method drastically outperforms all of the methods that do not apply a warm start. When applying a warm start, we still win in half of the cases. Note that, we outperform all of the other methods in terms of accuracy vs time. The same phenomenon is witnessed in the ImageNet experiment (Table 5) as our coreset achieves the highest accuracy. We refer the reader to the MNIST experiment (Table 6 in the appendix). We note that our sensitivity sampling vector is computed once during our experiments for each dataset. This vector can be used to sample coresets of different sizes, for different networks, at different epochs of training, in a time that is close to zero seconds. In all tables, the best results are highlighted in bold.
+
+Function Approximations. We now compare our coreset
+
+to uniform for function approximation. Specifically, we generate around 10,000 points in  $3D$ , while setting the third entry of each point to be a function of the first 2 entries,  $f(x) = e^{-\|x\|_2^2} + 0.2 \cos(4\|x\|_2)$ . We train an RBFNN to reproduce the function using only 400 points, where we saw that our coreset (Figure 2(c)) is closer visually to the true function (Figure 2(a)) using uniform sampling for reproducing the image (Figure 2(b)).
+
+RBF fitting problem. Finally, we show for the  $RBF$  fitting task on the HTRU and CreditCard datasets (Dua et al., 2017), where, our coreset is better than uniform sampling by at max a multiplicative factor of 1.5. In this experiment, we show the additive and multiplicative approximation associated with our method compared to uniform sampling with respect to the quality of solving the RBF fitting problem on the entire data. We refer the reader to Figure 3.
+
+# 8. Conclusion and Future Work
+
+In this paper, we have introduced a coreset that provably approximates any function that can be represented by RBFNN architectures. Our coreset construction can be used to approximate the gradients of any deep neural networks (DNNs), leading towards provable subset selection for learning/training DNNs. We also empirically demonstrate the value of our work by showing significantly better performances over various datasets and model architectures. As the first work on using coresets for data subset selection with respect to RBFNNs, our results lead to a number of interesting possible future directions. It is natural to ask whether there exist smaller coreset constructions that also provably give the same worst-case approximation guarantees. Furthermore, RKHS methods (Claici & Solomon, 2018; Claici et al., 2018) may be investigated in this context either by boosting their implementation or by merging ideas with this work. In addition, can our results be extended to more general classes of loss functions? Finally, we remark that although our empirical results significantly beat state-of-the-art, they nevertheless only serve as a proof-of-concept and have not been fully optimized with additional heuristics.
+
+# 9. Acknowledgements
+
+Support for this project has been provided in part by the Office of Naval Research (ONR) Grant N00014-18-1-2830. We are grateful for it.
+
+# References
+
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+
+# A. Coreset Constructions
+
+In what follows, we provide the necessary tools to obtain a coreset; see Definition 2.1.
+
+Definition A.1 (Query space). Let  $P$  be a set of  $n \geq 1$  points in  $\mathbb{R}^d$ ,  $w: P \to [0, \infty)$  be a non-negative weight function, and let  $f: P \times \mathbb{R}^d \to [0, \infty)$  denote a loss function. The tuple  $(P, w, \mathbb{R}^d, f)$  is called a query space.
+
+Definition A.2 (VC-dimension (Braverman et al., 2016)). For a query space  $(P,w,\mathbb{R}^d,f)$  and  $r\in [0,\infty)$ , we define
+
+$$
+\operatorname {r a n g e s} (x, r) = \left\{p \in P \mid w (p) f (p, x) \leq r \right\},
+$$
+
+for every  $x \in \mathbb{R}^d$  and  $r \geq 0$ . The dimension of  $(P, w, \mathbb{R}^d, f)$  is the size  $|S|$  of the largest subset  $S \subset P$  such that
+
+$$
+\left| \left\{S \cap \operatorname {r a n g e s} (x, r) \mid x \in \mathbb {R} ^ {d}, r \geq 0 \right\} \right| = 2 ^ {| S |},
+$$
+
+where  $|A|$  denotes the number of points in  $A$  for every  $A \subseteq \mathbb{R}^d$ .
+
+The following theorem formally describes how to construct an  $\varepsilon$ -coreset based on the sensitivity sampling framework.
+
+Theorem A.3 (Restatement of Theorem 5.5 in (Braverman et al., 2016)). Let  $(P, w, \mathbb{R}^d, f)$  be a query space as in Definition A.1. For every  $p \in P$  define the sensitivity of  $p$  as  $\sup_{x \in \mathbb{R}^d} \frac{w(p)f(p,x)}{\sum_{q \in P} w(q)f(q,x)}$ , where the sup is over every  $x \in \mathbb{R}^d$  such that the denominator is non-zero. Let  $s: P \to [0,1]$  be a function such that  $s(p)$  is an upper bound on the sensitivity of  $p$ . Let  $t = \sum_{p \in P} s(p)$  and  $d'$  be the VC dimension of the triplet  $(P, w, \mathbb{R}^d, f)$ ; see Definition A.2. Let  $c \geq 1$  be a sufficiently large constant,  $\varepsilon, \delta \in (0,1)$ , and let  $S$  be a random sample of
+
+$$
+| S | \geq \frac {c t}{\varepsilon^ {2}} \left(d ^ {\prime} \log t + \log \frac {1}{\delta}\right)
+$$
+
+i.i.d points from  $P$ , such that every  $p \in P$  is sampled with probability  $s(p) / t$ . Let  $v(p) = \frac{tw(p)}{s(p)|S|}$  for every  $p \in S$ . Then, with probability at least  $1 - \delta$ ,  $(S, v)$  is an  $\varepsilon$ -coreset for  $P$  with respect to  $f$ .
+
+Remark A.4 (VC dimension w.r.t. RBF/Laplacian fitting problems). In the context of RBF and Laplacian fitting problems, the corresponding VC dimension is  $O(d)$  where  $d$  denotes the dimensionality (number of features) of the queries.
+
+# B. Proofs for our Main Theorems
+
+# B.1. Proof of Claim 3.1.2
+
+Let  $p \in \mathbb{R}^d$  such that  $\| p \|_2 \leq 1$ , and let  $R > 0$  be positive real number. Then for every  $x \in \{x \in \mathbb{R}^d | \| x \|_2 \leq R\}$ ,  $\frac{1}{e^R(1 + R)} \left( 1 + |p^T x| \right) \leq e^{-|p^T x|} \leq |p^T x| + 1$ .
+
+Proof. Put  $x \in \{x \in \mathbb{R}^d \mid \| x \|_2 \leq R\}$  and note that if  $p^T x = 0$  then the claim is trivial. Otherwise, we observe that
+
+$$
+e ^ {- \left| p ^ {T} x \right|} \geq \frac {1}{e ^ {R}} \geq \frac {1 + \left| p ^ {T} x \right|}{(1 + R) e ^ {R}}.
+$$
+
+# B.2. Proof of Theorem 3.2
+
+Let  $a, b \geq 0$  be pair of nonnegative real numbers and let  $c, d > 0$  be a pair of positive real numbers. Then
+
+$$
+\frac {a + b}{c + d} \leq \frac {a}{c} + \frac {b}{d}.
+$$
+
+Proof. Observe that
+
+$$
+\frac {a + b}{c + d} = \frac {a}{c + d} + \frac {b}{c + d} \leq \frac {a}{c} + \frac {b}{d}
+$$
+
+where the inequality holds since  $c, d > 0$ .
+
+![](images/9b7b28ca4e2457622f647a1cb4dcd7a64ee674ada94a89e5c779563e733d6e45.jpg)
+
+Lemma B.1 (Sensitivity bound w.r.t. the RBF loss function). Let  $R \geq 1$  be a positive real number, and let  $X = \{x \in \mathbb{R}^d \mid \| x \|_2 \leq R\}$ . Let  $(P, w, \mathbb{R}^d, f)$  be query space as in Definition A.1 where for every  $p \in P$  and  $x \in \mathbb{R}^d$ ,  $f(p, x) = e^{-\| p - x \|_2^2}$ . Let  $P' := \{q_p = \left[ \| p \|_2^2, -2p^T, 1 \right]^T \mid \forall p \in P\}$  and let  $q^* \in \operatorname{argsup}_{q \in P'} e^{3R^2 \| q \|_2} (1 + 3R^2 \| q \|_2)$ . Let  $u(p) := \frac{w(p)}{e^{3R^2 \| q_p \|_2} (1 + 3R^2 \| q_p \|_2)}$  and let  $(U, D, V)$  be the  $\| \cdot \|_1$ -SVD of  $(P', u(\cdot))$ . Then for every  $p \in P$ ,
+
+$$
+s (p) \leq e ^ {3 R ^ {2} \| q _ {p} \| _ {2}} \left(1 + 3 R ^ {2} \| q _ {p} \| _ {2}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + u (p) \| U \left(q _ {p}\right) \| _ {1}\right),
+$$
+
+and
+
+$$
+\sum_ {q \in P} s (q) \leq e ^ {3 R ^ {2} \| q ^ {*} \| _ {2}} \left(1 + 3 R ^ {2} \| q ^ {*} \| _ {2}\right) \left(1 + (d + 2) ^ {1. 5}\right).
+$$
+
+Proof. Let  $X = \left\{x \in \mathbb{R}^d \mid \| x \|_2 \leq 1\right\}$ , and observe that for every  $p \in P$  and  $x \in \mathbb{R}^d$ , it holds that
+
+$$
+\left\| p - x \right\| _ {2} ^ {2} = \left| q _ {p} ^ {T} y \right|, \tag {3}
+$$
+
+where  $q_{p} = \left[\| p\|_{2}^{2}, - 2p^{T},1\right]^{T}$  and  $y = \left[1,x,\| x\| _2^2\right]^T$ .
+
+Let  $Y = \left\{\left[1,x,\| x\| _2^2\right]^T\bigg|x\in X\right\}$ . Following the definition of  $Y$ , for every  $y\in Y$ , we obtain that  $\| y\| _2\leq 3R^2$ . Hence, by plugging  $p\coloneqq \frac{q_p}{\|q_p\|}$  and  $R\coloneqq 3R^{2}\| q_{p}\|$  for every  $p\in P$  into Claim 3.1.2, we obtain that the for every  $y\in Y$  and  $p\in P$
+
+$$
+\frac {1}{e ^ {3 R ^ {2} \| q _ {p} \| _ {2}} \left(1 + 3 R ^ {2} \| q _ {p} \| _ {2}\right)} \left(1 + | q _ {p} ^ {T} y |\right) \leq e ^ {- | q _ {p} ^ {T} y |} \leq 1 + | q _ {p} ^ {T} y |. \tag {4}
+$$
+
+Note that for every  $p \in P$ ,  $u(p) := \frac{w(p)}{e^{3\|q_p\|_2}\left(1 + 3\|q_p\|_2\right)}$ . Thus
+
+$$
+\begin{array}{l} \sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \\ = \sup  _ {y \in Y} \frac {w (p) e ^ {- \left| q _ {p} ^ {T} y \right|}}{\sum_ {p ^ {\prime} \in P ^ {\prime}} w (q) \left| q _ {p ^ {\prime}} ^ {T} y \right|} \\ \leq e ^ {3 R ^ {2} \| q _ {p} \| _ {2}} \left(1 + 3 R ^ {2} \| q _ {p} \| _ {2}\right) \sup  _ {y \in Y} \frac {u (p) \left| q _ {p} ^ {T} y \right| + u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right) \left| q _ {p ^ {\prime}} ^ {T} y \right| + \sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} \tag {5} \\ \leq e ^ {3 R ^ {2} \| q _ {p} \| _ {2}} \left(1 + 3 R ^ {2} \| q _ {p} \| _ {2}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + \sup  _ {y \in Y} \frac {u (p) \left| q _ {p} ^ {T} y \right|}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right) \left| q _ {p ^ {\prime}} ^ {T} y \right|}\right), \\ \end{array}
+$$
+
+where the first inequality holds by Claim 3.1.2 and the second inequality is by Claim B.2.
+
+Let  $f: P' \times Y \to [0, \infty)$  be a function such that for every  $q \in P'$  and  $y \in Y$ ,  $f(q, y) = |q^T y|$ . Plugging in  $P := P'$ ,  $w := u$ ,  $d := d + 2$ , and  $f := f$  into Lemma 2.3 yields for every  $p \in P$
+
+$$
+\sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq e ^ {3 R ^ {2} \| q _ {p} \| _ {2}} \left(1 + 3 R ^ {2} \| q _ {p} \| _ {2}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + u (p) \| U \left(q _ {p}\right) \| _ {1}\right). \tag {6}
+$$
+
+Note that by definition,  $q^{*} \in \operatorname*{arg\sup}_{q \in P^{\prime}} e^{3R^{2}\|q\|_{2}} \left(1 + 3R^{2}\|q\|_{2}\right)$ . Then the total sensitivity is bounded by
+
+$$
+\sum_ {q \in P} \sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq e ^ {3 R ^ {2} \| q ^ {*} \| _ {2}} \left(1 + 3 R ^ {2} \| q ^ {*} \| _ {2}\right) \left(1 + (d + 2) ^ {1. 5}\right). \tag {7}
+$$
+
+Theorem 3.2 (Coreset for RBF). Let  $R \geq 1$  be a positive real number,  $X = \{x \in \mathbb{R}^d \mid \| x \|_2 \leq R\}$ , and let  $\varepsilon, \delta \in (0,1)$ . Let  $(P, w, X, f)$  be query space as in Definition A.1 such that every  $p \in P$  satisfies  $\| p \|_2 \leq 1$ . For every  $x \in X$  and  $p \in P$ , let  $f(p, x) := \rho(\| p - x \|_2)$ . Let  $(S, v)$  be a call to CORESET $(P, w, R, m)$  where  $S \subseteq P$  and  $v: S \to [0, \infty)$ . Then  $(S, v)$  is coreset of  $(P, w)$  with probability at least  $1 - \delta$ , if  $m = O\left(\frac{e^{12R^2} R^2 d^{1.5}}{\varepsilon^2} \left(R^2 + \log d + \log \frac{1}{\delta}\right)\right)$ .
+
+Proof. First, by plugging in the query space  $(P,w,\mathbb{R}^d,f)$  into Lemma B.1, we obtain that a bound on the sensitivities  $s(p)$  for every  $p\in P$  and a bound on the total sensitivities  $t\coloneqq e^{12R^2}\left(1 + 12R^2\right)\left(1 + (d + 2)^{1.5}\right)$ , since the  $\max_{q\in P}\| q\| _2\leq 1$ . Notice that the analysis done in Lemma B.1 is analogous to the steps done in Algorithm 1.
+
+By plugging the bounds on the sensitivities, the bound on the total sensitivity  $t$ , probability of failure  $\delta \in (0,1)$ , and approximation error  $\varepsilon \in (0,1)$  into Theorem A.3, we obtain a subset  $S' \subseteq Q$  and  $v': S' \to [0,\infty)$  such that the tuple  $(S', v')$  is an  $\varepsilon$ -coreset for  $(P,w)$  with probability at least  $1 - \delta$ .
+
+# B.3. Proof of Theorem 3.3
+
+Lemma B.2 (Sensitivity bound w.r.t. the Laplacian loss function). Let  $(P, w, \mathbb{R}^d, f)$  be query space as in Definition A.1 where for every  $p \in P$  and  $x \in \mathbb{R}^d$ ,  $f(p, x) = e^{-\| p - x \|_2}$ . Let  $P' := \left\{q_p = \left[ \| p \|_2^2, -2p^T, 1 \right]^T \mid \forall p \in P\right\}$  and let  $q^* \in \arg \sup_{q \in P'} e^{3\sqrt{\|q\|_2}}(1 + 3\sqrt{\|q\|_2})$ . Let  $u(p) := \frac{w(p)}{e^3\sqrt{\|q_p \|_2}(1 + 3\sqrt{\|q_p \|_2})}$  and let  $(U, D, V)$  be the  $\| \cdot \|_1$ -SVD of  $(P', u^2(\cdot))$ . Then for every  $p \in P$ ,
+
+$$
+s (p) \leq e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + u (p) \sqrt {\| U \left(q _ {p}\right) \| _ {1}}\right) + \frac {e ^ {\| p \| _ {2} + \sqrt {\| q ^ {*} \| _ {2}}} w (p)}{\sum_ {q \in P} w (q)},
+$$
+
+and
+
+$$
+\sum_ {q \in P} s (q) \leq 2 e ^ {3 \sqrt {\| q ^ {*} \| _ {2}}} + e ^ {3 \sqrt {\| q ^ {*} \| _ {2}}} \left(1 + 3 \sqrt {\| q ^ {*} \| _ {2}}\right) \left(1 + \sqrt {n} (d + 2) ^ {1. 2 5}\right).
+$$
+
+Proof. Let  $X = \{x \in \mathbb{R}^d \mid \| x \|_2 \leq 1\}$ , and observe that for every  $p \in P$  and  $x \in \mathbb{R}^d$ , it holds that
+
+$$
+\left\| p - x \right\| _ {2} = \sqrt {\left| q _ {p} ^ {T} y \right|}, \tag {8}
+$$
+
+where  $q_{p} = \left[\| p\|_{2}^{2}, - 2p^{T},1\right]^{T}$  and  $y = \left[1,x,\| x\| _2^2\right]^T$
+
+Let  $Y = \left\{\left[1,x,\| x\| _2^2\right]^T\bigg|x\in X\right\}$ . Hence, following Theorem A.3, the sensitivity of each point  $p\in P$ , can be rewritten as
+
+$$
+\sup  _ {x \in \mathbb {R} ^ {d}} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq \sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} + \sup  _ {x \in \mathbb {R} ^ {d} \backslash X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)}. \tag {9}
+$$
+
+From here, we bound the sensitivity with respect to subspaces of  $\mathbb{R}^d$ .
+
+Handling queries from  $X$ . Following the definition of  $Y$ , for every  $y \in Y$ , we obtain that  $\| y \|_2 \leq 3$ . Hence, by plugging  $p := q_p$  for every  $p \in P$  and  $R := 3 \| q_p \|_2$  into Claim 3.1.2, we obtain that the for every  $y \in Y$  and  $p \in P$ ,
+
+$$
+\frac {1}{e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right)} \left(1 + \sqrt {\left| q _ {p} ^ {T} y \right|}\right) \leq e ^ {- \sqrt {\left| q _ {p} ^ {T} y \right|}} \leq 1 + \sqrt {\left| q _ {p} ^ {T} y \right|}. \tag {10}
+$$
+
+Note that for every  $p \in P$ ,  $u(p) := \frac{w(p)}{e^3 \sqrt{\|q_p\|_2} \left(1 + 3 \sqrt{\|q_p\|_2}\right)}$ . Combining equation 9 and equation 10, yields that
+
+$$
+\begin{array}{l} \sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right) \sup  _ {y \in Y} \frac {u (p) \sqrt {\left| q _ {p} ^ {T} y \right|} + u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right) \sqrt {\left| q _ {p ^ {\prime}} ^ {T} y \right|} + \sum_ {p ^ {\prime} \in P} w \left(p ^ {\prime}\right)} \\ \leq e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + \sup  _ {y \in Y} \frac {u (p) \sqrt {\left| q _ {p} ^ {T} y \right|}}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right) \sqrt {\left| q _ {p ^ {\prime}} ^ {T} y \right|}}\right), \tag {11} \\ \end{array}
+$$
+
+where the first inequality holds by Claim 3.1.2 and the second inequality is by Claim B.2.
+
+By Cauchy-Schwartz inequality,
+
+$$
+\begin{array}{l} \sup _ {y \in Y} \frac {u (p) \sqrt {\left| q _ {p} ^ {T} y \right|}}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right) \sqrt {\left| q _ {p ^ {\prime}} ^ {T} y \right|}} = \sup _ {y \in Y} \frac {\sqrt {u (p) ^ {2} \left| q _ {p} ^ {T} y \right|}}{\sum_ {p ^ {\prime} \in P} \sqrt {u \left(p ^ {\prime}\right) ^ {2} \left| q _ {p ^ {\prime}} ^ {T} y \right|}} \\ \leq \sup _ {y \in Y} \frac {\sqrt {u (p) ^ {2} \left| q _ {p} ^ {T} y \right|}}{\sqrt {\sum_ {p ^ {\prime} \in P} u (q) ^ {2} \left| q _ {p ^ {\prime}} ^ {T} y \right|}} \\ \leq \sup  _ {y \in \mathbb {R} ^ {d + 2}} \frac {\sqrt {u (p) ^ {2} \left| q _ {p} ^ {T} y \right|}}{\sqrt {\sum_ {p ^ {\prime} \in P} u (q) ^ {2} \left| q _ {p ^ {\prime}} ^ {T} y \right|}}, \tag {12} \\ \end{array}
+$$
+
+where the last inequality follows from the properties associated with the supremum operation.
+
+Let  $u': P' \to [0, \infty)$  be a weight function such that for every  $p \in P$ ,  $u'(q_p) = u(p)^2$ ,  $f: P' \times Y \to [0, \infty)$  be a function such that for every  $q \in P'$  and  $y \in Y$ ,  $f(q, y) = |q^T y|$ . Plugging in  $P := P'$ ,  $w := u$ ,  $d := d + 2$ , and  $f := f$  into Lemma 2.3 yields for every  $p \in P$
+
+$$
+\sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + u (p) \sqrt {\| U \left(q _ {p}\right) \| _ {1}}\right). \tag {13}
+$$
+
+Note that by definition,  $q^{*} \in \operatorname*{arg\sup}_{q \in P^{\prime}} e^{3\sqrt{\|q_{p}\|_{2}}} \left(1 + 3\sqrt{\|q_{p}\|_{2}}\right)$ . Then the total sensitivity is bounded by
+
+$$
+\sum_ {q \in P} \sup  _ {x \in X} \frac {w (p) f (p , x)}{\sum_ {q \in P} w (q) f (q , x)} \leq e ^ {3 \sqrt {\| q ^ {*} \| _ {2}}} \left(1 + 3 \sqrt {\| q ^ {*} \| _ {2}}\right) \left(1 + \sqrt {n} (d + 2) ^ {1. 2 5}\right), \tag {14}
+$$
+
+where the  $\sqrt{n}$  follows from  $\sum_{p' \in P} \sqrt{\|U(q_{p'})\|_1} \leq \sqrt{n} \sqrt{\sum_{p' \in P} \|U(q_{p'})\|_1}$ , which is used when using Lemma 2.3. This inequality is a result of Cauchy-Schwartz's inequality.
+
+Handling queries from  $\mathbb{R}^d\setminus X$  . First, we observe that for any integer  $m\geq 1$  and  $x,y\in \mathbb{R}^{m}$
+
+$$
+- \| x \| _ {2} - \| y \| _ {2} \leq - \| x - y \| _ {2} \leq \| x \| _ {2} - \| y \| _ {2}, \tag {15}
+$$
+
+where the first inequality holds by the triangle inequality, and the second inequality follows from the reverse triangle inequality.
+
+Thus, by letting  $x_{p} \in \operatorname*{arg\sup}_{x \in \mathbb{R}^{d} \setminus X} \frac{w(p)f(p,x)}{\sum_{q \in P} w(q)f(q,x)}$  for every  $p \in P$ , we obtain that
+
+$$
+\frac {w (p) f \left(p , x _ {p}\right)}{\sum_ {q \in P} w (q) f \left(q , x _ {p}\right)} \leq \frac {w (p) e ^ {\| p \| _ {2} - \| x _ {p} \| _ {2}}}{\sum_ {q \in P} w (q) e ^ {- \| q \| _ {2} - \| x _ {p} \| _ {2}}} \leq \frac {w (p) e ^ {\| p \| _ {2} - \| x _ {p} \| _ {2}}}{\sum_ {q \in P} w (q) e ^ {- \sqrt {\| q ^ {*} \| _ {2}} - \| x _ {p} \| _ {2}}} = \frac {e ^ {\| p \| _ {2} + \sqrt {\| q ^ {*} \| _ {2}}} w (p)}{\sum_ {q \in P} w (q)}, \tag {16}
+$$
+
+where the first inequality holds by equation 15, and the second inequality holds since  $\sqrt{\|q^{*}\|_{2}}\geq \| p\|_{2}$  for every  $p\in P$  Combining equation 9, equation 13, equation 14, and equation 16, yields that for every  $p\in P$
+
+$$
+s (p) \leq e ^ {3 \sqrt {\| q _ {p} \| _ {2}}} \left(1 + 3 \sqrt {\| q _ {p} \| _ {2}}\right) \left(\frac {u (p)}{\sum_ {p ^ {\prime} \in P} u \left(p ^ {\prime}\right)} + u (p) \sqrt {\| U \left(q _ {p}\right) \| _ {1}}\right) + \frac {e ^ {\| p \| _ {2} + \sqrt {\| q ^ {*} \| _ {2}}} w (p)}{\sum_ {q \in P} w (q)}
+$$
+
+and
+
+$$
+\sum_ {p \in P} s (p) \leq 2 e ^ {3 \sqrt {\| q ^ {*} \| _ {2}}} + e ^ {3 \sqrt {\| q ^ {*} \| _ {2}}} \left(1 + 3 \sqrt {\| q ^ {*} \| _ {2}}\right) \left(1 + \sqrt {n} (d + 2) ^ {1. 2 5}\right).
