In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $\Omega(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal $O(\sqrt{T(1+P_T)})$ dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from $O(\log T)$ to $1$. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.

Nonstationarity is prevalent in sequential decision making, which poses a critical challenge to the vast majority of existing approaches developed offline.

InIRsystems, theretrievermodule directly influences theperformance ofdownstream tasks, such as retrieval-augmented generation [20, 29, 30] and knowledge-intensive question answering [34, 52]. However, these methods do not explicitly consider targeted optimization for tool usage or the impact on complex multi-stage tasks.

We study timestamped question answering over educational lecture videos under a single-GPU latency/memory budget. Given a natural-language query, the system retrieves relevant timestamped segments and synthesizes a grounded answer. We present CourseTimeQA (52.3 h, 902 queries across six courses) and a lightweight, latency-constrained cross-modal retriever (CrossFusion-RAG) that combines frozen encoders, a learned 512->768 vision projection, shallow query-agnostic cross-attention over ASR and frames with a temporal-consistency regularizer, and a small cross-attentive reranker. On CourseTimeQA, CrossFusion-RAG improves nDCG@10 by 0.10 and MRR by 0.08 over a strong BLIP-2 retriever while achieving approximately 1.55 s median end-to-end latency on a single A100. Closest comparators (zero-shot CLIP multi-frame pooling; CLIP + cross-encoder reranker + MMR; learned late-fusion gating; text-only hybrid with cross-encoder reranking and its MMR variant; caption-augmented text retrieval; non-learned temporal smoothing) are evaluated under matched hardware and indexing. We report robustness across ASR noise (WER quartiles), diagnostics for temporal localization, and full training/tuning details to support reproducible comparison.

Recently there has been a surge of interest in training neural networks to generate images.

We show our new slack "ALBO" compares favorably to the original.

In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $\Omega(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment (Ader), which achieves an optimal $O(\sqrt{T(1+P_T)})$ dynamic regret. The basic idea is to maintain a set of experts, each attaining an optimal dynamic regret for a specific path-length, and combines them with an expert-tracking algorithm. Furthermore, we propose an improved Ader based on the surrogate loss, and in this way the number of gradient evaluations per round is reduced from $O(\log T)$ to $1$. Finally, we extend Ader to the setting that a sequence of dynamical models is available to characterize the comparators.

We present a JIT PL semantics for ReLU-type networks that compiles models into a guarded CPWL transducer with shared guards. The system adds hyperplanes only when operands are affine on the current cell, maintains global lower/upper envelopes, and uses a budgeted branch-and-bound. We obtain anytime soundness, exactness on fully refined cells, monotone progress, guard-linear complexity (avoiding global $\binom{k}{2}$), dominance pruning, and decidability under finite refinement. The shared carrier supports region extraction, decision complexes, Jacobians, exact/certified Lipschitz, LP/SOCP robustness, and maximal causal influence. A minimal prototype returns certificates or counterexamples with cost proportional to visited subdomains.

We give a compositional, information-theoretic framework that turns short programs into locality on many independent blocks, and combine it with symmetry and sparsity of masked random Unique-SAT to obtain distributional lower bounds that contradict the self-reduction upper bound under $\mathsf{P}=\mathsf{NP}$. We work in the weakness quantale $w_Q=K_{\mathrm{poly}}(\cdot\mid\cdot)$. For an efficiently samplable ensemble $D_m$ made by masking random $3$-CNFs with fresh $S_m\ltimes(\mathbb{Z}_2)^m$ symmetries and a small-seed Valiant--Vazirani isolation layer, we prove a Switching-by-Weakness normal form: for any polytime decoder $P$ of description length $\le δt$ (with $t=Θ(m)$ blocks), a short wrapper $W$ makes $(P\circ W)$ per-bit local on a $γ$-fraction of blocks. Two ingredients then force near-randomness on $Ω(t)$ blocks for every short decoder: (a) a sign-invariant neutrality lemma giving $\Pr[X_i=1\mid \mathcal{I}]=1/2$ for any sign-invariant view $\mathcal{I}$; and (b) a template sparsification theorem at logarithmic radius showing that any fixed local rule appears with probability $m^{-Ω(1)}$. Combined with single-block bounds for tiny $\mathrm{ACC}^0$/streaming decoders, this yields a success bound $2^{-Ω(t)}$ and, by Compression-from-Success, $K_{\mathrm{poly}}\big((X_1,\ldots,X_t)\mid(Φ_1,\ldots,Φ_t)\big)\ge ηt$. If $\mathsf{P}=\mathsf{NP}$, a uniform constant-length program maps any on-promise instance to its unique witness in polytime (bit fixing via a $\mathrm{USAT}$ decider), so $K_{\mathrm{poly}}(X\midΦ)\le O(1)$ and the tuple complexity is $O(1)$, contradicting the linear bound. The proof is non-relativizing and non-natural; symmetry, sparsification, and switching yield a quantale upper-lower clash, hence $\mathsf{P}\ne\mathsf{NP}$.