+$$
+
+Theorem 3.3 (Coreset for the Laplacian loss function). Let  $(P, w, \mathbb{R}^d, f)$  be query space as in Definition A.1 such that every  $p \in P$  satisfies  $\| p \|_2 \leq 1$ . For  $x \in \mathbb{R}^d$  and  $p \in P$ , let  $f(p, x) \coloneqq e^{-\| p - x \|_2}$ . Let  $\varepsilon, \delta \in (0, 1)$ . Then there exists an algorithm which given  $P, w, \varepsilon, \delta$  return a weighted set  $(S, v)$  where  $S \subseteq P$  of size  $O\left(\frac{\sqrt{n} d^{1.25}}{\varepsilon^2} \left( \log n + \log d + \log \frac{1}{\delta} \right)\right)$  and a weight function  $v: S \to [0, \infty)$  such that  $(S, v)$  is an  $\varepsilon$ -coreset of  $(P, w)$  with probability at least  $1 - \delta$ .
+
+Proof. Plugging in the query space  $(P, w, \mathbb{R}^d, f)$  into Lemma B.2, we obtain that a bound on the sensitivities  $s(p)$  for every  $p \in P$  and a bound on the total sensitivities  $t := 2e^5 + 3e^5 (1 + 5) \left(1 + \sqrt{n} (d + 2)^{1.5}\right)$ , since the  $\max_{q \in P} \| q \|_2 \leq 1$ . By plugging the bounds on the sensitivities, the bound on the total sensitivity  $t$ , probability of failure  $\delta \in (0, 1)$ , and approximation error  $\varepsilon \in (0, 1)$  into Theorem A.3, we obtain a subset  $S' \subseteq Q$  and  $v': S' \to [0, \infty)$  such that the tuple  $(S', v')$  is an  $\varepsilon$ -coreset for  $(P, w)$  with probability at least  $1 - \delta$ .
+
+# B.4. Proof of Theorem 4.1
+
+To prove Theorem 4.1, we first prove the following theorem.
+
+Theorem B.3. There exists an algorithm that samples  $O\left(\frac{e^{8R^2}R^2d^{1.5}}{\varepsilon^2}\left(R^2 + \log d + \log \frac{2}{\delta}\right)\right)$  points to form weighted sets  $(S_1, w_1)$  and  $(S_2, w_2)$  such that with probability at least  $1 - 2\delta$ ,
+
+$$
+\frac {\left| \sum_ {p \in P} y (p) \rho (\| p - c ^ {(i)} \| _ {2}) - \left(\sum_ {p \in S _ {1}} w _ {1} (p) e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}} + \sum_ {q \in S _ {2}} w _ {2} (q) e ^ {- \left\| q - c ^ {(j)} \right\| _ {2} ^ {2}}\right) \right|}{\left| \phi^ {+} \left(c ^ {(i)}\right) + \phi^ {-} \left(c ^ {(i)}\right) \right|} \leq \varepsilon .
+$$
+
+Proof. We first construct strong coresets for both  $\phi^{+}$  and  $\phi^{-}$ . If  $\| \mathbf{c}_i\| _2\leq R$  for all  $i\in [n]$ , then by Theorem 3.2, it suffices to sample a weighted weight  $S$  of size  $O\left(\frac{e^{12R^2}R^2d^{1.5}}{\varepsilon^2}\left(R^2 +\log d + \log \frac{2}{\delta}\right)\right)$  to achieve a strong coreset  $S_{1}$  with corresponding weighting function  $w_{1}$  for  $\phi^{+}$  with probability at least  $1 - \frac{\delta}{2}$ . Similarly, we obtain a strong coreset for  $\phi^{-}$  with probability at least  $1 - \frac{\delta}{2}$  by sampling a set  $S_{2}$  with weights  $w_{2}$  of size  $O\left(\frac{e^{8R^2}R^2d^{1.5}}{\varepsilon^2}\left(R^2 +\log d + \log \frac{2}{\delta}\right)\right)$ .
+
+Hence by the definition of a strong coreset, we have that
+
+$$
+\begin{array}{l} \left|\sum_{\substack{p\in P\\ y(p) > 0}}y(p)\rho (\| p - c^{(i)}\|_{2}) - \sum_{p\in S_{1}}w_{1}(p)\rho (\| p - c^{(i)}\|_{2})\right|\leq \varepsilon \sum_{\substack{p\in P\\ y(p) > 0}}y(p)\rho (\| p - c^{(i)}\|_{2}) \\ = \varepsilon \phi^ {+} (c ^ {(i)}) \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} \left|\sum_{\substack{p\in P\\ y(p) <   0}}y(p)\rho (\| p - c^{(i)}\|_{2}) - \sum_{q\in S_{2}}w_{2}(q)\rho (\| q - c^{(i)}\|_{2})\right|\leq \varepsilon \sum_{\substack{p\in P\\ y(p) <   0}}|y(p)|\rho (\| p - c^{(i)}\|_{2}) \\ = \varepsilon \phi^ {-} (c ^ {(i)}) \\ \end{array}
+$$
+
+for any input  $\mathbf{x}\in \mathbb{R}^n$
+
+Thus by triangle inequality and a slight rearrangement of the inequality, we have that with probability at least  $1 - \delta$
+
+$$
+\frac {\left| \sum_ {p \in P} y (p) \rho (\| p - c ^ {(i)} \| _ {2}) - \left(\sum_ {p \in S _ {1}} w _ {1} (p) e ^ {- \left\| p - c ^ {(i)} \right\| _ {2} ^ {2}} + \sum_ {q \in S _ {2}} w _ {2} (q) e ^ {- \left\| q - c ^ {(j)} \right\| _ {2} ^ {2}}\right) \right|}{\left| \phi^ {+} \left(c ^ {(i)}\right) + \phi^ {-} \left(c ^ {(i)}\right) \right|} \leq \varepsilon .
+$$
+
+To prove Theorem 4.1, we split  $\alpha_{i}$  into the sets  $\{i|\alpha_i > 0\}$  and  $\{i|\alpha_i < 0\}$  and apply Theorem B.3 to each of the sets. We emphasize that this argument is purely for the purposes of analysis so that the algorithm itself does not need to partition the  $\alpha_{i}$  quantities (and so the algorithm does not need to recompute the coreset when the values of the weights  $\alpha_{i}$  change over time). We thus get the following guarantee:
+
+Theorem 4.1. There exists an algorithm that samples  $O\left(\frac{e^{8R^2}R^2d^{1.5}}{\varepsilon^2}\left(R^2 + \log d + \log \frac{2}{\delta}\right)\right)$  points to form weighted sets  $(S_1, w_1)$  and  $(S_2, w_2)$  such that with probability at least  $1 - 2\delta$ ,
+
+$$
+\frac{\left|\sum_{p\in P}y(p)\phi(p) - \left(\sum_{\substack{i\in[L]\\ \alpha_{i} > 0}}\alpha_{i}\gamma_{S_{1}} + \sum_{\substack{j\in[L]\\ \alpha_{j} <   0}}\alpha_{j}\gamma_{S_{2}}\right)\right|}{\sum_{i\in[L]}|\alpha_{i}|  \left(\phi^{+}  \left(c^{(i)}\right) + \phi^{-}  \left(c^{(i)}\right)\right)}\leq \varepsilon ,
+$$
+
+where  $\gamma_{S_1} := \sum_{p \in S_1} w_1(p) e^{-\left\| p - c^{(i)} \right\|_2^2}$  and  $\gamma_{S_2} := \sum_{q \in S_2} w_2(q) e^{-\left\| q - c^{(j)} \right\|_2^2}$ .
+
+# B.5. Proof sketch of Corollary 4.2
+
+Corollary B.4. Let RFBNN be a sufficiently large (RBF) neural network,  $P \subset \mathbb{R}^d$ , and let  $f(p): P \to \mathbb{R}$  be a function that can be approximated by a set of centers and weights  $q$  (the learned parameters that define the RBFNN) assigned to RFBNN such that, for every  $p \in P$ :  $RFBNN(p,q) \approx f(p)$ , and  $\left\| \sum_{p \in P} RFBNN(p,q) - \sum_{p \in P} f(p) \right\| \leq \varepsilon$ , where each center of the RBFNN lies in ball centered at the origin with radius  $R$  (to satisfy our coreset's assumption). Let  $(S,v)$  be the output of a call to Algorithm 1, i.e.,  $(S,v)$  is coreset for  $P$  is an RBF coreset; see Definition 2.1. Then,
+
+$$
+\left| \sum_ {p \in P} f (p) - \sum_ {s \in S} v (s) R B F N N (s, q) \right| \leq \varepsilon \sum_ {p \in P} f (p) + \varepsilon .
+$$
+
+Proof. First using our RBF coreset properties and the definition of  $RBFNN$  which is a weighted sum of RBFs, we easily show that  $\left|\sum_{s\in S}v(s)RBFNN(s,q) - \sum_{p\in P}RBFNN(p,q)\right|\leq \varepsilon \sum_{p\in P}RBFNN(p,q)$ .
+
+Combining the above with the fact that  $\sum_{p\in P}RBFNN(p,q)$  approximates  $\sum_{p\in P}f(p)$  yields our result.
+
+# C. Lower bound on the coreset size for the Gaussian loss function - illustration
+
+Here, we illustrate one dataset such that for any approximation  $\varepsilon$ , the  $\varepsilon$ -coreset must contain at least half the points to ensure the desired approximation from a theoretical point of view.
+
+![](images/79f93b37eb2deff6a87cd4ed9098a18e72b0565b27e66c8e8e5ba414ec878aff.jpg)  
+Figure 4. Evenly distributed points on some circle where the minimal distance between each point and any other point is at least  $\sqrt{\ln n}$ , where in this example  $n = 10$ .
+
+# D. Coresets and Function Approximation
+
+The weight function  $w$  in the query space refers to a function that assigns weights to each input in the input set. These weights can be used to indicate the relative importance of each input in the analysis or experiment.
+
+In traditional supervised learning, the weight function is not typically used, as the focus is on learning a function that maps inputs x to outputs y. Usually, all input examples are equally important. Notably, in some cases, such as when dealing with imbalanced datasets or when certain inputs are more informative than others, the use of weights can improve the performance of the learning algorithm. In general, the role of the weight function in analysis and experiments is to provide a mechanism for incorporating prior knowledge into the learning process. By assigning weights to the input set, we can guide the learning algorithm to focus on certain inputs of the input space (or certain features of the inputs in other cases), which can lead to improved performance.
+
+In the context of function approximation via coresets, the weight function is used to give greater weight to some chosen coreset points that represent a larger quantity of the input data for the task at hand. For example, if the function being approximated has many points (denoted by  $Q$ ) with a similar cost on the given queries, then, a point from this set  $Q$  will hopefully be chosen in the coreset as a representative for the whole set  $Q$ , and the weight function may assign a greater weight to this point to ensure that the approximation captures the other points not included in the coreset, that are close to this coreset point. In the deep learning experiments, once we sample coresets and compute corresponding weights, we can define the loss of the trained neural network, as the weighted mean of the losses of the computed coreset, instead of using uniform weights when updating each batch.
+
+# E. Gradient Approximation via Our Coreset for RBFNNs
+
+Radial basis function (RBF) neural networks are a type of neural network that use radial basis functions as activation functions. Given a "large enough" RBF network, it can approximate any continuous function to arbitrary precision. In other words, let  $RFBNN$  be a sufficiently large (RBF) neural network, and let  $f(p): P' \to \mathbb{R}$  where  $P' \subset \mathbb{R}^d$ , then, there exists a set of centers and weights (the learned parameters that define the RBFNN)  $q$  to assign  $RFBNN$  such that for every  $x \in X$ ,  $RFBNN(p, q) \approx f(p)$ .
+
+Focusing on the derivative of single parameters/weight in a given DNN in a given step of the training process, the derivative of this weight can be defined as function  $g(p)$ , where  $p$  here is a vector representation of an input example for the deep
+
+neural network. Thus, if  $RBFNN$  is large enough, then there exists a set of centers and weights  $q$  to assign  $RFBNN$  such that:
+
+- For every  $p \in P'$ :  $RFBNN(p, q) \approx g(p) - P'$  is the space where the input may come from. This leads to the following.  
+- For every  $\sum_{p\in P}RFBNN(p,q)\approx \sum_{p\in P}g(p) - P$  is the input training data of size  $n$
+
+Now, according to the definition of coreset for RBFNN, given a set of inputs  $P$  and an RBFNN (structure), a coreset for  $P$  with respect to RBFNN, is a pair  $(S, v)$ , where  $S \subset P$ , and  $v: S \to R$  is a weight function such that for any set of centers and weights  $q$  to assign  $RFBNN$ , we have
+
+$$
+\sum_ {p \in P} R F B N N (p, q) \approx \sum_ {s \in S} v (s) R F B N N (s, q).
+$$
+
+Finally, put  $q$  to be the set of centers and weights (of RBFNN) that approximates some desired derivative  $g(x)$  of a weight in the neural network. Combining all of the above yields
+
+$$
+\sum_ {p \in P} g (p) \approx \sum_ {p \in P} R F B N N (p, q) \approx \sum_ {s \in S} v (s) R F B N N (s, q).
+$$
+
+# F. Experimental Results - Extended
+
+# F.1. Intuition behind re-sampling the coreset
+
+Theoretically speaking, our coreset is independent of the model and can be computed once to all epochs. However, practically speaking, to improve the practical results we construct a new coreset after  $\Upsilon$  epochs to expose the model to new samples and thus ensure a model with better generalization capabilities. We note the following very important comments:
+
+- Once we compute the sensitivity, the sampling process takes  $\sim 0$  time. Thus, it is free to sample a new coreset, even after each epoch.  
+- Since we rely on sensitivity sampling, then our samples across different epochs should be similar, as high sensitivity points, will be sampled with a high probability. However, the sampled points which have low sensitivity may be changed across different samples/epochs, thus, making sure we still capture the real important samples, while we get a wider diversity in the less important samples.
+
+Finally, we note that we could have conducted experiments for sampling after each epoch, as it takes zero time to do that, and we would have achieved higher test accuracy results. However, we did not report these results to ensure a fair comparison with our competitors that cannot afford to compute a new coreset every epoch due to time constraints.
+
+# F.2. MNIST results
+
+In what follows, we present our subset selection results on the MNIST dataset at Table 6.
+
+# F.3. Standard deviation and statistical significance results
+
+Tables 7-9 show the standard deviation results over five training runs on CIFAR10, CIFAR100, and MNIST datasets, respectively.
+
+Table 6. Data Selection Results for MNIST using LeNet  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Top-1 Test accuracy of the Model(%)</td><td colspan="4">Model Training time(in hrs)</td></tr><tr><td>1%</td><td>3%</td><td>5%</td><td>10%</td><td>1%</td><td>3%</td><td>5%</td><td>10%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>99.35</td><td>99.35</td><td>99.35</td><td>99.35</td><td>0.82</td><td>0.82</td><td>0.82</td><td>0.82</td></tr><tr><td>RANDOM (skyline for training time)</td><td>94.55</td><td>97.14</td><td>97.7</td><td>98.38</td><td>0.0084</td><td>0.03</td><td>0.04</td><td>0.084</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>98.8</td><td>99.1</td><td>99.1</td><td>99.13</td><td>0.0085</td><td>0.03</td><td>0.04</td><td>0.085</td></tr><tr><td>GLISTER</td><td>93.11</td><td>98.062</td><td>99.02</td><td>99.134</td><td>0.045</td><td>0.0625</td><td>0.082</td><td>0.132</td></tr><tr><td>GLISTER-WARM</td><td>97.63</td><td>98.9</td><td>99.1</td><td>99.15</td><td>0.04</td><td>0.058</td><td>0.078</td><td>0.127</td></tr><tr><td>CRAIG</td><td>96.18</td><td>96.93</td><td>97.81</td><td>98.7</td><td>0.3758</td><td>0.4173</td><td>0.434</td><td>0.497</td></tr><tr><td>CRAIG-WARM</td><td>98.48</td><td>98.96</td><td>99.12</td><td>99.14</td><td>0.2239</td><td>0.258</td><td>0.2582</td><td>0.3416</td></tr><tr><td>CRAIGPB</td><td>97.72</td><td>98.47</td><td>98.79</td><td>99.05</td><td>0.08352</td><td>0.106</td><td>0.1175</td><td>0.185</td></tr><tr><td>CRAIGPB-WARM</td><td>98.47</td><td>99.08</td><td>99.01</td><td>99.16</td><td>0.055</td><td>0.077</td><td>0.0902</td><td>0.1523</td></tr><tr><td>GRADMATCH</td><td>98.954</td><td>99.174</td><td>99.214</td><td>99.24</td><td>0.05</td><td>0.0607</td><td>0.097</td><td>0.138</td></tr><tr><td>GRADMATCH-WARM</td><td>98.86</td><td>99.22</td><td>99.28</td><td>99.29</td><td>0.046</td><td>0.057</td><td>0.089</td><td>0.132</td></tr><tr><td>GRADMATCHPB</td><td>98.7</td><td>99.1</td><td>99.25</td><td>99.27</td><td>0.04</td><td>0.051</td><td>0.07</td><td>0.11</td></tr><tr><td>GRADMATCHPB-WARM</td><td>99.0</td><td>99.23</td><td>99.3</td><td>99.31</td><td>0.038</td><td>0.05</td><td>0.065</td><td>0.10</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>98.98</td><td>99.2</td><td>99.31</td><td>99.32</td><td>0.028</td><td>0.051</td><td>0.062</td><td>0.098</td></tr></table>
+
+Table 7. Data Selection Results for CIFAR10 using ResNet-18: Standard deviation of the Model (for 5 runs)  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Standard deviation of the Model(for 5 runs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>0.032</td><td>0.032</td><td>0.032</td><td>0.032</td></tr><tr><td>RANDOM (skyline for training time)</td><td>0.483</td><td>0.518</td><td>0.524</td><td>0.538</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>0.461</td><td>0.348</td><td>0.24</td><td>0.1538</td></tr><tr><td>GLISTER</td><td>0.453</td><td>0.107</td><td>0.046</td><td>0.345</td></tr><tr><td>GLISTER-WARM</td><td>0.325</td><td>0.086</td><td>0.135</td><td>0.129</td></tr><tr><td>CRAIG</td><td>0.289</td><td>0.2657</td><td>0.1894</td><td>0.1647</td></tr><tr><td>CRAIG-WARM</td><td>0.123</td><td>0.1185</td><td>0.1058</td><td>0.1051</td></tr><tr><td>CRAIGPB</td><td>0.152</td><td>0.1021</td><td>0.086</td><td>0.064</td></tr><tr><td>CRAIGPB-WARM</td><td>0.0681</td><td>0.061</td><td>0.0623</td><td>0.0676</td></tr><tr><td>GRADMATCH</td><td>0.192</td><td>0.123</td><td>0.112</td><td>0.1023</td></tr><tr><td>GRADMATCH-WARM</td><td>0.1013</td><td>0.1032</td><td>0.091</td><td>0.1034</td></tr><tr><td>GRADMATCHPB</td><td>0.0581</td><td>0.0571</td><td>0.0542</td><td>0.0584</td></tr><tr><td>GRADMATCHPB-WARM</td><td>0.0542</td><td>0.0512</td><td>0.0671</td><td>0.0581</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>0.25</td><td>0.2</td><td>0.17</td><td>0.13</td></tr><tr><td>RBFNN CORESET-WARM (OURS)</td><td>0.21</td><td>0.16</td><td>0.13</td><td>0.12</td></tr></table>
+
+Table 8. Data Selection Results for CIFAR100 using ResNet-18: Standard deviation of the Model (for 5 runs)  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Standard deviation of the Model(for 5 runs)</td></tr><tr><td>5%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>0.051</td><td>0.051</td><td>0.051</td><td>0.051</td></tr><tr><td>RANDOM (skyline for training time)</td><td>0.659</td><td>0.584</td><td>0.671</td><td>0.635</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>0.359</td><td>0.242</td><td>0.187</td><td>0.175</td></tr><tr><td>GLISTER</td><td>0.463</td><td>0.15</td><td>0.061</td><td>0.541</td></tr><tr><td>GLISTER-WARM</td><td>0.375</td><td>0.083</td><td>0.121</td><td>0.294</td></tr><tr><td>CRAIG</td><td>0.3214</td><td>0.214</td><td>0.195</td><td>0.187</td></tr><tr><td>CRAIG-WARM</td><td>0.18</td><td>0.132</td><td>0.125</td><td>0.115</td></tr><tr><td>CRAIGPB</td><td>0.12</td><td>0.134</td><td>0.123</td><td>0.115</td></tr><tr><td>CRAIGPB-WARM</td><td>0.1176</td><td>0.1152</td><td>0.1128</td><td>0.111</td></tr><tr><td>GRADMATCH</td><td>0.285</td><td>0.176</td><td>0.165</td><td>0.156</td></tr><tr><td>GRADMATCH-WARM</td><td>0.140</td><td>0.134</td><td>0.142</td><td>0.156</td></tr><tr><td>GRADMATCHPB</td><td>0.104</td><td>0.111</td><td>0.105</td><td>0.097</td></tr><tr><td>GRADMATCHPB-WARM</td><td>0.093</td><td>0.101</td><td>0.100</td><td>0.098</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>0.3</td><td>0.19</td><td>0.18</td><td>0.16</td></tr><tr><td>RBFNN CORESET-WARM (OURS)</td><td>0.19</td><td>0.14</td><td>0.11</td><td>0.1</td></tr></table>
+
+Table 9. Data Selection Results for MNIST using LeNet: Standard deviation of the Model (for 5 runs)  
+
+<table><tr><td rowspan="2">Budget(%)</td><td colspan="4">Standard deviation of the Model(for 5 runs</td></tr><tr><td>1%</td><td>3%</td><td>5%</td><td>10%</td></tr><tr><td>FULL (skyline for test accuracy)</td><td>0.012</td><td>0.012</td><td>0.012</td><td>0.012</td></tr><tr><td>RANDOM (skyline for training time)</td><td>0.215</td><td>0.265</td><td>0.224</td><td>0.213</td></tr><tr><td>RANDOM-WARM (skyline for training time)</td><td>0.15</td><td>0.121</td><td>0.110</td><td>0.103</td></tr><tr><td>GLISTER</td><td>0.256</td><td>0.218</td><td>0.145</td><td>0.128</td></tr><tr><td>GLISTER-WARM</td><td>0.128</td><td>0.134</td><td>0.119</td><td>0.124</td></tr><tr><td>CRAIG</td><td>0.186</td><td>0.178</td><td>0.162</td><td>0.125</td></tr><tr><td>CRAIG-WARM</td><td>0.0213</td><td>0.0223</td><td>0.0196</td><td>0.0198</td></tr><tr><td>CRAIGPB</td><td>0.021</td><td>0.0209</td><td>0.0216</td><td>0.0204</td></tr><tr><td>CRAIGPB-WARM</td><td>0.023</td><td>0.0192</td><td>0.0212</td><td>0.0184</td></tr><tr><td>GRADMATCH</td><td>0.156</td><td>0.128</td><td>0.135</td><td>0.12</td></tr><tr><td>GRADMATCH-WARM</td><td>0.087</td><td>0.084</td><td>0.0896</td><td>0.0815</td></tr><tr><td>GRADMATCHPB</td><td>0.0181</td><td>0.0163</td><td>0.0147</td><td>0.0129</td></tr><tr><td>GRADMATCHPB-WARM</td><td>0.0098</td><td>0.012</td><td>0.0096</td><td>0.0092</td></tr><tr><td>RBFNN CORESET (OURS)</td><td>0.18</td><td>0.13</td><td>0.11</td><td>0.1</td></tr><tr><td>RBFNN CORESET-WARM (OURS)</td><td>0.09</td><td>0.08</td><td>0.05</td><td>0.01</td></tr></table>

2023/Provable Data Subset Selection For Efficient Neural Networks Training/images.zip

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2023/Provable Dynamic Fusion for Low-Quality Multimodal Data/d904169e-d736-4803-8955-2211bdd0b904_origin.pdf

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2023/Provable Dynamic Fusion for Low-Quality Multimodal Data/full.md

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+# Provable Dynamic Fusion for Low-Quality Multimodal Data
+
+Qingyang Zhang $^{1}$  Haitao Wu $^{1}$  Changqing Zhang $^{12}$  Qinghua Hu $^{12}$  Huazhu Fu $^{3}$  Joey Tianyi Zhou $^{34}$  Xi Peng $^{5}$
+
+# Abstract
+
+The inherent challenge of multimodal fusion is to precisely capture the cross-modal correlation and flexibly conduct cross-modal interaction. To fully release the value of each modality and mitigate the influence of low-quality multimodal data, dynamic multimodal fusion emerges as a promising learning paradigm. Despite its widespread use, theoretical justifications in this field are still notably lacking. Can we design a provably robust multimodal fusion method? This paper provides theoretical understandings to answer this question under a most popular multimodal fusion framework from the generalization perspective. We proceed to reveal that several uncertainty estimation solutions are naturally available to achieve robust multimodal fusion. Then a novel multimodal fusion framework termed Quality-aware Multimodal Fusion (QMF) is proposed, which can improve the performance in terms of classification accuracy and model robustness. Extensive experimental results on multiple benchmarks can support our findings.
+
+# 1. Introduction
+
+Our perception of the world is based on multiple modalities, e.g., touch, sight, hearing, smell and taste. With the development of sensory technology, we can easily collect diverse forms of data for analysis. For example, multi-sensor in autonomous driving and wearable electrical devices (Xiao et al., 2020; Wen et al., 2022), or various examinations in
+
+$^{1}$ College of Intelligence and Computing, Tianjin University, Tianjin, China  $^{2}$ Tianjin Key Lab of Machine Learning, Tianjin University, Tianjin, China  $^{3}$ Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore  $^{4}$ Centre for Frontier AI Research (CFAR), Agency for Science, Technology and Research (A*STAR), Singapore  $^{5}$ College of Computer Science, Sichuan University, Chengdu, China. Correspondence to: Changqing Zhang <zhangchangqing@tju.edu.cn>.
+
+Proceedings of the  $40^{th}$  International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
+
+medical diagnosis and treatment (Qiu et al., 2022; Acosta et al., 2022). Intuitively, fusing information from different modalities offers the possibility of exploring cross-modal correlation and gaining better performance. However, conventional fusion methods have largely overlooked the unreliable quality of multimodal data. In real-world settings, the quality of different modalities usually varies due to unexpected environmental issues. Some recent studies have shown both empirically and theoretically that multimodal fusion may fail on low-quality multimodal data, e.g., imbalanced (Wang et al., 2020; Peng et al., 2022; Huang et al., 2022), noisy or even corrupted (Huang et al., 2021b) multimodal data. Empirically, it is recognized that multimodal models cannot always outperform unimodal models especially in a high noise (Scheunders & De Backer, 2007; Eitel et al., 2015; Silva et al., 2022) or imbalanced modality quality (Wu et al., 2022; Peng et al., 2022) regime. Theoretically, the previous study proves that the advantages of multimodal learning may vanish under the setting of limited data volume (Huang et al., 2021a) which implies the exploitation of cross-modal relationship is not a free lunch. To fully release the value of each modality and mitigate the influence of low-quality multimodal data, introducing dynamic fusion mechanism emerges as a promising way to obtain reliable predictions. As a concrete example, previous work (Guan et al., 2019) proposes a dynamic weighting mechanism to depict illumination condition of scenes. By introducing dynamics, they can integrate reliable cues from multi-spectral data for around-the-clock applications (e.g., pedestrian detection in security surveillance and autonomous driving). Dynamic fusion has been used in diverse real-world multimodal applications, including multimodal classification (Han et al., 2021; Geng et al., 2021; Han et al., 2022b), regression (Ma et al., 2021), object detection (Li et al., 2022a; Zhang et al., 2019; Chen et al., 2022b) and semantic segmentation (Tian et al., 2020). While dynamic multimodal fusion shows excellent power in practice, theoretical understanding is notably lack in this field with the following fundamental open problem: Can we realize reliable multimodal fusion in practice with theoretical guarantee?
+
+This paper tries to shed light upon the theoretical advantage and criterion of robust multimodal fusion. Following previous works in multimodal learning theory (Huang et al.,
+
+![](images/f2132478dd32b4b2a8e39f2cbe7b06dad5804b0142e7b17956616cc995339669.jpg)  
+Figure 1. Visualization of accuracy gap between multimodal learning methods (e.g., late fusion, align-based fusion, MMTM) and single-modal learning methods using the best modality on noisy multimodal data. Noted that the performance existing multimodal fusion methods degrade significantly of compared with their best unimodal counterparts in a high noise regime, while the proposed QMF consistently outperforms unimodal methods on low-quality data.
+
+2021b; Wang et al., 2020), the framework we study is also abstracted from decision-level multimodal fusion, which is one of the most fundamental research topics in multimodal learning (Baltrusaitis et al., 2018). In particular, we devise a novel Quality-aware Multimodal Fusion (QMF) framework for multimodal learning. Key to our framework, we leverage energy-based uncertainty to characterize the quality of each modality. Our contributions can be summarised as follows:
+
+- This paper provides a rigorous theoretical framework to understand the advantage and criterion of robust multimodal fusion as shown in Figure 2. Firstly, we characterize the generalization error bound of decision-level multimodal fusion methods from a Rademacher complexity perspective. Then, we identify under what conditions dynamic fusion outperforms static, i.e., when the fusion weights of multimodal fusion is negatively correlates to the unimodal generalization errors, dynamic fusion methods provably outperform static.  
+- Under the theoretical analysis, we proceed to reveal that the generalization ability of dynamic fusion coincides with the performance of uncertainty estimation. This directly implies a principle to design and evaluate new dynamic fusion algorithms.  
+- Directly motivated by the above analysis, we propose a novel dynamic multimodal fusion method termed Quality-aware Multimodal Fusion (QMF), which serves as a realization for provably better generalization ability. As shown in Figure 1, extensive experiments on commonly used benchmarks are carried out to empirically validate the theoretical observations.
+
+# 2. Related works
+
+# 2.1. Multimodal Fusion
+
+Multimodal fusion is one of the most original and fundamental topics in multimodal learning, which typically aims
+
+to integrate modality-wise features into a joint representation for downstream multimodal learning tasks. Multimodal fusion can be classified into early fusion, intermediate fusion and late fusion. Although studies in neuroscience and machine learning suggest that intermediate fusion could benefit representation learning (Schroeder & Foxe, 2005; Macaluso, 2006), late fusion is still the most widely used method for multimodal learning due to its interpretation and practical simplicity. By introducing modality-level dynamics based on various strategies, dynamic fusion practically improves overall performance. As a concrete example, the previous work (Guan et al., 2019) proposes a dynamic weighting mechanism to depict illumination conditions of scenes. By introducing dynamics, they can integrate reliable cues from multi-spectral data for around-the-clock applications (e.g., pedestrian detection in security surveillance and autonomous driving). Combining with additional dynamic mechanism (e.g., a simple weighting strategy or Dempster-Shafer Evidence Theory (Shafer, 1976)), recent uncertainty-based multimodal fusion methods show remarkable advantages in various tasks, including clustering (Geng et al., 2021), classification (Han et al., 2021; 2022b; Tellamekala et al., 2022; Subedar et al., 2019; Chen et al., 2022a), regression (Ma et al., 2021), object detection (Zhang et al., 2019; Li et al., 2022b) and semantic segmentation (Tian et al., 2020; Chang et al., 2022).
+
+# 2.2. Uncertainty Estimation
+
+Multimodal machine learning has achieved great success in various real-world application. However, the reliability of current fusion methods is still notably unexplored, which limits their application in safety-critic field (e.g., financial risk, medical diagnosis). The motivation of uncertainty estimation is to indicate whether the predictions given by machine learning models are prone to be wrong. Many uncertainty estimation methods have been proposed
+
+![](images/ddbc499462b5633d2d88e6e157211764e747040a9af7f4b681e125362546c9f2.jpg)  
+Figure 2. Left: The generalization error upper bound of multimodal fusion method  $f$  can be characterized by its performance on each modality in terms of empirical loss, model complexity and uncertainty awareness. Right: Dynamic vs Static multimodal fusion hypothesis space, where the latter is a subset of the former.  $f_{\mathrm{static}}$ ,  $f_{\mathrm{dynamic}}$  are the hypotheseses of static and dynamic fusion methods respectively and  $f^{*}$  is the true mapping. Informally, closer to the true mapping leads to less error. Under some certain conditions, dynamic multimodal fusion methods (e.g., the proposed QMF) can be well regularized and thus provably achieve better generalization ability.
+
+in the past decades, including Bayesian neural networks (BNNs) (Denker & LeCun, 1990; Mackay, 1992; Neal, 2012) and its varieties (Gal & Ghahramani, 2016; Han et al., 2022a), deep ensembles (Lakshminarayanan et al., 2017; Havasi et al., 2021), predictive confidence (Hendrycks & Gimpel, 2017), Dempster-Shafer thoery (Han et al., 2021) and energy score (Liu et al., 2020). Predictive confidence expects the predicted class probability to be consistent with the empirical accuracy, which is usually referred in classification tasks. Dempster-Shafer theory (DST) is a generalization of Bayesian theory to subjective probabilities and a general framework for modeling epistemic uncertainty. Energy score emerges as a promising way to capture Out-of-Distribution (OOD) uncertainty, which arises when a machine learning model encounters an input that differs from its training data, and thus the output from the model is unreliable. A plethora of recent researches have studied the issue of OOD uncertainty (Ming et al., 2022; Chen et al., 2021; Meinke & Hein, 2019; Hendrycks et al., 2019). In this paper, we investigate predictive confidence, the Dempster-Shafer theory and energy score due to their theoretical interpretability and effectiveness.
+
+# 3. Theory
+
+In this section, we first clarify the basic notations and the formal definition of multimodal fusion used in Section 3.1. Then we provide main theoretical results in Section 3.2 to rigorously demonstrate when and how dynamic fusion methods work from the perspective of generalization ability (Bartlett & Mendelson, 2002). Due to space constraints, we defer the full details to Appendix A and only present a brief summary of the proofs.
+
+# 3.1. Preliminaries
+
+We initialize by introducing the necessary notations for our theoretical frameworks. Considering a learning task on the data  $(x,y)\in \mathcal{X}\times \mathcal{Y}$ , where  $x = \{x^{(1)},\dots ,x^{(M)}\}$  has  $M$  modalities and  $y\in \mathcal{V}$  denotes the data label. The multimodal training data is defined as  $D_{\mathrm{train}} = \{x_i,y_i\}_{i = 1}^N$ . Specifically, we use  $\mathcal{X},\mathcal{Y}$  and  $\mathcal{Z}$  to denote the input space, target space and latent space. Similar to the previous work in multimodal learning theory (Huang et al., 2021a), we define  $h:\mathcal{X}\mapsto \mathcal{Z}$  is a multimodal fusion mapping from the input space to the latent space, and  $g:\mathcal{Z}\mapsto \mathcal{Y}$  is a task mapping. Our goal is to learn a reliable multimodal model  $f = g\circ h(x)$  performing well on the unknown multimodal test dataset  $D_{\mathrm{test}}$ .  $D_{\mathrm{train}}$  and  $D_{\mathrm{test}}$  are both drawn from joint distribution  $\mathcal{D}$  over  $\mathcal{X}\times \mathcal{Y}$ . Here  $f = g\circ h(x)$  represents the composite function of  $h$  and  $g$ .
+
+# 3.2. When and How Dynamic Multimodal Fusion Help
+
+For simplicity, we provide analysis of ensemble-like late fusion strategy using logistic loss function in two-class classification setting. Our analysis follows this roadmap: (1) we first characterize the generalization error bound of dynamic late fusion using Rademacher complexity (Bartlett & Mendelson, 2002) and then separate the bound into three components (Theorem 1); (2) base on above separation, we further prove that dynamic fusion achieves better generalization ability under certain conditions (Theorem 2). We initiate our analysis with the basic setting as follows.
+
+Basic setting. Under a  $M$  input modalities and two-class classification scenario, we define  $f^{m}$  as the unimodal classifier on modality  $x^{(m)}$ . The final prediction of late fusion multimodal method is calculated by weighting decisions
+
+from different modalities:  $f(x) = \sum_{m=1}^{M} w^{m} \cdot f^{m}(x^{(m)})$ , where  $f(x)$  denotes the final prediction. In contrast to static late fusion, the weights in dynamic multimodal fusion are generated dynamically and vary for different samples. For clarity, we use subscript to distinguish them, i.e.,  $w_{\text{static}}^{m}$  refers to the ensemble weight of modality  $m$  in static late fusion and  $w_{\text{dynamic}}^{m}$  refers to the weight in dynamic fusion. Specifically,  $w_{\text{static}}^{m}$  is a constant and  $w_{\text{dynamic}}^{m}(\cdot)$  is a function of the input sample  $x$ . The generalization error of two-class multimodal classifier  $f$  is defined as:
+
+$$
+\operatorname {G E r r o r} (f) = \mathbb {E} _ {(x, y) \sim \mathcal {D}} [ \ell (f (x), y) ], \tag {1}
+$$
+
+where  $\mathcal{D}$  is the unknown joint distribution, and  $\ell$  is logistic loss function. For convenience, we simplify unimodal classifier loss  $\ell(f^m(x^m), y)$  as  $l^m$  and omit the inputs in the following analysis. Now we present our first main result regarding multi-modal fusion.
+
+Theorem 1 (Generalization Bound of Multimodal Fusion). Let  $D_{\mathrm{train}} = \{x_i, y_i\}_{i=1}^N$  be a training dataset of  $N$  samples,  $\hat{E}(f^m)$  is the unimodal empirical errors of  $f^m$  on  $D_{\mathrm{train}}$ . Then for any hypothesis  $f$  in  $\mathcal{H}$  (i.e.,  $\mathcal{H}: \mathcal{X} \to \{-1,1\}$ ,  $f \in \mathcal{H}$ ) and  $1 > \delta > 0$ , with probability at least  $1 - \delta$ , it holds that
+
+$$
+\begin{array}{l} \operatorname {G E r r o r} (f) \leq \underbrace {\sum_ {m = 1} ^ {M} \mathbb {E} \left(w ^ {m}\right) \hat {E} \left(f ^ {m}\right)} _ {\text {T e r m - L (a v e r a g e e m p i r i c a l l o s s)}} + \underbrace {\sum_ {m = 1} ^ {M} \mathbb {E} \left(w ^ {m}\right) \mathfrak {R} _ {m} \left(f ^ {m}\right)} _ {\text {T e r m - C (a v e r a g e c o m p l e x i t y)}} \\ + \underbrace {\sum_ {m = 1} ^ {M} C o v \left(w ^ {m} , l ^ {m}\right)} _ {\text {T e r m - C o v (c o v a r i a n c e)}} + M \sqrt {\frac {\ln (1 / \delta)}{2 N}}, \tag {2} \\ \end{array}
+$$
+
+where  $\mathbb{E}(w^m)$  is the expectations of fusion weights on joint distribution  $\mathcal{D}$ ,  $\Re_m(f^m)$  is Rademacher complexity,  $Cov(w^{m},\ell^{m})$  is the covariance between fusion weight and loss.
+
+Intuitively, Theorem 1 demonstrates that the generalization error of multimodal classifier is bounded by the weighted average performances of all the unimodal classifiers in terms of empirical loss, model complexity and the covariance between fusion weight and unimodal loss. Having established the general error bound, our next goal is to verify when dynamic multimodal late fusion indeed achieves tighter bound than that of static late fusion. Informally, in Eq. 1, Term-Cov measures the joint variability of  $w^{m}$  and  $\ell^{m}$ . Remember that in static multimodal fusion  $w_{\mathrm{static}}^{m}$  is a constant, which means Term-Cov = 0 for any static fusion methods. Thus the generalization error bound of static fusion methods re
+
+duces to
+
+$$
+\begin{array}{l} \operatorname {GError}\left(f_{\text{static}}\right)\leq \underbrace{\sum_{m = 1}^{M}w_{\text{static}}^{m}\hat{E}\left(f^{m}\right)}_{\text{Term - L (average empirical loss)}} \\ + \underbrace {\sum_ {m = 1} ^ {M} w _ {\text {s t a t i c}} ^ {m} \Re_ {m} \left(f ^ {m}\right)} _ {\text {T e r m - C (a v e r a g e c o m p l e x i t y)}} + M \sqrt {\frac {\ln (1 / \delta)}{2 N}}. \tag {3} \\ \end{array}
+$$
+
+So when summation of Term-L, Term-C is invariant or smaller in dynamic fusion and Term-Cov  $\leq 0$ , we can ensure that dynamic fusion provably outperforms static fusion. This theorem is formally presented as
+
+Theorem 2. Let  $\mathcal{O}(\mathrm{GError}(f_{\mathrm{dynamic}}))$ ,  $\mathcal{O}(\mathrm{GError}(f_{\mathrm{static}}))$  be the upper bound of generalization error of multimodal classifier using dynamic and static fusion strategy respectively.  $\hat{E}(f^m)$  is the unimodal empirical errors of  $f^m$  on  $D_{\mathrm{train}}$  defined in Theorem 1. Then for any hypothesis  $f_{\mathrm{dynamic}}$ ,  $f_{\mathrm{static}}$  in  $\mathcal{H}: \mathcal{X} \to \{-1,1\}$  and  $1 > \delta > 0$ , it holds that
+
+$$
+\mathcal {O} (\operatorname {G E r r o r} (f _ {\text {d y n a m i c}})) \leq \mathcal {O} (\operatorname {G E r r o r} (f _ {\text {s t a t i c}})) \tag {4}
+$$
+
+with probability at least  $1 - \delta$ , if we have
+
+$$
+\mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) = w _ {\text {s t a t i c}} ^ {m} \tag {5}
+$$
+
+and
+
+$$
+r \left(w _ {\text {d y n a m i c}} ^ {m}, \ell \left(f ^ {m}\right)\right) \leq 0 \tag {6}
+$$
+
+for all input modalities, where  $r$  is the Pearson correlation coefficient which measures the correlation between fusion weights  $w_{\text{dynamic}}^m$  and unimodal loss  $\ell^m$ .
+
+Remark. Theoretically, optimizing over the same function class efficiently results in the same empirical loss. Suppose for each modality  $m$ , the unimodal classifier  $f^{m}$  we used in dynamic and static fusion are of the same architecture, then the intrinsic complexity of unimodal classifier  $\Re_{m}(f^{m})$  and empirical risk  $\hat{E}(f^{m})$  can be invariant. Thus in this case, it holds that
+
+$$
+\sum_ {m = 1} ^ {M} \mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) \hat {E} \left(f ^ {m}\right) \leq \sum_ {m = 1} ^ {M} w _ {\text {s t a t i c}} ^ {m} \hat {E} \left(f ^ {m}\right), \tag {7}
+$$
+
+and
+
+$$
+\sum_ {m = 1} ^ {M} \mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) \Re_ {m} \left(f ^ {m}\right) \leq \sum_ {m = 1} ^ {M} w _ {\text {s t a t i c}} ^ {m} \Re_ {m} \left(f ^ {m}\right), \tag {8}
+$$
+
+if Eq. 5 is satisfied for any modality  $m$ . According to Theorem 2, it is easy to derive the conclusion that the main
+
+challenge of achieving reliable dynamic multimodal fusion is to learn a reasonable  $w_{\mathrm{dynamic}}^m (x)$  for each modality that satisfies Eq. 5 and Eq. 6.
+
+# 4. Method
+
+Now we proceed to answer "How to realize robust dynamic fusion?" In this section, we theoretically identify the connection between dynamic multimodal fusion and uncertainty estimation. Then, a unified dynamic multimodal fusion framework termed Quality-aware Multimodal Fusion (QMF) is proposed. We next show how to realize this framework in decision-level late fusion and classification tasks to support our findings.
+
+# 4.1. Coincidence with Uncertainty Estimation
+
+Firstly, we focus on how to satisfy Eq. 6. As we discuss in Section 2.2, the common motivation of various uncertainty estimation methods is to provide an indicator of whether the predictions given by models are prone to be wrong. This motivation is inherently close to obtaining weights that satisfy Eq. 6. We formulate this claim with the following assumption
+
+Assumption 1. Given an effective uncertainty estimator  $u^m: \mathcal{X} \to \mathbb{R}$  on modality  $m$ , the estimated uncertainty  $u^m(x)$  is positively correlated with its modal-specific loss  $\ell^m(x): r(u^m, \ell^m(x)) \geq 0$ , where  $r$  is the Pearson correlation coefficient.
+
+This insight offers opportunity to explore novel dynamic fusion methods provably outperform conventional static fusion methods. Similar to previous dynamic fusion methods (Blundell et al., 2015; Zhang et al., 2019; Han et al., 2022b), we deploy modal-level weighting strategy to introduce dynamics.
+
+Uncertainty-aware weighting. The uncertainty-aware fusion weighting  $w^{m}:\mathcal{X}\to \mathbb{R}$  is a function that linearly and negatively relates to the corresponding uncertainty
+
+$$
+w ^ {m} (x) = \alpha^ {m} u ^ {m} (x) + \beta^ {m}, \tag {9}
+$$
+
+where  $\alpha^m < 0$ ,  $\beta^m \geq 0$  are modal-specific hyperparameters.  $u^m(x)$  is the uncertainty of modality  $m$ . By tuning hyper-parameters  $\alpha^m, \beta^m$ , we can ensure dynamic fusion weights satisfied Eq. 5 and 6 simultaneously. This lemma is formally presented as
+
+Lemma 1 (Satisfiability). With Assumption 1, for any  $w_{\mathrm{static}}^m \in \mathbb{R}$ , there always exist  $\beta^m \in \mathbb{R}$  such that
+
+$$
+\mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) = w _ {\text {s t a t i c}} ^ {m}, r \left(w _ {\text {d y n a m i c}} ^ {m}, \ell \left(f ^ {m}\right)\right) \leq 0. \tag {10}
+$$
+
+Once we obtain the fusion weights, we can perform uncertainty-aware weighting fusion in decision-level according to the following rule
+
+$$
+f (x) = \sum_ {m = 1} ^ {M} w ^ {m} (x) \cdot f ^ {m} (x), \tag {11}
+$$
+
+where  $f^{m}(x)$  defined in Section 3.2 denotes unimodal prediction on modality  $m$ .
+
+# 4.2. Enhance Correlation by Additional Regularization
+
+With the above analysis, the core challenges of robust dynamic multimodal fusion present in Section. 3.2 have been reduced to obtain an effective uncertainty estimator in Assumption 1. In our implementation, we leverage energy score (Liu et al., 2020), which is a widely accepted metric in the literature of uncertainty learning. Energy score bridges the gap between the Helmholtz free energy of a given data point and its density. For multimodal data, the density functions of different modalities can be estimated by the corresponding energy function:
+
+$$
+\log p \left(x ^ {(m)}\right) = - \operatorname {E n e r g y} \left(x; f ^ {m}\right) / \mathcal {T} ^ {m} - \log Z ^ {m}, \tag {12}
+$$
+
+where  $x^{(m)}$  is the  $m$ -th input modality and  $f^m$  is the unimodal classification model. Energy(\cdot) is the energy function and  $Z^m$  is an intractable constant for all  $x^m$ . The above equation suggests that  $-\mathrm{Energy}(x^{(m)};f^{m})$  is linearly aligned with density  $p(x^{(m)})$ . The energy score for the  $m$ -th modality of input  $x$  can be calculated as
+
+$$
+\operatorname {E n e r g y} \left(x ^ {(m)}\right) = - \mathcal {T} ^ {m} \cdot \log \sum_ {k} ^ {K} e ^ {f _ {k} ^ {m} \left(x ^ {(m)}\right) / \mathcal {T} ^ {m}}, \tag {13}
+$$
+
+where  $f_{k}^{m}(x^{(m)})$  is the output logits of classifier  $f^{m}$  corresponding to the  $k$ -th class label and  $T^{m}$  is a temperature parameter. Intuitively, more uniformly distributed prediction leads to higher estimated uncertainty.
+
+However, it has been shown experimentally that the uncertainty estimated in this way without additional regularization is not well enough to satisfy our Assumption 1. To address this, we propose a sampling-based regularization technology to enhance the original method in terms of correlation. The most simple and straightforward way to improve the correlation between estimated uncertainty and respective loss is to leverage the sample-wise loss during training stage as supervision information. However, due to the overparameterization phenomenon of deep neural networks, the
+
+# Algorithm 1 Training Pseudo Code of Quality-aware Multimodal Fusion (QMF)
+
+Input : Multimodal training dataset  $D_{\mathrm{train}}$  , the number of sampling T, hyperparameters  $\lambda$  , temperature parameters  $\{\mathcal{T}^m\}_{m = 1}^M$  , unimodal predictors  $\{f^{m}(\cdot)\}_{i = m}^{M}$
+
+Output: The multimodal classifier  $f$ ;
+
+1 for each iteration do
+
+2 Obtain training sample  $(x_{i},y_{i})$  from dataset  $D_{\mathrm{train}}$  and the decisions on each modality  $f^{m}(x)$  
+3 Calculate uncertainty-aware fusion weights  $[w_i^1,\dots ,w_i^m ]$  defined in Eq. 9;  
+4 Update the average training loss  $\kappa_{i}^{m}$  of each modalities;  
+5 Obtain the multimodal decision by weighting unimodal predictions dynamically according to Eq. 11;  
+6 Update model parameters of each unimodal predictor by minimizing  $\mathcal{L}_{\mathrm{overall}}$  in Eq. 18.
+
+7 end
+
+losses constantly reduce to zero during training. Inspired by recent works in Bayesian learning (Maddox et al., 2019) and uncertainty estimation (Moon et al., 2020; Han et al., 2022a), we propose to leverage the information from historical training trajectory to regularize the fusion weights. Specifically, given the  $m$ -th modality of a sample  $(x_{i},y_{i})$ , the training average loss for  $x_{i}^{m}$  is calculated as:
+
+$$
+\kappa_ {i} ^ {m} = \frac {1}{T} \sum_ {\mathrm {t} = T _ {\mathrm {s}}} ^ {T _ {\mathrm {s}} + T} \ell \left(y _ {i}, f _ {\theta_ {t}} ^ {m} \left(x _ {i}\right)\right), \tag {14}
+$$
+
+where  $f_{\theta_t}^m$  is the unimodal classifier on each iteration epoch  $t$  with parameters  $\theta_t$ . After training  $T_s - 1$  epochs, we sample  $T$  times and calculate the average training loss.
+
+Empirically, recent works (Geifman et al., 2019) show that easy-to-classify samples are learned earlier during training compared to hard-to-classify samples (e.g., noise samples (Arazo et al., 2019)). It is desirable to regularize a dynamic fusion model by learning the following relationship during training
+
+$$
+\kappa_ {i} ^ {m} \geq \kappa_ {j} ^ {m} \Longleftrightarrow w _ {i} ^ {m} \leq w _ {j} ^ {m}. \tag {15}
+$$
+
+We now present the full definition of our regularization term as follows
+
+$$
+\mathcal {L} _ {\text {r e g}} = \max  \left(0, g \left(w _ {i} ^ {m}, w _ {j} ^ {m}\right) \left(\kappa_ {i} ^ {m} - \kappa_ {j} ^ {m}\right) + \left| w _ {i} ^ {m} - w _ {j} ^ {m} \right|\right), \tag {16}
+$$
+
+where
+
+$$
+g \left(w _ {i} ^ {m}, w _ {j} ^ {m}\right) = \left\{ \begin{array}{l} 1 \text {i f} w _ {i} ^ {m} > w _ {j} ^ {m}, \\ 0 \text {i f} w _ {i} ^ {m} = w _ {j} ^ {m}, \\ - 1 \text {o t h e r w i s e .} \end{array} \right. \tag {17}
+$$
+
+Inspired by multi-task learning, we define the total loss function as a summation of standard cross-entropy classification losses of multiple modalities and the regularization term
+
+$$
+\mathcal {L} _ {\text {o v e r a l l}} = \mathcal {L} _ {\mathrm {C E}} (y, f (x)) + \sum_ {m = 1} ^ {M} \mathcal {L} _ {\mathrm {C E}} \left(y, f ^ {m} \left(x ^ {m}\right)\right) + \lambda \mathcal {L} _ {\mathrm {r e g}}, \tag {18}
+$$
+
+where  $\lambda$  is a hyperparameter which controls the strength of regularization,  $\mathcal{L}_{\mathrm{CE}}$  and  $\mathcal{L}_{\mathrm{reg}}$  are the cross-entropy loss and regularization term respectively. The whole training process is shown in Algorithm 1.
+
+Intuitive explanation of the effectiveness of QMF. Without loss of generality, we assume modality  $x^A$  is clean and modality  $x^B$  is noisy due to unknown environmental factors or sensor failure. At this time,  $x^A$  is in the distribution of clean training data but  $x^B$  deviates significantly from it. Accordingly, we have  $u(x^A) \leq u(x^B)$  and thus  $w^A \geq w^B$ . Therefore, for our QMF, the multimodal decision will tend to more rely on the high-quality modality  $x^A$  than the other modality  $x^B$ . By dynamically determining the fusion weights of each modality, the influence of the unreliable modalities can be alleviated.
+
+# 5. Experiment
+
+In this section, we conduct experiments on multiple datasets of diverse applications  ${}^{2}$  . The main questions to be verified are highlighted here:
+
+- Q1 Effectiveness I. Does the proposed method has better generalization ability than its counterparts? (Support Theorem 1)  
+- Q2 Effectiveness II. Under what conditions does uncertainty-aware dynamic multimodal fusion work? (Support Theorem 2)  
+- Q3 Reliability. Does the proposed method have an effective perception for the uncertainty of modality? (Support Assumption 1)  
+- Q4 Ablation study. What is the key factor of performance improvement in our method?
+
+# 5.1. Experimental Setup
+
+We briefly present the experimental setup here, including the experimental datasets and comparison methods. Please
+
+![](images/244e402c766463db61395167b7cf53257be058d0fd2d406a2ec474f86d13d4a6.jpg)  
+(a) NYU Depth V2
+
+![](images/44875d2156afcf86653397a7458e82ed308ab1aa20852ac5d1f2396d56698b6e.jpg)  
+(b) SUN RGB-D  
+Figure 3. Test accuracy and Pearson correlation coefficient achieved by different fusion methods over 10 times random experiments. The average and worst-case accuracy are highly consistency with uncertainty estimation ability.
+
+refer to Appendix B for more detailed setup.
+
+Tasks and datasets. We evaluate our method on two multimodal classification tasks. ○ Scenes Recognition: NYU Depth V2 (Silberman et al., 2012) and SUN RGB-D (Song et al., 2015) are two public indoor scenes recognition datasets, which are associated with two modalities, i.e., RGB and depth images. ○ Image-text classification: The UPMC FOOD101 dataset (Wang et al., 2015) contains (possibly noisy) images obtained by Google Image Search and corresponding textual descriptions. MVSA sentiment analysis dataset (Niu et al., 2016) includes a set of image-text pairs with manual annotations collected from social media. Although the datasets above are all under the condition that  $M = 2$ , it is intuitive and easy to generalize to  $M \geq 3$ .
+
+Evaluation metrics. Due to the randomness involved, we report the mean accuracy, standard deviation and worst-case accuracy on NYU Depth V2 and SUN RGB-D over 10 different seeds. To be consistent with existing works (Han et al., 2022c; Kiela et al., 2019; Yadav & Vishwakarma, 2023), we repeat experiments over 3 times on UMPC FOOD101 and 5 times on MVSA.
+
+Compared methods. For scene recognition task, we compare the proposed method with three static fusion methods: Late fusion, Concatenate-based fusion, Alignment-based fusion methods (Wang et al., 2016) and two representative dynamic fusion methods, i.e., MMTM (Joze et al., 2020) and  $\mathrm{TMC}^3$  (Han et al., 2021). For image-text classification, we compare against strong unimodal baselines (i.e., Bow, Bert and ResNet-152) as well as sophisticated multimodal fusion methods, including Late fusion, ConcatBow,
+
+3There are two variants in (Han et al., 2021): TMC and ETMC (with additional concatenated-based multimodal fusion strategy). TMC has comparable performance and is a more fair comparison.
+
+ConcatBERT and recent sota MMBT (Kiela et al., 2019).
+
+# 5.2. Experimental Results
+
+Classification robustness (Q1). To validate the robustness of the uncertainty-aware weighting fusion, we evaluate QMF and the compared methods in terms of average and worst-case accuracy under Gaussian noise (for image modality) and blank noise (for text modality) following previous works (Han et al., 2021; Ma et al., 2021; Verma et al., 2021; Hu et al., 2019; Xie et al., 2017). More results under different types of noise (e.g. Salt-Pepper Noise) can be found in Appendix C.2. The experimental results are presented in Table 1. It is observed that QMF usually performs in the top three in terms of both average and worst-case accuracy. This observation indicates that QMF has better generalization ability than their counterparts experimentally. It is also noteworthy that the QMF outperforms the prior state-of-the-art methods (i.e., MMBT and TMC) on large-scale benchmark UPMC FOOD101, which illustrates the superiority of the proposed method.
+
+Connection to uncertainty estimation (Q2). We further conduct comparisons with QMF realized by various uncertainty estimation algorithms, i.e., prediction confidence (Hendrycks & Gimpel, 2017) and Dempster-Shafer evidence theory (DST) (Han et al., 2021). According to comparison results shown in the Figure 3, it is clear that (i) the generalization ability (i.e., average and worst-case accuracy) of dynamic fusion methods coincide with their uncertainty estimation ability and (ii) our QMF achieves the best performance in terms of classification accuracy and uncertainty estimation in the meantime. This comparison reveals the underlying reason of why QMF outperforms other fusion methods and supports Theorem 2. We show the results on NYU Depth V2 and SUN RGB-D under Gaussian
+
+Table 1. Classification comparison when  ${50}\%$  of the modalities are corrupted with Gaussian noise i.e., zero mean with variance of  $\epsilon$  . The best three results are in bold brown and the best results are highlighted in bold blue. Full results with standard deviation are in Appendix.  
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Dynamic</td><td rowspan="2">Method</td><td colspan="2">ε = 0.0</td><td colspan="2">ε = 5.0</td><td colspan="2">ε = 10.0</td></tr><tr><td>Avg.</td><td>Worst.</td><td>Avg.</td><td>Worst.</td><td>Avg.</td><td>Worst.</td></tr><tr><td rowspan="8">NYU Depth V2</td><td>X</td><td>RGB</td><td>63.30</td><td>62.54</td><td>53.12</td><td>50.31</td><td>45.46</td><td>42.20</td></tr><tr><td>X</td><td>Depth</td><td>62.65</td><td>61.01</td><td>50.95</td><td>42.81</td><td>44.13</td><td>35.93</td></tr><tr><td>X</td><td>Late fusion</td><td>69.14</td><td>68.35</td><td>59.63</td><td>53.98</td><td>51.99</td><td>44.95</td></tr><tr><td>X</td><td>Concat</td><td>70.30</td><td>69.42</td><td>59.97</td><td>55.89</td><td>53.20</td><td>47.71</td></tr><tr><td>X</td><td>Align</td><td>70.31</td><td>68.50</td><td>59.47</td><td>56.27</td><td>51.74</td><td>44.19</td></tr><tr><td>✓</td><td>MMTM</td><td>71.04</td><td>70.18</td><td>60.37</td><td>56.73</td><td>52.28</td><td>46.18</td></tr><tr><td>✓</td><td>TMC</td><td>71.06</td><td>69.57</td><td>61.04</td><td>58.72</td><td>53.36</td><td>49.23</td></tr><tr><td>✓</td><td>Ours</td><td>70.09</td><td>68.81</td><td>61.62</td><td>58.87</td><td>55.60</td><td>51.07</td></tr><tr><td rowspan="8">SUN RGB-D</td><td>X</td><td>RGB</td><td>56.78</td><td>56.51</td><td>48.40</td><td>47.16</td><td>42.94</td><td>41.02</td></tr><tr><td>X</td><td>Depth</td><td>52.99</td><td>51.32</td><td>37.81</td><td>35.63</td><td>33.07</td><td>30.41</td></tr><tr><td>X</td><td>Late fusion</td><td>62.09</td><td>60.55</td><td>52.44</td><td>50.83</td><td>47.33</td><td>44.60</td></tr><tr><td>X</td><td>Concat</td><td>61.90</td><td>61.19</td><td>52.69</td><td>50.61</td><td>45.64</td><td>42.95</td></tr><tr><td>X</td><td>Align</td><td>61.12</td><td>60.12</td><td>50.05</td><td>47.63</td><td>44.19</td><td>38.12</td></tr><tr><td>✓</td><td>MMTM</td><td>61.72</td><td>60.94</td><td>51.86</td><td>50.80</td><td>46.03</td><td>44.28</td></tr><tr><td>✓</td><td>TMC</td><td>60.68</td><td>60.31</td><td>51.24</td><td>49.45</td><td>45.66</td><td>41.60</td></tr><tr><td>✓</td><td>Ours</td><td>62.09</td><td>61.30</td><td>53.40</td><td>52.07</td><td>48.58</td><td>47.50</td></tr><tr><td rowspan="9">FOOD 101</td><td>X</td><td>Bow</td><td>82.50</td><td>82.32</td><td>61.68</td><td>60.98</td><td>41.95</td><td>41.41</td></tr><tr><td>X</td><td>Img</td><td>64.62</td><td>64.22</td><td>34.72</td><td>34.19</td><td>33.03</td><td>32.67</td></tr><tr><td>X</td><td>Bert</td><td>86.46</td><td>86.42</td><td>67.38</td><td>67.19</td><td>43.88</td><td>43.56</td></tr><tr><td>X</td><td>Late fusion</td><td>90.69</td><td>90.58</td><td>68.49</td><td>65.05</td><td>58.00</td><td>55.77</td></tr><tr><td>X</td><td>ConcatBow</td><td>70.77</td><td>70.68</td><td>38.28</td><td>37.95</td><td>35.68</td><td>34.92</td></tr><tr><td>X</td><td>ConcatBert</td><td>88.20</td><td>87.81</td><td>61.10</td><td>59.25</td><td>49.86</td><td>47.79</td></tr><tr><td>✓</td><td>MMBT</td><td>91.52</td><td>91.38</td><td>72.32</td><td>71.78</td><td>56.75</td><td>56.21</td></tr><tr><td>✓</td><td>TMC</td><td>89.86</td><td>89.80</td><td>73.93</td><td>73.64</td><td>61.37</td><td>61.10</td></tr><tr><td>✓</td><td>Ours</td><td>92.92</td><td>92.72</td><td>76.03</td><td>74.68</td><td>62.21</td><td>61.76</td></tr><tr><td rowspan="9">MVSA</td><td>X</td><td>Bow</td><td>48.79</td><td>35.45</td><td>42.20</td><td>32.56</td><td>41.57</td><td>32.18</td></tr><tr><td>X</td><td>Img</td><td>64.12</td><td>62.04</td><td>49.36</td><td>45.67</td><td>45.00</td><td>39.31</td></tr><tr><td>X</td><td>Bert</td><td>75.61</td><td>74.76</td><td>69.50</td><td>65.70</td><td>47.41</td><td>45.86</td></tr><tr><td>X</td><td>Late fusion</td><td>76.88</td><td>74.76</td><td>63.46</td><td>58.57</td><td>55.16</td><td>47.78</td></tr><tr><td>X</td><td>ConcatBow</td><td>64.09</td><td>62.04</td><td>49.95</td><td>45.28</td><td>45.40</td><td>40.95</td></tr><tr><td>X</td><td>ConcatBert</td><td>65.59</td><td>64.74</td><td>50.70</td><td>44.70</td><td>46.12</td><td>41.81</td></tr><tr><td>✓</td><td>MMBT</td><td>78.50</td><td>78.04</td><td>71.99</td><td>69.94</td><td>55.35</td><td>52.22</td></tr><tr><td>✓</td><td>TMC</td><td>74.88</td><td>71.10</td><td>66.72</td><td>60.12</td><td>60.36</td><td>53.37</td></tr><tr><td>✓</td><td>Ours</td><td>78.07</td><td>76.30</td><td>73.85</td><td>71.10</td><td>61.28</td><td>57.61</td></tr></table>
+
+Table 2. Ablation study on NYU Depth V2. Full results with standard deviation are in Appendix C.1.  
+
+<table><tr><td rowspan="2">UAW</td><td rowspan="2">Lreg</td><td colspan="2">ε = 0.0</td><td colspan="2">ε = 5.0</td><td colspan="2">ε = 10.0</td><td colspan="2">ε = 20.0</td></tr><tr><td>Avg.</td><td>Worst.</td><td>Avg.</td><td>Worst.</td><td>Avg.</td><td>Worst.</td><td>Avg.</td><td>Worst.</td></tr><tr><td>X</td><td>X</td><td>69.14</td><td>68.35</td><td>59.62</td><td>53.98</td><td>51.94</td><td>44.95</td><td>43.76</td><td>36.85</td></tr><tr><td>X</td><td>✓</td><td>69.68</td><td>67.74</td><td>61.35</td><td>58.26</td><td>55.44</td><td>51.53</td><td>47.32</td><td>42.97</td></tr><tr><td>✓</td><td>X</td><td>70.06</td><td>69.11</td><td>61.59</td><td>57.49</td><td>55.14</td><td>50.15</td><td>47.46</td><td>42.05</td></tr><tr><td>✓</td><td>✓</td><td>70.09</td><td>68.81</td><td>61.62</td><td>58.87</td><td>55.81</td><td>51.07</td><td>48.26</td><td>43.73</td></tr></table>
+
+Table 3. Pearson correlation coefficient  $r$  between losses and fusion weights of test samples (a higher  $|r|$  indicates a better uncertainty estimation).  
+
+<table><tr><td></td><td>ε = 0.0</td><td>ε = 5.0</td><td>ε = 10.0</td></tr><tr><td>MSP</td><td>0.391</td><td>0.433</td><td>0.486</td></tr><tr><td>Energy score</td><td>0.272</td><td>0.429</td><td>0.510</td></tr><tr><td>Entropy</td><td>0.397</td><td>0.420</td><td>0.452</td></tr><tr><td>Evidence</td><td>0.157</td><td>0.136</td><td>0.265</td></tr><tr><td>Ours</td><td>0.498</td><td>0.652</td><td>0.735</td></tr></table>
+
+noise with zero mean and variance of 10.
+
+Reliability of QMF (Q3). We calculate the fusion weights defined in Eq. 9 of different modalities in Table 3 on UPMC FOOD-101. It is observed that the fusion weights of QMF have the most effective perception of modal quality compared with other uncertainty estimation methods (in terms of correlation). This observation justifies our expectation of uncertainty-aware weights in Eq. 9.
+
+Ablation study (Q4). We compare different combinations of components (i.e., uncertainty-aware weighting and the regularization term  $\mathcal{L}_{\mathrm{reg}}$ ). Here we also employ Gaussian noise on NYU Depth V2 in Table 2, and more results can be found in the Appendix C.1. It is easy to conclude that 1) adding  $\mathcal{L}_{\mathrm{reg}}$  is beneficial to obtaining more reasonable fusion weights; 2) the best performance could be expected with the full QMF. Please refer to Table. 4 in Appendix C.1 for full results with standard deviation.
+
+In summary, the empirical results can support our theoretical findings. These works identify the causes and conditions of performance gains of dynamic multimodal fusion methods. The proposed method can help to improve robustness on multiple datasets.
+
+# 6. Limitations
+
+Even though the proposed method achieves superior performance, there are still some potential limitations. Firstly, the fusion weights of QMF are based on uncertainty estimation, which can be a challenging task in the real world.
+
+For example, in our experiments, we can only achieve mild Pearson's  $r$  on NYU Depth V2 and SUN RGB-D dataset. Therefore, it is important and valuable to explore novel uncertainty estimation methods in the future work. Secondly, though we characterize the generalization error bound of the proposed method, our theoretical justifications are based on Assumption 1. However, previous work (Fang et al., 2022) reveals that OOD detection is not learnable under some scenarios. Thus it's still a challenging open problem to further characterize the generalization ability of dynamic multimodal fusion.
+
+# 7. Conclusions and Future works
+
+Introducing dynamics in multimodal fusion has yielded remarkable empirical results in various applications, including image classification, object detection and semantic segmentation. Many state-of-the-art multimodal models introduce dynamic fusion strategies, but the inductive bias provided by this technique is not well understood. In this paper, we provide rigorous analysis towards understanding when and what dynamic multimodal fusion methods are more robust on multimodal data in the wild. These findings demonstrate the connection between uncertainty learning and robust multimodal fusion, which further implies a principle to design novel dynamic multimodal fusion methods. Finally, we perform extensive experiments on multiple benchmarks to support our findings. In the work, the energy-based weighting strategy is devised, and other uncertainty estimation ways could be explored. Another interesting direction is proving the dynamic fusion under a more general setting.
+
+# Acknowledgments
+
+This work is partially supported by the National Natural Science Foundation of China (Grant No. 61976151) and A*STAR Central Research Fund. We gratefully acknowledge the support of MindSpore and CAAI. The authors would like to thank Zhipeng Liang (Hong Kong University of Science and Technology) for checking on math details and Zongbo Han, Huan Ma (Tianjin University) for their comments on writing. The authors also appreciate the suggestions from ICML anonymous peer reviewers.
+
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+
+# Appendix
+
+# A. Proofs
+
+# A.1. Proof of Theorem 1
+
+Proof. Let  $(x,y)\sim \mathcal{D}$  denotes the multimodal sample, then we have
+
+$$
+\ell (f (x), y) = \ell \left(\sum_ {m = 1} ^ {M} w ^ {m} f ^ {m} \left(x ^ {(m)}\right), y\right). \tag {19}
+$$
+
+Noted that  $\ell$  is a convex logistic loss function, which indicates that
+
+$$
+\ell (f (x), y) = \ell \left(\sum_ {m = 1} ^ {M} w ^ {m} f ^ {m} \left(x ^ {(m)}\right), y\right) \leq \sum_ {m = 1} ^ {M} w ^ {m} \ell \left(f ^ {m} \left(x ^ {(m)}\right), y\right). \tag {20}
+$$
+
+Then we take the expectation on both sides of the above equation
+
+$$
+\mathbb {E} _ {(x, y) \sim \mathcal {D}} \ell (f (x), y) \leq \mathbb {E} _ {(x, y) \sim \mathcal {D}} \sum_ {m = 1} ^ {M} w ^ {m} \ell \left(f ^ {m} \left(x ^ {(m)}\right), y\right), \tag {21}
+$$
+
+since expectation is a linear operator and the expected value of the product is equal to the product of the expected values plus the covariance, we can further decompose the right-hand side of the equation into
+
+$$
+\begin{array}{l} \mathbb {E} _ {(x, y) \sim \mathcal {D}} \ell (f, y) \leq \sum_ {m = 1} ^ {M} \mathbb {E} _ {(x, y) \sim \mathcal {D}} [ w ^ {m} \ell (f ^ {m}, y) ] (22) \\ = \sum_ {m = 1} ^ {M} \mathbb {E} _ {(x, y) \sim \mathcal {D}} \left(w ^ {m}\right) \mathbb {E} _ {(x, y) \sim \mathcal {D}} \left(\ell \left(f ^ {m}, y\right)\right) + C o v \left(w ^ {m}, \ell \left(f ^ {m}, y\right)\right) (23) \\ \end{array}
+$$
+
+Next, we recap the Rademacher complexity measure for model complexity. We use complexity-based learning theory (Bartlett & Mendelson, 2002) (Theorem 8) to quantify the generalization error of unimodal models.
+
+Let  $D_{\mathrm{train}} = \{x_i, y_i\}_{i=1}^N$  be the training dataset of  $N$  samples,  $\hat{E}(f^m)$  is the unimodal empirical error of  $f^m$ . Then for any hypothesis  $f^m$  in  $\mathcal{H}$  (i.e.,  $\mathcal{H}: \mathcal{X} \to \{-1, 1\}$ ,  $f \in \mathcal{H}$ ) and  $1 > \delta > 0$ , with probability at least  $1 - \delta$ , we have
+
+$$
+\mathbb {E} _ {(x, y) \sim \mathcal {D}} (f ^ {m}) \leq \hat {E} (f ^ {m}) + \Re_ {m} (\mathcal {H}) + \sqrt {\frac {\ln (1 / \delta)}{2 N}},
+$$
+
+where  $\Re_m(f^m)$  is the Rademacher complexities.
+
+Finally, it holds with probability at least  $1 - \delta$  that
+
+$$
+\operatorname {G E r r o r} (f) \leq \sum_ {m = 1} ^ {M} \mathbb {E} \left(w ^ {m}\right) \hat {E} \left(f ^ {m}\right) + \mathbb {E} \left(w ^ {m}\right) \Re_ {m} (\mathcal {H}) + C o v \left(w ^ {m}, \ell \left(f ^ {m}, y\right)\right) + M \sqrt {\frac {\ln (1 / \delta)}{2 N}}. \tag {24}
+$$
+
+# A.2. Proof of Theorem 2
+
+Proof. Let  $\mathcal{O}(\mathrm{GError}(f_{\mathrm{dynamic}}))$ ,  $\mathcal{O}(\mathrm{GError}(f_{\mathrm{static}}))$  be the upper bound of generalization error of multimodal classifier using dynamic and static fusion strategy respectively,  $\hat{E}(f^m)$  is the unimodal empirical errors of  $f^m$  on  $D_{\mathrm{train}}$  defined in Theorem 1. Theoretically, optimizing over the same function class results in the same empirical risk. Therefore
+
+$$
+\hat {E} \left(f _ {\text {s t a t i c}} ^ {m}\right) = \hat {E} \left(f _ {\text {d y n a m i c}} ^ {m}\right). \tag {25}
+$$
+
+Additionally, the intrinsic complexity of unimodal classifier  $\Re_m(f^m)$  is also invariant
+
+$$
+\Re_ {m} \left(f _ {\text {s t a t i c}} ^ {m}\right) = \Re_ {m} \left(f _ {\text {d y n a m i c}} ^ {m}\right). \tag {26}
+$$
+
+Thus in this special case, it holds that
+
+$$
+\sum_ {m = 1} ^ {M} \mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) \hat {E} \left(f ^ {m}\right) \leq \sum_ {m = 1} ^ {M} w _ {\text {s t a t i c}} ^ {m} \hat {E} \left(f ^ {m}\right), \tag {27}
+$$
+
+and
+
+$$
+\sum_ {m = 1} ^ {M} \mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) \Re_ {m} \left(f ^ {m}\right) \leq \sum_ {m = 1} ^ {M} w _ {\text {s t a t i c}} ^ {m} \Re_ {m} \left(f ^ {m}\right), \tag {28}
+$$
+
+if  $\mathbb{E}(w_{\mathrm{dynamic}}^m) = w_{\mathrm{static}}^m$
+
+Since the covariance and correlation coefficient have the same sign, when  $r(w^{m},l^{m})\leq 0$ , the covariance  $Cov(w^{m},l^{m})$  is also less than or equal to 0. Therefore, it holds that
+
+$$
+\mathcal {O} (\operatorname {G E r r o r} (f _ {\text {d y n a m i c}})) \leq \mathcal {O} (\operatorname {G E r r o r} (f _ {\text {s t a t i c}})) \tag {29}
+$$
+
+with probability at least  $1 - \delta$ , if we have
+
+$$
+\mathbb {E} \left(w _ {\text {d y n a m i c}} ^ {m}\right) = w _ {\text {s t a t i c}} ^ {m} \tag {30}
+$$
+
+and
+
+$$
+r \left(w _ {\text {d y n a m i c}} ^ {m}, \ell \left(f ^ {m}\right)\right) \leq 0 \tag {31}
+$$
+
+for all input modality  $m$ .
+
+![](images/46496a936e203f904d2bdd982100b719875eb1ae9ee17855658bfdb525cf6f0e.jpg)
+
+# B. Experimental details
+
+# B.1. Datasets details
+
+- Senses recognition. For NYUD-V2, following the standard split, we reorganize the 27 categories into 10 categories with 9 usual senes and one "others" category. For SUN RGB-D, following the previous work (Han et al., 2021), we use the 19 major scene categories of SUN RGB-D, each of which contains at least 80 images.  
+- Image-text classification. For FOOD-101, following the previous work (Kiela et al., 2019), there are 60101 image-text pairs in the training set, 5000 image-text pairs in the validation set, and 21695 image-text pairs in the test set. For MVSA, we conduct the division strategy presents in (Kiela et al., 2019). There are 1555 image-text pairs in the training set. The validation set contains 518 image-text pairs, and the test set contains 519 image-text pairs.
+
+# B.2. Implementation details
+
+Senses recognition. For senses recognition task, we compare the proposed method with diverse multimodal fusion methods, including late fusion, align-based fusion, concatenated-based fusion and recent SOTA MMTM (attention-based fusion). Regarding late fusion, align-based fusion, concatenated-based fusion, we adopt the architecture of ResNet (He et al., 2016) pretrained on ImageNet (Deng et al., 2009) as the backbone network for each modality.  $\circ$  Concatenate-based fusion For concatenate-based fusion, we concatenate the representations extracted from different modalities by ResNet. Then a fully connection layer is deployed to map the multimodal representation to the target space. The dimensions of unimodal representation and common representation are 128 and 256 respectively.  $\circ$  Align-based fusion The alignment fusion method is a re-implementation of (Wang et al., 2016). We deploy cosine distance to measure the similarity of representations.  $\circ$  MMTM We follow the authors' implementation, where the squeeze ratio is set to 4. For all compared methods, we use Adam optimizer to train all above models with  $\mathcal{L}_2$  regularization and dropout (Srivastava et al., 2014). The learning rate is 1e-4 and the dropout rate is 0.1.
+
+Image-text classification. For image-text classification, we compare the proposed method with diverse multimodal fusion methods, including late fusion, concatenated-bow fusion, concatenated-bert fusion and MMTM. For late fusion and concatenated-bert fusion, we adopt the architecutre of ResNet (He et al., 2016) pretrained on ImageNet (Deng et al., 2009) as
+
+the backbone network for image modality and pre-trained Bert(Devlin et al., 2018) for text modality. For concatenated-Bow fusion, we use the Bow (Pennington et al., 2014) to replace BERT for text modality. For the Bert models, we use BertAdam and regular Adam for the other models. The learning rate is 1e-4 with a warmup rate of 0.1. We adopt the early stop strategy based on validation accuracy.
+
+For all above experiments, we conduct sampling during the whole training phase  $(\mathrm{T_s} = 1)$ . The hyperparameter  $\lambda$  is set to 0.1. Temperature parameters  $\{\mathcal{T}^m\}_{m=1}^M$  are set to 1.
+
+# C. Additional results
+
+# C.1. Full results with standard deviation
+
+In this section, we present the full results with standard deviation in Tab. 5, and Tab. 4.
+
+# C.2. Different type of noise
+
+We provide more results with different type of noise (i.e., salt-pepper noise with varying noise rate  $\epsilon$ ) in Tab. 6. The results validate that the proposed method can improve the performance of multimodal fusion methods under different type of noise.
+
+Table 4. Full ablation study on NYU Depth V2.  
+
+<table><tr><td>UAW</td><td>Lreg</td><td>ε = 0.0</td><td>ε = 5.0</td><td>ε = 10.0</td><td>ε = 20.0</td></tr><tr><td>X</td><td>X</td><td>69.14 ± 0.69</td><td>68.35 ± 0.82</td><td>59.62 ± 1.17</td><td>53.98 ± 1.08</td></tr><tr><td>X</td><td>✓</td><td>69.68 ± 0.39</td><td>67.74 ± 0.40</td><td>61.35 ± 0.34</td><td>58.26 ± 0.1.13</td></tr><tr><td>✓</td><td>X</td><td>70.06 ± 0.1.03</td><td>69.11 ± 0.82</td><td>61.59 ± 0.0.72</td><td>57.49 ± 1.41</td></tr><tr><td>✓</td><td>✓</td><td>70.09 ± 0.38</td><td>68.81 ± 0.62</td><td>61.62 ± 0.31</td><td>58.87 ± 0.40</td></tr></table>
+
+Table 5. Full comparison results when  ${50}\%$  of the modalities are corrupted with Gaussian noise.  
+
+<table><tr><td>Dataset</td><td>Dynamic</td><td>Method</td><td>ε = 0.0</td><td>ε = 5.0</td><td>ε = 10.0</td></tr><tr><td rowspan="8">NYU Depth V2</td><td>X</td><td>RGB</td><td>62.65 ± 1.22</td><td>50.95 ± 3.38</td><td>44.13 ± 3.80</td></tr><tr><td>X</td><td>Depth</td><td>63.30 ± 0.48</td><td>53.12 ± 1.52</td><td>45.46 ± 2.07</td></tr><tr><td>X</td><td>Late fusion</td><td>69.14 ± 0.67</td><td>59.63 ± 2.44</td><td>51.99 ± 3.11</td></tr><tr><td>X</td><td>Concat</td><td>70.31 ± 0.80</td><td>59.97 ± 2.14</td><td>53.20 ± 3.50</td></tr><tr><td>X</td><td>Align</td><td>70.31 ± 1.28</td><td>59.47 ± 1.84</td><td>51.74 ± 3.41</td></tr><tr><td>✓</td><td>MMTM</td><td>71.04 ± 0.41</td><td>60.37 ± 2.61</td><td>52.28 ± 3.77</td></tr><tr><td>✓</td><td>TMC</td><td>71.06 ± 0.76</td><td>61.04 ± 1.66</td><td>53.36 ± 2.76</td></tr><tr><td>✓</td><td>Ours</td><td>70.09 ± 0.97</td><td>61.62 ± 1.84</td><td>55.60 ± 2.09</td></tr><tr><td rowspan="8">SUN RGB-D</td><td>X</td><td>RGB</td><td>52.99 ± 0.88</td><td>37.81 ± 1.14</td><td>33.07 ± 1.81</td></tr><tr><td>X</td><td>Depth</td><td>56.78 ± 0.19</td><td>48.40 ± 1.11</td><td>42.94 ± 1.63</td></tr><tr><td>X</td><td>Late fusion</td><td>62.00 ± 0.15</td><td>52.52 ± 0.67</td><td>47.48 ± 1.40</td></tr><tr><td>X</td><td>Concat</td><td>62.48 ± 0.50</td><td>53.30 ± 0.39</td><td>48.01 ± 0.96</td></tr><tr><td>X</td><td>Align</td><td>61.12 ± 0.61</td><td>50.05 ± 1.59</td><td>44.19 ± 2.18</td></tr><tr><td>✓</td><td>MMTM</td><td>61.72 ± 0.67</td><td>51.86 ± 1.14</td><td>46.03 ± 1.47</td></tr><tr><td>✓</td><td>TMC</td><td>60.68 ± 0.24</td><td>51.24 ± 0.96</td><td>45.66 ± 2.06</td></tr><tr><td>✓</td><td>Ours</td><td>62.09 ± 0.56</td><td>53.40 ± 0.89</td><td>48.58 ± 0.82</td></tr><tr><td rowspan="9">UMPC FOOD101</td><td>X</td><td>Bow</td><td>82.50 ± 0.18</td><td>61.68 ± 0.71</td><td>41.95 ± 0.54</td></tr><tr><td>X</td><td>Img</td><td>64.62 ± 0.40</td><td>34.72 ± 0.53</td><td>33.03 ± 0.37</td></tr><tr><td>X</td><td>Bert</td><td>86.46 ± 0.05</td><td>67.38 ± 0.19</td><td>43.88 ± 0.32</td></tr><tr><td>X</td><td>Late fusion</td><td>90.69 ± 0.12</td><td>68.49 ± 3.37</td><td>57.99 ± 1.59</td></tr><tr><td>X</td><td>Concatbow</td><td>70.77 ± 0.09</td><td>38.28 ± 0.26</td><td>35.68 ± 0.69</td></tr><tr><td>X</td><td>Concatbert</td><td>88.20 ± 0.34</td><td>61.10 ± 2.02</td><td>49.86 ± 2.05</td></tr><tr><td>✓</td><td>MMBT</td><td>91.52 ± 0.10</td><td>72.32 ± 0.34</td><td>56.75 ± 0.33</td></tr><tr><td>✓</td><td>TMC</td><td>89.86 ± 0.07</td><td>73.93 ± 0.34</td><td>61.37 ± 0.21</td></tr><tr><td>✓</td><td>Ours</td><td>92.92 ± 0.11</td><td>76.03 ± 0.70</td><td>62.21 ± 0.25</td></tr><tr><td rowspan="9">MVSA</td><td>X</td><td>Bow</td><td>48.79 ± 7.05</td><td>42.20 ± 6.40</td><td>41.57 ± 6.28</td></tr><tr><td>X</td><td>Img</td><td>64.12 ± 1.23</td><td>49.36 ± 2.02</td><td>45.00 ± 2.63</td></tr><tr><td>X</td><td>Bert</td><td>75.61 ± 0.53</td><td>69.50 ± 1.50</td><td>47.41 ± 0.79</td></tr><tr><td>X</td><td>Late fusion</td><td>76.88 ± 1.30</td><td>63.46 ± 3.46</td><td>55.16 ± 3.60</td></tr><tr><td>X</td><td>ConcatBow</td><td>64.08 ± 1.54</td><td>49.95 ± 2.29</td><td>45.39 ± 3.03</td></tr><tr><td>X</td><td>Concatbert</td><td>65.59 ± 1.33</td><td>50.70 ± 2.65</td><td>46.12 ± 2.44</td></tr><tr><td>✓</td><td>MMBT</td><td>78.50 ± 0.40</td><td>71.99 ± 1.04</td><td>55.34 ± 2.84</td></tr><tr><td>✓</td><td>TMC</td><td>74.87 ± 2.24</td><td>66.72 ± 4.55</td><td>60.35 ± 2.79</td></tr><tr><td>✓</td><td>Ours</td><td>78.07 ± 1.10</td><td>73.85 ± 1.42</td><td>61.28 ± 2.12</td></tr></table>
+
+Table 6. Full comparison results when  ${50}\%$  of the modalities are corrupted with Salt-pepper noise.  
+
+<table><tr><td>Dataset</td><td>Dynamic</td><td>Method</td><td>ε = 0.0</td><td>ε = 5.0</td><td>ε = 10.0</td></tr><tr><td rowspan="8">NYU Depth V2</td><td>X</td><td>RGB</td><td>62.61 ± 1.21</td><td>49.14 ± 1.40</td><td>34.76 ± 1.59</td></tr><tr><td>X</td><td>Depth</td><td>63.32 ± 0.50</td><td>50.99 ± 1.41</td><td>38.56 ± 2.16</td></tr><tr><td>X</td><td>Late fusion</td><td>69.16 ± 0.68</td><td>56.27 ± 2.40</td><td>41.22 ± 2.78</td></tr><tr><td>X</td><td>Concat</td><td>70.44 ± 0.89</td><td>57.98 ± 2.08</td><td>44.51 ± 2.91</td></tr><tr><td>X</td><td>Align</td><td>70.31 ± 1.28</td><td>57.54 ± 2.50</td><td>43.01 ± 2.66</td></tr><tr><td>✓</td><td>MMTM</td><td>71.04 ± 0.41</td><td>59.45 ± 1.38</td><td>44.59 ± 2.49</td></tr><tr><td>✓</td><td>TMC</td><td>71.01 ± 0.75</td><td>59.34 ± 1.03</td><td>44.65 ± 2.30</td></tr><tr><td>✓</td><td>Ours</td><td>70.06 ± 0.81</td><td>58.50 ± 2.05</td><td>45.69 ± 2.79</td></tr><tr><td rowspan="8">SUN RGB-D</td><td>X</td><td>RGB</td><td>52.63 ± 0.89</td><td>40.42 ± 0.99</td><td>28.15 ± 1.00</td></tr><tr><td>X</td><td>Depth</td><td>56.81 ± 0.57</td><td>46.36 ± 0.82</td><td>35.66 ± 1.44</td></tr><tr><td>X</td><td>Late fusion</td><td>61.79 ± 0.57</td><td>51.54 ± 2.12</td><td>39.35 ± 2.89</td></tr><tr><td>X</td><td>Concat</td><td>62.06 ± 0.53</td><td>51.09 ± 1.91</td><td>38.61 ± 3.07</td></tr><tr><td>X</td><td>Align</td><td>61.02 ± 0.54</td><td>50.45 ± 0.82</td><td>38.70 ± 1.46</td></tr><tr><td>✓</td><td>MMTM</td><td>61.80 ± 0.40</td><td>51.09 ± 0.77</td><td>38.38 ± 1.56</td></tr><tr><td>✓</td><td>TMC</td><td>61.02 ± 0.39</td><td>50.88 ± 1.28</td><td>39.61 ± 2.30</td></tr><tr><td>✓</td><td>Ours</td><td>61.89 ± 0.49</td><td>52.49 ± 1.81</td><td>40.53 ± 2.79</td></tr><tr><td rowspan="9">UMPC FOOD101</td><td>X</td><td>Bow</td><td>82.43 ± 0.18</td><td>60.83 ± 0.54</td><td>41.56 ± 0.33</td></tr><tr><td>X</td><td>Img</td><td>64.53 ± 0.47</td><td>50.75 ± 0.44</td><td>36.83 ± 0.92</td></tr><tr><td>X</td><td>Bert</td><td>86.44 ± 0.02</td><td>67.41 ± 0.20</td><td>43.89 ± 0.33</td></tr><tr><td>X</td><td>Late fusion</td><td>90.66 ± 0.16</td><td>77.99 ± 0.54</td><td>58.75 ± 0.99</td></tr><tr><td>X</td><td>Concatbow</td><td>70.68 ± 0.12</td><td>55.01 ± 0.33</td><td>38.81 ± 0.62</td></tr><tr><td>X</td><td>Concatbert</td><td>88.22 ± 0.36</td><td>72.49 ± 0.75</td><td>52.10 ± 0.97</td></tr><tr><td>✓</td><td>MMBT</td><td>91.51 ± 0.10</td><td>76.27 ± 0.22</td><td>54.98 ± 0.55</td></tr><tr><td>✓</td><td>TMC</td><td>89.86 ± 0.07</td><td>77.86 ± 0.41</td><td>60.22 ± 0.43</td></tr><tr><td>✓</td><td>Ours</td><td>92.90 ± 0.13</td><td>80.87 ± 0.40</td><td>61.60 ± 0.20</td></tr><tr><td rowspan="9">MVSA</td><td>X</td><td>Bow</td><td>48.82 ± 7.08</td><td>42.23 ± 6.43</td><td>41.60 ± 6.31</td></tr><tr><td>X</td><td>Img</td><td>64.12 ± 1.23</td><td>56.72 ± 1.92</td><td>50.71 ± 3.20</td></tr><tr><td>X</td><td>Bert</td><td>75.61 ± 0.53</td><td>69.50 ± 1.50</td><td>47.41 ± 0.79</td></tr><tr><td>X</td><td>Late fusion</td><td>76.88 ± 1.30</td><td>67.88 ± 1.87</td><td>55.43 ± 1.94</td></tr><tr><td>X</td><td>ConcatBow</td><td>64.08 ± 1.54</td><td>56.66 ± 1.73</td><td>49.35 ± 2.44</td></tr><tr><td>X</td><td>Concatbert</td><td>65.59 ± 1.33</td><td>58.69 ± 2.25</td><td>51.16 ± 2.99</td></tr><tr><td>✓</td><td>MMBT</td><td>78.50 ± 0.40</td><td>74.07 ± 1.12</td><td>51.26 ± 5.65</td></tr><tr><td>✓</td><td>TMC</td><td>74.87 ± 2.24</td><td>68.02 ± 3.07</td><td>56.62 ± 3.67</td></tr><tr><td>✓</td><td>Ours</td><td>78.07 ± 1.10</td><td>73.90 ± 1.89</td><td>60.41 ± 2.63</td></tr></table>

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2023/Provable Reset-free Reinforcement Learning by No-Regret Reduction/full.md

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+# Provable Reset-free Reinforcement Learning by No-Regret Reduction
+
+Hoai-An Nguyen<sup>1</sup> 2 Ching-An Cheng<sup>1</sup>
+
+# Abstract
+
+Reinforcement learning (RL) so far has limited real-world applications. One key challenge is that typical RL algorithms heavily rely on a reset mechanism to sample proper initial states; these reset mechanisms, in practice, are expensive to implement due to the need for human intervention or heavily engineered environments. To make learning more practical, we propose a generic no-regret reduction to systematically design reset-free RL algorithms. Our reduction turns the reset-free RL problem into a two-player game. We show that achieving sublinear regret in this two-player game would imply learning a policy that has both sublinear performance regret and sublinear total number of resets in the original RL problem. This means that the agent eventually learns to perform optimally and avoid resets. To demonstrate the effectiveness of this reduction, we design an instantiation for linear Markov decision processes, which is the first provably correct reset-free RL algorithm.
+
+# 1. Introduction
+
+Reinforcement learning (RL) enables an artificial agent to learn problem-solving skills directly through interactions. However, RL is notorious for its sample inefficiency. Successful stories of RL so far are mostly limited to applications where a fast and accurate simulator of the world is available for collecting large amounts of samples (like in games). Real-world RL, such as robot learning, remains a challenging open question.
+
+One key obstacle to scaling up data collection in real-world RL problems is the need for resetting the agent. The ability to reset the agent to proper initial states plays an important role in typical RL algorithms, as it affects which region the
+
+agent can explore and whether the agent can recover from its past mistakes (Kakade & Langford, 2002). In most settings, completely avoiding resets without prior knowledge of the reset states or environment is impossible. In the absence of a reset mechanism, agents may get stuck in absorbing states, such as those where it has damaged itself or irreparably altered the learning environment. For instance, a robot learning to walk would inevitably fall before perfecting the skill, and timely intervention is needed to prevent damaging the hardware and to return the robot to a walkable configuration. Another example is a robot manipulator learning to stack three blocks on top of each other. Unrecoverable states would include the robot knocking a block off the table, or the robot smashing itself forcefully into the table. Reset would then reconfigure the scene to a meaningful initial state that is good for the robot to learn from.
+
+Although resetting is necessary to real-world RL, it is nontrivial to implement. Unlike in simulation, a real-world agent (e.g., a robot) cannot be reset to an arbitrary initial state with a click of a button. Resetting in the real world is expensive as it usually requires constant human monitoring and intervention. Normally, a person would need to oversee the entire learning process and manually reset the agent (e.g., a robot) to a meaningful starting state before it enters an unrecoverable state where the problem can no longer be solved. Sometimes automatic resetting can be implemented by cleverly engineering the physical learning environment (Gupta et al., 2021), but it is not always feasible.
+
+An approach we can take to make real-world RL more cost-efficient is through reset-free RL. The goal of reset-free RL is not to completely remove resets, but to have an agent learn how to perform well while minimizing the amount of external resets required. Some examples of problems that have been approached in reset-free RL include agents learning dexterity skills, such as picking up an item or inserting a pipe, and learning how to walk (Ha et al., 2020; Gupta et al., 2021). While there has been numerous works proposing reset-free RL algorithms using approaches such as multitask learning (Gupta et al., 2021; Ha et al., 2020), learning a reset policy (Eysenbach et al., 2018; Sharma et al., 2022), and skill-space planning (Lu et al., 2020), to our knowledge, there has not been any work with provable guarantees.
+
+In this work, we take the first step by providing a provably
+
+correct framework to design reset-free RL algorithms. Our framework is based on the idea of a no-regret reduction. First, we reduce the reset-free RL problem to a sequence of constrained Markov decision processes (CMDPs) with an adaptive initial state sequence. Using the special structure of reset-free RL, we establish the existence of a common saddle-point for these CMDPs. Interestingly, we show such a saddle-point exists without using the typical Slater's condition for strong duality and despite the fact that CMDPs with different initial states in general do not share a common Markovian optimal policy. Then, we derive our main no-regret reduction, which further turns this sequence into a two-player game between a primal player (updating the Markovian policy) and a dual player (updating the Lagrange multiplier function of the CMDPs) to solve for the common saddle-point. We show that if no regret is achieved in this game, then the regret of the original RL problem and the total number of required resets are both provably sublinear. This means the agent eventually learns to perform optimally and avoid resets. Using this reduction, we design a reset-free RL algorithm instantiation under the linear MDP assumption, using learning with upper confidence bound as the baseline algorithm for the primal player and projected gradient descent for the dual player. Our algorithm achieves  $\tilde{O} (\sqrt{d^3H^4K})$  regret and  $\tilde{O} (\sqrt{d^3H^4K})$  resets with high probability, where  $d$  is the feature dimension,  $H$  is the length of an episode, and  $K$  is the total number of episodes. $^1$
+
+# 2. Related Work
+
+Reset-free RL, despite being a promising avenue to tackle real-world RL, is a relatively new concept in the literature. The work thus far, to our knowledge, has been limited to approaches without theoretical guarantees but with only empirical verification. One such work takes the approach of learning a reset policy (Eysenbach et al., 2018; Sharma et al., 2022). The idea is to learn two policies concurrently: one to maximize reward, and one to bring the agent back to a reset-free initial state if they encounter a reset state (a state which normally requires human intervention). This approach prevents the need for manual resets; however, it requires the knowledge of the reset states (Eysenbach et al., 2018). Sharma et al. (2022) avoid this assumption but assume given demonstrations on how to accomplish the goal and a fixed initial state distribution.
+
+Using multi-task learning is another way to perform resets without human intervention. Here, the agent learns to solve multiple tasks instead of just maximizing the reward of one task. The hope is that a combination of the learned tasks can achieve the main goal, and some tasks can perform natural
+
+resets for others. This approach breaks down the reset process and (possibly) makes it easier to learn. However, the order in which tasks should be learned needs to be provided manually (Gupta et al., 2021; Ha et al., 2020).
+
+A related problem is infinite-horizon non-episodic RL with provable guarantees (see Wei et al. (2020; 2019); Dong et al. (2019) and the references within) as this problem is also motivated by not using resets. In this setting, there is only one episode that goes on indefinitely. The objective is to maximize cumulative reward, and progress is usually measured in terms of regret with the comparator being an optimal policy. However, compared with the reset-free RL setting we study here, extra assumptions, such as the absence or knowledge of absorbing states, are usually required to achieve sublinear regret. In addition, the objective does not necessarily lead to a minimization of resets as the agent can leverage reset transitions to maximize reward. In reset-free RL, minimizing/avoiding resets is a priority. Learning in infinite-horizon CMDPs (where one possible constraint could be minimizing resets) has been studied (Zheng & Ratliff, 2020; Jain et al., 2022), but to our knowledge, all such works make strong assumptions such as a fixed initial state distribution or known dynamics. None of these assumptions are feasible for most real-world RL settings. Enforcing a fixed initial state distribution or removing absorbing states in theory oftentimes requires physically resetting a real-world agent to satisfy those desired mathematical conditions. In this paper, we focus on an episodic setting of reset-free RL (see Section 3); a non-episodic formulation of reset-free RL could be an interesting one for further research.
+
+Another related problem is safe RL, which involves solving the standard RL problem while adhering to some safety constraints. We can think of reset states in reset-free RL as unsafe states in safe RL. There has been a lot of work in safe RL, with approaches such as utilizing a baseline safe (but not optimal) policy (Huang et al., 2022; Garcia Polo & Fernandez Rebollo, 2011), pessimism (Amani & Yang, 2022), and shielding (Alshiekh et al., 2018; Wagener et al., 2021). These works have promising empirical results but usually require extra assumptions such as a given baseline policy or knowledge of unsafe states. There are also provable safe RL algorithms. To our knowledge, all involve framing safe RL as a CMDP. Here, the safety constraints are modeled as a cost, and the overall goal is to maximize performance while keeping the cost below a threshold. Some of the works explicitly study safe RL while others study learning in CMDPs more generally. They commonly have provable guarantees of either sublinear regret and constraint violations, or sublinear regret with zero constraint violation (Wei et al., 2021; HasanzadeZonuzy et al., 2021; Qiu et al., 2020; Wachi & Sui, 2020; Efroni et al., 2020; Ghosh et al., 2022; Ding et al., 2021). However, most works (including all the aforementioned ones), consider the episodic case
+
+where the initial state distribution of each episode is fixed. This prevents a natural extension to reset-free learning as human intervention would be required to reset the environment at the end of each episode. In technical terms, this is the difference between solving a sequence of the same CMDP versus solving a sequence of different CMDPs. Works that allow for arbitrary initial states require fairly strong assumptions, such as knowledge (and the existence) of safe actions from each state (Amani et al., 2021). In our work, we utilize some techniques from provable safe RL for reset-free RL. However, it is important to note that safe RL and reset-free RL are fundamentally different, albeit related, problems. Safe RL aims to ensure the safety of an agent. On the other hand, reset-free RL aims to avoid reset states, which encompass not only unsafe states but all undesirable ones, for a varying sequence of initial states as the agent cannot be reset freely.
+
+We weaken typical assumptions of current approaches in empirical reset-free RL, infinite-horizon RL, and safe RL by dropping any requirements on knowledge of undesirable states or for demonstrations, and by allowing arbitrary initial state sequences that admit reset-free policies. Considering arbitrary initial state sequences where initial states potentially are correlated with past behaviors is not only necessary to the reset-free RL setting, but also allows for extensions to both lifelong and multi-task learning. We achieve this important relaxation on the initial state sequence with a key observation that identifies a shared Markovian policy saddle-point across CMDPs where the constraint imposes zero resets. This observation is new to our knowledge, and it is derived from the particular structure of reset-free RL; we note that generally, CMDPs with different initial states do not admit shared Markovian policy saddle-points. Finally, by the analogy between safe states and reset states, on the technical side, our framework and algorithm can also be viewed as the first provable safe RL algorithm that allows for arbitrary initial state sequences without strong assumptions.
+
+While our main contribution is a generic reduction technique to design reset-free RL algorithms, we also instantiate the framework and achieve regret and constraint violation bounds that are still comparable to the above works when specialized to their setting. Under the linear MDP assumption, our algorithm achieves  $\tilde{O} (\sqrt{d^3H^4K})$  regret and violation (equivalently, the number of resets in reset-free RL), which is asymptotically equivalent to Ghosh et al. (2022) and comparable to the bounds of  $\tilde{O} (\sqrt{d^2H^6K})$  from Ding et al. (2021) for a fixed initial state.
+
+In summary, our contributions are as follows.
+
+1. We create a framework to design provably correct reset-free RL algorithms via a reduction first to a sequence of CMDPs with an adaptive initial state sequence, and
+
+then to a two-player game. We prove that achieving sublinear regret in this two-player game implies learning a policy that achieves sublinear performance regret and sublinear number of resets in the original problem.
+
+2. On the technical side, we show that such a reduction can be constructed without using the typical Slater's condition for strong duality and despite the fact that CMDPs with different initial states in general do not share a common Markovian optimal policy.  
+3. We instantiate the framework under the linear MDP setting as a proof of concept, creating the first provably correct reset-free RL algorithm to our knowledge that achieves sublinear regret and resets.
+
+# 3. Preliminary
+
+We consider episodic reset-free RL: in each episode, the agent aims to optimize for a fixed-horizon return starting from the last state of the previous episode or some state that the agent was reset to if reset occurred.
+
+Problem Setup and Notation Formally, we can define episodic reset-free RL as a Markov decision process (MDP),  $(S, A, P, r, H)$ , where  $S$  is the state space,  $A$  is the action space,  $P = \{P_h\}_{h=1}^H$  is the transition dynamics,  $r = \{r_h\}_{h=1}^H$  is the reward function, and  $H$  is the task horizon. We assume  $P$  and  $r$  are unknown. We allow  $S$  to be large or continuous but assume  $A$  is relatively small so that  $\max_{a \in A}$  can be performed. We designate the set of reset states, or any states that human intervention normally would have been required, as  $S_{\text{reset}} \subseteq S$ ; we do not assume that the agent has knowledge of  $S_{\text{reset}}$ . We also do not assume that there is a reset-free action at each state, as opposed to Amani et al. (2021). Therefore, the agent needs to plan for the long-term to avoid resets. We assume  $r_h : S \times A \to [0, 1]$ , and for simplicity, we assume  $r_h$  is deterministic. However, we note that it would be easy to extend this to the setting where rewards are stochastic.
+
+The agent interacts with the environment for  $K$  total episodes. Following the convention of episodic problems, we suppose the state space  $S$  is layered, and a state  $s_t \in S$  at time  $t$  is factored into two components  $s_t = (\bar{s}, t)$  where  $\bar{s}$  denotes the time-invariant part. Reset happens at some time  $t$  if the time-invariant part of  $s_t$ ,  $\bar{s} \in S_{\text{reset}}$ . The initial state of the next episode will be  $s_1 = (\bar{s}', 1)$  where  $\bar{s}'$  is sampled from an unknown state distribution. In a given episode, if reset happens, we designate the time step this occurs as  $t = \tau$ . If there is no reset in an episode, the initial state of the next episode is the last state of the current episode, i.e., for episode  $k + 1$ ,  $s_1^{k + 1} = (\bar{s}, 1)$  if  $s_H^k = (\bar{s}, H)$  in episode  $k$ . Therefore, the initial state sequence is adaptive. This
+
+sequence is necessary to consider since in reset-free RL, we want to avoid resetting, including after each episode.
+
+We denote the set of Markovian policies as  $\Delta$ , and a policy  $\pi \in \Delta$  as  $\pi = \{\pi_h(a_h|s_h)\}_{h=1}^H$ . We define the state value function and the state-action value function under  $\pi$  as
+
+$$
+V _ {r, h} ^ {\pi} (s) := \mathbb {E} _ {\pi} \left[ \sum_ {t = h} ^ {\min  (H, \tau)} r _ {t} \left(s _ {t}, a _ {t}\right) | s _ {h} = s \right] \tag {1}
+$$
+
+$$
+Q _ {r, h} ^ {\pi} (s, a) := r _ {h} (s, a) + \mathbb {E} \left[ V _ {r, h + 1} ^ {\pi} \left(s _ {h + 1}\right) \mid s _ {h} = s, a _ {h} = a \right],
+$$
+
+where  $h \leq \tau$ , and we recall  $\tau$  is the time step when the agent enters  $S_{\text{reset}}$  (if at all).
+
+Objective The overall goal is for the agent to learn a Markovian policy to maximize its cumulative reward while avoiding resets. Therefore, our performance measures are as follows (we seek to minimize both quantities):
+
+$$
+\operatorname {R e g r e t} (K) = \max  _ {\pi \in \Delta_ {0} (K)} \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \tag {2}
+$$
+
+$$
+\operatorname {R e s e t s} (K) = \sum_ {k = 1} ^ {K} \mathbb {E} _ {\pi^ {k}} \left[ \sum_ {h = 1} ^ {\min  (H, \tau)} \mathbb {1} \left[ s _ {h} \in \mathcal {S} _ {\text {r e s e t}} \right] \mid s _ {1} = s _ {1} ^ {k} \right] \tag {3}
+$$
+
+where  $\Delta_0(K) \subseteq \Delta$  denotes the set of Markovian policies that avoid resets for all episodes, and  $\pi^k$  is the policy used by the agent in episode  $k$ . Note that by the reset mechanism  $\sum_{h=1}^{\min(H, \tau)} \mathbb{1}[s_h \in S_{\text{reset}}] \in \{0, 1\}$ .
+
+Notice that the initial states in our regret and reset measures are determined by the learner. Given the motivation behind reset-free RL (see Section 1), we can expect that the initial states here are meaningful for performance comparison by construction. Otherwise, a reset would have occurred to set the learner to a meaningful state. Note that this means by the reset mechanism, all "bad" absorbing states are in  $S_{\text{reset}}$ , and hence, the agent cannot hide in a "bad" absorbing state to achieve small regret. In addition, since the learner does not receive any reward within an episode after being reset, the learner cannot leverage resets to gain an advantage over the optimal policy (which never resets) to minimize or even achieve negative regret.
+
+To make the problem feasible, we assume achieving no resets is possible. We state this assumption formally below.
+
+Assumption 3.1. For any sequence  $\{s_1^k\}_{k = 1}^K$ , the set  $\Delta_0(K)$  is not empty. That is, there is a Markovian policy  $\pi \in \Delta$  such that  $\mathbb{E}_{\pi}[\sum_{h = 1}^{H}\mathbb{1}[s_h\in \mathcal{S}_{reset}]|s_1 = s_1^k ] = 0$ .
+
+This assumption is simply stating that every episode of learning admits a reset-free policy. As discussed in Section 2, this assumption is weaker than existing assumptions in the
+
+literature. We note that an alternate assumption that only  $s_1^1$  (i.e., the initial state of the first episode) admits a reset-free policy is insufficient to make reset-free RL feasible; since the transition dynamics are unknown to the agent, under this assumption alone, for any algorithm, there is a problem such that the number of resets must be linear.
+
+We highlight that Assumption 3.1 is a reasonable assumption in practice. It does not require a fixed initial state. In addition, if reset happens, in practice, the agent is usually set to a state where it can continue to operate in without reset; if the agent is at a state where no such reset-free policy exists, reset should happen. This assumption is similar to the assumption on the existence of a perfectly safe policy in safe RL literature, which is a common and relatively weak assumption (Ghosh et al., 2022; Ding et al., 2021). If there were to be initial states that inevitably lead to a reset, the problem would be infeasible and does not follow from the motivation of reset-free RL.
+
+# 4. A No-Regret Reduction for Reset-Free RL
+
+In this section, we present our main reduction of reset-free RL to regret minimization in a two-player game. In the following, we first show that reset-free RL can be framed as a sequence of CMDPs with an adaptive initial state sequence. Then we design a two-player game based on a primal-dual analysis of this sequence of CMDPs. Finally, we show achieving sublinear regret in this two-player game implies sublinear regret and resets in the original reset-free RL problem in (2), and we discuss how to leverage this framework to systematically design reset-free RL algorithms.
+
+Our reduction differs from standard reductions involving bounding the Nash gap with the sum of two players' regret in the literature of constrained optimization and online learning. First, we show a reduction for a sequence of saddle-point problems instead of for a single fixed saddle-point problem. There are CMDP methods (e.g. (Ghosh et al., 2022) that implicitly use the two players' regret to bound the Nash gap of a CMDP in their analysis. However, those proofs are applicable to only a single CMDP with a fixed initial state distribution. And importantly, they fundamen
+
+tally rely on Slater's condition assumption (i.e., requiring a strictly feasible policy), which does not hold for the CMDPs considered here. Additionally, unlike the typical bound for a convex-concave saddle-point problem in the optimization literature, (Wang & Li, 2020; Kovalev & Gasnikov, 2022; Boyd & Vandenberghe, 2014) the saddle-point problem of a CMDP is non-concave in terms of the policy (and convex in terms of the dual function  $\lambda$ ). In this paper, we take a different analysis to bypass the interplaying complexities due to CMDP sequences, non-concavity, and the lack of Slater's condition. The complete proofs for this section are in Appendix A.1.
+
+# 4.1. Reset-free RL as a Sequence of CMDPs
+
+The first step of our reduction is to cast the reset-free RL problem in Section 3 to a sequence of CMDP problems which share the same rewards, constraints, and dynamics, but have different initial states. Each problem instance in this sequence corresponds to an episode of the reset-free RL problem, and its constraint describes the probability of the agent entering a state that requires reset.
+
+Specifically, we denote these constrained MDPs as  $\{(S, A, P, r, H, c, s_1^k)\}_{k=1}^K$ : in episode  $k$ , the CMDP problem is defined as
+
+$$
+\max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} \left(s _ {1} ^ {k}\right), \text {s . t .} V _ {c, 1} ^ {\pi} \left(s _ {1} ^ {k}\right) \leq 0 \tag {4}
+$$
+
+where we define the cost as
+
+$$
+c _ {h} (s, a) := \mathbb {1} [ s \in \mathcal {S} _ {\text {r e s e t}} ]
+$$
+
+and  $V_{c,1}^{\pi}$ , defined similarly to (1), is the state value function with respect to the cost  $c$ . We note that the initial state,  $s_1^k$ , depends on the past behaviors of the agent, and that Assumption 3.1 ensures each CMDP in (4) is a feasible problem (i.e., there is a Markovian policy satisfying the constraint). The objective of the agent in each CMDP is to be reward-maximizing while adhering to the constraint, namely that the number of resets should be 0. Requiring zero constraint violation instead of a small constraint violation will be crucial for our reduction.
+
+Since CMDPs are typically defined without early episode termination unlike (1), with abuse of notation, we extend the definitions of  $P$ ,  $S$ ,  $r$ ,  $c$  as follows so that the CMDP definition above is consistent with the common literature. We introduce a fictitious absorbing state denoted as  $s^{\dagger}$  in  $S$ , where  $r_h(s^\dagger, a) = 0$  and  $c_h(s^\dagger, a) = 0$ ; once the agent enters  $s^{\dagger}$ , it stays there until the end of the episode. We extend the definition  $P$  such that, after the agent is in a state  $s \in S_{\text{reset}}$ , any action it takes brings it to  $s^{\dagger}$  in the next time step. In this way, we can write the value function, e.g. for reward, as  $V_{r,h}^{\pi}(s) = \mathbb{E}_{\pi}\left[\sum_{t=h}^{H} r_t(s_t, a_t) | s_h = s\right]$  in terms of this extended dynamics. We note that these two
+
+formulations are mathematically the same for the purpose of learning; when the agent enters  $s^{\dagger}$ , it means that the agent is reset in the episode.
+
+By the construction above, we can write
+
+$$
+\operatorname {R e s e t s} (K) = \sum_ {k = 1} ^ {K} V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right)
+$$
+
+which is the same as the number of total constraint violations across the CMDPs. Because we do not make any assumptions about the agent's knowledge of the constraint function (e.g., the agent does not know states  $\in S_{\text{reset}}$ ), we allow the agent to reset during learning while minimizing the total number of resets over all  $K$  episodes.
+
+# 4.2. Reduction to Two-Player Game
+
+From the problem formulation above, we see that the major difficulty of reset-free RL is the coupling between an adaptive initial state sequence and the constraint on reset probability. If we were to remove either of them, we could use standard algorithms, since the problem would become a single CMDP problem (Altman, 1999) or an episodic RL problem with varying initial states (Jin et al., 2019).
+
+We propose a reduction to systematically design algorithms for this sequence of CMDPs and therefore for reset-free RL. The main idea is to approximately solve the saddle-point problem of each CMDP in (4), i.e.,
+
+$$
+\max  _ {\pi \in \Delta} \min  _ {\lambda \geq 0} V _ {r, 1} ^ {\pi} \left(s _ {1} ^ {k}\right) - \lambda V _ {c, 1} ^ {\pi} \left(s _ {1} ^ {k}\right) \tag {5}
+$$
+
+where  $\lambda$  denotes the dual variable (i.e., the Lagrange multiplier). Each CMDP can be framed as a linear program (Hernandez-Lerma & Lasserre, 2002) whose primal and dual optimal values match (see section 8.1 in Hazan et al. (2016)). Therefore, for each CMDP,  $\max_{\pi \in \Delta} \min_{\lambda \geq 0} V_{r,1}^{\pi}(s_1^k) - \lambda V_{c,1}^{\pi}(s_1^k) = \min_{\lambda \geq 0} \max_{\pi \in \Delta} V_{r,1}^{\pi}(s_1^k) - \lambda V_{c,1}^{\pi}(s_1^k)$ .
+
+While using a primal-dual algorithm to solve for the saddlepoint of a single CMDP is known, using this approach for a sequence of CMDPs is not obvious. This is because in general, each CMDP's optimal policy (and Lagrange multiplier) can be a function of its initial state distribution (Altman, 1999). An easy way to see this is that an optimal policy satisfying a constraint on an initial state  $s_1$  may be infeasible for a constraint on another initial state  $s_1'$ , if, e.g., running this policy starting from  $s_1$  does not reach  $s_1'$  and therefore can have arbitrary behavior at  $s_1'$ . This behavior is different from that of unconstrained MDPs, where there is an optimal Markovian policy across all the initial states.
+
+Therefore, in general, a common saddle-point of Markovian polices and Lagrange multipliers does not necessarily exist for a sequence of CMDPs with varying initial states. As a
+
+result, it is unclear if there exists a primal-dual algorithm to solve this sequence, especially given that the initial states here are adaptively chosen.
+
+Existence of a Shared Saddle-Point Fortunately, we show that there is a shared saddle-point with a Markovian policy across all the CMDPs considered in (4) due to the special structure of reset-free RL. It is a proof that does not use Slater's condition for strong duality, unlike similar literature, but attains the desired property. Instead, it follows from the fact that we impose a constraint that requires zero constraint violations. In addition, we also use Assumption 3.1 and the fact that the cost  $c$  is non-negative. We formalize this below.
+
+Theorem 4.1. There exists a function  $\hat{\lambda} (\cdot)$  where for each  $s$
+
+$$
+\hat {\lambda} (s) \in \arg \min  _ {y \geq 0} \left(\max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s) - y V _ {c, 1} ^ {\pi} (s)\right),
+$$
+
+and a Markovian policy  $\pi^{*} \in \Delta$ , such that  $(\pi^{*}, \hat{\lambda})$  is a saddle-point to the CMDPs
+
+$$
+\max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}), s. t. V _ {c, 1} ^ {\pi} (s _ {1}) \leq 0
+$$
+
+for all initial states  $s_1 \in S$  such that the CMDP is feasible. That is, for all  $\pi \in \Delta$ ,  $\lambda : S \to \mathbb{R}$ , and  $s_1 \in S$ ,
+
+$$
+\begin{array}{l} V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \\ \geq V _ {r, 1} ^ {\pi} \left(s _ {1}\right) - \hat {\lambda} \left(s _ {1}\right) V _ {c, 1} ^ {\pi} \left(s _ {1}\right). \tag {6} \\ \end{array}
+$$
+
+Corollary 4.2. For  $\pi^*$  in Theorem 4.1, it holds that  $\operatorname{Regret}(K) = \sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k) - V_{r,1}^{\pi^k}(s_1^k)$ .
+
+We prove for ease of construction that the pair  $(\pi^{*},\lambda^{*})$  where  $\lambda^{*}(\cdot) = \hat{\lambda} (\cdot) + 1$  is also a saddle-point.
+
+Corollary 4.3. For any saddle-point to the CMDPs
+
+$$
+\max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}), s. t. V _ {c, 1} ^ {\pi} (s _ {1}) \leq 0
+$$
+
+of  $(\pi^{*},\hat{\lambda})$  from Theorem 4.1,  $(\pi^{*},\lambda^{*})\eqqcolon (\pi^{*},\hat{\lambda} +1)$  is also a saddle-point as defined in (6).
+
+Therefore, the pair  $(\pi^{*},\lambda^{*})$  in Corollary 4.3 is a saddlepoint to all the CMDPs the agent faces. This makes potentially designing a two-player game reduction possible. Now we give the details of our construction.
+
+Two-Player Game Our two-player game proceeds iteratively in the following manner: in episode  $k$ , a dual player determines a state value function  $\lambda^k : S \to \mathbb{R}$ , and then, a primal player determines a policy  $\pi^k$  which can depend on  $\lambda^k$ . The primal and dual players then receive losses  $\mathcal{L}^k(\pi^k, \lambda)$  and  $-\mathcal{L}^k(\pi, \lambda^k)$ , respectively, where  $\mathcal{L}^k(\pi, \lambda)$  is a Lagrangian function defined as
+
+$$
+\mathcal {L} ^ {k} (\pi , \lambda) := V _ {r, 1} ^ {\pi} \left(s _ {1} ^ {k}\right) - \lambda \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi} \left(s _ {1} ^ {k}\right). \tag {7}
+$$
+
+The regret of these two players are defined as follows.
+
+exists on the other hand.
+
+Definition 4.4. Let  $\pi_c$  and  $\lambda_c$  be comparators. The regret of the primal and the dual players are
+
+$$
+\begin{array}{l} R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi_ {c}\right) := \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi_ {c}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right), \\ R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, \lambda_ {c}\right) := \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda_ {c}\right). \\ \end{array}
+$$
+
+We present our main reduction theorem below.
+
+Theorem 4.5. Under Assumption 3.1, for any sequences  $\{\pi^k\}_{k=1}^K$  and  $\{\lambda^k\}_{k=1}^K$ , it holds that
+
+$$
+\begin{array}{l} R e g r e t (K) \leq R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, 0\right) \\ \operatorname {R e s e t s} (K) \leq R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, \lambda^ {*}\right) \\ \end{array}
+$$
+
+where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+Designing Reset-free RL Algorithms by Reduction By Theorem 4.5, if both players have sublinear regret in the two-player game, then the resulting policy sequence will have sublinear performance regret and a sublinear number of resets in the original RL problem. Minimization of regret in a two-player game and solving saddle-point problems have been very well studied (see, e.g., (Ho- Nguyen & Klinç-Karzan, 2018; Benzi et al., 2005; Hazan et al., 2016)). Therefore, our reduction gives a constructive and provably correct template for designing reset-free RL algorithms. It should be noted however, that the resultant two-player game here is slightly different from the classical full-information online learning problems. Unlike in these problems, the learning agent needs to actively explore the environment in each episode to collect information about the unknown dynamics, reward, and cost (i.e., which states are reset-free). Specifically Theorem 4.5 needs the primal (policy) player to solve an online MDP sequence (with the same transition dynamics but varying reward functions due to changing  $\lambda$ ), and the dual player to solve an online linear problem. In the next section, we will give an example algorithm under this reduction for linear MDPs.
+
+# 4.3. Proof Sketches
+
+Proof Sketch of Theorem 4.1 Let  $Q_{c}^{*}(s,a) = \min_{\pi \in \Delta}Q_{c}^{\pi}(s,a)$  and  $V_{c}^{*}(s) = \min_{\pi \in \Delta}V_{c}^{\pi}(s)$ . We define  $\pi^{*}$  in Theorem 4.1 as the optimal policy to the following MDP:  $(S,\overline{A},P,r,H)$ , where we define a state-dependent action space  $\overline{A}$  as
+
+$$
+\overline {{\mathcal {A}}} _ {s} = \{a \in \mathcal {A}: Q _ {c} ^ {*} (s, a) \leq V _ {c} ^ {*} (s) \}.
+$$
+
+By definition,  $\overline{A}_s$  is non-empty for all  $s$ .
+
+We also define a shorthand notation: we write  $\pi \in \overline{\mathcal{A}}(s)$  if  $\mathbb{E}_{\pi}[\sum_{t=1}^{H} \mathbb{1}\{a_t \notin \overline{\mathcal{A}}_{s_t}\}|s_1 = s] = 0$ . We have the following lemma, which is an application of the performance difference lemma (see Lemma 6.1 in (Kakade & Langford, 2002) and Lemma A.1 in (Cheng et al., 2021)).
+
+Lemma 4.6. For any  $s_1 \in S$  such that  $V_c^*(s_1) = 0$  and any  $\pi \in \Delta$ , it is true that  $\pi \in \overline{\mathcal{A}}(s_1)$  if and only if  $V_c^\pi(s_1) = 0$ .
+
+We prove our main claim, (6), below. Because  $V_{c,1}^{\pi^{*}}(s_{1}) = 0$  the first inequality is trivial:  $V_{r,1}^{\pi^{*}}(s_{1}) - \lambda(s_{1})V_{c,1}^{\pi^{*}}(s_{1}) = V_{r,1}^{\pi^{*}}(s_{1}) = V_{r,1}^{\pi^{*}}(s_{1}) - \hat{\lambda}(s_{1})V_{c,1}^{\pi^{*}}(s_{1})$ .
+
+To prove the second inequality, we use Lemma 4.6:
+
+$$
+\begin{array}{l} V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \\ \leq \max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \\ = \min  _ {y \geq 0} \max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}) - y V _ {c, 1} ^ {\pi} (s _ {1}) \\ = \max  _ {\pi \in \overline {{\mathcal {A} _ {c}}} (s _ {1})} V _ {r, 1} ^ {\pi} (s _ {1}) \quad (\text {B y}) \\ = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}). \\ \end{array}
+$$
+
+Proof Sketch of Theorem 4.5 We first establish the following intermediate result that will help us with our decomposition.
+
+Lemma 4.7. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}(\mathcal{L}^{k}(\pi^{*},\lambda^{\prime}) - \mathcal{L}^{k}(\pi^{k},\lambda^{k}))\leq R_{p}(\{\pi \}_{k = 1}^{K},\pi^{*})$  where  $(\pi^{*},\lambda^{\prime})$  is the saddle-point defined in either Theorem 4.1 or Corollary 4.3.
+
+Then we upper bound  $\operatorname{Regret}(K)$  and  $\operatorname{Resets}(K)$  by  $R_{p}(\{\pi^{k}\}_{k=1}^{K}, \pi_{c})$  and  $R_{d}(\{\lambda^{k}\}_{k=1}^{K}, \lambda_{c})$  for suitable comparators. This decomposition is inspired by the techniques used in Ho-Nguyen & Kılınc-Karzan (2018). We first bound Resets $(K)$ .
+
+Lemma 4.8. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}V_{c,1}^{\pi^k}(s_1^k)\leq R_p(\{\pi \}_{k = 1}^K,\pi^*) + R_d(\{\lambda \}_{k = 1}^K,\lambda^*),$  where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+Proof. Notice  $\sum_{k=1}^{K} V_{c,1}^{\pi^k}(s_1^k) = \sum_{k=1}^{K} \mathcal{L}^k(\pi^k, \hat{\lambda}) - \mathcal{L}^k(\pi^k, \lambda^*)$  where  $(\pi^*, \hat{\lambda})$  is the saddle-point defined in Theorem 4.1. By (6), and adding and subtracting  $\sum_{k=1}^{K} \mathcal{L}^k(\pi^k, \lambda^k)$ , we can bound this difference by
+
+$$
+\sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {*}, \hat {\lambda}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) + \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {*}).
+$$
+
+Using Lemma 4.7 and Definition 4.4 to upper bound the above, we get the desired result.
+
+Lastly, we bound  $\operatorname{Regret}(K)$  with the lemma below and Corollary 4.2.
+
+Lemma 4.9. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}(V_{r,1}^{\pi^{*}}(s_{1}^{k}) - V_{r,1}^{\pi^{k}}(s_{1}^{k}))\leq R_{p}(\{\pi \}_{k = 1}^{K},\pi^{*}) +$ $R_{d}(\{\lambda \}_{k = 1}^{K},0)$  , where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+Proof. Note that  $\mathcal{L}(\pi^{*},\lambda^{*}) = \mathcal{L}(\pi^{*},0)$  since  $V_{c,1}^{\pi^{*}} = 0$  for all  $k\in [K] = \{1,\dots,K\}$ . Since by definition, for any  $\pi$ ,  $\mathcal{L}^k (\pi ,0) = V_{r,1}^\pi (s_1^k)$ , we have the following:
+
+$$
+\sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi^ {*}} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {*}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, 0\right)
+$$
+
+$$
+= \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {*}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) + \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, 0\right)
+$$
+
+$$
+\leq R _ {p} \left(\left\{\pi \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda \right\} _ {k = 1} ^ {K}, 0\right)
+$$
+
+where the last inequality follows from Lemma 4.7 and Definition 4.4.  $\square$
+
+# 5. Reset-Free Learning for Linear MDP
+
+To demonstrate the utility of our reduction, we design a provably correct reset-free algorithm instantiation for linear MDP. This example serves to ground our abstract framework and to illustrate concretely how an algorithm instantiating our framework might operate. We also aim to show that despite generality, our framework does not lose any optimality in the rates of regret and total number of resets when compared to similar but specialized works.
+
+We consider a linear MDP setting, which is common in the RL theory literature (Jin et al., 2019; Amani et al., 2021; Ghosh et al., 2022).
+
+Assumption 5.1. We assume  $(S, A, P, r, c, H)$  is linear with a known feature map  $\phi : S \times A \to \mathbb{R}^d$ : for any  $h \in [H]$ , there exists  $d$  unknown signed measures  $\mu_h = \{\mu_h^1, \dots, \mu_h^d\}$  over  $S$  such that for any  $(s, a, s') \in S \times A \times S$ , we have
+
+$$
+P _ {h} \left(s ^ {\prime} \mid a\right) = \langle \phi (s, a), \mu_ {h} \left(s ^ {\prime}\right) \rangle ,
+$$
+
+and there exists unknown vectors  $\omega_{r,h},\omega_{c,h}\in \mathbb{R}^d$  such that for any  $(s,a)\in S\times A$
+
+$$
+r _ {h} (s, a) = \langle \phi (s, a), \omega_ {r, h} \rangle , \quad c _ {h} (s, a) = \langle \phi (s, a), \omega_ {c, h} \rangle .
+$$
+
+We assume, for all  $(s,a,h)\in S\times A\times [H],||\phi (s,a)||_2\leq 1$  and  $\max \{||\mu_h(s)||_2,||\omega_{r,h}||_2,||\omega_{c,h}||_2\} \leq \sqrt{d}$
+
+Note that the above assumption on the cost function does not imply knowledge of the reset states nor some hidden structure of the reset states. At the high level, it merely asks that the feature is expressive enough to separate the reset states and reset-free states in a classification problem.
+
+In addition, we make a linearity assumption on the function  $\lambda^{*}$  defined in Theorem 4.1.
+
+Assumption 5.2. We assume the knowledge of a feature  $\xi : S \to \mathbb{R}^d$  such that  $\forall s \in S, ||\xi(s)||_2 \leq 1$  and  $\lambda^*(s) = \langle \xi(s), \theta^* \rangle$  for some unknown vector  $\theta^* \in \mathbb{R}^d$ . In addition, we assume the knowledge of a convex set  $S \subseteq \mathbb{R}^d$  such that  $\theta^*, 0 \in S$  and  $\forall s \in S, ||\xi(s)||_2 \leq B$  and  $\langle \xi(s), \theta \rangle \geq 0$ .
+
+For a linear CMDP with a fixed initial state distribution, it is standard to assume that the optimal Lagrange multiplier (i.e.,  $\lambda^{*}(s_{1})$ ) is bounded and non-negative. In this regard, our assumption is a mild and natural linear extension of this boundedness assumption for varying initial states, and is true when the feature map is expressive enough.
+
+# 5.1. Algorithm
+
+The basis of our algorithm lies between the interaction between the primal and dual players. We adopt the common no-regret plus best-response approach (Wang et al., 2021) to the no-regret two-player game in designing algorithms for these two players. We let the dual player perform (no-regret) projected gradient descent and the primal player update policies based on upper confidence bound (UCB) with knowledge of the dual player's decision (i.e., the best-response scheme to the dual player).
+
+As discussed above, the environment is unknown to the agent. Therefore, as discussed in Section 4, we need to modify the actions of the primal player as compared to in the generic saddle-point problem to instantiate our framework. We do this using UCB, where the agent views unknown actions/states in a more positive ("optimistic") way than the ones that have been observed to drive exploration in unknown environments.
+
+Specifically, in each episode, upon receiving the initial state, we execute actions according to the policy based on a softmax (lines 5-8). Then, we perform the dual update through projected gradient descent. The dual player plays for the next round,  $k + 1$ , after observing its loss after the primal player plays for the current round,  $k$ . The projection is to a  $l_{2}$  ball containing  $\lambda^{*}(\cdot)$  (lines 9-11). Finally, we perform the update of the primal player by computing the  $Q$ -functions for both the reward and cost with a bonus to encourage exploration (lines 12-20).
+
+This algorithm uses Ghosh et al. (2022) as the baseline for the primal player. However, we emphasize that our algorithm can handle the adaptive initial state sequence that is seen in reset-free RL Theorems 4.1 and 4.5.
+
+# 5.2. Analysis
+
+We show below that our algorithm achieves regret and number of resets that are sublinear in the total number of time steps,  $KH$ , using Theorem 4.5. This result is asymptotically equivalent to Ghosh et al. (2022) and comparable to the bounds of  $\tilde{O}(\sqrt{d^2H^6K})$  from Ding et al. (2021). Therefore, we do not make sacrifices in terms of the regret and total number of resets when specializing our abstract framework.
+
+Theorem 5.3. Under Assumptions 3.1, 5.1, and 5.2, with
+
+high probability, Regret  $(K)\leq \tilde{O} ((B + 1)\sqrt{d^3H^4K})$  and Resets  $(K)\leq \tilde{O} ((B + 1)\sqrt{d^3H^4K})$
+
+Proof Sketch of Theorem 5.3 We provide a proof sketch here and defer the complete proof to Appendix A.2. We first bound the regret of  $\{\pi^k\}_{k=1}^K$  and  $\{\lambda^k\}_{k=1}^K$  and then use this to prove the bounds on our algorithm's regret and number of resets with Theorem 4.5.
+
+We first bound the regret of  $\{\lambda^k\}_{k = 1}^K$
+
+Lemma 5.4. Consider  $\lambda_c(s) = \langle \xi(s), \theta_c \rangle$  for some  $\theta_c \in \mathcal{U}$ . Then it holds that  $R_d(\{\lambda^k\}_{k=1}^K, \lambda_c) \leq 1.5B\sqrt{K} + \sum_{k=1}^{K} (\lambda^k (s_1^k) - \lambda_c (s_1^k)) (V_{c,1}^k (s_1^k) - V_{c,1}^{\pi^k} (s_1^k))$ .
+
+Proof. We notice first an equality.
+
+$$
+\begin{array}{l} R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, \lambda_ {c}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda_ {c}\right) \\ = \sum_ {k = 1} ^ {K} \lambda_ {c} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \\ = \sum_ {k = 1} ^ {K} \left(\lambda^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right)\right) \left(- V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right)\right) \\ + \sum_ {k = 1} ^ {K} \left(\lambda^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right)\right) \left(V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right)\right). \\ \end{array}
+$$
+
+We observe that the first term is an online linear problem for  $\theta^k$  (the parameter of  $\lambda^k (\cdot)$ ). In episode  $k\in [K]$ ,  $\lambda^k$  is played, and then the loss is revealed. Since the space of  $\theta^k$  is convex, we use standard results (see Lemma 3.1 in Hazan et al. (2016)) to show that updating  $\theta^k$  through projected gradient descent results in an upper bound for  $\sum_{k = 1}^{K}(\lambda^{k}(s_{1}^{k}) - \lambda_{c}(s_{1}^{k}))(-V_{c,1}^{k}(s_{1}^{k}))$ .
+
+We now bound the regret of  $\{\pi\}_{k=1}^{K}$ .
+
+Lemma 5.5. Consider any  $\pi_c$ . With high probability,  $R_p(\{\pi\}_{k=1}^K, \pi_c) \leq 2H(1 + B + H) + \sum_{k=1}^K V_{r,1}^k(s_1^k) - V_{r,1}^{\pi^k}(s_1^k) + \lambda^k(s_1^k)(V_{c,1}^{\pi^k}(s_1^k) - V_{c,1}^k(s_1^k))$ .
+
+Proof. First we expand the regret into two terms.
+
+$$
+\begin{array}{l} R _ {p} \left(\left\{\pi \right\} _ {k = 1} ^ {K}, \pi_ {c}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi_ {c}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) \\ = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \left[ V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \right] \\ = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \left[ V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) \right] \\ + \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {k} (s _ {1} ^ {k}) - V _ {r, 1} ^ {\pi^ {k}} (s _ {1} ^ {k}) + \lambda^ {k} (s _ {1} ^ {k}) \bigl (V _ {c, 1} ^ {\pi^ {k}} (s _ {1} ^ {k}) - V _ {c, 1} ^ {k} (s _ {1} ^ {k}) \bigr). \\ \end{array}
+$$
+
+To bound the first term, we use Lemma 3 from Ghosh et al. (2022), which characterizes the property of upper confidence bound.
+
+Lastly, we derive a bound on  $R_{d}(\{\lambda^{k}\}_{k = 1}^{K},\lambda_{c}) + R_{p}(\{\pi^{k}\}_{k = 1}^{K},\pi_{c})$ , which directly implies our final upper bound on  $\operatorname{Regret}(K)$  and  $\operatorname{Resets}(K)$  in Theorem 5.3 by Theorem 4.5. Combining the upper bounds in Lemma 5.4 and Lemma 5.5, we have a high-probability upper bound of
+
+$$
+\begin{array}{l} R _ {d} (\{\lambda^ {k} \} _ {k = 1} ^ {K}, \lambda_ {c}) + R _ {p} (\{\pi^ {k} \} _ {k = 1} ^ {K}, \pi_ {c}) \\ \leq 1. 5 B \sqrt {K} + 2 H (1 + B + H) + \\ + \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {k} (s _ {1} ^ {k}) - V _ {r, 1} ^ {\pi^ {k}} (s _ {1} ^ {k}) + \lambda_ {c} (s _ {1} ^ {k}) (V _ {c, 1} ^ {\pi^ {k}} (s _ {1} ^ {k}) - V _ {c, 1} ^ {k} (s _ {1} ^ {k})) \\ \end{array}
+$$
+
+Algorithm 1 Primal-Dual Reset-Free RL Algorithm for Linear MDP with Adaptive Initial States  
+1: Input: Feature maps \(\phi\) and \(\xi\). Failure probability \(p\). Some universal constant \(\mathcal{C}\).  
+2: Initialization: \(\theta^1 = 0\), \(w_{r,h} = 0\), \(w_{c,h} = 0\), \(\alpha = \frac{\log(|A|)K}{2(1 + B + H)}\), \(\beta = \mathcal{C}dH\sqrt{\log(4\log|\mathcal{A}|dKH / p)}\)  
+3: for episodes \(k = 1, \ldots, K\) do  
+4: Observe the initial state \(s_1^k\).  
+5: for step \(h = 1, \ldots, H\) do  
+6: Compute \(\pi_{h,k}(a|\cdot) \gets \frac{\exp(\alpha(Q_{r,h}^k(\cdot,a) - \lambda^k(s_1^k)Q_{c,h}^k(\cdot,a)))}{\sum_a\exp(\alpha(Q_{r,h}^k(\cdot,a) - \lambda^k(s_1^k)Q_{c,h}^k(\cdot,a)))}\).  
+7: Take action \(a_h^k \sim \pi_{h,k}(\cdot | s_h^k)\) and observe \(s_{h+1}^k\).  
+8: end for  
+9: \(\eta_k \gets B / \sqrt{k}\)  
+10: Update \(\theta^{k+1} \gets \mathrm{Proj}_{\mathcal{U}}(\theta^k + \eta_k \cdot \xi(s_1^k)V_{c,1}^k(s_1^k))\)  
+11: \(\lambda^{k+1}(\cdot) \gets \langle \theta^{k+1}, \xi(\cdot) \rangle\)  
+12: for step \(h = H, \ldots, 1\) do  
+13: \(\Lambda_h^{k+1} \gets \sum_{i=1}^{k} \phi(s_h^i, a_h^i) \phi(s_h^i, a_h^i)^T + \lambda \mathbb{I}\).  
+14: \(w_{r,h}^{k+1} \gets (\Lambda_h^{k+1})^{-1}[\sum_{i=1}^{k} \phi(s_h^i, a_h^i)[r_h(s_h^i, a_h^i) + V_{r,h+1}^{k+1}(s_h^{i+1})]]\)  
+15: \(w_{c,h}^{k+1} \gets (\Lambda_h^{k+1})^{-1}[\sum_{i=1}^{k} \phi(s_h^i, a_h^i)[c_h(s_h^i, a_h^i) + V_{c,h+1}^{k+1}(s_h^{i+1})]]\)  
+16: \(Q_{r,h}^{k+1}(\cdot,\cdot) \gets \max\{\min\{\langle w_{r,h}^{k+1}, \phi(\cdot,\cdot) \rangle + \beta(\phi(\cdot,\cdot)^T(\Lambda_h^{k+1})^{-1}\phi(\cdot,\cdot)\)^{-1}\), \(H-h+1\}, 0\}\)  
+17: \(Q_{c,h}^{k+1}(\cdot,\cdot) \gets \max\{\min\{\langle w_{c,h}^{k+1}, \phi(\cdot,\cdot) \rangle - \beta(\phi(\cdot,\cdot)^T(\Lambda_h^{k+1})^{-1}\phi(\cdot,\cdot)\)^{-1}\), \(H-h+1\}, 0\}\)  
+18: \(V_{r,h}^{k+1}(\cdot) = \sum_a \pi_{h,k}(a|\cdot) Q_{r,h}^{k+1}(\cdot,a)\)  
+19: \(V_{c,h}^{k+1}(\cdot) = \sum_a \pi_{h,k}(a|\cdot) Q_{c,h}^{k+1}(\cdot,a)\)  
+20: end for  
+21: end for
+
+where the last term is the overestimation error due to optimism. Note that for all  $k \in [K]$ ,  $V_{r,1}^{k}(s_{1}^{k})$  and  $V_{c,1}^{k}(s_{1}^{k})$  are as defined in Algorithm 1 and are optimistic estimates of  $V_{r,1}^{\pi^{*}}(s_{1}^{k})$  and  $V_{c,1}^{\pi^{*}}(s_{1}^{k})$ . To bound this term, we use Lemma 4 from (Ghosh et al., 2022).
+
+# 5.3. Other Possible Instantiations
+
+We demonstrated above that our general framework can be instantiated to achieve sublinear regret and total number of resets. Importantly, our algorithm serves as an example of how our general framework can be used to systematically design new algorithms for reset-free RL. We can leverage the multitude of existing algorithms that aim to minimize regret in a two-player game. An example of a different strategy is using a no-regret algorithm like optimistic mirror descent (Mertikopoulos et al., 2018) for the dual player. We can also replace UCB for the primal player with an online MDP no-regret algorithm such as a variation of PO-LITEX (Abbasi-Yadkori et al., 2019). Further studying different combinations of baseline algorithms is an interesting future research direction, which perhaps could even improve existing algorithms for more specialized settings.
+
+# 6. Conclusion
+
+We propose a generic no-regret reduction for designing provable reset-free RL algorithms. Our reduction casts reset-free RL into the regret minimization problem of a two-player game, for which many existing no-regret algorithms are available. As a result, we can reuse these techniques, and future better techniques, to systematically build new reset-free RL algorithms. In particular, we design a reset-free RL algorithm for linear MDPs using our new reduction techniques, taking the first step towards designing provable reset-free RL algorithms. Extending these techniques to nonlinear function approximators and verifying their effectiveness empirically are important future research directions.
+
+# Acknowledgements
+
+Part of this work was done during Hoai-An Nguyen's internship at Microsoft Research.
+
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+
+# A. Appendix
+
+# A.1. Missing Proofs for Section 4
+
+# A.1.1. PROOF OF THEOREM 4.1
+
+Theorem 4.1. There exists a function  $\hat{\lambda} (\cdot)$  where for each  $s$
+
+$$
+\hat {\lambda} (s) \in \arg \min  _ {y \geq 0} \left(\max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s) - y V _ {c, 1} ^ {\pi} (s)\right),
+$$
+
+and a Markovian policy  $\pi^{*}\in \Delta$ , such that  $(\pi^{*},\hat{\lambda})$  is a saddle-point to the CMDPs
+
+$$
+\max _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}), s. t. V _ {c, 1} ^ {\pi} (s _ {1}) \leq 0
+$$
+
+for all initial states  $s_1 \in S$  such that the CMDP is feasible. That is, for all  $\pi \in \Delta$ ,  $\lambda : \mathcal{S} \to \mathbb{R}$ , and  $s_1 \in S$ ,
+
+$$
+\begin{array}{l} V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \\ \geq V _ {r, 1} ^ {\pi} \left(s _ {1}\right) - \hat {\lambda} \left(s _ {1}\right) V _ {c, 1} ^ {\pi} \left(s _ {1}\right). \tag {6} \\ \end{array}
+$$
+
+For policy  $\pi^{*}$ , we define it by the following construction (we ignore writing out the time dependency for simplicity): first, we define a cost-based MDP  $\mathcal{M}_c = (\mathcal{S},\mathcal{A},P,c,H)$ . Let  $Q_{c}^{*}(s,a) = \min_{\pi \in \Delta}Q_{c}^{\pi}(s,a)$  and  $V_{c}^{*}(s) = \min_{\pi \in \Delta}V_{c}^{\pi}(s)$  be the optimal values, where we recall  $V_{c}^{\pi}$  and  $Q_{c}^{\pi}$  are the state and state-action values under policy  $\pi$  with respect to the cost. Now we construct another reward-based MDP  $\overline{\mathcal{M}} = (\mathcal{S},\overline{\mathcal{A}},P,r,H)$ , where we define the state-dependent action space  $\overline{\mathcal{A}}$  as
+
+$$
+\overline {{\mathcal {A}}} _ {s} = \left\{a \in \mathcal {A}: Q _ {c} ^ {*} (s, a) \leq V _ {c} ^ {*} (s) \right\}.
+$$
+
+By definition,  $\overline{\mathcal{A}}_s$  is non-empty for all  $s$ . We define a shorthand notation: we write  $\pi \in \overline{\mathcal{A}}(s)$  if  $\mathbb{E}_{\pi}[\sum_{t=1}^{H} \mathbb{1}\{a_t \notin \overline{\mathcal{A}}_{s_t}\}|s_1 = s] = 0$ . Then we have the following lemma, which is a straightforward application of the performance difference lemma.
+
+Lemma 4.6. For any  $s_1 \in \mathcal{S}$  such that  $V_c^*(s_1) = 0$  and any  $\pi \in \Delta$ , it is true that  $\pi \in \overline{\mathcal{A}}(s_1)$  if and only if  $V_c^\pi(s_1) = 0$ .
+
+Proof. By performance difference lemma (Kakade & Langford, 2002), we can write
+
+$$
+V _ {c} ^ {\pi} (s _ {1}) - V _ {c} ^ {*} (s _ {1}) = \mathbb {E} _ {\pi} \left[ \sum_ {t = 1} ^ {H} Q _ {c} ^ {*} (s _ {t}, a _ {t}) - V _ {c} ^ {*} (s _ {t}) | s _ {1} = s _ {1} \right].
+$$
+
+If for some  $s_1 \in S$ ,  $\pi \in \overline{\mathcal{A}}(s_1)$ , then  $\mathbb{E}_{\pi}\left[\sum_{t=1}^{H} Q_c^*(s_t, a_t) - V_c^*(s_t)\right] \leq 0$ , which implies  $V_c^\pi(s_1) \leq V_c^*(s_1)$ . But since  $V_c^*$  is optimal,  $V_c^\pi(s_1) = V_c^*(s_1)$ . On the other hand, suppose  $V_c^\pi(s_1) = 0$ . It implies  $\mathbb{E}_{\pi}\left[\sum_{t=1}^{H} Q_c^*(s_t, a_t) - V_c^*(s_t)\right] = 0$  since  $V_c^*(s_1) = 0$ . Because by definition of optimality  $Q_c^*(s_t, a_t) - V_c^*(s_t) \geq 0$ , this implies  $\pi \in \overline{\mathcal{A}}(s_1)$ .
+
+We set our candidate policy  $\pi^{*}$  as the optimal policy of this  $\overline{\mathcal{M}}$ . By Lemma 4.6, we have  $V_{c}^{\pi^{*}}(s) = V_{c}^{*}(s)$ , so it is also an optimal policy to  $\mathcal{M}_c$ . We prove our main claim of Theorem 4.1 below:
+
+$$
+V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}).
+$$
+
+Proof. Because  $V_{c,1}^{\pi^{*}}(s_{1}) = 0$  (for an initial state  $s_1$  such that the CMDP is feasible), the first inequality is trivial:
+
+$$
+V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}).
+$$
+
+For the second inequality, we use Lemma 4.6:
+
+$$
+\begin{array}{l} V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \leq \max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \\ = \min  _ {y \geq 0} \max  _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}) - y V _ {c, 1} ^ {\pi} (s _ {1}) \\ = \max  _ {\pi \in \overline {{\mathcal {A} _ {c}}} (s _ {1})} V _ {r, 1} ^ {\pi} (s _ {1}) \quad (\text {B y}) \\ = V _ {r, 1} ^ {\pi^ {*}} \left(s _ {1}\right) \\ = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}). \\ \end{array}
+$$
+
+![](images/cbba8bcd1666e58a88d8dc9b7fc4a4a89c13d59ffc40522a721659a6f0ecbaff.jpg)
+
+# A.1.2. PROOF OF COROLLARY 4.2
+
+Corollary 4.2. For  $\pi^*$  in Theorem 4.1, it holds that  $\operatorname{Regret}(K) = \sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k) - V_{r,1}^{\pi^k}(s_1^k)$ .
+
+Proof. To prove  $\operatorname{Regret}(K) = \sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k) - V_{r,1}^{\pi^k}(s_1^k)$ , it suffices to prove  $\sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k) = \max_{\pi \in \Delta_0(K)} \sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k)$ . By Lemma 4.6 and under Assumption 3.1, we notice that  $\max_{\pi \in \Delta_0(K)} \sum_{k=1}^{K} V_{r,1}^{\pi}(s_1^k) = \max_{\pi \in \overline{\mathcal{A}}(s_1^k), \forall k \in [K]} \sum_{k=1}^{K} V_{r,1}^{\pi}(s_1^k)$ . This is equal to  $\sum_{k=1}^{K} V_{r,1}^{\pi^*}(s_1^k)$  by the definition of  $\pi^*$  in the proof of Theorem 4.1.
+
+# A.1.3. PROOF OF COROLLARY 4.3
+
+Corollary 4.3. For any saddle-point to the CMDPs
+
+$$
+\max _ {\pi \in \Delta} V _ {r, 1} ^ {\pi} (s _ {1}), s. t. V _ {c, 1} ^ {\pi} (s _ {1}) \leq 0
+$$
+
+of  $(\pi^{*},\hat{\lambda})$  from Theorem 4.1,  $(\pi^{*},\lambda^{*})\eqqcolon (\pi^{*},\hat{\lambda} +1)$  is also a saddle-point as defined in (6).
+
+Proof. We prove that eq (6) holds for  $(\pi^{*},\lambda^{*})$ , that is
+
+$$
+V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda^ {*} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \geq V _ {r, 1} ^ {\pi} (s _ {1}) - \lambda^ {*} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}).
+$$
+
+Because  $V_{c,1}^{\pi^{*}}(s_{1}) = 0$  (for an initial state  $s_1$  such that the CMDP is feasible), the first inequality is trivial:
+
+$$
+V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda^ {*} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}).
+$$
+
+For the second inequality, we use Theorem 4.1:
+
+$$
+\begin{array}{l} V _ {r, 1} ^ {\pi} (s _ {1}) - \lambda^ {*} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \leq V _ {r, 1} ^ {\pi} (s _ {1}) - \hat {\lambda} (s _ {1}) V _ {c, 1} ^ {\pi} (s _ {1}) \\ \leq V _ {r, 1} ^ {\pi^ {*}} \left(s _ {1}\right) - \hat {\lambda} \left(s _ {1}\right) V _ {c, 1} ^ {\pi^ {*}} \left(s _ {1}\right) \\ = V _ {r, 1} ^ {\pi^ {*}} (s _ {1}) - \lambda^ {*} (s _ {1}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1}) \\ \end{array}
+$$
+
+where the first step is because  $V_{c,1}^{\pi}(s_1)$  by definition is in [0, 1] and  $\lambda^{*} = \hat{\lambda} + 1$ , and the second step is by Theorem 4.1.
+
+# A.1.4. PROOF OF THEOREM 4.5
+
+Theorem 4.5. Under Assumption 3.1, for any sequences  $\{\pi^k\}_{k=1}^K$  and  $\{\lambda^k\}_{k=1}^K$ , it holds that
+
+$$
+\operatorname {R e g r e t} (K) \leq R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, 0\right)
+$$
+
+$$
+\operatorname {R e s e t s} (K) \leq R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, \lambda^ {*}\right)
+$$
+
+where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+We first establish the following intermediate result that will help us with our decomposition.
+
+Lemma 4.7. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}(\mathcal{L}^{k}(\pi^{*},\lambda^{\prime}) - \mathcal{L}^{k}(\pi^{k},\lambda^{k}))\leq R_{p}(\{\pi \}_{k = 1}^{K},\pi^{*})$  where  $(\pi^{*},\lambda^{\prime})$  is the saddle-point defined in either Theorem 4.1 or Corollary 4.3.
+
+Proof. We derive this lemma by Theorem 4.1 and Corollary 4.3. First notice by Theorem 4.1 and Corollary 4.3 that for  $\lambda' = \lambda^*, \hat{\lambda}$ ,
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {*}, \lambda^ {\prime}) = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi^ {*}} (s _ {1} ^ {k}) - \lambda^ {\prime} (s _ {1} ^ {k}) V _ {c, 1} ^ {\pi^ {*}} (s _ {1} ^ {k}) \\ \leq \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi^ {*}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {*}} \left(s _ {1} ^ {k}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {k}\right). \\ \end{array}
+$$
+
+Then we can derive
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} \left(\mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {\prime}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right)\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {\prime}\right) - \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {k}\right) + \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) \\ \leq \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) = R _ {p} \left(\left\{\pi \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) \\ \end{array}
+$$
+
+which finishes the proof.
+
+![](images/057147c4ae3e0de4c190c47e4e1a35981434aa843a0086728a5a55319d36754c.jpg)
+
+Then we upper bound  $\operatorname{Regret}(K)$  and  $\operatorname{Resets}(K)$  by  $R_{p}(\{\pi^{k}\}_{k=1}^{K}, \pi_{c})$  and  $R_{d}(\{\lambda^{k}\}_{k=1}^{K}, \lambda_{c})$  for suitable comparators. This decomposition is inspired by the techniques used in Ho- Nguyen & Klinç-Karzan (2018).
+
+We first bound  $\operatorname{Resets}(K)$ .
+
+Lemma 4.8. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}V_{c,1}^{\pi^k}(s_1^k)\leq R_p(\{\pi \}_{k = 1}^K,\pi^*) + R_d(\{\lambda \}_{k = 1}^K,\lambda^*)$ , where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+Proof. Notice  $\sum_{k=1}^{K} V_{c,1}^{\pi^k}(s_1^k) = \sum_{k=1}^{K} \mathcal{L}^k(\pi^k, \hat{\lambda}) - \mathcal{L}^k(\pi^k, \lambda^*)$  where  $(\pi^*, \hat{\lambda})$  is the saddle-point defined in Theorem 4.1. This is because, as defined,  $\lambda^* = \hat{\lambda} + 1$ . Therefore, we bound the RHS. We have
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {k}, \hat {\lambda}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {*}) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {k}, \hat {\lambda}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) + \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {*}) \\ \leq \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {*}, \hat {\lambda}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) + \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {*}) \\ \leq R _ {p} \left(\left\{\pi \right\} _ {k = 1} ^ {K}, \pi^ {*}\right) + R _ {d} \left(\left\{\lambda \right\} _ {k = 1} ^ {K}, \lambda^ {*}\right) \\ \end{array}
+$$
+
+where second inequality is because  $\sum_{k=1}^{K} \mathcal{L}^k(\pi^*, \hat{\lambda}) \geq \sum_{k=1}^{K} \mathcal{L}^k(\pi^k, \hat{\lambda})$  by Theorem 4.1, and the first inequality follows from Lemma 4.7 and Definition 4.4.
+
+Lastly, we bound  $\operatorname{Regret}(K)$  with the lemma below and Corollary 4.2.
+
+Lemma 4.9. For any primal-dual sequence  $\{\pi^k,\lambda^k\}_{k = 1}^K$ $\sum_{k = 1}^{K}(V_{r,1}^{\pi^*}(s_1^k) - V_{r,1}^{\pi^k}(s_1^k))\leq R_p(\{\pi \}_{k = 1}^K,\pi^*) + R_d(\{\lambda \}_{k = 1}^K,0)$ , where  $(\pi^{*},\lambda^{*})$  is the saddle-point defined in Corollary 4.3.
+
+Proof. Note that  $\mathcal{L}(\pi^{*},\lambda^{*}) = \mathcal{L}(\pi^{*},0)$  since  $V_{c,1}^{\pi^{*}}(s_{1}^{k}) = 0$  for all  $k\in [K]$ . Since by definition, for any  $\pi$ ,  $\mathcal{L}^k (\pi ,0) = V_{r,1}^{\pi}(s_1^k)$ , we have the following:
+
+$$
+\begin{array}{l} \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi^ {*}} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {*}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, 0\right) \\ = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi^ {*}, \lambda^ {*}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) + \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, 0\right) \\ \leq R _ {p} (\{\pi \} _ {k = 1} ^ {K}, \pi^ {*}) + R _ {d} (\{\lambda \} _ {k = 1} ^ {K}, 0) \\ \end{array}
+$$
+
+where the last inequality follows from Lemma 4.7 and Definition 4.4.
+
+![](images/3ff55d330dd92706ac8efc8b1b3ad2a0512a65314fc19db3cb98c8126fbec5a6.jpg)
+
+# A.2. Missing Proofs for Section 5
+
+# A.2.1. PROOF OF THEOREM 5.3
+
+Theorem 5.3. Under Assumptions 3.1, 5.1, and 5.2, with high probability,  $\text{Regret}(K) \leq \tilde{O}((B + 1)\sqrt{d^3H^4K})$  and  $\text{Resets}(K) \leq \tilde{O}((B + 1)\sqrt{d^3H^4K})$ .
+
+We first bound the regret of  $\{\pi^k\}_{k=1}^K$  and  $\{\lambda^k\}_{k=1}^K$ , and then use this to prove the bounds on our algorithm's regret and number of resets with Theorem 4.5.
+
+We first bound the regret of  $\{\lambda^k\}_{k = 1}^K$
+
+Lemma 5.4. Consider  $\lambda_c(s) = \langle \xi(s), \theta_c \rangle$  for some  $\theta_c \in \mathcal{U}$ . Then it holds that  $R_d(\{\lambda^k\}_{k=1}^K, \lambda_c) \leq 1.5B\sqrt{K} + \sum_{k=1}^{K} (\lambda^k(s_1^k) - \lambda_c(s_1^k))(V_{c,1}^k(s_1^k) - V_{c,1}^{\pi^k}(s_1^k))$ .
+
+Proof. We notice first an equality.
+
+$$
+\begin{array}{l} R _ {d} (\{\lambda^ {k} \} _ {k = 1} ^ {K}, \lambda_ {c}) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} (\pi^ {k}, \lambda^ {k}) - \mathcal {L} ^ {k} (\pi^ {k}, \lambda_ {c}) \\ = \sum_ {k = 1} ^ {K} \lambda_ {c} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \\ = \sum_ {k = 1} ^ {K} \lambda_ {c} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \\ + \sum_ {k = 1} ^ {K} \lambda_ {c} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) + \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) \\ = \sum_ {k = 1} ^ {K} \left(\lambda^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right)\right) \left(- V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right)\right) + \sum_ {k = 1} ^ {K} \left(\lambda^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right)\right) \left(V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right)\right). \\ \end{array}
+$$
+
+We observe that the first term is an online linear problem for  $\theta^k$  (the parameter of  $\lambda^k(\cdot)$ ). In episode  $k \in [K]$ ,  $\lambda^k$  is played, and then the loss is revealed. Since the space of  $\theta^k$  is convex, we use standard results (Lemma 3.1 (Hazan et al., 2016)) to show that updating  $\theta^k$  through projected gradient descent results in an upper bound for  $\sum_{k=1}^{K} (\lambda^k(s_1^k) - \lambda_c(s_1^k)) (-V_{c,1}^k(s_1^k))$ . We restate the lemma here.
+
+Lemma A.1 (Lemma 3.1 (Hazan et al., 2016)). Let  $\mathcal{S} \subseteq \mathbb{R}^d$  be a bounded convex and closed set in Euclidean space. Denote  $D$  as an upper bound on the diameter of  $\mathcal{S}$ , and  $G$  as an upper bound on the norm of the subgradients of convex cost functions  $f_k$  over  $\mathcal{S}$ . Using online projected gradient descent to generate sequence  $\{x_k\}_{k=1}^K$  with step sizes  $\{\eta_k = \frac{D}{G\sqrt{k}}, k \in [K]\}$  guarantees, for all  $K \geq 1$ :
+
+$$
+R e g r e t _ {K} = \max  _ {x ^ {*} \in \mathcal {K}} \sum_ {k = 1} ^ {K} f _ {k} \left(x ^ {k}\right) - f _ {k} \left(x ^ {*}\right) \leq 1. 5 G D \sqrt {K}.
+$$
+
+Let us bound  $D$ . By Assumption 5.2,  $\lambda^{*} = \langle \xi (s),\theta^{*}\rangle$  and  $||\theta^{*}||_{2}\leq B$ . Since the comparator we use is  $\lambda^{*}$ , we can set  $D$  to be  $B$ . To bound  $G$ , we observe that the subgradient of our loss function is  $\xi (s)V_{c,1}^{k}(s_{1}^{k})$  for each  $k\in [K]$ . Therefore, since  $V_{c,1}^{k}(s_{1}^{k})\in [0,1]$  and  $||\xi (s)||_2\leq 1$  by Assumption 5.2, we can set  $G$  to be 1.
+
+We now bound the regret of  $\{\pi\}_{k=1}^{K}$ .
+
+Lemma 5.5. Consider any  $\pi_c$ . With high probability,  $R_p(\{\pi\}_{k=1}^K, \pi_c) \leq 2H(1 + B + H) + \sum_{k=1}^K V_{r,1}^k(s_1^k) - V_{r,1}^{\pi^k}(s_1^k) + \lambda^k(s_1^k)(V_{c,1}^{\pi^k}(s_1^k) - V_{c,1}^k(s_1^k))$ .
+
+Proof. First we expand the regret into two terms.
+
+$$
+\begin{array}{l} R _ {p} \left(\left\{\pi \right\} _ {k = 1} ^ {K}, \pi_ {c}\right) = \sum_ {k = 1} ^ {K} \mathcal {L} ^ {k} \left(\pi_ {c}, \lambda^ {k}\right) - \mathcal {L} ^ {k} \left(\pi^ {k}, \lambda^ {k}\right) \\ = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \left[ V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \right] \\ = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \left[ V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) \right] \\ + \sum_ {k = 1} ^ {K} \left[ V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) \right] - \left[ V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) \right] \\ = \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {\pi_ {c}} \left(s _ {1} ^ {k}\right) - \left[ V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - \lambda^ {k} \left(s _ {1} ^ {k}\right) V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) \right] \\ + \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) + \lambda^ {k} \left(s _ {1} ^ {k}\right) \left(V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right)\right). \\ \end{array}
+$$
+
+To bound the first term, we use Lemma 3 from Ghosh et al. (2022), which characterizes the property of upper confidence bound. For completeness, we re-write the lemma here.
+
+Lemma A.2 (Lemma 3 (Ghosh et al., 2022)). With probability  $1 - p / 2$ , it holds that  $\mathcal{T}_1 = \sum_{k=1}^{K}\left(V_{r,1}^{\pi_c}(s_1^k) - \lambda^k V_{c,1}^{\pi_c}(s_1^k)\right) - \left(V_{r,1}^k(s_1^k) - \lambda^k V_{c,1}^k(s_1^k)\right) \leq KH\log(|\mathcal{A}|)/\alpha$ . Hence, for  $\alpha = \frac{\log(|\mathcal{A}|)K}{2(1 + C + H)}$ ,  $\mathcal{T}_1 \leq 2H(1 + C + H)$ , where  $C$  is such that  $\lambda^k \leq C$ .
+
+In our problem setting, we can set  $C = B$  in the lemma above. Therefore, the first term is bounded by  $2H(1 + B + H)$ .
+
+Lastly, we derive a bound on  $R_{d}(\{\lambda^{k}\}_{k = 1}^{K},\lambda_{c}) + R_{p}(\{\pi^{k}\}_{k = 1}^{K},\pi_{c})$ , which directly implies our final upper bound on  $\operatorname{Regret}(K)$  and  $\operatorname{Resets}(K)$  in Theorem 5.3 by Theorem 4.5.
+
+Lemma A.3. For any  $\pi_c$  and  $\lambda_c(s) = \langle \xi(s), \theta_c \rangle$  such that  $\|\theta_c\| \leq B$ , we have with probability  $1 - p$ ,  $R_d(\{\lambda^k\}_{k=1}^K, \lambda_c) + R_p(\{\pi^k\}_{k=1}^K, \pi_c) \leq 1.5B\sqrt{K} + 2H(1 + B + H) + O((B + 1)\sqrt{d^3H^4K\iota^2})$  where  $\iota = \log[\log(|\mathcal{A}|)4dKH/p]$ .
+
+Proof. Combining the upper bounds in Lemma 5.4 and Lemma 5.5, we have an upper bound of
+
+$$
+\begin{array}{l} R _ {d} \left(\left\{\lambda^ {k} \right\} _ {k = 1} ^ {K}, \lambda_ {c}\right) + R _ {p} \left(\left\{\pi^ {k} \right\} _ {k = 1} ^ {K}, \pi_ {c}\right) = 1. 5 B \sqrt {K} + \sum_ {k = 1} ^ {K} \left(\lambda^ {k} \left(s _ {1} ^ {k}\right) - \lambda_ {c} \left(s _ {1} ^ {k}\right)\right) \left(V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right)\right) \\ + 2 H (1 + B + H) + \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) + \lambda^ {k} \left(s _ {1} ^ {k}\right) \left(V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right)\right) \\ = 1. 5 B \sqrt {K} + 2 H (1 + B + H) + \\ + \sum_ {k = 1} ^ {K} V _ {r, 1} ^ {k} \left(s _ {1} ^ {k}\right) - V _ {r, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) + \lambda_ {c} \left(s _ {1} ^ {k}\right) \left(V _ {c, 1} ^ {\pi^ {k}} \left(s _ {1} ^ {k}\right) - V _ {c, 1} ^ {k} \left(s _ {1} ^ {k}\right)\right) \\ \end{array}
+$$
+
+where the last term is the overestimation error due to optimism. To bound this term, we use Lemma 4 from Ghosh et al. (2022). We re-write the lemma here.
+
+Lemma A.4 (Lemma 4 (Ghosh et al., 2022)). With probability at least  $1 - p / 2$ , for any  $\lambda \in [0,C]$ ,  $\sum_{k = 1}^{K}\left(V_{r,1}^{k}(s_{1}^{k}) - V_{r,1}^{\pi^{k}}(s_{1}^{k})\right) + \lambda \sum_{k = 1}^{K}\left(V_{c,1}^{\pi^{k}}(s_{1}^{k}) - V_{c,1}^{k}(s_{1}^{k})\right) \leq O((\lambda + 1)\sqrt{d^{3}H^{4}K\iota^{2}})$  where  $\iota = \log [\log (|\mathcal{A}|)4dKH / p]$ .
+
+Since we have a bound on all  $\lambda_c(s_1^k)$  of  $B$  for all  $k\in [K]$ , we have a bound of  $O((B + 1)\sqrt{d^3H^4Kt^2})$ .

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